LECTURE SUPPLEMENT #3 . . . [LS #3]
CHAPTER #03
Bipolar Junction Transistor
Models and Biasing Circuits
Dr. John Choma
Professor of Electrical Engineering
University of Southern California
Ming Hsieh Department of Electrical Engineering
University Park: Mail Code: 0271
Los Angeles, California 90089–0271
213–740–4692 [USC Office]
213–740–7581 [USC Fax]
818–384–1552 [Cell]
johnc@usc.edu
PRELUDE:
In this chapter, we develop the circuit level models and study the associated volt-ampere
characteristics of the bipolar junction transistor (BJT). Although these models are not definitively derived, the physical properties on which they are premised are discussed thoroughly.
These discussions are couched in a largely engineering fashion that exploits many of the precepts set forth in the preceding chapter on semiconductor PN junction diodes. The models and
their engineering implications are then used to forge practical network topologies and
corresponding design guidelines for bipolar technology biasing circuits.
June 2013
Chapter 3
BJT Models & Biasing
3.1.0. INTRODUCTION
The successful realization of reliable and reproducible analog signal processors that
boast predictable I/O performance relies on the availability of meaningful and mathematically
tractable circuit level models for the active devices utilized in these networks. The electrical
properties that these models must emulate include both the static and high frequency volt-ampere
characteristics, as well as the impact exerted on these characteristics by device operating
temperature. The formulation and application of accurate models capable of relating observable
electrical performance to pertinent physical phenomenology comprise daunting challenges in
light of the complexities of semiconductor device physics and the somewhat vagarious nature of
integrated circuit processing. The use of supremely accurate device models can be relegated to
definitive computer-aided circuit investigations. Nevertheless, the models exploited by circuit
designers in pre-computer-aided studies must be sufficiently comprehensive to convey an insightful understanding of the relationship between the performance characteristics of the devices
and the circuit topologies configured in a design exercise.
In this chapter, we focus our attention on the silicon monolithic bipolar junction transistor (BJT). Our circuit level ruminations span understanding BJT performance at high signal
frequencies through formulating efficient and reliable biasing for linear signal processing
applications.
 Ve 
 Vc 
C
E
 Vb 

Ve

 
Vc
E
B
E

Ve
 Vb 
 
(a).

Vc
Emitter
(N+)
Base
(P)
C
Xw
B
Xe
Xb
Xc
Emitter
(P+)
Base
(N)
Collector
(P)
C
E
 Ve 
B
Collector
(N)
C
Xw
 Vc 
B
(b).
Figure (3.1). (a). The schematic symbol and simplified cross section of the NPN bipolar junction transistor
(BJT). (b). Schematic symbol and simplified physical abstraction of the PNP BJT. The cross
section diagrams are not drawn to scale.
BJTs come in two flavors: NPN and PNP. In Figure (3.1a), we display the schematic
symbol and simplified cross section of the NPN bipolar junction transistor, which is the principle
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type of BJT exploited in the signal flow paths of broadband analog networks. Figure (3.1b) abstracts the PNP transistor. In each of these devices, a BJT is seen as an amalgam of two PN
junction diodes, which arguably suggests the prudence of the time we devoted to studying the PN
junction diode in the preceding chapter. As is evident in the figure at hand, one of the two diodes implicit to a BJT is formed by the emitter and base semiconductor volumes, while the other
is fashioned with the collector and base regions.
In each of the transistors depicted in Figure (3.1), the emitter region is very heavily
doped to an impurity concentration approaching the solid solubility limit of silicon; that is, the
doping concentration is very nearly the maximum that can be received by silicon. The average
dopant concentration in the base layer is typically five to six orders of a magnitude less than that
of the emitter, while the collector is doped to an impurity concentration that is one or two orders
of a magnitude less than that of the base. For minimal geometry transistors intended for high frequency circuit applications, the emitter length, Xe, is generally under a micron, the base width,
Xb, is two-tenths of a micron or smaller, and the collector length, Xc, is often as large as tens of
microns. On the other hand, the feature thickness, Xw, of the transistor is generally under a micron, and the thickness, say Wb (dimension perpendicular to the page face), which is a designable
geometric parameter in integrated circuit design, is at least as large as Xw. In monolithic
technologies, the geometric parameter, Wb, is commonly referred to as the emitter finger length.
It is worthwhile recanting that a micron (μm) is 10−4 cm, and the average thickness of a human
hair is around 75 μm. Thus, the width if a typical BJT base is less than (1/375)th the thickness of
an average human hair, which this author used to have in abundance.
The PN junction diode studied in the preceding chapter is a two-terminal element for
which only a single current and a single voltage are necessary to define its volt-ampere properties. In contrast, the BJT is fundamentally a three-terminal device (actually four terminals in its
monolithic embodiment, which entails a substrate that is not shown in the subject figure) for
which at least two independent voltages and two independent currents are required to characterize its electrical behavior. In the NPN structure of Figure (3.1a), one of the current variables is
the emitter current, Ie, which is displayed as a positive current flowing out of the transistor. Positive collector current and base current, Ic and Ib, respectively, flow into the device. On the other
hand, the PNP device shows positive emitter current as flowing into the transistor and positive
collector and base currents flowing out of the device. In both cases, Kirchhoff happily radiates
Ie  Ic  Ib ,
(3-1)
which affirms that only two of the three transistor terminal currents are independent electrical
variables.
The diagrams in Figure (3.1) also delineate an internal emitter-base junction voltage,
Ve, and an internal base-collector junction voltage, Vc. Both of these voltages are interpreted as
positive variables when the potential on the p-side of the applicable junction is larger than the
potential on the n-side. The resultant internal collector to emitter voltage, Vb, of the NPN transistor is
Vb  Ve  Vc .
(3-2)
An identical voltage relationship applies for the emitter to collector voltage of the PNP device.
3.2.0. EBERS-MOLL MODEL
To first order, the static volt-ampere characteristics of either an NPN or a PNP bipolar
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junction transistor follow the dual Ebers-Moll relationships[1],
I c  I cc  I rr  I bc
,
I b  I be  I bc
which by (3-1) implies an emitter current, Ie, of
I e  I c  Ib  I cc  I rr  Ibe .
In (3-3), current Icc is given by
 V n V

 V n V

 A J  e e f T  1   I  e e f T  1  ,
I
cc
e s
s




where
A  W X
e
b w
(3-3)
(3-4)
(3-5)
(3-6)
is the carrier injection area of the emitter-base junction. Like the injection area of a PN junction
diode, Ae is the cross section area pierced by mobile charge carriers that are motivated to penetrate the junction interface between the emitter and base volumes. Continuing with (3-5), Js is
the saturation current density of the transistor, Is = AeJs is the corresponding saturation current,
and nf is the injection coefficient of the emitter-base junction. Finally,
kT j
VT 
(3-7)
q
is our familiar Boltzmann voltage, with Tj representing the absolute temperature of the emitterbase junction. The saturation current density, Js in (3-5), and hence the actual saturation current,
Is, is strongly affected by the operating temperature of the junction. As witnessed in conjunction
with PN junction diodes, the numerical value of Js nominally doubles to quadruples for each ten
degrees centigrade rise in the operating temperature of the emitter-base PN junction.
The current component, Icc, in (3-3) through (3-5) is known as the forward transport
current of the BJT. When the emitter-base junction of an NPN device is suitably forward biased
(Ve > 0), electrons from the emitter are injected into the base region. In tandem with this electron transport, holes are injected from the base to the emitter. However, the junction current
manifested by injected holes is smaller than is the junction current arising from injected electrons
because the emitter region is doped to a concentration that is far larger than the impurity
concentration indigenous of the base dopant. The phrase, “suitably forward biased,” is taken to
mean that in addition to the requirement of a positive junction voltage, Ve must be at least as
large as the turn on, or threshold, voltage, Veon, of the emitter-base junction. We recall that for a
junction of p-type and n-type silicon semiconductors, this threshold voltage is about 700 mV.
Our tacit examination of (3-3) and (3-4) confirms that Icc is a component of collector
current Ic and emitter current Ie, but it does not contribute to base current Ib. We can therefore
attribute the forward transport current of an NPN transistor to those electrons that are injected
into the base region from the emitter site and thence swept from the base and into the collector.
Thus, once the injected charges are transported to the collector volume, no net change in charge
is incurred in the base due to the forward transport current. Accordingly, the physical mechanisms contributing to current Icc do not influence the measurable base current, Ib.
The transport from emitter to collector of injected minority carriers in the base is rendered plausible and likely by the relatively low impurity concentration and narrowness of the
base region. In particular, low base region doping tends to limit the level to which base region
holes can recombine with injected electrons therein. There simply is insufficient numbers of free
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base region holes to recombine with a significant percentage of electrons injected from the emitter. Such global transport is further encouraged by the built-in potential manifested at the basecollector junction, which effectively acts as an intrinsic junction reverse bias. This potential
gives rise to an intrinsic field, directed from the collector to the base, which attracts to the collector the minority carriers (electrons in an NPN device) within the narrow confines of the base volume. An additional reverse bias (Vc < 0) imposed by external means across the base-collector
junction serves to further promote the likelihood of emitter to collector charge transport.
Most of the carrier injection across the forward biased emitter-base junction of an NPN
BJT is comprised of electrons. Nonetheless, a comparatively low level of hole injection from
base to emitter prevails, thereby giving rise to a base and emitter current manifested by injected
holes. Analogous base and emitter current components are realized in NPN transistors from the
recombination of injected electrons with base region holes. And like the former current component, this recombination component is limited by the relatively low dopant concentration in, and
narrowness of, the base volume. The net current arising from hole injection and hole-electron
recombination mechanisms arising from a forward biased emitter-base junction is Ibe, which is
given by
I
AJ
I
V n V
V n V
I be  cc  e s e e f T  1  s e e f T  1 .
(3-8)
βf
βf
βf




The fact that the form of this expression mirrors the static volt-ampere relationship of a traditional diode is hardly surprising in that the emitter and base volumes shown in Figure (3.1) coalesce to form a classic PN junction diode. The effective saturation current of this intrinsic diode
is Is /βf. In concert with our discussion herewith, we note in (3-3) and (3-4) that current Ibe affects only the BJT base and emitter currents. In (3-8), βf, the forward short circuit (meaning Vc =
0) base to collector transport current transfer ratio, is generally of the order of one hundred in
conventional bipolar processing technologies. It is important that we understand that whether
through hole injection or via hole-electron recombination, current Ibe reduces the number of injected carriers available for ultimate transport into the collector. Thus, for example, the implication of βf = 100 in (3-8) is a net emitter to collector transport current loss, or reduction, of 1%.
Returning to (3-3) and (3-4), the current component, Irr, is given by




I rr  Ae J s eVc nrVT  1  I s eVc nrVT  1 ,
(3-9)
while current Ibc is
AJ
I
I
I bc  rr  e s eVc nrVT  1  s eVc nrVT  1 .
(3-10)
βr
βr
βr
With Vc  Vcon, which denotes the threshold voltage of the base-collector junction diode, majority
carriers in the collector region are injected into the base region where they become minority
carriers. Recombination in this so-called reverse direction is more probable than in the formerly
considered forward direction because the base region is doped to a higher impurity concentration
than is the collector. Current Ibc reflects this reverse recombination phenomena, as well as the
effects of holes injected into the collector across a forward biased base-collector junction.
Parameter βr is the reverse short circuit (meaning Ve = 0) base to emitter transport current transfer ratio. Owing to the likelihood of enhanced recombination in reverse operation, βr is substantially smaller than is its forward counterpart, βf. The current, Irr, represents the effects of
collector-injected minority carriers that are swept from the base and into the emitter by the builtin potential at the emitter-base junction or by an imposed reverse bias therein. In (3-9) and (310), nr is the injection coefficient of the base-collector PN junction. Generally, nr and nf in (3-5)



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
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BJT Models & Biasing
and (3-9) are taken as identical constants. Both of these metrics are only slightly larger than unity.
Similar statements can be proffered for the current components, Icc, Irr, Ibe, and Ibc, of a
PNP bipolar junction transistor. In the PNP device, Icc and Irr are measures of injected holes that
are transported from emitter to collector and from collector to emitter, respectively. Similar to
the NPN device, Ibe represents the recombination and base to emitter electron injection current.
In the PNP case, recombination is attributed to base region electrons that recombine with holes
injected from the emitter. On the other hand, Ibc is the current arising from base region electrons
that recombine with holes injected into the base from the collector. Current Ibc also reflects electron injection into the collector across a forward biased base-collector junction.
3.2.1. PHYSICAL INTERPRETATION OF THE EBERS-MOLL MODEL
Before proceeding further, an engineering discussion of the salient physical features of
the four bipolar currents, Icc, Ibe, Irr, and Ibc may prove useful to the goal of assimilating a thorough understanding of BJT operation. To this end, we shall focus on the NPN embodiment in
Figure (3.1a), with the understanding that subject to an interchange of the charge terms, “hole”
and “electron,” and a few other clarion changes, our discourse applies equally well to PNP BJTs.
(1). In the so-called normal forward, or normal active, mode of BJT operation, we forward bias
the emitter-base junction through the application of an external static voltage that forces the
emitter-base junction potential, Ve, to exceed its threshold level, Veon. We should interject
that prior to any applied biasing voltage, the emitter, base, and collector regions lie in an
equilibrium state that is characterized by charge neutrality. In addition to forward biasing
the emitter-base junction, we apply a zero or reverse bias across the base-collector PN junction. Even at zero base-collector junction bias, we know from our PN junction diode studies
that a built-in junction potential and corresponding electric field, directed from the collector
to the base, is established at equilibrium. The intensity of this field, which increases with
imposed reverse bias across the base-collector junction, exerts an attractive force on those
electrons that are injected into the base volume from the emitter region. Thus, most of the
electrons that find themselves proximate to the geometric base-collector junction are likely
to be sucked into, and ultimately transported through, the collector.
(2). A forward bias applied across the emitter-base junction facilitates the injection of electrons
from the emitter volume to the base region. Accompanying this electron injection is hole
injection from the base to the emitter volumes. However, electron injection dominates over
its hole counterpart due to the fact that the emitter volume is far more heavily doped with
donor impurities than the base is doped with acceptor atoms. Nevertheless, for every hole
that relocates from the base to the emitter, the initial charge neutrality is upset in both the
base and the emitter regions. The departed positively charged holes leave behind a net
negative ionic charge owing to acceptor donor atoms that have graciously allowed their valence band holes to vacate and seek their electronic fortune elsewhere. We may restore
charge neutrality in the base by extracting electrons from the base through the base terminal.
This electron extraction gives rise to a current flowing into the base. In the emitter, each injected hole increases the charge therein by an amount that is effectively equal to a positive
electron. We may therefore restore emitter charge neutrality by providing the emitter region
with a suitable number of electrons, thereby establishing a current flowing out of the emitter.
(a). However, it is far more likely that hole-electron recombination occurs in the emitter,
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because the emitter is doped so heavily with donor atoms. Under the recombination
scenario, injected holes are effectively eaten by electrons, thereby neutralizing the injected hole charge. But nonetheless, the electrons that recombine with the injected hole
comprise lost negative charge that leaves in its wake positively charged, ionized donor
atoms. This positive charge can be neutralized in a fashion that is similar to the
aforementioned electrons being inserted into the emitter volume through the emitter
terminal. Of course, the current resulting from these inserted electrons flows out of the
emitter lead.
(b). In summary, hole injection from the base to the emitter manifests a base current flowing into the base region. This current reestablishes charge neutrality in the base by
extracting electrons to cancel the negative ionic charge left in the wake of hole injection. Once the holes reach the emitter volume, emitter current flows out of the emitter
to sustain charge neutrality. This situation is true whether recombination, which is the
likely event, takes place or whether the hole avoids recombination and magically survives in the emitter volume. In either case, electrons must be brought in either to replace the electrons that recombine with injected holes or to offset the positive charge of
holes that steadfastly elude being eaten by electrons.
(3). For electrons that find themselves injected into the base, three issues are immediately
interesting. The first and most obvious of these issues is that the emitter volume is now
charged positively by the loss of those injected electrons. This charge must be neutralized
by inserting electrons into the emitter, thereby bringing about a current that flows out of the
emitter terminal. In the base region, where there are injected electrons, two possible
circumstances prevail. First, electrons can recombine with holes in the base, thereby requiring, in the interest of sustaining charge neutrality, electrons to be extracted through the base
contact and lead. Resultantly, a base current is caused to flow into the base region. But the
second circumstance is more likely. This is to say, that as opposed to recombination, injected electrons are likely to be transported across the base and thence into the collector volume. Our view that this second situation is more likely than hole-electron recombination
derives from the fact that there are simply not a lot of holes in the base that can capture swift
moving electrons (electrons boast high mobility) for recombination purposes. Moreover, the
base is a very thin region, which is to say, that as soon as injected electrons navigate through
the emitter-base transition layer, they likely experience the influence of the attractive force
associated with the collector to base electric field established in the base-collector depletion
zone. In either the recombination or the transport case of emitter-injected electrons, a current flows out of the emitter. Under the less likely recombination scenario, charge neutrality
is restored by extracting electrons from the base volume, thereby incurring base current flow
into the base region. In the transport situation, no base current is needed to sustain charge
neutrality in the base, since injected electrons are themselves extracted by their field-induced transport into the collector. In this latter case, electrons must be extracted from the
collector, which means that collector current flows into the collector volume.
(4). The preceding two items pertain exclusively to the forward active situation that is defined
by a forward biased emitter-base junction and a base-collector junction that is not forward
biased. The engineering consideration of this particular operational state leads to the
following conclusions.
(c). First, current Ibe is a current that, in concert with (3-3) and (3-4), affects only the net
emitter and base currents. It is caused by hole recombination with emitter electrons
that are injected from the emitter to the base, electron recombination with holes injected
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from the base to the emitter, and/or holes injected from the base into the emitter, where
they survive without recombining with emitter region electrons.
(d). Second, Icc is a forward transport current that, in accord with (3-3) and (3-4), affects
only the emitter and the collector currents. It is manifested by electrons that are injected from the emitter to the base and are then transported completely through the base
volume and into the collector.
(e). We can now appreciate the significance of the forward beta, βf. Since, Ibe = Icc/βf, (1/βf)
is indicative of the percentage of the forward transport current that is effectively lost
through hole-electron recombination.
(5). A similarly experiential discovery of the significance of current components Irr and Ibc requires the consideration of a transistor operated in so-called inverse mode. This mode is the
converse of the forward active condition in that now, we forward bias the base-collector
junction, while applying zero or reverse bias across the emitter-base junction. In other
words, the inverse mode can be thought of as the operating condition that happens when the
collector and emitter leads are interchanged. The current conditions that apply in inverse
mode are simply the reverse of those that have been delineated for the forward active condition. In other words, instead of electron injection from emitter to base, we now observe
electron injection from the collector to the base. The details are left as an exercise for the
reader but in summary, the following statements can be made.
(f). Current Ibc is a current that, in concert with (3-3) and (3-4), affects only the collector
and base currents. It is caused by hole recombination with electrons that are injected
from the collector to the base and/or electron recombination with holes injected from
the base to the collector. Hole recombination with injected electrons is the more likely
scenario in that the base region is more heavily doped with acceptor atoms than the
collector volume is doped with donor impurities. Stated quite simply, electrons coming
across from the lightly doped collector encounter a relatively dense sea of base region
holes that are hungry for oppositely charged electrons.
(g). Current Irr is a reverse transport current that, in accord with (3-3) and (3-4), affects only
the collector current and the emitter current. It is manifested by electrons that are injected from the collector -to- the base and are then transported completely through the
base volume and into the emitter. However, the efficiency of this reverse transport is
dubious owing to the relatively high density of free holes in the base. In addition, the
extremely dense concentration of free electrons in the emitter sets up a barrier for base
region electrons that effectively retards and discourages their attraction by the emitter.
(h). Since, Ibc = Irr/βr, (1/βr) is indicative of the factor of the reverse transport current that is
lost through hole-electron recombination. Because of the dense hole population in the
base and the extremely dense population of free electrons in the emitter volume, reverse
beta βr is invariably much smaller than βf. Indeed, βr can be smaller than unity, which
is indicative of a recombination current component that is larger than the reverse transport current component.
It is worth underscoring that each of the four current components in the Ebers-Moll
model of a bipolar junction transistor is directly proportional to the saturation current density, Js.
Recall that Js increases dramatically with the operating temperature of the emitter-base junction.
Thus, all three transistor currents −emitter current Ie, collector current Ic, and base current Ib− are
vulnerable to device temperature increases if the emitter-base and base-collector junction potentials, Ve and Vc, respectively, are held fast over temperature. The design lesson we learn from
this argument is that temperature insensitivity in bipolar networks requires circuit designs for
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which Ve and/or Vc are allowed to decrease appropriately to compensate for the increased currents incurred by junction temperature rises.
We should additionally note that all of the Ebers-Moll current components are directly
proportional to the emitter-base junction area, Ae. This circumstance means that for fixed values
of the junction potentials, Ve and Vc, the emitter, base, and collector currents scale linearly with
emitter-base junction area. Assume, for example, that the second of two identical transistors has
a junction area that is K-times that of the first transistor and that both transistors are excited by
identical emitter-base and base-collector junction potentials. Both transistors therefore exhibit
the same current densities despite the fact that the second transistor conducts emitter, base, and
collector currents that are respectively K-times larger than the corresponding currents of the first
device. Since the internal self-heating caused by current conduction is intimately related to current density, as opposed to actual current level, the proper scaling of monolithic transistors can
circumvent performance problems that may be incurred by excessive on-chip thermal gradients.
In short, we need fat transistors when large currents must be conducted, just as we require thick
wires to power high current appliances, such as microwave ovens and electric clothes dryers.
3.2.2. SILICON GERMANIUM HETEROJUNCTION BJT
The bipolar transistors addressed at this juncture are exclusively conventional, so-called
homojunction, BJTs. A popular alternative to the BJT homojunction is the silicon germanium
(SiGe) heterojunction bipolar transistor (HBT)[2],[3]. Its popularity stems fundamentally from its
ability to process ultra high frequency signals. Indeed, SiGe HBTs offer short circuit, or Norton,
current gains whose magnitudes do not degrade to unity until several hundreds of gigahertz1.
This capability compares quite favorably with the frequency attributes of homojunction devices
for which unity gain frequencies are at best several tens of gigahertz.
In an NPN SiGe HBT, silicon semiconductor is deployed for the emitter and collector
regions, while the base volume is formed of germanium. The basic Ebers-Moll model remains
applicable for these new generation bipolar transistors. However, the physical implications of
the emitter-base heterojunction give rise to significant performance ramifications at the circuit
level. On notable performance difference between an NPN BJT homojunction and an NPN SiGe
heterojunction derives from the fact that silicon boasts a larger bandgap potential than does
germanium. The bandgap disparity between the emitter and base volumes serves to lower the
potential barrier that electrons must overcome if they are to succeed in being injected into the
base when a suitable forward bias is applied to the emitter-base junction. Simultaneously, the
subject bandgap disparity actually raises the potential barrier that holes must overcome if they
are to be injected into the emitter. In a word, the emitter-base heterojunction facilitates emitter to
base electron injection, but it makes it harder to effect hole injection from the germanium base
volume into the silicon emitter region. The relevant immediate impact is that for a given emitterbase junction forward bias, current Ibe is reduced because hole injection from the base to the
emitter is virtually eliminated. Current Ibe therefore becomes almost exclusively a recombination
current. But since recombination is minimal for a thin base, parameter βf in (3-8) is rendered
very large. In particular, it is two to five times larger than the nominal 100 or so quoted for
parameter βf in a conventional transistor. This parametric enhancement is important because, as
we shall learn shortly, βf is a measure of the achievable gain (and of numerous other performance
metrics) for an electronic network realized in bipolar technology.
1
We shall have much more to say about the unity gain frequency metric of a BJT later in this chapter.
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A second feature of a SiGe HBT is that the germanium base is not doped uniformly
with acceptor impurities. Instead, the base impurity profile is monotonically graded from a
maximum level at the emitter-base junction to a minimum concentration at the base-collector
junction. This impurity grading manifests a drift field within the base that enhances the mobility
of electrons injected into the base from the emitter. This mobility enhancement supports the
very high, short circuit, unity gain frequencies of SiGe HBTs. Aside from facilitating very wideband frequency responses in SiGe HBT circuits, the aiding of injected electron transport further
supports the minimization of base region hole-electron recombination and therefore a maximization of the value of the gain measure, βf.
The advantages of a SiGe heterojunction over a counterpart BJT homojunction transistor continue. The fact that electron injection from the emitter to base is facilitated in SiGe means
that the average dopant concentration in the base, which is kept relatively low to enable efficient
carrier injection from the emitter to the base, can be made an order of magnitude or two larger
than it is in a conventional BJT. The immediate impact of a large average dopant level in the
base is that it reduces the base resistance of a BJT. The last contention follows from the fact that
resistance in general is directly proportional to resistivity, which in turn is inversely related to
dopant concentration. A small base resistance is a significant attribute because it postures at
least four circuit level advantages. First, small base resistance reduces the equivalent input noise
voltage, which quantifies the smallest possible signal voltage amplitude that can be distinguished
by the amplifier from electrical noise and therefore reliably processed. Second, it improves voltage gain in that the available gain of several BJT configurations is inversely related to the sum
of the Thévenin signal source resistance and the transistor base resistance. Third, small base
resistance supports higher bandwidth frequency responses whenever the net effective capacitance
observed between the base terminal and ground significantly limits the observed bandwidth. As
we explore later in this text, circuit bandwidth is intimately related to the sum of the capacitive
constants established in the considered network. At the base port of a BJT, the capacitive time
constant turns out to be the product of the capacitance developed at this port and a resistance
function that is proportional to the aforementioned sum of Thévenin signal source and base resistances. Finally, a small base resistance minimizes an annoying phenomenon known as base
conductivity modulation. We shall ultimately demonstrate that such modulation produces a
resistance across the output ports of many amplifiers realized in BJT technology. This incurred
resistance serves to reduce the achievable voltage gain, as well as incur other performance detriments.
The general impression we should carry away from the foregoing discourse is that SiGe
HBT technology offers significant performance advantages over most conventional BJT
homojunction networks. This opinion is particularly true in the modern epitaxial processing era
that succeeds in realizing SiGe HBTs with insignificant manufacturing cost penalties. But there
is at least one disadvantage to SiGe technology. In particular, the larger potential barrier effected
for hole injection from base to emitter, along with the larger average impurity concentration in
the germanium base gives rise to an emitter-base junction turn on potential that is at least 800 mV
or so. This 100 mV or so larger than conventional silicon junction turn on voltage means that
supply line voltages for SiGe devices may need to be larger than the supply voltages adopted for
conventional BJTs. While this detriment may prove insignificant for many prudently designed
circuits, it does generally bode poorly for the electronic portability culture in which we are all
immersed.
- 196 -
Chapter 3
BJT Models & Biasing
3.2.3. CIRCUIT LEVEL INTERPRETATION OF EBERS-MOLL MODEL
In consideration of the fact that both the emitter-base and base-collector junctions of a
BJT(inclusive of SiGe) are PN junction diodes, the circuit level interpretation of (3-3) and (3-4)
are the NPN and PNP model topologies offered in Figure (3.2). In these topologies, diode DBE
represents the emitter-base junction diode, while DBC recognizes the base-collector junction as a
semiconductor diode. A trivial model modification, which derives from (3-8) and (3-10), appears in Figure (3.3). In particular, (3-8) shows that the forward transport current, Icc, can be
written as Icc = βf Ibe. The forward transport component, Icc, of the collector current can therefore
be represented as a current controlled current source that is controlled by the current, Ibe, which
is conducted by diode DBE. Analogously, (3-10) implies Irr = βr Ibc, thereby suggesting that the
reverse transport component, Irr, of the collector current can be represented as a current controlled current source that is controlled by the reverse base current component, Ibc, which flows
through diode DBC.
Base

Ib
Collector
Base
Ib


Vb
Ve
Ve
Ie


Ic
Emitter
Base
Collector
Ib
Ve
Ve

Icc
Ic

Ibe

DBE
Icc
Ie
Emitter

Ibc 
(b).
Collector
Irr Vb
(a).
Vb
Ie
DBE
Ie




Ibe

DBC
Base

Ib
Ic
Ibc 
DBC
Ic
Emitter
Collector
Irr Vb

Emitter
Figure (3.2). (a). Ebers-Moll model of the NPN bipolar junction transistor. (b). Ebers-Moll model of the PNP
bipolar junction transistor.
It is instructive to study the Ebers-Moll equations for the two special cases of Ve = 0
and Vc = 0, which implies a short circuit imposed across the base-collector junction. For example, with Vc = 0, (3-9) and (3-10) give Irr = 0 and Ibc = 0, whence by (3-3) and (3-4),
I b  I be
I c  I cc  β f I be  β f I b


.


(3-11)
I e  I cc  I be  β f  1 I be  β f  1 I b
Equation (3-11) produces the circuit interpretation in Figure (3.4a), where the voltage, Ve, of the
indicated battery is presumed larger than the threshold voltage, Veon, of the emitter-base PN junc- 197 -
Chapter 3
BJT Models & Biasing
tion. The condition, Ve > Veon, ensures measurable and positive emitter and base currents in the
transistor. The diagram in Figure (3.4a) underscores the logic of referring to parameter βf as a
short circuit (meaning Vc = 0) base to collector current transfer ratio in that the ratio, Ic /Ib, of
collector to base currents in the subject circuit is clearly βf Ibe/Ibe = βf. We also note a resultant
ratio of collector current to emitter current of
Base

Ib
Collector
Base
Ib


Vb
Ve
Ve
Ie


Ic
Emitter
Base
Collector
Ib
Ve
Ve

DBE
f Ibe

Ic
Ibc 

Ibe

DBE
f Ibe
Emitter
Collector
r Ibc Vb
Ie
Emitter
Collector
r Ibc Vb
(a).
Vb
Ie

Ie




Ibe
DBC
Base

Ib
Ic
Ibc 
DBC
Ic

(b).
Emitter
Figure (3.3). (a). Alternative Ebers-Moll model of the NPN bipolar junction transistor. (b). Alternative
Ebers-Moll model of the PNP bipolar junction transistor.
βf
Ic

 αf ,
I e V 0
βf 1
c
(3-12)
f Ibe
r Ibc
Ibe
Ibc


Ve

Vc

(f +1)Ibe
(a).
(r +1)Ibc
(b).
Figure (3.4). (a). An NPN bipolar junction transistor operated with a forward biased
emitter-base junction and a short circuited base-collector junction. Voltage
Ve is presumed to be at least as large as the threshold voltage of the emitterbase diode. (b). An NPN BJT operated with a forward biased basecollector junction and a short circuited emitter-base junction. Voltage Vc is
at least as large as the threshold voltage of the base-collector diode.
- 198 -
Chapter 3
BJT Models & Biasing
where parameter αf is commonly referred to as the short circuit emitter to collector current transfer ratio. This parameter is a measure of the extent of forward recombination in that (3-12)
makes clear that the amount of emitter current that does not materialize as collector current and
is therefore “lost” in recombination, is (1−αf )Ie. For large βf and therefore for αf approaching unity, forward recombination in the base volume is minimal.
In contrast, with Ve = 0, Icc in (3-5) and Ibe in (3-8) are null and resultantly,
I b  I bc
(3-13)
I c   I rr  I bc    βr  1 I bc    βr  1 I b .
I e   I rr   βr I bc   βr I b
The last result begets the circuit level representation in Figure (3.4b) in which the voltage, Vc,
applied from the base to the collector is presumed to be at least as large as Vcon, the threshold
voltage of the base-collector diode. We observe in (3-13) that while the base current remains
positive and therefore flows into the base of the NPN transistor, the collector and emitter currents
are negative quantities. Consequently, the collector current now flows out of the collector, while
emitter current flow is into the transistor emitter terminal. In the circuit diagram at hand,
parameter βr is recognized as the magnitude of the base to emitter current transfer ratio. The
indicated positive value of the applied voltage Vc serves to inject carriers from the collector to
the emitter. Most of these injected carriers recombine with holes in the base, while others are
transported to the emitter to establish the current, −βrIb. The resultant ratio of emitter current to
collector current is
Ie
βr

 αr .
(3-14)
I c V 0
βr  1
e
Since βr is of the order of one or even smaller, at least half of the aforementioned injected carrier
population is lost to recombination within the base volume. In (3-14), parameter αr is the short
circuit collector -to- emitter current transfer ratio.
3.2.3.1. Forward Active Operation
A bipolar junction transistor is said to operate in its forward active, or linear, regime
when its emitter-base junction voltage exceeds the threshold potential (Ve ≥ Veon) of the emitterbase junction and its base-collector voltage is non-positive (Vc ≤ 0); that is the base-collector
junction operates at either zero bias or is reverse biased. The transistors in almost all bipolar
technology circuits that are designed to operate as viable approximations of linear networks are
biased in this operating domain.
Using (3-3),
I c  β f Ibe   βr  1 Ibc  β f Ib  β f  βr  1 Ibc ,
(3-15)
where the base current component, Ibe, is positive for Ve ≥ Veon. Recalling (3-10),
 β f  βr  1 
V nV
V nV
Ic  β f Ib  
 Ae J s e c r T  1  β f I b  I co e c r T  1 , (3-16)
βr


where
 β f  βr  1 
I co  
(3-17)
 Ae J s
βr






- 199 -


Chapter 3
BJT Models & Biasing
is termed the collector leakage current. Specifically, Ico is the collector current fashioned by an
open circuited base (Ib = 0) and a base-collector junction that is strongly reverse biased (Vc <<
0). This current is small since it derives from the minority carrier injection promoted by strong
reverse biasing of the base-collector junction.
There are at least two interesting aspects to (3-16). First, with Vc ≤ 0,
Ic
 β f I b  I co  β f I b ,
(3-18)
Vc 0
Base
Ib

Ve

Ic
Vc0 
Ibe

Collector
Base

DBE
f Ib Vb
Ie
(a).
Ib

Ve

Ic
Vc0 
Ibe

DBE
f Ib Vb
Ie
Emitter
Collector
(b).

Emitter
Figure (3.5). (a). Approximate equivalent circuit of an NPN BJT for the case in which the emitter-base junction
voltage, Ve, is at least as large as the threshold voltage of this junction, and the base-collector junction voltage, Vc, is negative or no larger than zero. (b). Approximate equivalent circuit of a PNP
BJT for the case in which Ve is at least as large as the threshold voltage of the junction, and Vc is no
larger than zero.
which suggests that to the extent that parameter f is voltage and current invariant and Ico <<
fIb, the collector current of a BJT operated in the forward active region is directly proportional
to its base current; that is, the BJT emulates input base current to output collector current linearity. To this end, the BJT might be modeled for Vc ≤ 0 by the simple equivalent circuits provided
in Figure (3.5). Moreover, since f is a transistor metric that is significantly greater than one, the
linearity between base and collector currents that is inferred by (3-18) establishes a vehicle for
the realization of I/O gain from the input base terminal to the output collector terminal. A
second point pertinent to (3-18) is that for Vc ≤ 0, current Ibc in (3-10) is essentially zero, which
means that the base current component, Ibe, dominantly determines the observed base current, Ib.
Accordingly, (3-8) and (3-10) combine to yield
Ic

Ve n f VT
 β f I b  β f I be  I cc  Ae J s e
Vc 0


Ve n f VT
 1  Is e

1 .
(3-19)
The last result is interesting in that it shows that in the linear domain, the collector current of a BJT is determined almost exclusively by the emitter-base junction voltage, Ve. Actually, since Ic ≈ Icc ≈ βf Ibe, Ib ≈ Ibe, and Ie ≈ Icc + Ibe = (βf +1)Ibe for Vc ≤ 0, all three transistor
currents, as opposed to the collector current alone, are determined almost exclusively by emitterbase junction voltage Ve. This state of affairs is cause for jubilation since a design scenario in
which all three transistor currents are predicated on a single device voltage is easier to embrace
analytically than is a situation in which said currents are fixed by a particular combination of two
transistor voltages. It is important to understand that the transistor currents are independent of
base-collector junction voltage only if the base collector junction is not forward biased. From (32), this requirement translates to an internal collector-emitter voltage (or emitter-collector voltage in PNP devices), Vb, that must equal, and likely exceed, the emitter-base junction voltage Ve.
In short, forward linear active BJT operation requires Vb ≥ Ve ≥ Veon. Thus, for a homojunction
BJT, Vb must be at least 700 mV, while a SiGe heterojunction BJT requires that Vb be at least as
- 200 -
Chapter 3
BJT Models & Biasing
large as 800 mV or so.
In summary, the vast majority of BJTs deployed in linear signal processing systems are
biased to operate in the forward active domain. Operation in this regime requires an emitter-base
junction voltage, Ve, which exceeds the threshold potential of said junction and a base-collector
junction voltage, Vc, which is no larger than zero. Equivalently, Vc ≤ 0 implies that the collector
to emitter bias, Vb, for an NPN functioning in its forward active region is at least as large as is
Veon. We now understand that Ve > Veon and Vc ≤ 0 gives rise to a collector current (Ic) given by
(3-19). To the extent that gain parameter βf is a constant, (3-19) confirms that the collector current is linearly related to base current Ib. Despite this laudable linearity, however, the same equation underscores the unfortunate fact that the collector current remains a nonlinear function of
junction voltage Ve.
3.2.3.2. Reverse Active Operation
A bipolar junction transistor operating in its reverse, or inverse, active regime is the
converse of forward active transistor operation. In particular, the base-collector junction potential exceeds the junction threshold potential (Vc ≥ Vcon), while the emitter-base junction voltage is
non-positive (Ve ≤ 0) in the inverse operating mode Transistors embedded in linear signal
processing networks rarely operate as inverse devices. Indeed, device operation in this domain is
invariably the result of errors either in biasing or in the electrical connections made to the transistor terminals. The most common connection error, especially in discrete component networks, is
an inadvertent interchange of collector and emitter terminals. With Vc ≥ Vcon and Ve ≤ 0, (3-3),
(3-4), and (3-8) deliver
 β f  βr  1 
V n V
V nV
I e   βr I b  
 Ae J s e e f T  1   βr I b  I eo e e r T  1 , (3-20)


βf


where, with an appeal to (3-17),
 β   1
 
f
r


(3-21)
I

A J   r I
eo

 e s
 β  co
β
f


 f 
is termed the emitter leakage current. In particular, Ieo is the emitter current observed when the
base terminal is open circuited (Ib = 0) and the emitter-base junction is strongly back biased (Ve
<< 0). We note that Ieo is significantly smaller than the collector leakage current, Ico, in that
parameter f is much larger than inverse gain metric r.




With voltage Ve ≤ 0, current Ibe in (3-8) becomes negligible. Accordingly, the base current, Ib, in reverse active mode is largely determined by the inverse base current component, Ibc.
It then follows that the emitter current in (3-20) is set by the base-collector voltage, Vc, and is
essentially independent of the emitter-base voltage, Ve. But since Ib ≈ Ibc and Ic = Ie − Ib ≈ −(βr
+ 1)Ibc with Ve ≤ 0, all currents in a reverse active transistor are effectively determined by only
the base-collector junction voltage Vc. Moreover, the pertinent magnitude of the collector current to base current ratio is seen to be (βr +1), which is far less than the base to collector transfer
ratio, βf, evidenced in forward mode. In inverse mode, we note that the collector current to base
current ratio is a magnitude that equates to βr. This small ratio (very small in SiGe heterojunction BJTs) is reason enough to forsake reverse active operation in favor of forward active operation.
- 201 -
Chapter 3
BJT Models & Biasing
3.2.3.3. Transistor Saturation
A BJT operates in saturation when both its emitter-base and base-collector junctions are
forward biased. Because of the dual junction forward bias, the base of a transistor is injected
with electrons (for NPN) or holes (for PNP) from both the emitter and the collector. All three
transistor currents are resultantly influenced by both of the junction voltages, Ve and Vc. Using
(3-3), the collector -to- base current ratio in saturation is
 1   βr  1  Ibc I cc  
I   βr  1 Ibc
Ic
(3-22)
 cc
 βf 
.
Ib
Ibe  Ibc
1   Ibc Ibe 


Since the base current components, Ibe and Ibc, are each positive in saturation, the collector current to base current ratio in (3-22) is clearly smaller than the gain metric, βf. In other words, the
collector current to base current ratio always falls below the short circuit current gain, βf, when a
BJT operates in saturation. Aside from a dependence of all transistor currents on both transistor
junction voltages, the degradation in I/O gain is ample reason to avoid saturation in linear BJT
signal processing networks. Another reason to avoid saturation is poor frequency response since
a significant amount of time is generally required to transport or otherwise displace the high density of free charge that populates the saturated base. Additionally, and as we shall discuss
shortly, saturation incurs current flow in the substrate body of a monolithic transistor, which can
cause potentially damaging substrate heating and other detriments.
Finally, the intrinsic collector-emitter voltage in (3-2) is the difference of two positive
junction voltages when the subject transistor enters its saturation regime. Because of the doping
disparity between the emitter and collector volumes, this difference is always positive. It is normally referenced as the collector-emitter saturation voltage, Vcesat, which is dependent on transistor currents and is typically in the range of 50 mV to 300 mV. We can determine this saturation
voltage by solving (3-22) for the current ratio, Ibc/Ibe. Denoting this ratio by Xce,
β f   Ic Ib 
I
(3-23)
X ce  bc 
,
I be
1  βr   I c I b 
which is zero if (Ic/Ib) = βf. The latter disclosure is emblematic of our engineering expectations
in that (Ic/Ib) = βf in only the forward active regime of operation where the reverse current, Ibc,
approaches zero. Since (Ic/Ib) is less than βf in saturation, positive Xce accompanies device
saturation. The result at hand, coupled with (3-3) and (3-8), delivers
1  X ce  Ae J s Ve n f VT
I b   1  X ce  I be 
e
(3-24)
1 ,
βf


whence an emitter-base junction voltage of
β f Ib


(3-25)
Ve  n f VT ln 1 
.

1
X
A
J


ce
e
s


It is interesting that operation in the saturation domain, where Xce > 0, reduces the emitter-base
junction voltage from the level evidenced for a similar base current conducted by a transistor in
its forward active domain. This effect is reasonable in light of the fact that in saturation, the base
current is partitioned between emitter-base and base-collector current components while in the
active domain, the base current is exclusively determined by emitter-base injection phenomena.
Returning to (3-23), (3-3) and (3-10) yield
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Chapter 3
BJT Models & Biasing
 1  X ce 
1  X ce  Ae J s Vc nrVT
(3-26)
1 ,
Ib  
e
 I bc 
X ce βr
 X ce 
which establishes a base-collector junction voltage of

X ce βr I b 
Vc  nrVT ln 1 
(3-27)
.

1
X
A
J


ce e s 

Recalling (3-2) and assuming Xce > 0, the collector-emitter saturation voltage, Vcesat, in an NPN
unit (or emitter to collector saturation voltage, Vecsat, in a PNP transistor) is
n

β f Ib
 f 
 1 

 
1  X ce  Ae J s  
Vcesat  Ve  Vc  VT ln 
, X ce  0 .
(3-28)
nr 

X ce βr I b  
 1 
1  X ce  Ae J s  
 
Using (3-23), it is a simple matter to confirm that gain parameter f is always larger than the
product, Xcer, and indeed, it is never equal to Xcer. It follows that the collector-emitter saturation voltage in (3-28) is always a nonzero and positive voltage. For the case of identical junction
injection coefficients, say nf = nr = n, the small transistor saturation current, AeJs, simplifies (328) to
 βf 
(3-29)
Vcesat  Ve  Vc  nVT ln 
 , X ce  0 ,
 X ce βr 
which suggests that the saturation voltage is nominally dependent on only the current ratio, (Ic/Ib)
(because of parameter Xce), and not on the individual transistor currents, Ic or Ib.


3.2.3.4. Cutoff
There is nothing fundamentally wrong with asserting that a transistor is cutoff when either its collector current or its emitter current is zero. Formally, however, a transistor is said to
be in cutoff when its emitter current, Ie is zero. You are asked to investigate this useless operating case in Problem #3.3.
EXAMPLE #3.1:
In a bipolar transistor technology integrated circuit, PN junction diodes are realized as appropriate connections of BJTs. Three plausible realizations appear in
Figure (3.6). For each of these “diodes,” determine the volt-ampere characteristic equation, Id -versus- Vd, and give expressions for their effective saturation
currents. Assume that the two junction injection coefficients of the transistor are
identical and equal to n.
SOLUTION #3.1:
(1).
In the interconnection of Figure (3.6a), the base and collector terminals are connected together, thereby constraining base-collector junction voltage Vc to zero and rendering voltage
Vd equal to the emitter-base junction voltage, Ve. With Vc = 0, Ibc and Irr in (3-3) are null.
Thus, (3-3) and (3-8) provide
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Chapter 3
BJT Models & Biasing
Id

Id


Vd
Vd

Id
(a).
Vd


(b).
(c).
Figure (3.6). (a). A diode formed by short-circuiting the base to the collector of an NPN
BJT. (b). A diode formed by short-circuiting the base to the emitter of a PNP
BJT. (c). A diode formed by short-circuiting the collector to the emitter of an
NPN BJT.
I d  I b  I c  I be  I cc 
 β f  1 Ibe .
(E1-1)
Using (3-8) once again, it follows that the volt-ampere characteristic of the subject diode
interconnection is
 βf  1
 βf 1
V nV
V nV
Id  
 Ae J s e d T  1  
I e d T  1 .
(E1-2)
 βf 
 βf  s




In this relationship, we witness an effective saturation current, say Io, of
 βf  1
AJ
I
Io  
A J  e s  s .
(E1-3)
 βf  e s
αf
αf


In the preceding two equations, Is = AeJs is, of course, the BJT saturation current corresponding to a saturation current density of Js and an emitter-base junction injection area of Ae.

(2).



In Figure (3.6b), the base and emitter terminals are tied together, thereby constraining emitter-base junction voltage Ve to zero and rendering voltage Vd equal to base-collector junction
voltage Vc. With Ve = 0, Ibe and Icc in (3-3) are zero, whence (3-3) and (3-10) yield
I d   I c  I bc  I rr   βr  1 I bc .
(E1-4)
Using (3-10) the volt-ampere characteristic of the subject diode interconnection is
 β  1
 βr  1 
V nV
V nV
Id   r
(E1-5)
 Ae J s e d T  1  
 Is e d T  1 ,
 βr 
 βr 




and the saturation current follows as
 β 1
Ae J s
I
Io   r
 s .
 Ae J s 
αr
αr
 βr 
(3).
(E1-6)
In Figure (3.6c), the collector and emitter terminals are interconnected, thereby rendering Vd
= Ve = Vc. Accordingly, (3-3), (3-8) and (3-10) combine to give
I d  I b  I be  I bc ,
(E1-7)
and the volt-ampere characteristic of the subject diode interconnection is
 1
 1
1 
1 
V nV
V nV


Id  
 Ae J s e d T  1  
 Is e d T  1 .
(E1-8)
 βf



β
β
β
r 
r 

 f
Clearly, the corresponding saturation current, Io, is
 1
 1
1 
1 
Io  


 Ae J s  
 Is .
(E1-9)
 βf
 βf
βr 
βr 




- 204 -


Chapter 3
BJT Models & Biasing
ENGINEERING COMMENTARY:
The only diode interconnection of the three considered topologies that precludes a forward
biased base-collector junction is the configuration of Figure (3.6a). As is demonstrated
subsequently, a forward biased base-collector junction can be detrimental in integrated bipolar circuits and consequently, the diode configuration of Figure (3.6a) is rightfully deemed
the most efficacious of the three considered interconnections.
EXAMPLE #3.2:
A PNP bipolar junction transistor has a forward short circuit current gain, f, of
100 amps/amp, a reverse short circuit current gain, r, of 1.0 amp/amp, an
emitter-base junction injection coefficient, nf, of 1.025, and a base-collector
junction injection coefficient, nr, of 1.0. When operated at room temperature (27
°C), the collector current, Ic, in a particular circuit is measured to be 2 mA, while
the base current is determined to be 200 A. Calculate the emitter-collector
saturation voltage, Vecsat. Assume temperature saturation current, Is, of 2 fA at
room temperature.
SOLUTION #3.2:
(1).
Since Ic/Ib in this example is 2 mA/200A = 10, which is smaller than f, the subject transistor
is saturated. For the quoted parameters and Ic /Ib = 10, equation (3-23), which is applicable
for NPN and PNP units alike, delivers a recombination current ratio of Xce = Ibc/Ibe = 7.50.
At a junction temperature of Tj = 27 C, the Boltzmann voltage, VT, is VT = 25.89 mV. With
Xce = 7.50, (3-28) resultantly delivers an emitter-collector saturation voltage of Vecsat = 85.05
mV.
ENGINEERING COMMENTARY:
As confirmed above, the emitter-collector saturation voltage is a small, positive number. We
should appreciate, however, that the measured value of this metric is somewhat larger owing
to ohmic resistances in the collector and emitter leads of the transistor. The nature and effect
of these resistances are addressed shortly.
The fact that Xce is 7.50 indicates that the reverse, largely recombination, current precipitated
by base region mobile carriers and collector-injected carriers is 7.5-times larger than the
forward recombination current attributed to base region and emitter-injected carriers. In
general, Xce > 1 is the norm in that the base region doping concentration is larger than the
collector impurity concentration, which in turn is several orders of magnitude smaller than
the impurity concentration in the emitter volume.
(2).
3.2.4. ENHANCEMENTS TO THE EBERS-MOLL MODEL
Despite its better than half century age, the Ebers-Moll model remains a viable design
tool for manually estimating bipolar circuit performance. Viability notwithstanding, numerous
enhancements to the model demonstrably improve its accuracy, particularly for scaled, minimal
geometry, monolithic bipolar devices. A definitive discussion, yet alone a satisfying mathematical derivation, of all of the modeling improvements promoted by the archival literature in the almost six decades since the original publication of the seminal Ebers and Moll paper is beyond
the scope of this chapter. However, four specific modeling enhancements are potentially vital to
support realistic design-oriented analyses of monolithic circuits that exploit state of the art bipo- 205 -
Chapter 3
BJT Models & Biasing
lar devices in linear circuit applications. These improvements entail corrections for the transport
current in the face of high levels of emitter to base carrier injection, adjustments to the transport
current necessitated by base conductivity modulation, parasitic ohmic resistances in the base,
emitter, and collector, and incorporation of high frequency effects on the I/O transfer characteristics. The consideration of these enhancements in the subsections that follow pertain only to the
linear forward active regime of BJT operation (Ve ≥ Veon and Vc ≤ 0) since the focus of this chapter, and indeed the majority of this text, is circuit analysis that underpins the design and realization of linear active networks.
3.2.4.1. High Injection Phenomena
The Ebers-Moll model implicitly presumes low carrier injection conditions. For forward active operation, low injection is tantamount to restricting the concentration of injected
carriers from the emitter to the base to a level that is significantly smaller than the background
impurity concentration in the base volume. As the injected charge level rises toward the level of
the base dopant concentration, a high injection condition is initiated, which manifests impaired
carrier mobility owing to the traffic jam of minority carriers that arises in the base. One upshot
of degraded carrier mobility is an increased likelihood of carrier recombination, thereby resulting
in diminished charge transport into the collector volume and hence, a reduced collector current.
For the large geometry transistors of decades past and for present day discrete component, off the
shelf, transistors, low injection presumptions are suitable. But low injection presuppositions become progressively more questionable as transistor geometries shrink to the submicron scale that
is commonplace in the monolithic state of the art.
The Gummel-Poon model of a BJT accounts for the foregoing high injection effects by
supplanting the transport current, Icc, in (3-5) with a corrected, and indeed reduced, transport current, Ict. This corrected transport current is given by[4]
I cc
I cc
I ct 

,
(3-30)
I cc
I cc
1
1
I kf
Ae J kf
where Ikf is termed the forward knee current of the bipolar device, and Jkf is the corresponding
density of the forward knee current[5]. The knee current parameter is nominally proportional to
the average impurity concentration in the base volume. It assumes a barometric stature of
measuring the onset of high injection phenomena in the sense that Icc << Ikf is indicative of low
injection circumstances. When Icc, which remains given by (3-5), is significantly smaller than Ikf,
we see that Ict in (3-30) collapses to the conventional transport current advanced by Ebers and
Moll. Engineering logic alone dictates this result. For minimal geometry homojunction transistors, current Ikf is in the neighborhood of a few to low tens of milliamperes. Because the base
volume of a SiGe HBT is doped higher than is the base volume of a conventional BJT, SiGe
heterojunction BJTs boast larger values of parameter Ikf.
We can glean a fundamental understanding of the impact of high injection phenomena
on bipolar transistor performance by examining the model in Figure (3.7), which depicts the corrected form of the large signal bipolar model. Only an NPN model is drawn since the PNP
model is little more than its NPN counterpart with the current directions and voltage polarities
reversed. The model clearly establishes a collector current, Ic, of[6]
- 206 -
Chapter 3
BJT Models & Biasing
I c  I ct 
I cc
I cc
1
Ae J kf
.
(3-31)
At low injection levels,
Base
Ib

Ve

Ic

Vc0
Ibe

DBE
Ict
Collector
Vb
Ie

Emitter
Figure (3.7). Linear active region model of an NPN BJT with a
correction adopted for high injection effects on carrier
transport. The current, Ict, is given by (3-30). The indicated junction voltage, Ve, must be at least as large as the
threshold voltage, Veon, of the emitter-base junction.
V n V
I c  I ct  I cc  Ae J s e e f T ,
(3-32)
where the unity term in the parenthesized quantity on the right hand side of (3-5) is ignored in
deference to the requirement that Ve be at least as large as the emitter-base junction threshold
potential. On the other hand, at the high levels of carrier injection that attempt to manifest large
forward transport currents, Icc, we see that (3-31) approaches
I c  I ct 
Ve 2n f VT
Ae J kf I cc  Ae J kf J s e
.
(3-33)
Note that while the natural logarithm of the low injection collector current in (3-32) rises linearly
with Ve /nfVT, the logarithm of the high injection collector current in (3-33) rises less dramatically
with voltage Ve. Indeed, the subject logarithmic rate of rise is linear with Ve /2nfVT. It follows
that the sensitivity of the collector current to forward emitter-base junction voltage is reduced at
the onset of high injection phenomena. This reduced sensitivity at high injection makes
engineering sense. Since the traffic jam in the base of injected electrons impairs electron mobility, the prospects for base region recombination are enhanced. Even for those electrons that escape recombination, the impaired carrier mobility increases the average time required to
transport electrons across a strongly injected base. In effect, we can summarily argue that the
transport efficiency of electrons across the base-collector junction under high injection conditions no longer matches the emitter to base electron injection efficiency evidenced at low injection.
A related high injection phenomenological effect is an attenuation of the static current
transfer ratio, Ic /Ib. Recalling (3-3) and (3-8), the Ebers-Moll, or low injection, BJT model predicts Ic/Ib ≈ f in the linear domain. But noting that Icc = f Ib in this regime, (3-31) predicts
Ic
βf

.
(3-34)
Ib
I cc
1
Ae J kf
- 207 -
Chapter 3
BJT Models & Biasing
This current transfer relationship is traditionally symbolized as hFE, which is commonly called
the DC beta, or static beta, of a transistor. Somewhat distressingly, we now note that this current
transfer ratio is no longer a constant but is, in fact, a nonlinear function of the forward transport
current. At this stage of our model refinement, we have
Ic
βf
βf
(3-35)
hFE 


.
Ib
1  I cc Ae J kf
1  I cc I kf
As is demonstrated in discussions surrounding the design of suitable biasing networks for bipolar
technology circuits, hFE is a troublesome parameter largely because of its direct dependence on
parameter βf. In particular, the numerical value of βf suffers a numerical variance of at least three
to one owing to process uncertainties surrounding the realization of a deep submicron base
width. The problem stems from the fact that βf is inversely dependent on the base width, whose
control at submicron dimensions proves to be a daunting challenge. The early design lesson we
should grasp here is that bipolar circuit design should be accomplished to ensure that critical
performance metrics are not directly dependent on the vagarious parameter, hFE.
3.2.4.2. Base Conductivity Modulation
The BJT model in Figure (3.7) represents the collector-emitter port of the transistor as
an ideal controlled current source, Ict. This collector current is independent of the internal collector-emitter voltage, Vb, and thus, it is therefore independent of the base-collector junction voltage, Vc. But the transport efficiency of emitter-injected charge through the base and on into the
collector is intuitively promoted by applying a reverse bias across the base-collector PN junction.
This junction reverse bias can be achieved through increases in the internal collector-emitter voltage, Vb. We assert this recommendation because of (3-2). In that relationship, an increase in
voltage Vb presumably incurs an increase in voltage Ve, the forward biasing potential across the
intrinsic emitter-base PN junction. To be sure, a modest increase in Ve can be expected. But
large changes above the junction threshold level, Veon, are not possible because of the exponential volt-ampere nature of a PN junction diode. Thus, most of the increase afforded to voltage Vb
is transferred to increasing (−Vc), which is to say that an increase in Vb is tantamount to an enhanced reverse bias across the base-collector junction.
The enhanced transport current promoted by increases in voltage Vb is known as the
Early effect[7]2. Two engineering reasons serve to explain this transport current phenomenon.
First, an increased Vb results in increased reverse bias across the base-collector junction, as noted
above. This boost in reverse bias establishes stronger electric fields in the depletion layer
surrounding the base-collector junction. In turn, these robust fields exert attractive forces on the
mobile carriers that are injected into the base volume from the emitter region. Aside from
encouraging charge transport through the base volume and hence, increased collector current,
such attractive forces serve to inhibit recombination between emitter-injected and base region
carriers. Second, the increased reverse bias at the base-collector junction widens the junction
depletion layer about that junction. The immediate result of this depletion zone widening is a
narrowing of the neutral base width, which further decreases the likelihood of recombination of
injected carriers with majority carriers in the base. Stated quite simply, the charge neutral base
volume, where recombination is most likely to occur, is reduced to a base width −commonly
called the electrical base width− that is smaller than the geometric base width. Collectively, the
2
The “Early effect” has nothing to do with time. The name of this high injection effects derives from the work of
Dr. James Early, who was renowned technical creativity and leadership at Fairchild Semiconductor Corporation.
- 208 -
Chapter 3
BJT Models & Biasing
two reasons we have articulated herewith argue that increased base-collector reverse bias embellishes the conductivity of the neutral base so that charge transport through the entire base is
correspondingly promoted.
A physically sound model for the foregoing base conductivity modulation, or Early effect, proves too cumbersome for manual estimates of static BJT performance. Instead, an
empirical representation of the form,

V 
I cc  1  b 

Vaf 

V 
Vb

 I ct  1  b   I ct 
Ic 
,
(3-36)


V
V
I
1  I cc I kf
af
af
ct


is adopted for circuit level modeling, where Ict is given by (3-31) and voltage Vaf is an
experimentally discerned Early voltage parameter, whose value for minimal geometry transistors
is typically in the mid tens of volts. In contrast, Vaf can be in the neighborhood of the mid tens to
low hundreds of volts in SiGe HBTs. We hasten to interject that the modified volt-ampere
model postured by (3-36) is embedded in the basic BJT circuit models that are commonly embraced by SPICE simulation software. The resultant electrical model for the NPN device is
shown in Figure (3.8), where we note that the appended parenthesized term on the right hand
side of (3-36) establishes a current-dependent resistance. This resistance, whose value is inversely proportional to current Ict, is in shunt with the controlled source, Ict, which renders the
current source signifying current Ict a non-ideal source. In effect, the collector-emitter port is a
Norton equivalent circuit that boasts finite shunt resistance.
Ib
Ic
Base
Collector

Vc0


Vaf
DBE
Ict
Ve
Ict Vb

Ie

Emitter
Figure (3.8). Linear forward active region model of an NPN BJT with an account made of both high injection and base conductivity modulation (Early effect) phenomena.
We must now update the static current gain of the transistor to the slightly more
complicated form,


V 
V 
β f 1  b 
β f 1  b 


Vaf 
Vaf 
Ic


(3-37)
hFE 


.
Ib
I cc
I cc
1
1
Ae J kf
I kf
Observe that the behavioral expression for the Early effect in (3-36) and in the current gain
expression at hand somewhat mitigates the deleterious impact of high injection phenomena, at
least insofar as the current transfer ratio is concerned. This observation synergizes with the common circuit design practice of invoking only the Ebers-Moll model to deduce a quick and
reasonable first order estimate of static BJT performance.
- 209 -
Chapter 3
BJT Models & Biasing
3.2.4.3. Parasitic Resistances
The circuit in Figure (3.8) models only intrinsic transistor action in that the voltages
appearing at the terminals of the transistor are identical to internal emitter-base junction voltage,
Ve, internal base-collector junction voltage, Vc, and internal collector-emitter voltage, Vb. In
practical transistors, these internal voltages couple to the accessible external terminals of the device through series ohmic resistances that are associated with the charge neutral regions of the
emitter, base, and collector volumes. In other words, the internal junction voltages link to the
corresponding external transistor terminals through resistances.
rb
rc
Ib
Ic
Base
Vc0
Collector





Vaf
DBE
Ve
Ict Vb
Ict


Ibe
Vbe
Vce
Ie
re

Base

Ib
(a).
rb
Vc0



Ve

Veb
rc

Ict Vb

DBE
Ibe
Ic

Emitter
Collector
Vaf
Ict
Vec
Ie
re


(b).
Emitter
Figure (3.9). (a). The NPN BJT model of Figure (3.8) embellished by the incorporation of series
resistances re, rb, and rc in the emitter, base, and collector leads, respectively. For most
modeling circumstances, resistances re and rc can be ignored. (b). The PNP model
counterpart to the NPN model in (a).
The extremely high impurity concentration in the emitter volume, together with its
short geometric length, promotes a series emitter resistance, re, that is very small. Values of
resistance re that are larger than a few tenths of an ohm are rare. At the collector, a resistance, rc,
appears as depicted in the revised model of Figure (3.9), in which both NPN and PNP models are
offered in the interest of completeness. Because of the length of the collector region and its low
impurity concentration, resistance rc is relatively large. For minimal geometry devices destined
for broadband circuit applications, rc is in the range of mid tens of ohms to as much as low hundreds of ohms. Both rc and re scale inversely with the emitter-base junction area, Ae, and since
both are physical ohmic resistances, both contribute thermal noise to the total integrated output
- 210 -
Chapter 3
BJT Models & Biasing
noise generated by the biased transistor.
The base resistance, rb, like emitter resistance re and collector resistance rc, is an ohmic
element that generates thermal noise and scales inversely with the emitter-base junction area.
Owing to the inherently two-dimensional nature of the volt-ampere characteristics in the base
region and emitter crowding, rb, is current dependent[8],[9]. The phrase, “emitter crowding,” refers to an inability to establish uniformly constant voltage bias across the width of the emitterbase junction of a monolithic device. As a result, carrier injection densities crowd to those portions of the emitter-base junction that support comparatively larger forward biasing.
A commonly invoked empirical expression for the base resistance of a bipolar device
subjected to emitter crowding is
1
Norm. Base Resistance, (r b /r bb )
r bb /R bm = 2
r bb /R bm = 4
0.8
0.6
0.4
r bb /R bm = 20
r bb /R bm = 10
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Norm. Collector Current, (I c /I rb )
Figure (3.10). The dependence of the internal base resistance, rb, on the collector current flowing in a bipolar
junction transistor.
rbb  Rbm
(3-38)
,
1   I c I rb 
where Ic is the collector current, Irb is an experimentally deduced constant that scales with emitter-base junction area, Rbm is the minimum base resistance (typically tens of ohms as extrapolated
from high current measurements), and rbb is the zero bias (meaning Ic = 0) value of the measured
base resistance. For minimal geometry transistors, current parameter Irb is generally of the order
of one-half the transistor forward knee current, Ikf, while resistance rbb can be as large as a few
hundred ohms. For the same emitter-base cross section area, Rbm and rbb are smaller in SiGe
structures than they are in homojunction transistors since SiGe HBTs boast relatively large dopant concentrations in their base volumes. Figure (3.10) displays a plot of the normalized base
resistance, rb/rbb, versus the normalized collector current, Ic/Irb, for various resistor ratios, rbb/Rbm.
Notice that a better than two factor between the actual base resistance, rb, and its zero current
value, rbb, is possible for even reasonable values of collector current. Large base resistance
rb  Rbm 
- 211 -
Chapter 3
BJT Models & Biasing
generally reflects worst case operating conditions, in the senses of incurring reduced I/O voltage
gain, compromised bandwidth, and increased thermal noise contribution to the net integrated output noise of a transistor. For this reason, designers executing a pre-computer-aided design
assessment typically adopt the conservative worst case condition of allowing the base resistance
to equal its zero current value, which by (3-38) and our mathematical powers, is rbb.
3.2.4.4. Transistor Capacitances
A realistic assessment of the impact exerted by high signal frequencies on transistor circuit performance is a challenging undertaking that requires a careful consideration of charge storage and related memory effects that prevail within the semiconductor body of a BJT. At the circuit level, at least three capacitances must be incorporated into the transistor model to establish
an upper bound on the deleterious effects that high signal frequencies have on BJT circuit responses. These capacitances are the emitter-base capacitance, Cπ, the base-collector junction
depletion capacitance, Cμ, and the substrate depletion capacitance, Cs, as we depict in the high
frequency NPN BJT model set forth in Figure (3.11). An analogous model prevails for PNP devices, wherein all current directions, all voltage polarities, and all branch diodes are reversed.
rbb
C
rc
Ib
Ic
Base
Collector



Vc0



Vaf
DBE
C
Cs
Ve
Ict Vb
Vsc
DSC
Ict



Ibe
Vbe
Vce
Ie
re

Substrate

Emitter
Figure (3.11). High frequency model of the NPN BJT. The high frequency model of a PNP BJT is topologically
similar, with the provisos that the connections of both diodes are reversed, all voltage polarities are
reversed, and the directions of all currents are reversed. Ordinarily, the substrate terminal is
incident with signal ground.
As in the case of a forward biased PN junction diode, a shunt interconnection of two
capacitances model the high frequency effects at the forward biased emitter-base junction of a
BJT operated in the linear active mode. The dominant component of these two capacitances is
the emitter-base diffusion capacitance, Cbe. It is reminiscent of the diffusion component of the
net PN junction diode capacitance in that it models the effects of charge stored temporarily in the
volume prior to either its transport through the base volume or recombination with free holes that
reside in the base volume. This diffusion component, Cbe, of emitter-base junction capacitance is
given by
Cbe  τ fe g m ,
(3-39)
where τfe is the average time required to transport minority carriers through the base region and
gm represents the forward transconductance of the BJT. Parameter τfe is naturally affected by the
average mobility of the minority carriers that prevail in base region subsequent to their injection
from the emitter volume. Since minority carrier mobility diminishes with the increasingly large
carrier concentrations that necessarily underpin substantial forward transport current in a BJT, τfe
rises with progressively larger collector currents. In short, increased device capacitances and the
- 212 -
Chapter 3
BJT Models & Biasing
potentially slower circuit response speeds that may accompany this increase are expected when
BJTs are conduct relatively large collector currents. The degradation of transport time induced
by pronounced carrier concentrations in the base is somewhat mitigated by the imposed basecollector junction reverse bias, which exerts an attractive force on carriers injected into the base
from the emitter. Accordingly, a plausible semi-empirical relationship for the effective minority
carrier transport time, τfe, in (3-39) is
 1  I c I kf 
τ fe  τ fo 
(3-40)
,
 1  Vb Vaf 


where τfo is the low current (small Ic) and Vc = 0 value of the minority carrier transport time in
the BJT. Observe that the expression in question allows τfe to increase with progressive increases
in the collector current, Ic, which is given by (3-31). On the other hand, (3-40) permits τfe to decrease with increasing Vb, which accounts implicitly for the charge attraction ramifications of
reverse bias imposed across the base-collector junction. An important, engineering lesson surfaces herewith. In particular, increasing Vb reduces τfe, and hence device capacitance Cbe, which
conflates with the design goal of faster circuit responses. But Vb increases only if a suitably
larger voltage, Vce, biases the collector-emitter terminals of an NPN transistor (Vec for its PNP
brethren). In turn and for a given collector current, increased Vce increases the power dissipated
in a BJT, which upholds the philosophical banner that the likely price paid for improved circuit
response speeds or wider bandwidth is increased circuit power dissipation. In a word, if you
want more performance, the cost is more power dissipation, which translates directly to decreased battery life in portable electronics. Get your battery chargers ready!
Continuing with (3-40), parameter τfo is typically in the range of tens to hundreds of
picoseconds for minimal geometry transistors. For silicon-germanium HBTs, fo can be as small
as only a few tenths of a picosecond. Since parameter τfo is nominally proportional to the square
of base width Xb in Figure (3.1), a progressive narrowing of the base theoretically returns dramatic dividends insofar as reducing the emitter-base diffusion capacitance, Cbe. There are, however, limits to the tolerable base region narrowness. In particular, too narrow a base promotes a
voltage breakdown condition known as punch through. In this undesirable situation, a sufficiently lsrge reverse bias imposed at the base-collector junction allows the base side boundary of
the base-collector depletion layer to encroach upon, and even coalesce with, the base-side boundary of the emitter-base depletion layer.
Finally, the transconductance metric in (3-39) is, using (3-36),

I c
Ic 
Ic
V 
1 
1  b  .
gm 

(3-41)
Ve
n f VT 
Vaf 
2 I cc I kf  


Assuming that voltage Vb is much smaller than the Early voltage, Vaf, as is often the case in welldesigned, high performance BJT networks, (3-36) combines with (3-41) to deliver
I c  1  I cc 4I kf 

.
gm 
(3-42)
n f VT  1  I cc I kf 


The last result portrays the forward transconductance as increasing linearly with collector current
Ic with a slope, dgm/dIc, of roughly 1/nfVT at the low injection levels implied by Icc << Ikf. Under
high injection conditions, transconductance gm continues to increase with Ic, but at a slope,
dgm/dIc, that is a factor of two smaller than that which we witness at low current levels. We are
lucky, for yet another electronics lesson surfaces for us without additional financial charge.
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Chapter 3
BJT Models & Biasing
Specifically, transconductance gm is a measure of attainable I/O gain in a BJT network. Increased gain potential accrues with increased gm, which mandates robust collector current. But
the aforementioned slopes suggest that diminishing returns accrue as currents approach levels
commensurate with the high injection state. And, of course, increased collector currents, just
like increased collector-emitter voltage, promote the dubious distinction of a circuit that burns
more power.
Although the emitter-base junction is forward biased in the linear active mode of BJT
operation, a narrow depletion layer nonetheless prevails at this PN junction. Accordingly, the
second component of the net emitter-base junction capacitance is the depletion capacitance, Cje,
of the junction depletion layer. Under forward biased circumstances, this capacitance, which is
analogous to the depletion component of capacitance prevailing in a forward biased PN junction
diode, is given approximately by
C jeo
C je 
,
(3-43)
me


V
 1  eon 
V je 

where Cjeo is the zero bias (meaning Ve = 0) value of the junction depletion capacitance, Vje is the
built-in potential of the emitter-base junction, and me is the grading coefficient of the subject
junction. The potential, Vje, is typically in the range of 800 mV to 900 mV, while parameter me is
usually close to 0.5. It is understood that capacitance Cjeo scales linearly with the emitter-base
junction area, Ae. In summary, the net emitter-base junction capacitance, Cπ, as delineated in
Figure (3.11), is, recalling (3-39), (3-40), (3-41), and (3-43),

τ I 
C
I 
I
V 
fo c 
jeo

1  b  
c
c
 1
C  C C 
1
. (3-44)

be
je


m




n V
I
V
2
I
I


e


f T 
kf  
af 
cc kf  
V
 1  eon 

V 
je 

Since the base-collector junction of a BJT is reverse biased under forward active
circumstances, the capacitance, Cμ, in Figure (3.11) is exclusively a depletion capacitance.
Accordingly, it is given by the familiar mathematical form,
C jco
(3-45)
Cμ 
.
mc


V
 1  c 
V jc 

In (3-35), Cjco is the zero bias (meaning Vc = 0) value of the base-collector junction depletion
capacitance, Vjc is the built in potential of the base-collector junction (of the order of 700 mV to
800 mV), and mc is the grading coefficient of the junction. Typically, 1/3 ≤ mc ≤ 1/2. Capacitance Cjco, like Cjeo, scales linearly with the emitter-base junction area.
3.2.4.5. Substrate Capacitance and Other Monolithic Parasitics
Monolithic technology has swallowed most of the recently developed, low power, state
of the art electronic systems. It is easy to understand the popularity −indeed dominance− of integrated circuits in light of the increased response speed and enhanced bandwidth, radically diminished size, improved reliability, and a plethora of other advantages that pervade the monolithic
- 214 -
Chapter 3
BJT Models & Biasing
art. But while the positive attributes of integrated circuit technologies outweigh their negative
aspects, there are, to be sure, potential shortfalls that must be carefully assessed, understood, and
mitigated. One such downside is the high cost of integrated circuit realization. The high cost of
manufacturing that first wafer of acceptable high performance chips might not be justified unless
the market for the realized circuit chips is sufficiently large. Testing issues are also demanding,
largely because the small surface footprints associated with a monolithic realization of an electronic network tempts us to incorporate progressively more features on chip. Many of these
additional features commonly address enhanced information processing and transmission
requirements, which mandate significantly wide bandwidths that may prove challenging to
achieve reproducibly and reliably. These wide bandwidth circuits often prove difficult to test
satisfactorily. Other integrated circuit shortfalls, as we shall discuss herewith, embrace substrate
parasitics that, in addition to incurring bandwidth degradation, can actually be so severe as to
cause catastrophic, thermally induced chip failures. Fortunately, thoughtful design approaches
premised on insightfully understood circuit and system concepts usually produce viable, if not
creative, solutions to the dilemmas encountered.
Substrate Collector
Base
Emitter
Intrinsic
Transistor
N+ Collector Contact
N+ Type
Emitter
P-Type
Base
N-Type
Collector
P+ Substrate Contact
P+ Substrate Contact
Silicon Dioxide
N+ Buried Layer
P-Type Substrate
Figure (3.12). Simplified cross section representation of a monolithic NPN bipolar junction transistor. The diagram is not drawn to scale. The section of the diagram enclosed by the dotted rectangle is the
intrinsic transistor. It corresponds to the cross section abstraction appearing in Figure (3.1a).
We initiate our focus on substrate issues by examining carefully the substrate diode,
DSC in Figure (3.11), and its associated substrate capacitance, Cs. To this end, a simplified cross
section of an NPN monolithic BJT is provided in Figure (3.12). The vertical part of the diagram
enclosed by the dotted rectangle lying directly under the emitter terminal and consisting of the
n+-emitter, the p-type base, and the n-type collector is the intrinsic transistor and corresponds to
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Chapter 3
BJT Models & Biasing
the simplified BJT structure in Figure (3.1a). Since all terminal contacts are necessarily formed
at the surface of the structure, the collector metal contact is incident with a heavily doped diffused or implanted collector contact to reduce the parasitic series resistance associated with the
collector. A further reduction in this series resistance is fomented by assuring that this n+-collector contact is deep enough to coalesce with the n+ buried layer that abuts the lightly doped intrinsic collector region. The entire structure rests atop a lightly doped p-type substrate to which
electrical accessibility is provided at the surface through dual p+-substrate contacts. Aside from
allowing an electrical contact to the substrate foundation of the NPN transistor, these deep substrate contacts create a mechanism to isolate the subject NPN device from proximately located
other chip transistors. The isolation proves effective if the substrate terminal at the transistor surface is incident with the most negative potential afforded by the circuit in which the NPN transistor is electrically embedded. This electrical connection ensures nominally non-conductive, reverse biased PN junctions around the periphery of the NPN unit. In effect, the reverse bias at the
collector-substrate junctions renders the NPN transistor a virtual electrical island in a sea of
potentially many other active and passive devices that are incorporated on the subject chip. Electrical contact is similarly made to the p-type base by doping the base region adjacent to the emitter region sidewalls at a level that is higher than the impurity concentration of the intrinsic base
immediately below the emitter. In addition to mitigating the effects of work function phenomena
between metal contact and semiconductor and reducing the series base resistance, this doping
gradient minimizes carrier injection from the emitter to the base along the emitter sidewalls, thereby rendering the simplistic diagram in Figure (3.1a) reasonably reflective of the transistor action implied by the cross section under present consideration.
Our study of the diagram in Figure (3.12) reveals that in addition to realizing the desired vertical NPN transistor, a parasitic PNP device, QP, is forged with the p-type substrate, the
n-type collector and the p-type base. While a parasitic PNP transistor is unquestionably formed,
questions can be raised as to whether its emitter is the NPN substrate or the base region of the
desired NPN unit. These questions can be answered only when the nature of the electrical
connection made to the substrate contact is clarified.
Collector
Collector
Base
Base
QP
Emitter
QP
Substrate
Emitter
(a).
Substrate
(b).
Figure (3.13). (a). Behavioral representation of a monolithic NPN transistor for the case in which the substrate is biased to a potential that is larger than the potential applied to the collector of the
NPN device. (b). Behavioral representation of a monolithic NPN transistor for the case in
which the substrate is returned to the most negative circuit potential afforded by the circuit
into which the NPN transistor is embedded.
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Chapter 3
BJT Models & Biasing
To the foregoing end, consider the atypical case in which the substrate is not connected
to the most negative of available circuit potentials and instead supports a voltage with respect to
ground that is larger than the voltage developed at the collector of the NPN transistor. Because
the substrate-collector junction is forward biased, the substrate can rationally be viewed as the
emitter of parasitic transistor QP, and the resultant circuit level abstraction becomes the structure
we show in Figure (3.13a). The forward bias across the substrate-collector junction allows holes
from the substrate to be injected into the NPN collector region, particularly in the substratecollector junction neighborhood lying to the right of the buried layer. Given that the base-collector junction of the NPN device is reverse biased for forward active operation, the base-collector
junction of QP is correspondingly reverse biased. Traditional transistor action through the PNP
device is therefore fostered, which means that the base current conducted by the vertical NPN
device is perturbed by the collector current of the parasitic PNP transistor. In extreme cases, the
observed base current of the vertical NPN transistor may actually flow out of the device, as opposed to its normal directional flow, which is into the NPN base terminal. Assuming that the DC
beta of QP is greater than one, the immediate impact of this enhanced NPN base current is a
reduction of the available current transfer ratio, hFE, for the NPN unit. Obviously, the power
dissipation of the overall structure increases because the substrate now conducts the current that
is demanded by the emitter of transistor QP. This substrate flow can cause potentially damaging
heating effects in the relatively high resistivity substrate volume.
When, as is routinely required in linear signal processing applications, the substrate is
connected to the most negative circuit potential, the substrate-collector PN junction is reverse
biased. Accordingly, the substrate is precluded from serving as a PNP transistor emitter that injects holes into the NPN collector, or equivalently, the PNP base. Instead, the substrate acts as
the collector region of transistor QP, thereby permitting the base of the NPN structure to be delineated as the emitter of QP, as shown in Figure (3.13b). The routine reverse biasing of the basecollector junction of the NPN transistor operated in its forward active mode imposes a reverse
bias across the emitter-base junction of transistor QP. Consequently, QP conducts only leakage
current and therefore exerts minimal impact on the operation of the target NPN device, least at
low signal frequencies. At high signal frequencies, the base-collector depletion capacitance of
QP can be absorbed into capacitance Cμ of the NPN transistor. Moreover, the reverse biased PN
junction diode observed at the QP emitter-base junction shunts the reverse biased diode evidenced at the NPN base-collector junction. The reversed biased substrate-collector junction can
be modeled as a shunt interconnection of a reverse biased diode, DSC, and the depletion capacitance, Cs, associated with the substrate-collector junction. The foregoing capacitance item assumes its usual depletion capacitance form,
Cso
(3-46)
Cs 
,
ms


V
 1  sc 
V js 

where Vsc is the reverse biased, and therefore negative, voltage established across the substratecollector junction, Vjs is the built in potential of said junction, ms is the grading coefficient of the
subject junction, and Cso is the Vsc = 0 value of the substrate-collector depletion capacitance. For
a properly biased, minimal geometry, high speed NPN transistor, Vjs is in the range of 550 mV to
700 mV, ms is of the order of one-third, and Cso is generally a few femtofarads to tens of femtofarads.
- 217 -
Chapter 3
BJT Models & Biasing
The base-collector junction of the NPN transistor is forward biased when this device
enters its saturation domain. Correspondingly, the emitter-base junction of the parasitic PNP device in Figure (3.13b) mirrors this forward biasing, which, along with the imposed reverse bias at
the substrate-collector junction, forces QP to conduct and actually to function in its forward active domain. The result is a degraded short circuit current gain in the NPN transistor, whose base
must now supply current to the PNP emitter. Aside from degraded current gain, the power
dissipation of the entire structure increases and undesirable heating of the substrate is incurred
because of the collector current manifested in transistor QP. These observations support our earlier contention that BJT operation in saturation must be avoided.
3.3.0. SMALL SIGNAL MODEL
We now know that in order for a bipolar junction transistor to function nominally as a
linear signal processor, it must be biased in its forward active domain. As we have previously
asserted, the prerequisites that cultivate operation in this linear domain are that the emitter-base
junction voltage, Ve, must be at least as large as the junction threshold voltage, Veon, and the basecollector junction voltage, Vc, must be negative or zero. The latter requirement is equivalent to
stipulating an internal collector-emitter (or emitter-collector in PNP devices) voltage, Vb, which
is equal to or greater than the emitter-base junction voltage, Ve. For monolithic devices, an additional necessity is that the substrate terminal be returned to the most negative potential available
in the circuit into which the transistor is embedded. These biasing constraints coalesce to establish a quiescent collector current, say IcQ, and corresponding base and emitter quiescent currents,
IbQ and IeQ, respectively. Specifically,
IbQ  I cQ hFE
I eQ  IbQ  I cQ 
 hFE  1 IbQ

I cQ
 I cQ 
 
αFE
 hFE 
 hFE  1 
where hFE is given by (3-37), and
I cQ
hFE

αFE 
I eQ
hFE  1
,
(3-47)
(3-48)
is nearly unity because hFE >> 1. These three static BJT currents are determined largely by the
quiescent emitter-base junction voltage, VeQ, and hence, they are essentially stipulated by the Qpoint value of the voltage, VbeQ, applied to the emitter-base terminal. The three BJT currents are
far less sensitive to the Q-point collector-emitter terminal voltage, VceQ. From Figure (3.11), we
see that voltages VbeQ and VceQ can be expressed as,
VbeQ  I bQ rbb  VeQ  I eQ re
(3-49)
.
VceQ  I cQ rc  VbQ  I eQ re
Since VeQ > Veon and all transistor currents are nonzero and positive in the forward active regime,
VbeQ > VeQ > Veon. In actual practice, the Q-point base current and series emitter resistance are so
small that VbeQ can be approximated by VeQ. Moreover, recall that the requirement that VcQ, the
Q-point value of the internal base-collector voltage, not exceed zero gives rise to the static
operating constraint, VceQ  VbeQ.
In the forward active domain, the small signal model of either an NPN or a PNP bipolar
junction transistor is the structure depicted in Figure (3.14). The subject model is burdened by
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Chapter 3
BJT Models & Biasing
two caveats. First, it is presumed that the model is typically exploited to arrive at conservative
estimates of bipolar circuit performance. Thus, the current-dependent resistance, rb in (3-38), is
supplanted by the maximum anticipated base resistance, rbb, which simplifies relevant circuit
analyses in that resistance rbb is current invariant. Second, the substrate is presumed connected
to the most negative of available circuit potentials. This means that the substrate-collector PN
junction diode, DSC in Figure (3.11), can be ignored for small signal analyses since it is reverse
biased and therefore conducts only small leakage currents. In light of the fact that the substrate
is connected purposefully to a suitable constant potential that is either circuit ground or a static
voltage lying below circuit ground, the substrate lies at signal ground potential. In other words,
no signal is supported by the substrate terminal. As witnessed in the small signal model of a PN
junction diode, the BJT model at hand is capable of establishing only the appropriate linear
interrelationships among the signal components of all node voltage and branch current variables
in the transistor model. It cannot predict Q-point voltages and currents. Indeed, the parameters
of the small signal model rely a priori on the numerical values of these Q-point currents and voltages.
rbb
rc
C
Base

V

Collector
r
C
I
gmV
ro
Cs
Substrate
re
Emitter
Figure (3.14). Approximate small signal model of either an NPN or a PNP bipolar junction transistor. The model presumes that the transistor undergoing study is biased in its forward
active domain, which in turn presupposes that the substrate terminal is incident with
the signal ground of the circuit into which the transistor is embedded. In the absence
of specific model information, resistances rc and re are often ignored.
In the model of Figure (3.14), rbb, re, and rc, are physical resistances that account for
voltage drops incurred by currents flowing in the charge neutral regions of the base, emitter, and
collector, respectively. On the other hand, the resistance, rπ, is a mathematical artifact arising
from a linear representation of the volt-ampere characteristics indigenous to the emitter-base
junction diode. In particular, if the dependence of the forward base current component, Ibe, on
internal emitter-base junction voltage is approximated by only the linear terms in its Taylor series expansion about the transistor operating point established in part by Ve = VeQ,
I
AJ
I
V n V
Ibe  cc  e s e e f T  1  IbeQ  be
Ve  VeQ .
(3-50)
βf
βf
Ve Q




With reference to Figure (3.14) and our experience with the small signal modeling of the PN
junction diode, (Ve − VeQ) in this relationship is the signal-induced change, V, across the internal
emitter-base junction terminals and (Ibe − IbeQ) is the corresponding signal induced change, I, in
the forward base current. Accordingly, resistance rπ in the subject model derives from
I beQ
Ae J s Ve n f VT
I be
1
.

e
(3-51)


rπ
n f VT β f
n f VT
Ve Q
Q
Given that current IbeQ closely approximates the quiescent transistor base current, IbQ, in the linear forward region of operation, (3-47) combines with (3-51) to deliver
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Chapter 3
rπ 
BJT Models & Biasing
n f VT
I beQ

n f VT
I bQ

hFE n f VT
I cQ
(3-52)
.
We see that for a given quiescent collector current, IcQ, the small signal resistance, rπ, is directly
proportional to the gain metric, hFE, whose measured value suffers significant variance due to
processing uncertainties. Additionally, the inverse dependence of rπ on Q-point collector current, along with the direct dependence of Boltzmann voltage VT on absolute temperature, renders
it dependent on junction temperature Tj. It is possible to mitigate the temperature dependence of
resistance rπ through a biasing design that ensures a quiescent collector current proportional to
absolute temperature. This specialized type of bias design is commonly referred to as a PTAT
design.
Like rπ, the Early resistance, ro, in Figure (3.14) is also a mathematical artifact. But in
contrast to rπ, which derives from a consideration of the volt-ampere characteristics of the emitter-base junction, ro pertains to a different volt-ampere consideration. In particular, ro is related
to a coefficient in a linear truncation of the Taylor series expansion of the function that relates
transistor collector current to internal emitter-base junction voltage, Ve, and collector-emitter voltage, Vb. From (3-36) and (3-5), the pertinent two-variable Taylor series expansion of the collector current, Ic, is
I c
I c
I c  I cQ 
Ve  VeQ 
Vb  VbQ .
(3-53)
Ve Q
Vb Q




The coefficient of (Ve − VeQ) on the right hand side of this relationship is the Q-point value of the
transconductance, gm, which we introduced in (3-41); that is,

I cQ 
I cQ
V 
I c
1 
  1  bQ  .
gm 

(3-54)
n f VT 
Vaf 
Ve Q
2 I ccQ I kf  


On the other hand, the coefficient of (Vb − VbQ) produces an expression for the Early resistance
ro. In particular, (3-36) yields
I cQ
1
I c
I
 ct

,

(3-55)
ro
Vb Q
Vaf
VbQ  Vaf
Q
whence
ro 
VbQ  Vaf
.
I cQ
(3-56)
In most cases, voltage VbQ in (3-54) and (3-56) can be supplanted by the Q-point collector-emitter voltage, VceQ, since the quiescent voltage drops in both the collector resistance, rc, and the
emitter resistance, re, are usually small enough to justify their tacit neglect. Thus,


I cQ 
I cQ
V
1 
  1  ceQ 
gm 
n f VT 
Vaf 
2 I ccQ I kf  


.
(3-57)
VceQ  Vaf
ro 
I cQ
Clearly, resistance ro in (3-57) is large for very small collector bias currents, IcQ, as well
as for large Early voltages, Vaf. With reference to the small signal model of Figure (3.14), it fol- 220 -
Chapter 3
BJT Models & Biasing
lows that at low signal frequencies, where all device capacitances can be ignored, the collectoremitter port of a BJT emulates an ideal voltage controlled current source (VCCS) for low current
biasing. However and despite this low frequency VCCS persona of the collector-emitter port,
the transistor does not behave as an ideal transconductance element because of the finite input
impedance evidenced between the base and emitter terminals. In other words, this VCCS postures finite, as opposed to infinitely large, input impedance. This situation encourages
representing the collector-emitter output port as a current controlled current source (CCCS) that
is controlled by the signal current, I, flowing as shown through resistance rπ. Since the model at
hand verifies V = rπ I,
gmV  gm rπ I  βac I ,
(3-58)
where, by (3-57) and (3-52), the so-called signal beta, or AC beta, βac, of a bipolar junction
transistor is


I cQ
V 
  1  bQ  .
βac  gm rπ  hFE  1 
(3-59)

Vaf 
2 I ccQ I kf  


rbb
rc
C
Base

V

Collector
r
I
C
ac I
re
ro
Cs
Substrate
Emitter
Figure (3.15). Alternative small signal model of the BJT. Note herein that in contrast to the small
signal model in Figure (3.14), the collector-emitter port contains the current controlled current source, βacI, whose controlling current, I, is conducted by the model
resistance, rπ.
We should take note of the fact that βac, is directly proportional to its static gain counterpart, hFE,
which in turn is directly dependent on the vagarious short circuit gain parameter, bf.
Figure (3.15) depicts the alternative CCCS model, for which prudence dictates our
underscoring that current I is not the base current at all signal frequencies. In particular, the net
signal base current must supply current to the branch represented by resistance r, and the currents conducted by the capacitive formed of Cπ and Cμ. In short, current I in the βacI generator is
always only the signal base current component that flows through the resistance, rπ.
EXAMPLE #3.3:
The diode-connected transistor in Figure (3.6a) operates in a circuit that allows
the silicon technology transistor to conduct a quiescent collector current, IcQ, of 4
mA. A room temperature (27 °C) characterization of the transistor at hand
reveals a maximum base resistance, rbb, of 140 Ω, a nominal series collector
resistance, rc, of 20 Ω, and a nominal emitter resistance, re, of 0.9 Ω. Moreover,
the DC beta, hFE, of the considered device is found to be at least 120 amps/amp,
the knee current, Ikf, is 12 mA, the Early voltage, Vaf, is 45 volts, and the
saturation current, Is, is 10 fA. Use these numerical data to compute the low
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Chapter 3
BJT Models & Biasing
frequency small signal parameters of the transistor. Exploit these parameters to
determine the low frequency, small signal terminal resistance, say rd, of the
diode-connected structure.
Assume an emitter-base junction injection
coefficient of nf = 1.
SOLUTION #3.3:
(1).
In the diode interconnection of Figure (3.6a), the measurable quiescent collector-emitter
voltage, Vd, of the transistor is identical to the quiescent base-emitter voltage, say VbeQ. In
silicon, VbeQ, and hence voltage Vd, is rarely larger than 800 mV. Owing to the voltages
dropped across the internal collector and emitter resistances, the internal Q-point collectoremitter voltage, VbQ, in Figure (3.9a) is therefore in the neighborhood of 800 mV. This
collector-emitter potential is assuredly smaller than the forward Early voltage, Vaf = 45 volts.
It follows that the collector current in (3-36) is essentially the net transport current, Ict.
Setting IcQ = Ict in (3-30), the Q-point Ebers-Moll transport current, IccQ, can therefore be
determined. Unfortunately, a quadratic relationship must be solved to arrive at IccQ. The
relevant fruit of this admittedly annoying exercise is

I cQ 
4I kf  
1  1 
   7.07 mA .
I ccQ  I cQ 1 
(E3-1)
2I kf 
I cQ  



(2).
With nf = 1, Is = 10 fA, IccQ = 7.07 mA, and, recalling (3-7), VT = 25.89 mV for a junction
temperature of Tj = 300.16 °K, (3-5) gives a Q-point voltage, VeQ, developed across the
intrinsic emitter-base junction diode of VeQ = 706.36 mV. Now, for IcQ = 4 mA and hFE = 120
amps/amp, the Q-point base current, IbQ, is IbQ = IcQ /hFE = 33.33 A, while the Q-point
emitter current, IeQ, is IeQ = (hFE+1)IbQ = 4.03 mA. These currents, the computed internal
emitter-base junction voltage of VeQ = 706.36 mV, the given base resistance of rbb = 140 Ω,
and the emitter resistance of re = 0.9 Ω imply, via (3-49), a quiescent base-emitter terminal
voltage, VbeQ, of
VbeQ  IbQ rbb  VeQ  I eQ re  714.66 mV .
(E3-2)
This voltage is only slightly more than 8 mV above its intrinsic junction voltage counterpart.
Since VbeQ is identical to voltage Vd in Figure (3.6a), the internal collector-emitter voltage,
VbQ, at the quiescent operating point of the diode-connected transistor is
VbQ  Vd  I cQ rc  I eQ re  631.03 mV .
(E3-3)
Note that voltage VbQ is more than 71-times smaller than the forward Early voltage, Vaf, of the
transistor, which assuredly validates our previously invoked approximation of IcQ  Ict in (330).
(3).
The foregoing calculations set the table for evaluating the small signal parameters of the
considered BJT. Using (3-52), the small signal emitter-base junction resistance, rπ, is rπ =
776.66 Ω. From (3-54) the forward transconductance, gm, is 122.66 mS, and since βac = gmrp,
the AC beta of the transistor is βac = 95.26 amps/amp. Finally, VbQ = 631.03 mV, Vaf = 45 V,
and IcQ = 4 mA in (3-56) yield an Early resistance, ro, of ro = 11.41 KΩ.
(4).
If we lean on Figure (3.15), we can posture Figure (3.16) as the low frequency, small signal
model of the diode-connected transistor in Figure (3.6a). All of the device capacitances in
the model of Figure (3.15) are tacitly ignored because at present, our interest lies on only low
frequency diode characteristics. The desired small signal terminal resistance, rd, derives from
the “ohmmeter” voltage to current ratio, Vx/Ix. An analysis of the structure at hand provides
Vx   rbb  rπ  I  re I x
(E3-4)
and
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Chapter 3
BJT Models & Biasing
I
rbb

V

Ix

Vx
Ix

rc
r
ro
ac I
Ix  I

Vx
Ix  (ac+1)I
re
I
Ix

Ix
Figure (3.16). Low frequency, small signal model of the diode connected transistor in Figure
(3.6a). The effective, or small signal terminal resistance, rd, of the diode
connection is determined via the “ohmmeter” method, which entails a calculation
of the voltage to current ratio, Vx /Ix, of our imposed mathematical ohmmeter.
(E3-5)
Vx  rc  I x  I   ro  I x   βac  1 I   re I x .
Upon elimination of current I in these two relationships, the terminal resistance, rd, of the
diode is readily shown to be
rd 
 ro  rc   rbb  rπ 
Vx
 re 
 10.43 ohms .
βac ro
Ix
1
ro  rc  rbb  rπ
(E3-6)
Since ro is a large resistance, (E3-6) can be approximated as
V
r r
rd  x  re  bb π  10.42 ohms ,
(E3-7)
Ix
βac  1
which differs insignificantly from the “exact” result in (E3-6).
ENGINEERING COMMENTARY:
The computed small signal diode resistance of 10.43 Ω is comparable to the resistance
presented to a circuit by a traditional PN junction diode. This resistance is small because of
the large forward transconductance, and hence large AC beta, afforded by a BJT operated at
reasonable biasing levels. In stark contrast, the resistance presented by a diode-connected,
deep submicron technology MOSFET is ultimately shown to be considerably larger owing to
the relatively small forward transconductance typically afforded by a MOS technology
transistor.
rbb
Ix

Vx

Ix

V

I
r
I
rc
Ix  I
ac I

Vx
re
Ix

Ix = (ac+1)I
Figure (3.17). The equivalent circuit in Figure (3.16) with the Early resistance, ro, ignored.
The approximate diode resistance stemming from the tacit neglect of the Early resistance is
worthy of further exploration. To this end, the pertinent small signal model is offered in
Figure (3.17), wherein the removal of resistance ro postures the controlled current source,
βacI, as an ideal current generator that projects infinitely large resistance across the collector-
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Chapter 3
BJT Models & Biasing
emitter terminals of the transistor. The series interconnection of the internal collector
resistance, rc, with this infinite resistance current source renders rc inconsequential, regardless
of its value. Moreover, since the resistance, re, is in series with the port at which the
mathematical ohmmeter is applied to evaluate the diode resistance, the resistance, (rbb + rπ)
evidenced in the base circuit, is necessarily referred to the emitter. In turn, we know that the
emitter conducts a signal current that is larger than the signal current in the base by a factor of
(βac + 1). Accordingly, it is reasonable to expect, as is confirmed by (E3-7), that the effective
resistance seen in series with re is the resistance observed in the base, scaled by a factor of the
base current to emitter current ratio, 1/(ac + 1).
3.3.1. UNITY GAIN FREQUENCY
IcQ + Ics
Sophisticated communication and data processing systems require large bandwidths to
process and transmit perpetually increasing amounts of audio, video, or digitized information. It
is only natural that engineers tasked with realizing these high performance systems insist on being provided with convenient circuit level metrics that quantify the high frequency capabilities of
the active devices available for possible deployment in these systems. A commonly adopted, but
not necessarily the meaningful or relevant, metric is the short circuit, unity gain frequency, fT,
which is commonly referenced as simply the unity gain frequency.
rbb
rc
C
VceQ
Ibs

V

r
C
gmV
re
IeQ + Ies
Vbe 
IbQ
Cs
Ics

Ibs
ro
Ies
(a).
(b).
rbb
Ibs
C

V

r
I
C
gmV
Ics
Ies
(c).
Figure (3.18). (a). Test circuit used in the measurement of the unity gain frequency, fT, of a BJT. (b). High frequency model of the test structure in (a). (c). The model in (b) simplified by ignoring the collector
resistance, rc, and the emitter resistance, re.
In order to facilitate our understanding of the engineering implications of metric fT, we
consider the test bipolar cell in Figure (3.18a). In this structure, the transistor, whose emitter is
connected to electrical ground, is biased in its linear regime by the quiescent base current, IbQ,
and the static voltage source, VceQ. In the interest of analytical simplicity, both the static current
source and the static voltage source are treated as ideal sources of energy; that is, no Thévenin
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Chapter 3
BJT Models & Biasing
impedances are identified in conjunction with these energy sources. Transistor operation in its
forward active domain relies on selecting voltage VceQ to ensure that it exceeds the quiescent
emitter-base voltage, VbeQ. In turn, VbeQ is presumably rendered larger than the turn on voltage
of the emitter-base junction by static current IbQ. A sinusoidal test signal with small current
amplitude Ibs is applied in shunt with the aforementioned source of static base current. The combined action of IbQ and Ibs incurs a net collector current comprised of the superposition of a Qpoint current, IcQ, and a signal component, Ics, as we highlight in Figure (3.18a). Similarly, a Qpoint current, IeQ, superimposes with a signal current, Ies, in the emitter lead of the transistor.
Before proceeding further, it may be instructive to address our earlier allegation to the
effect that the unity gain frequency lacks meaning and relevance as a circuit level figure of merit.
To this end the fT test cell in Figure (3.18a) itself offers cause for skepticism. In particular, no
rational design engineer is going to design a grounded emitter amplifier in which the collector of
the utilized BJT is grounded through the supply line voltage. Aside from offering no possibility
of an output signal voltage response at the collector, note that short circuiting the collector to signal ground all but shorts out the transistor substrate capacitance, which can establish a significant
time constant at the output port. This shorting, along with remanding the base-collector junction
capacitance, Cμ, as a simple base to ground capacitance, has the effect of unrealistically increasing the theoretically achievable 3-dB bandwidth of the common emitter stage. A second reason
for skepticism associated with the viability of the fT metric, is the applied signal source in the
subject test cell is a current, as opposed to the voltage signal that commonly activates a common
emitter amplifier. Currents are conducted instantaneously by capacitances; in this case, the basecollector and net emitter-base junction capacitances. Thus, the immediate effect of an input current signal is to deemphasize the critical importance of the time required to charge these transistor capacitances to appropriate signal voltage levels, which in turn liberalizes bandwidth estimates. In a word, fT is indeed a measure of bandwidth but unfortunately, it is an overly generous
bandwidth measure.
Using the model of Figure (3.14), the small signal equivalent circuit of the test structure
in question is the network shown in Figure (3.18b). Since IbQ in Figure (3.18a) is a constant current source, it does not appear in the small signal model. In effect, IbQ is replaced by an open circuit to reflect the fact that IbQ embraces no signal fluctuations and therefore offers zero signal
current value. Similarly, the collector in the equivalent circuit is grounded to the emitter because
VceQ is a constant. It is therefore a source of zero signal voltage. Because IbQ and VceQ are
necessarily set to zero for small signal analysis purposes, the only current flowing in the collector
of the equivalent circuit is the signal component, Ics, of the net collector current. Similarly, the
signal emitter current, Ies, is the lone current flowing in the emitter lead of the model.
If we ignore the series collector resistance, rc, and the series emitter resistance, re, we
can collapse the model in Figure (3.18b) to the simplified topology presented in Figure (3.18c).
Observe that setting rc to zero short circuits the substrate capacitance, Cs, while setting both rc
and re to zero imposes a short circuit across the Early resistance, ro. Moreover, ignoring resistances rc and re allows the inherently three-pole structure in Figure (3.18b) to be emulated by the
simple single pole, or single time constant, circuit in Figure (3.18c). To the latter end, note that
the short circuit imposed at the collector port effectively connects capacitance Cμ from base to
signal ground and in parallel with capacitance Cπ.
Obviously, the frequency response of the I/O current transfer function, Ics /Ibs, can be
extracted from the measurement test cell in Figure (3.18a). Indeed, this transfer function, which
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Chapter 3
BJT Models & Biasing
is denoted herewith as β(j), can be determined straightforwardly from the simplified model
given in Figure (3.18c). In particular, the model advances
V
 jω Cπ  C μ V
I bs 
rπ
(3-60)
,
I cs  g mV  jωC μV


whence



βac 1  jωC μ gm
gm  jωC μ
I cs


,
(3-61)
1
I bs
1  jωrπ Cπ  C μ
 jω Cπ  C μ
rπ
where (3-59) has been used. This result confirms the anticipated low frequency, short circuit
current gain of βac. It also projects a right half plane zero at a frequency of gm /Cμ. The frequency of this zero is invariably large owing to the relatively small base-collector depletion
capacitance, Cμ, and the usually substantial forward transconductance, gm, of a small geometry
BJT that is biased in its forward active domain. For that matter, the frequency, gm /C, is often so
large (high tens to low hundreds of giga-radians/sec) as to cast aspersions on the propriety of the
simplified high frequency model we have used to generate (3-61). Consequently, we are justified to approximate the last result by the expression,
I
βac
β(jω)  cs 
.
(3-62)
Ibs
1  jω ωβ
β(jω) 



In (3-62), the so-called beta cutoff frequency,
1
ωβ 
,
rπ Cπ  C μ

(3-63)

(j) (in dB)
ac (dB)
e=
op
Sl
3 dB
ec
/d
dB
20

0

T

Figure (3.19). Asymptotic approximation of the frequency response of the
short circuit, grounded emitter, current gain of a bipolar junction transistor.
is recognized as the resultant 3-dB bandwidth of the short circuit, small signal current gain,
β(j). The unity gain frequency can now be extrapolated as the gain-bandwidth product, βacωβ,
as suggested by the asymptotic frequency response plot given in Figure (3.19). This assertion
derives from the fact that at high signal frequencies, (3-62) collapses to
βacωβ
β(jω) ω ω 
,
(3-64)
β
jω
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Chapter 3
BJT Models & Biasing
from which it is apparent that the magnitude of the current gain degrades to one at an approximate radial frequency, ωT, which corresponds to frequency fT, of
βacωβ
gm
ω
fT  T 
.
(3-65)

2π
2π
2π Cπ  C μ


EXAMPLE #3.4:
Under room temperature operating conditions, the high frequency silicon transistor in Figure (3.18a) is biased at a collector current, IcQ, of 800 A and a collector-emitter voltage, VceQ, of 2 volts. The transistor has a maximum base resistance, rbb, of 360 Ω, a series collector resistance, rc, of 60 Ω, and a series emitter
resistance, re, of 1.5 Ω. The short circuit current gain, βf, of the device is 140
amps/amp, the knee current, Ikf, is 10 mA, the Early voltage, Vaf, is 25 volts, and
the saturation current, Is, is 6 fA. For the base-collector junction, the capacitance-related parameters are Cjco = 5 fF, mc = 1/3, and Vjc = 660 mV; for the
emitter-base junction, Cjeo = 1.5 fF, me = 1/2, and Vje = 920 mV. Moreover, the
low current, short circuit transit time, τfo, for base region minority carriers is τfo =
2.5 pSEC. The emitter-base junction injection coefficient, nf, can be taken to be
one. Use these numerical data to compute the unity gain frequency, fT, at the
specified operating point.
SOLUTION #3.4:
(1).
Equation (E3-1) in the preceding example defines the quiescent transport current, IccQ, in
terms of the Q-point collector current, IcQ. For IcQ = 800 A and Ikf = 10 mA, this relationship
gives

I cQ 
4I kf  
1  1 
   1.06 mA .
I ccQ  I cQ 1 
(E4-1)
2I kf 
I cQ  



With nf = 1, Is = 6 fA, and, recalling (3-7), VT = 25.89 mV for a junction temperature of Tj =
300.16 K, (3-5) gives a Q-point voltage, VeQ, developed across the intrinsic emitter-base
junction of VeQ = 670.47 mV.
(2).
Since parameter β, to which the static beta, hFE, is directly proportional is large, the quiescent
base current, IbQ = IcQ/hFE, is correspondingly small. Thus, the Q-point emitter current, IeQ, is
virtually identical to its collector current counterpart, IcQ. Taking IeQ ≈ IcQ, it follows from
Figure (3.11) that the internal collector-emitter voltage, VbQ, is, with VceQ = 2 volts, rc = 60 Ω,
and re = 1.5 Ω,
VbQ  VceQ  I cQ rc  I cQ re  1.95 V .
(E4-2)
In the interest of completeness, we can use (3-37) to enumerate hFE. In particular, for VbQ =
1.95 volts, and βf = 140, hFE = 113.85 amps/amp, which is slightly more than 81% of the
Ebers-Moll parameter, βf.
(3).
We compute the forward transconductance, gm, with the help of (3-41). For the given and
calculated variables,

I cQ 
V 
Ic
1 
  1  bQ   29.22 m .
gm 
(E4-3)
n f VT 
Vaf 
2 I ccQ I kf  


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Chapter 3
BJT Models & Biasing
(4).
Recalling (3-40), the effective value, τfe, of the minority carrier transit time is
I cQ 

1

I kf 
 2.505 pSEC
(E4-4)
τ fe  τ fo 

VbQ 
 1 

Vaf 

for τfo = 2.50 pSEC. It follows by (3-39) that the emitter-base diffusion capacitance, Cbe, is,
for gm = 29.22 mS and τfe = 2.505 pSEC,
Cbe  τ fe g m  73.19 fF .
(E4-5)
(5).
The remaining device capacitances derive from (3-43) and (3-45). In the former of these two
relationships, the emitter-base junction threshold potential, Veon, can be taken as 700 mV, despite the fact that the computed value of the emitter-base junction voltage, VeQ, is about 30
mV smaller. Equation (3-43) is, after all, only an approximation of the depletion capacitance
evidenced at the forward biased junction. Additionally, capacitance Cjeo is so small that the
depletion component of the net emitter-base junction capacitance is doubtlessly inconsequential to the present exercise. Accordingly, with Cjeo = 1.5 fF, Vje = 920 mV, Veon = 700 mV,
and me = 1/2,
C jeo
C je 
 3.07 fF .
(E4-6)
me


V
 1  eon 

V je 

In order to garner a value of capacitance Cμ in (3-45), the intrinsic base-collector junction
voltage, VcQ, must first be computed. Appealing to Figure (3.11),
VcQ  VeQ  VbQ   1.28 volts .
(E4-7)
Thus, (3-45) yields
C jco
Cμ 
 3.49 fF ,
m

VcQ  c
1 


V jc 

with Cjco = 5 fF, Vjc = 660 mV, and mc = 1/3.
(6).
(E4-8)
Recalling that capacitance C is simply the sum of capacitances Cbe in (E4-5) and Cje in (E46), (3-65) finally delivers
gm
gm
fT 

 58.32 GHz .
(E4-9)
2π Cπ  C μ
2π τ fe gm  C je  C μ




Those who may be concerned with the tacit neglect of the right half plane zero in (3-61) can
verify that for the stipulated operating point conditions, gm /C  1,332 GHz, which is almost
23-times larger than fT in (E4-9). Consequently, this right half plane zero bodes no substantive engineering significance.
ENGINEERING COMMENTARY:
The procedure underlying the numerical evaluation of the unity gain frequency of a BJT is
computationally intensive. In order to allay any trepidation that may ensue from this
computational intensity, the procedure is documented herewith merely to demonstrate the
utility of all of the modeling relationships derived to this point. In practice, and as you will
be thrilled to learn, operating point printouts accompanying SPICE simulations of BJT circuits include a numerical delineation of fT.
The critical issue at hand is the practical utility of the unity gain frequency metric. In brief,
its principle value is to serve as a measure of comparison among devices proposed for
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Chapter 3
BJT Models & Biasing
deployment in broadband analog or high-speed digital circuit applications. The rationale surrounding this contention is that transistors exemplifying very large values of fT for a specified
quiescent operating point can generally be expected to perform more admirably at high signal
frequencies than can competitive devices exuding smaller values of the unity gain metric.
For example, a transistor boasting fT = 100 GHz is arguably capable of processing more faithfully very high signal frequencies than can transistors offering fT = 40 GHz.
While the foregoing statement reflects solid engineering rationale, it is important for us to
appreciate that with the exception of very specialized circuit topologies, fT is an incredibly
optimistic measure of achievable circuit performance at high signal frequencies. It is even
more vital that we comprehend the logic that underlies this asserted negativity. In particular,
fT is a frequency measure for only the short circuit current gain of a transistor cell whose
emitter is grounded. It pertains to a circuit topology whose base is driven by an ideal signal
current source, the response to which is a current delivered to a short circuited load termination. As such, the fT metric is effectively a measure of only the time required to transport
charge carriers across the base region. To wit, the net applied base current, (IbQ + Ibs), serves
to forward bias the emitter-base junction of the subject transistor. This forward biasing promotes the injection of charge carriers from the emitter to the base. In turn, such injection
gives rise to a net emitter signal current, (IeQ + Ies). Because of the narrowness of the base
volume and the reverse bias imposed at the base-collector junction, the preponderance of
emitter-injected carriers is transferred to the collector region where they incur a net collector
current of (IcQ + Ics). Once the carriers injected into the base are indeed delivered to the
collector, the short circuited load imposed at the collector simplifies an assessment of frequency response. In particular, the gain metric, β(jω) in (3-61), and frequency metric fT in (365) inherently ignore the time required to charge the substrate capacitance and any other
capacitances implicit to the actual load that we ultimately append to the collector port. In
other words, fT fails to account for the frequency response of the output voltage produced by
the applied input current since in effect, the output voltage is clamped to zero by the short circuit imposed at the output port. Moreover, and especially for the case of a first stage in a
high performance electronic system, fT is likewise oblivious to the time required to charge input port capacitances associated with the utilized transistor. The tacit neglect of capacitive
charging times is significant in practical circuits. For example, in a circuit designed to deliver
an output voltage response to an applied input signal, the achievable 3-dB bandwidth is rarely
larger than about an order of a magnitude smaller than fT.
Potentially notable exceptions to the foregoing assertions are current mode circuits, which extract output responses as signal currents that are applied as currents to succeeding stages.
Current mode signal processing is addressed later in this text. But for the moment, observe
that in the case of the grounded emitter configuration, the collector is a relatively high impedance port owing to the large Early resistance that prevails therein. Accordingly, the collector
can be modeled accurately as an almost ideal Norton equivalent network. If the Norton output current is then applied to a subsequent amplification stage that features relatively low input impedance, a short circuit is effectively approximated at the collector port of the
predecessor stage. To the extent that the input port of the first stage boasts low input impedance so that large port voltages are precluded, fT becomes a meaningful barometer of high
frequency circuit performance. Of course, short circuited input ports in either predecessor or
follow on load stages are never precisely realizable. As a result, the achievable bandwidths in
prudently designed current mode configurations remain themselves rarely larger than 50% to
60% of the published fT value.
In summary, fT is arguably a viable, and indeed convenient, metric for purposes of comparative shopping of transistors earmarked for deployment in high performance, broadband networks. But except for current mode configurations, which can be adopted only at circuit
interstages −stages that include neither the input first stage nor the output last stage− it is a
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Chapter 3
BJT Models & Biasing
virtually useless circuit level figure of merit. Its demonstrable circuit impropriety as a high
frequency performance measure stems from the fact that the fT measure is oblivious to the
time required to charge device and critical circuit capacitances at both input and output amplifier ports.
Although transistor vendors and process foundries are anxious to quote impressive fTvalues to skew the comparative shopping to which our foregoing comments allude, they are reticent to underscore the fact that fT varies with collector current. Indeed, fT is a notably sensitive
function of the quiescent collector current, IcQ, when IcQ is small, as is typical of circuits constrained to function at low power levels. In the true spirit of aggressive marketing, quoted unity
gain frequency values invariably reflect maximum attainable values, which may unfortunately
correspond to collector currents deemed inappropriate for specific applications. To wit, the
transistor parameterized in the preceding example generates the fT -versus- IcQ characteristic displayed in Figure (3.20). A careful examination of the data underpinning this curve reveals that
the maximum value, say fTM, of the unity gain frequency is 58.3 GHz, which is achieved at a
collector current of IcM = 831.8 μA. Fortunately, the curve at hand displays a rather broad maximum, thereby negating the need to set the quiescent collector current precisely to achieve a unity
gain frequency that is within a few percent of its theoretical maximum. Note, however, that it
makes little sense to bias the transistor of interest to the right of current IcM in that similar fT-values are achievable at the lower circuit power dissipation associated with current biasing below
IcM.
Unity Gain Frequency, fT (GHz)
60
fTM
50
40
30
20
10
0
0.010
IcM
0.100
0.962
9.618
Collector Current, IcQ (mA)
Figure (3.20). The unity gain frequency, fT, plotted as a function of the quiescent collector current,
IcQ, for the transistor studied in Example #3.4. The quiescent collector-emitter voltage
of the subject transistor is held at 2 volts.
An analytical delineation of the optimal collector current associated with maximal unity
gain frequency is a challenge owing to the dependence of numerous relevant modeling parameters on collector current. Nonetheless, we can deduce acceptable approximations to the optimal
current and corresponding maximal unity gain frequency. To this end, return to (3-65) and note
that the net capacitance, say Cnet, in the denominator on the right hand side is
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Chapter 3
BJT Models & Biasing
Cnet  Cπ  C je  C μ  τ fe g m  C j ,
(3-66)
where (3-39) and (3-44) have been used. Moreover, we have introduced the sum total, Cj, of
transistor depletion capacitances as
C j  C je  C μ .
(3-67)
In the last expression, Cje and Cμ are given by (3-43) and (3-45), respectively. While capacitance
Cje is a constant, Cμ is a function of the intrinsic base-collector junction reverse bias, Vc, which
depends somewhat on the collector current owing to the presence of series resistances in the
collector and emitter leads of the transistor. Because these series resistances are small, this
collector current dependence can be ignored, thereby establishing Cj as a constant, independent
of quiescent collector current. In particular, the tacit neglect of intrinsic series resistances implies that the intrinsic collector-emitter voltage is simply the externally applied collector-emitter
voltage, VceQ, whence VcQ equates to the voltage difference, (Veon − VceQ). Moreover, voltage VbQ
is presumed to be significantly smaller than the Early voltage, Vaf, so that fe in (3-39) becomes
I cQ 

1


I kf 
I cQ 

(3-68)
τ fe  τ fo
 τ fo  1 
.



VbQ 
I
kf


 1 

V
af 

The relationship for the unity gain frequency in (3-65) now becomes
gm
gm
(3-69)
fT 
.

 

2π Cπ  C μ
I cQ 
2π  τ fo  1 
 gm  C j 
I kf 
 



At small collector currents, the transconductance parameter, gm in (3-54) collapses to

I cQ 
I cQ
V 
I cQ

  1  bQ  
(3-70)
gm 
1
,
n f VT 
Vaf 
n f VT
2 I ccQ I kf  


where we continue to presume that voltage VbQ is much smaller than the Early metric, Vaf.
Specifically, transconductance gm, which increases monotonically for all values of the quiescent
collector current, IcQ, is seen as increasing linearly with small quiescent collector current IcQ.
This simplified transconductance relationship, allows us to couch (3-69) in the form
I cQ
gm
(3-71)
fT 
.

 

 τ fo 

I cQ 
I cQ 
2π  τ fo  1 
2πn f VT 
 gm  C j 
1 
 I cQ  C j 
I kf 
I kf 
 

 n f VT 

Equation (3-71) is arguably valid through the immediate neighborhood of maximum fT in that
Figure (3.20) suggests maximal fT is achieved at a collector current that is substantively smaller
than the forward knee current of a BJT.
Equation (3-71) affirms the original allegation of a current dependence ascribed to the
unity gain frequency metric. Aside from such affirmation, it also highlights two additionally
important, design-oriented issues. First, it validates the nominally linear rise of fT with current
IcQ that is observed in Figure (3.20) for low collector currents. From (3-71), we note that the
slope of this low current rise converges to 1/2πnfVTCj, which means that progressively larger
depletion capacitances at either or both transistor junctions slows the current rate of fT ascent to
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Chapter 3
BJT Models & Biasing
its maximum value. Second, since (3-71) delivers positive fT for all considered values of IcQ and
further yields fT = 0 at both IcQ = 0 and IcQ = ∞, it foretells the existence of an optimal current,
say IcM, commensurate with securing maximal unity gain frequency. This optimal current derives from constraining to zero the derivative, dfT /dIcQ, in (3-71). The result of this somewhat
annoying exercise is
n f VT I kf C j
I cM 
.
(3-72)
τ fo
Now, if we set IcQ = IcM in (3-71) we arrive at maximum unity gain frequency, say fTM, of
1 2π τ fo
fTM 
.
(3-73)
n f VT C j
1 2
τ fo I kf
We should probably tweet home the fact that the foregoing expression articulates 1/fo as the
largest possible radial value of the maximum unity gain frequency. We note further that this
maximum fT value is approached when the knee current, Ikf, is large. The last observation suggests that high injection breeds diminished frequency response capabilities. This suggestion is
reasonable in that high injection creates the proverbial traffic jam of carriers in the base, whereupon the mobility of those carriers is degraded.
When (3-72) and (3-73) are applied to the transistor studied in Example #3.4, IcM =
823.3 μA, and fTM = 54.66 MHz. These computations differ from their simulated values by
slightly more than 1% and about 6.7%, respectively, which is good enough for government work,
while confounding marketing practices.
3.3.2. UNITY POWER GAIN FREQUENCY
The unity gain frequency discussed in the preceding subsection liberally brackets the
high frequency response capabilities of a grounded emitter transistor in terms of the frequency
response of its short circuit, and therefore maximum, current gain. Although mathematical
tractability and ease dictate that the small signal, frequency domain performance of electronic
networks be evaluated in terms of either current gain or voltage gain transfer functions, we
should record that the primary purpose of active circuits is to amplify signal power. It is therefore meaningful and sensible to assess high frequency transistor response capabilities in terms of
the maximum power gain afforded by the utilized transistor.
To the foregoing end, consider the grounded emitter amplifier of Figure (3.21a) in
which the static supply voltages, Vcc and Vbb, combine to ensure that the utilized BJT operates in
its forward active regime. The signal source, Vs, is applied to the amplifier input port through a
complex series impedance, Zs(jω), which is understood to include the Thévenin impedance of the
signal source itself. In response to Vs, which is presumably a small amplitude sinusoid, a signal
component, Vos, of the net output port voltage, Vo, is generated. In addition, signal voltage Vs
manifests a signal component, Vis, to the net input port voltage, Vi. The amplifier, whose output
port is terminated to signal ground in the complex load admittance, Yl(jω), can be represented as
a voltage controlled current source, as we suggest in Figure (3.21b). In this Norton representation, Yfe(jω)Vis represents the short circuit, output signal current, while Yoe(jω) is the shunt output
admittance associated with the amplifier output port. At the amplifier input port, it is convenient
to separate the intrinsic base resistance, rbb, from the net input impedance or admittance. Accordingly, the admittance, Yie(jω), is the load presented to the signal source circuit by only the net
- 232 -
Chapter 3
BJT Models & Biasing
admittance that is pervasive of the branch that shunts the intrinsic emitter-base junction.
rbb
Io
+Vcc
Zs(j)
Vis
Yie(j)
Yl(j)

Vs

Vo
Ios


Yfe(j)Vis
Vbb

(a).
Yoe(j)
Vs
Vos
Yl(j)

Vi
Zs(j)
(b).
Figure (3.21). (a). A grounded emitter amplifier. (b). Voltage controlled current source representation
of the small signal characteristics postured by the I/O ports of the amplifier in (a).
Assuming that the transistor in Figure (3.21a) derives from a monolithic fabrication
process, the device equivalent circuit shown in Figure (3.14) launches the small signal amplifier
model depicted in Figure (3.22a). Since the Norton equivalent output current, Yfe(jω)Vis in Figure (3.21a), is the current conducted by a short circuited load, the model appropriate to determining Yfe(jω) is the structure given in Figure (3.22b). Observe that the only difference between the
topologies of Figures (3.22a) and (3.22b) is that load admittance Yl(jω) in the former diagram is
supplanted by a short circuit in the latter figure. From Figure (3.22b), the signal current, Ios, conducted by the short circuited load satisfies
I os  g mV  jωC μV  g m  jωC μ V .
(3-74)


On the other hand, signal voltage V relates to the signal component, Vis, at the network input port
as
rπ
rπ
1
jωr
C
C


V
rπ  rbb
π
π
μ
(3-75)


.
rπ
Vis
1  jω  rπ rbb  Cπ  C μ
 rbb
1  jωrπ Cπ  C μ






Using the last two expressions, we find that the Norton transadmittance, Yfe(jω), is
jωC μ 
g m rπ 
1 

rπ  rbb 
gm 
I os
Y fe (jω) 

.
Vis
1  jω  rπ rbb  Cπ  C μ


(3-76)
The right half plane zero at gm /Cμ is invariably so large as to lie outside the frequency range over
which the transistor model in Figure (3.14) can be presumed valid and is therefore ignored
henceforth. Thus, the transadmittance in (3-76) can be approximated at high signal frequencies
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Chapter 3
BJT Models & Biasing


V

Vs

r
C
gmV
ro
Cs
Yl(j)
(but not so high as to encroach on the frequency of the aforementioned right half plane zero) as
rbb
C
Vis
Vos
Ios
Zs(j)
(a).
Zs(j)
Vis
rbb

V


Vs
C
r
C
Ios
gmV
ro
Cs

(b).
Vis

V

r
C
Vx
gmV
ro
Cs
Ix
(c).
Yie(j)
C
Vx
Ix
Yoe(j)
C

V = Vx

r
C
Vos
gmV
ro
Cs
Yl(j)
Zs(j)
rbb
(d).
Figure (3.22). (a). Small signal, high frequency equivalent circuit of the amplifier in Figure (3.21a). (b).
Small signal model used in the evaluation of the forward transadmittance function, Yfe(jω).
(c). Small signal model for evaluating the Norton admittance, Yoe(jω), established at the amplifier output port. (d). Small signal model for evaluating the Norton admittance, Yie(jω),
established at the intrinsic emitter-base junction of the utilized transistor.
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Chapter 3
BJT Models & Biasing
Y fe (jω) 
I os
gm

Vis
jωrbb Cπ  C μ



ωT
,
jωrbb
(3-77)
where
ωT 
gm
Cπ  C μ
(3-78)
is the transistor unity gain frequency introduced in (3-65). It is interesting to note that the form
of (3-77) implies that the high frequency transadmittance of a grounded emitter BJT amplifier
reflects the response properties of an ideal integrator because of the presence of (jω), which
corresponds to Laplace operator “s,” in the denominator.
Figure (3.22c) diagrams the equivalent circuit appropriate to evaluating the shunt output admittance, Yoe(jω). This structure is the original model in Figure (3.22a) with two modifications. First, the independent signal source, Vs, is set to zero, as is mandated by a Thévenin
impedance computation at any network port. Second, the load admittance, Yl(jω), is supplanted
by a mathematical ohmmeter that is simulated by the independent current source, Ix. This current
forges a voltage, Vx, across, and in disassociated polarity with, current Ix so that the target output
admittance, Yoe(j), is simply Ix/Vx. Conventional circuit analysis reveals
1

(3-79)
I x    jω Cs  C μ Vx  g m  jωC μ V ,
 ro

and


1
(3-80)
 jω Cπ  C μ V  jωC μVx  0 .

 rπ  rbb  Rs 

Inserting the solution for voltage V in (3-80) into (3-79) leads to
jω  rπ  rbb  Rs   C μ g m  jωC μ
I
1
(3-81)
 jω Cs  C μ 
Yoe (jω)  x 
.
Vx
ro
1  jω  rπ  rbb  Rs   Cπ  C μ
For high frequency signal environments where transconductance gm can still be presumed to be
substantively larger than the capacitive susceptance, ωCμ, (3-81) reduces to the more compact
relationship,
1
Yoe (jω) 
 ωT C μ  jω Cs  C μ .
(3-81)
ro
We note with considerable interest that the base-collector depletion capacitance, Cμ, generates a
conductance (real) component, ωTCμ, to the net shunt output admittance. We may be able to exploit this result when a specific circuit application warrants a lossless reduction in the amplifier
output resistance. In particular, placing a capacitor across the base-collector ports of a common
source amplifier enhances the value of the base-collector junction depletion capacitance, Cμ. For
given or known ωT, we may choose this expanded capacitance value to contrive the additional
shunt output conductance (lowered shunt output resistance) that our design environment requires.
Observe that the additional conductance has the laudable attribute of burning no signal power
since it derives exclusively as the magnitude of a capacitive admittance at frequency ωT. In
asserting this claim, we assume, of course, that the capacitance we use is ideal; that is, the
capacitance is divorced of any shunt terminal resistance.














Figure (3.22d) is the small signal model for calculating the shunt input admittance,
Yie(jω) for the subject grounded emitter amplifier. We recognize that the voltage, Vx, which is
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Chapter 3
BJT Models & Biasing
developed across the imposed mathematical ohmmeter, is identical to the signal voltage, V,
appearing across the emitter-base junction. A conventional nodal circuit analysis delivers
1

I x    jω Cπ  C μ V  jωC μVos
 rπ

(3-82)
,


1
 jω Cs  C μ Vos  gm  jωC μ V  0

 ro Z l (jω)







where Zl(jω) is the impedance corresponding to the load admittance, Yl(jω). The elimination of
voltage Vos in these two independent relationships leads to


 
gm  ro Z l (jω)
Ix
1

 Cμ  ,
(3-83)
Yie (jω) 

 jω Cπ  1 
V
rπ
1  jω  ro Z l (jω) Cs  C μ  


 
where the indicated approximation exploits the presumption that for all signal frequencies of
interest, gm >> ωCμ. At high signal frequencies that do not contradict the last inequality, (3-83)
collapses to
 Cμ 
I
1
(3-84)
Yie (jω)  x 
 gm 
  jω Cπ  C μ .
 Cs  C μ 
V
rπ


rbb
Vis
Zs(j)



Vs
r



C s + C
gmC
C
C
Yie(j)
T Vis
jrbb
ro
1
T C
C
Vos
Yl(j)
Ios
Cs
Yoe(j)
Figure (3.23). Approximate high frequency, Norton equivalent I/O port model for the grounded emitter amplifier in Figure (3.21a).
As is the case with the shunt output admittance, the base-collector depletion capacitance, C,
spawns a small shunt input conductance component in the form of the second term on the right
hand side of (3-84). Equation (3-84), along with (3-81) and (3-77), gives rise to the high frequency I/O port model we present in Figure (3.23). This topology clearly identifies the net shunt
resistances and capacitances at both the input and output ports of the amplifier. In the interest of
clarity, we observe that while Yoe(jω) is the net admittance facing the terminating load admittance, Yie(j) is not the net input admittance faced by the signal source at the base terminal of the
transistor. Instead, the input impedance driven by the applied input signal is [rbb + 1/Yie(jω)].
- 236 -
Chapter 3
BJT Models & Biasing
The prerequisite for maximum power gain in any type of amplifier is the successful
realization of conjugate impedance or admittance matches at both the input and the output ports.
In particular, the source impedance, Zs(jω), must be the complex conjugate of the net input
impedance it faces at signal frequency ω. Concurrently, the load admittance, Yl(jω), must be a
conjugate match to the net output admittance that it shunts. Achieving these conjugate matches,
particularly over a broad passband of signal frequencies, is hardly a casual walk in the proverbial
park on a sunny weekend morning. The requisite engineering tasks entail the design and
implementation of appropriate I/O filters that respectively couple the signal source to the input
port and the load to the amplifier output port to produce the desired matching. These filters are
generally nominally lossless structures, which is to say that their intrinsic branch elements are
exclusively inductances and capacitances. As a result, coupling filters do not waste significant
amounts of applied signal power prior to being processed for delivery to the load termination.
The lossless nature of coupling filters looms significant in many modern communication systems
whose received radio frequency signal often projects anemic amplitude. Another issue that we
need to address with an incorporation of I/O matching filters is overall network stability. In
particular, impedance matching at both the input and output ports for certain types of transistor
amplifiers can result in an unstable network in the sense that one or more network poles end up
lying in the right half complex frequency plane.
The conjugate admittance match at the output port implies two operational requirements. First, the shunt conductance component, say Gl, of the load admittance must equate to
the real part, or conductance component, of Yoe(j). In particular,
1  ωT roC μ
1
1
 ωT C μ 

(3-85)
Gl  Re Yoe (jω) 
.
ro
ro
Roe
Second, the susceptive component of load admittance Yl(jω) must be inductive to enable its
resonance with the net shunt output capacitance, (Cμ + Cs), at the radial signal frequency, ω. The
proper selection of a load that satisfies these two design requirements resultantly compacts the
output port section shown in Figure (3.23) to the simple topology of Figure (3.24a), which
projects an output signal voltage, Vos, of
rbb
rbb
Vis
Vos
Ios

T Vis
jrbb
Roe
Roe
(a).
Vs

(b).
Figure (3.24). (a). The high frequency output port model of the grounded emitter amplifier in Figure
(3.21a) when the load admittance, Yl(jω), is selected to be a conjugate match to the
Norton output admittance, Yoe(jω), in Figure (3.23). (b). High frequency input port
model of the grounded emitter amplifier under the condition of a source impedance
[Zs(jω)] conjugate match to the net input impedance, rbb + 1/Yie(jω) in Figure (3.23).
ω V R 
Vos   T is  oe  .
jωrbb  2 
(3-86)
- 237 -
Chapter 3
BJT Models & Biasing
The corresponding maximum signal power, say Pomax, delivered to the resistive component of the
load that terminates the output port is
Pomax 
2
Vos
Roe
2
ω V 
  T is  Roe .
 2ωrbb 
(3-87)
At the input port, the real part, say Rs, of the effective source impedance, Zs(j), must
be set to effect a conjugate match between the source impedance and amplifier input impedance,
[rbb + 1/Yie(j)]. With reference to Figure (3.23),


1
1
 1  jωRie Cπ  C μ 

 
R , (3-88)
2  ie
Yie (jω)


C
1
μ
 1  ωRie Cπ  C μ  
 gm 
  jω Cπ  C μ

 


rπ
 Cs  C μ 
where
βac g m
1
Rie 
.
(3-89)

 Cμ 
 Cμ 
1
1  βac 
 gm 
 Cs  C μ 
 Cπ  C μ 
rπ




Interestingly, the impedance, 1/Yie(j), is purely imaginary at very high signal frequencies. In
particular,


1

Yie (jω) ω large


2
  jωRie  Cπ  C μ  


 jωRie2 Cπ  C μ




1
jω Cπ  C μ



.
(3-90)
Thus, the magnitude of the very high frequency current conducted by resistance r in Figure
(3.23) is insignificant to the current conducted by the net shunt input port capacitance, (Cπ + Cμ).
This revelation hardly warrants prize paper recognition in view of our understanding that
capacitances in the sinusoidal steady state emulate short circuited elements when they are compelled to conduct high frequency currents. It follows that the real part of 1/Yie(jω) approaches
zero at high frequencies, whence the source resistance, Rs, commensurate with input port impedance matching becomes

 1 
1 
(3-91)
Rs  Re  rbb 
  rbb  Re 
  rbb .
Yie (jω) 

 Yie (jω) 
Figure (3.24b) shows the immediate ramification of a high frequency input port impedance
match, where it is to be understood that the reactive component of the original source impedance,
Zs(jω), has been selected to resonate with the net shunt capacitance, (Cπ + Cμ). Such a design
tack is tantamount to ensuring that the sum of the reactances associated with Zs(jω) and 1/Yie(jω)
is zero at the signal frequency, ω, of interest.
For the maximum power transfer condition implied by (3-91), the maximum signal
power, Pimax, delivered to the input port of the grounded emitter amplifier is obviously
Pimax 
Vis
2
.
rbb
A combination of this result with (3-87) leads to a maximum power gain, Ap, of
- 238 -
(3-92)
Chapter 3
BJT Models & Biasing
2
Pomax
 ωT   Roe 
Ap 
 
 ,
 
Pimax
 2ω   rbb 
and recalling (3-85),
(3-93)
2
ωT
ω  R 
(3-94)
Ap   T   oe  
.
 2ω   rbb 
4ω 2 rbbC μ 1  1 ωT roC μ
Somewhat disconcertingly, (3-94) shows that the high frequency power gain of a grounded emitter amplifier degrades sharply with increasing signal frequency. This observation explains the
debilitating pain we experience when we are tasked with the responsibility of achieving substantial signal power gain at very high frequencies. Indeed, power gain Ap is inversely proportional
to the square of the radial frequency, ω, which means that the power gain frequency response
rolls off with increasing frequency at a rate of 40 dB/decade.


A maximum power gain of one effectively defines the practical utility of an amplifier.
In other words, the frequency at which unity power gain is achieved effectively defines the maximum frequency for which the amplifier behaves as an active network capable of greater than unity gain. This frequency, say ωmax, at which Ap degrades to one is, by (3-94),
ωT
ωmax 
,
(3-95)

1 
4rbbC μ  1 


ωT roC μ 

or in units of hertz,
ωmax
fT
fT
f max 
,
(3-96)


2π
8πrbbC μ


1
8πrbbC μ  1 


2πfT roC μ 

where the indicated approximation reflects the presumption, 2fTroCμ >> 1. For the transistor
examined in Example #3.4, which at the stipulated operating point has rbb = 360 Ω, Cμ = 3.49 fF,
and fT = 58.32 GHz, the preceding equation delivers fmax = 42.86 GHz, which is more than 26%
smaller than the fT rating of the device.
When rbbCμ > 1/4ωT, which is a commonly satisfied condition, particularly for SiGe
heterojunction bipolar transistors, the unity power gain frequency, fmax, more conservatively
brackets the high frequency performance ceiling of a transistor than does the unity current gain
frequency, fT. Despite this arguably laudable conservatism, device selection decisions premised
on fmax must still be weighed carefully and tempered in light of the myriad of analytical
approximations we invoked to preserve our sanity while formulating (3-96). More importantly,
the engineering implications implicit to the concept of the unity power gain frequency must be
placed into proper perspective. In particular, fmax represents the approximate input signal frequency for which the maximum possible power gain of a grounded emitter amplifier degrades to
unity. But maximum power gain is realized if and only if conjugate impedance matches prevail
between terminating load and amplifier output impedances and between signal source and amplifier input impedances. Except for certain classes of narrowband, tuned amplifiers, such conjugate matches are rarely achievable without filtering heroics in broadband architectures. Indeed,
most broadband amplifiers, and particularly lowpass amplifiers, do not operate with conjugate
impedance matches at I/O ports, which means that these amplifiers do not deliver maximum
power gain. In fact, they may be designed expressly for the delivery of maximum voltage or cur- 239 -
Chapter 3
BJT Models & Biasing
rent gains or perhaps specific values of these gains. Thus, just like the short circuit figure of merit, fT, does not reflect realistic operating conditions, fmax may likewise be somewhat impertinent
to a broad range of amplifier topologies. In these cases, the actual achievable bandwidths are
generally found to lie in the range of one-fifth to one-third of fmax.
An even more insidious problem is that all grounded emitter amplifiers cannot be designed to deliver match terminated maximum power gain. As we briefly mentioned above, some
amplifiers are potentially unstable, which is to say that there exists a range of passive load
terminations for which self-sustaining oscillations can be supported by the considered network
under zero input signal conditions. Potential instability does not surface in the foregoing discourse because the high frequency models invoked are simplified to an extent that the actual
potential instability mechanisms of common emitter units are masked. More complicated high
frequency models, which embrace the response ramifications of time delays associated with the
forward transconductance and the dynamics observed at junction sidewalls in the complicated
two dimensional monolithic macromodel of Figure (3.12), may imply potential instability for
certain quiescent operating points. It follows that grounded emitter amplifiers represented by
these high level models may be incapable of delivering maximum power gain without risking
instability. In these cases, the pertinence of fmax in (3-96) is dubious.
3.4.0. BJT BIASING
As we have already noted, we are compelled to bias analog electronic circuits in order
to facilitate nominally linear signal processing of applied input voltage or current signals. In the
case of bipolar junction transistors, the biasing problem fundamentally addresses two designoriented issues. The first of these issues is the selection of a quiescent collector current, IcQ,
which is appropriate to the satisfaction of the I/O specifications targeted for the proposed circuit.
Such selection must be mindful of the fact that a necessary condition for nominally linear BJT
operation that BJTs must be biased in their linear, forward active regimes. Recall that in this
operating domain, the base-emitter voltage, Vbe, of each BJT must be sufficiently large to establish, for all pertinent signal levels, an intrinsic junction voltage that is at least as large as the threshold potential of the emitter-base junction. Moreover, we must ensure that the intrinsic basecollector junction voltage is negative or at most zero for all time. The latter voltage constraint is
satisfied automatically in NPN devices if collector-emitter voltages, Vce, are at least as large as
base-emitter voltages, Vbe. In practice, we find it prudent to set the Q-point value of the collector-emitter voltage above the base-emitter voltage by an amount that at least equals the anticipated signal swing across the collector-emitter voltage. Assuming a small or zero impedance in
the emitter lead, this swing is proportional to the product of the gain magnitude and applied input
signal amplitude. Analogous constraints apply to PNP units whose emitter-collector voltages,
Vec, must be larger than the emitter-base voltages, Veb, which support the desired quiescent
collector currents.
The selection of suitable quiescent collector currents is hardly a visceral engineering
exercise. These currents are typically premised on desired circuit gains, since gain is a function
of current-dependent device forward transconductance. The quiescent collector current supported by the biasing design might also be chosen in light of considerations surrounding network
bandwidth, noise, distortion, power dissipation, or other circuit performance barometers deemed
by the circuit designer as critical to a satisfying realization of system performance specifications.
In addition to the junction voltage constraints underlying linear domain operation of a BJT, an
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Chapter 3
BJT Models & Biasing
important practical issue must be appreciated and addressed by the circuit designer during the
collector current selection process. In particular, it makes no sense to bias a BJT at a collector
current that exceeds the device knee current. One problem associated with high current biasing
is needlessly enhanced power dissipation. Another problem implicit to biasing at or beyond the
transistor knee current is that the DC beta, hFE, decreases monotonically with high collector current levels, thereby compromising the current gain of the network into which the subject transistor is connected. While forward transconductance increases monotonically with collector Qpoint current, the unity gain frequency, fT peaks at a current that is smaller than the device knee
current. The immediate implication of this fact is that biasing the collector at a current to the
right of the fT peak produces an fT value that is identical to that forged slightly to the left of the fT
peak. Thus, power dissipation budgets, thermal concerns, and even electrical noise considerations render high current biasing foolish when suitably lower current biasing results in similar
device and circuit performance. Although design generalities in electronics are fraught with
engineering peril, the portability culture renders one assertion abundantly clear. In particular, the
quiescent collector current of each active device within an electronic network should be chosen
as the smallest current level commensurate with nominally linear device operation and both the
predictable and reliable satisfaction of all stipulated circuit operating specifications.
The second of the two-step biasing design task derives from the previously espoused
fact that the quiescent collector current conducted by a BJT fixes the numerical values of several
branch elements in the small signal models of Figures (3.14) and (3.15). It follows that the biasing structure must establish each device collector current in a reliable and predictable fashion. In
most cases, these collector currents are desirably constant, although in a few, carefully controlled, design situations, we may wish them to be linear functions of junction operating temperature. For example, recall from (3-54) that the forward transconductance at low collector currents
is proportional to the ratio of quiescent collector current to Boltzmann voltage. Since the
Boltzmann voltage is directly proportional to absolute temperature, a temperature invariant gm,
which translates to a gain invulnerability to operating temperature, requires a collector current
that is likewise proportional to absolute temperature (PTAT).
But temperature is not the only environmental factor that determines design quality.
Suppose, for example, that device biasing currents fluctuate as a function of noise coupled
parasitically to the static voltage supply line of a circuit. The resultant contamination of such
small signal parameters as rπ, Cπ, gm, and ro produce a time varying model for which the challenges associated with executing tractable small signal analytical analysis all but preclude the
generation of predictable and reproducible results.
In summary, the biasing problem is seen as generally entailing the implementation of
collector currents that are predictable constants. Implicit to the requirement of predictable and
reproducible biasing current is that the current in question be rendered as insensitive as possible
to the impact exerted on device characterization metrics by the vagarious nature of semiconductor device processing and manufacturing. Stated quite simply, the quiescent collector current,
IcQ, of each BJT must be reliably predictable in terms of designable circuit elements. It must
usually be appropriately desensitized to temperature, and it must be independent of signal voltages, signal currents, and the parametric uncertainties that comprise the implicit baggage of active monolithic devices.
3.4.1. PASSIVE BIASING NETWORKS
The most straightforward of biasing structures engaged for use in linear signal
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processing networks employ only passive elements to realize a desired quiescent collector current. In the subsections that follow, we examine two passive networks from four distinct
perspectives. The first of these design-oriented perspectives is the penchant of the proposed network to achieve and sustain a predictable quiescent operating current. Second, we attempt to
illuminate the sensitivity of the quiescent current to key transistor parameters. Third, we determine the degree to which the quiescent collector current is affected by increases in the junction
operating temperature of the subject bipolar devices. Finally, the dependence of Q-point collector current on signal swings manifested at the collector port is investigated. The last investigation is tantamount to examining the small signal impedance established at the collector port.
Since we generally desire a constant collector current and since sources of constant current are
characterized by infinitely large terminal impedances, we wish to establish a small signal impedance seen looking into the collector terminal of the subject transistor that is ideally infinitely
large.
3.4.1.1. World’s Worst BJT Biasing Circuit
The task of developing biasing strategies begins by acquiring an understanding of the
two major issues that accompany the task of engineering a bias circuit. These issues, which embrace quiescent collector current dependence on the static gain parameter, hFE, and the inherently
positive temperature coefficient of collector current, are highlighted by the simple biasing cell
offered in Figure (3.25a). The circuit at hand uses the voltage divider formed of resistances R1
and R2 to set the quiescent base-emitter voltage, VbeQ, which in turn supports the desired quiescent collector current, IcQ. Since IcQ is largely determined by the emitter-base junction bias when
the transistor operates in its linear domain, it is nominally independent of the collector load resistance, Rl, which is selected to satisfy small signal gain or other requirements. However, Rl does
set the quiescent collector-emitter voltage, VceQ in accordance with
+V cc
+Vcc
R1
Rl
Rl
IcQ

IcQ /hFE
VbeQ


IcQ /hFE
VceQ

R2
IcQ
VceQ


( hh +1 ) I
R1||R2
FE
cQ
FE


R2
( R +R )V
1
VbeQ
2
(a).

( hh +1 ) I
FE
cQ
FE
cc

(b).
Figure (3.25). (a). Simple biasing circuit for a bipolar junction transistor. (b). An equivalent representation of the structure in (a).
Vcc  Rl I cQ  VceQ .
(3-97)
For a given power supply voltage, Vcc, and a desired quiescent collector current, IcQ, this KVL
relationship demonstrates that in view of the fact that VceQ must be sufficiently larger than VbeQ to
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BJT Models & Biasing
promote BJT operation in its linear regime, large values of Rl must be avoided. In concert with
the linear domain requirement, observe that the base current is delineated in Figure (3.25a) as
IcQ/hFE, whence an emitter current that is a factor of (hFE +1) larger than the base current.
The application of Thévenin’s theorem to the circuit in Figure (3.25a) results in the
equivalent circuit representation that we contrive in Figure (3.25b). In the latter diagram,
 I cQ 
 R2 
(3-98)
  VbeQ ,

Vcc   R1 R2  
 R1  R2 
 hFE 
which delivers


h 
R 
I cQ  FE Vcc   1  1 VbeQ  .
(3-99)
R1 
R2 


Prior to using this result to assess the quality of the subject biasing circuit, it is important to
understand the reason underlying the decision to apply KVL to the base-emitter loop of the circuit, as opposed to using (3-97), which is the fruit of KVL applied to the collector-emitter loop.
In particular, (3-97) delivers the irrefutably simpler collector current expression,
Vcc  VceQ
I cQ 
.
(3-100)
Rl
While this equilibrium relationship is valid, it conveys the incorrect impression that the quiescent
collector-emitter voltage, VceQ, is an independent voltage variable on which the quiescent collector current, IcQ, depends. But (3-33) and (3-36) demonstrate that the collector current of a BJT is
only weakly dependent on the internal collector-emitter voltage, Vb, and thus its extrinsic
counterpart, VceQ, since the Early voltage, Vaf, is a relatively large voltage metric. Theoretically,
the collector current is independent of VceQ if the idealized case of infinitely large Early voltage
is adopted. In contrast, (3-33) and (3-36) show a pronounced, exponential dependence of collector current on internal emitter-base junction voltage, Ve, which relates intimately to the baseemitter quiescent voltage, VbeQ. Accordingly, (3-99) propounds a design-oriented, and thus far
more relevant, expression for Q-point collector current than does (3-100). In a word, collectoremitter voltage VceQ in (3-100) exerts minimal impact on the quiescent collector current, IcQ,
while IcQ in (3-99) is intimately related to the quiescent base-emitter voltage, VbeQ.
Equation (3-99) offers two reasons for nominating the biasing circuit of Figure (3.25a)
as the worst in the world. The first of these reasons is the direct dependence of quiescent collector current on the gain metric, hFE. We recall that hFE is directly proportional to the Ebers-Moll
current transfer parameter, βf, which, because of its inverse dependence on the narrow width of
the base in a bipolar junction transistor, can neither be controlled nor predicted accurately and
reliably. Thus, the precise value of hFE for a given transistor is elusive, which renders an accurate prediction of IcQ dubious. The upshot of the matter is that conventional monolithic
processing, which advances a vagarious nature to hFE, renders impossible the accurate numerical
delineation of IcQ in Figure (3.25a).
A second reason motivating a veto of the biasing circuit at hand is unacceptably large
temperature sensitivity of the quiescent collector current. Recall from the discussion in Section
(3.2.0) and the temperature sensitivity disclosures in the preceding chapter on PN junction diodes
that in order to sustain constant current in the face of junction temperature increases, the internal
emitter-base (diode) junction voltage must decrease with junction temperature at a nominally
constant rate. This requisite decrease in intrinsic emitter-base junction voltage can be semiempirically translated to a decrease in terminal base-emitter voltage, VbeQ, as
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Chapter 3
BJT Models & Biasing
ΔVbeQ   Sbe ΔT j ,
(3-101)
where Sbe is a constant of the order of 2 mV/C, and Tj is the change observed in the operating
temperature, Tj, of the emitter-base junction. Accordingly, (3-98) provides a collector current
temperature coefficient, say i, (since this text is slotted for the university student market, the
liberal use of Greek symbols is compelled) of
ΔI cQ
h S
σi 
 FE be .
(3-102)
ΔT j
R1 R2
The expression at hand shows that the temperature sensitivity factor, Sbe, is amplified by hFE,
which while precisely unpredictable, is nonetheless an assuredly large number. The actual
temperature coefficient of the quiescent collector current can be expected to be somewhat larger
than i in (3-102) owing to the slight dependence on temperature of parameter hFE and the nonzero temperature coefficients associated with the circuit resistors used in the biasing network.
It is important that we supplement, through mere circuit inspection, the temperature
sensitivity implications of (3-102) with an engineering appreciation of this important issue. In
particular, if hFE is very large, which is to say that the base current in the network shown in Figure (3.25a) approaches zero for finite collector current IcQ, voltage Vbe derives as a simple resistive divider off the power line voltage, Vcc. Indeed, this observation is confirmed by (3-98). To
the extent that the resistance ratio, R2/R1, is accurately controlled and nominally temperature
invariant, and assuming that voltage Vcc is a temperature invariant constant, the base-emitter terminal voltage, VbeQ, and hence, the internal emitter-base junction voltage, Ve, are held constant.
But we have already witnessed that in order to mitigate the of a BJT saturation current that increases with increasing junction temperature, voltage Ve (and hence Vbe) must decrease at
roughly a temperature rate of 2 mV/°C. Unfortunately, the divider in the subject circuit sustains
constant VbeQ. This means that the collector current is free to increase with temperature in accordance with the degree to which the transistor saturation current is sensitive to junction temperature.
Aside from providing a qualitative measure of the alleged ineffectiveness of the biasing
cell depicted in Figure (3.25a), the foregoing discourse offers a clue as to how we might improve
the biasing design. In particular, we need to allow Vbe to decrease appropriately with temperature. As we shall see in the next section of material, this decrease can be achieved by employing
feedback that senses the undesirable temperature-induced increase in quiescent collector current.
Once sensed, this increased collector current can be transformed to a voltage, probably through
use of nothing more complicated than a resistor. The transformed voltage (voltage across the
deployed resistance) can then be used to subtract from the voltage used to drive the base-emitter
terminals of the transistor. In the process, collector current IcQ is forced to decrease in response
to its original thermally induced increase. We can even propose that the quality of our compensated bias network can be meaningfully quantified by comparing the original current increase to
its feedback-controlled decrease. Obviously, we achieve nirvana if the decreased collector current matches the original, temperature-induced, increased current.
EXAMPLE #3.5:
The silicon transistor in the biasing circuit of Figure (3.25a) has a minimum hFE
of 100. Using a 3 volt power supply, design the circuit so that it dissipates no
more than 8 mW of power, while delivering a quiescent collector current of 2 mA
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Chapter 3
BJT Models & Biasing
at a collector-emitter voltage of 1 volt at room temperature. Assume that the
transistor delivers hFE = 100 and that the required quiescent base-emitter voltage, Vbe, is 720 mV. For this design, estimate the resultant change in quiescent
collector current when the emitter-base junction temperature rises by 30 C. To
the latter end, take Sbe = 1.8 mV/C.
SOLUTION #3.5:
(1).
The simplest calculation pertains to the collector load resistance, Rl. From either Figure
(3.25a) or (3-100), IcQ = 2 mA, Vcc = 3 volts, and VceQ = 1 volt requires
Vcc  VceQ
Rl 
 1 KΩ .
(E5-1)
I cQ
(2).
The current, say IR1, conducted by resistance R1 in Figure (3.25a) is
VbeQ
I cQ
(E5-2)
I R1 

.
R2
hFE
The subject figure also confirms that the power supply must deliver a current of IcQ + IR1 to
the biasing circuit. It follows that the power delivered to the circuit by voltage source Vcc,
which is the power, Pdis, dissipated by the biasing circuit, is
VbeQ 
 h  1 
(E5-3)
Pdis  Vcc I cQ  I R1  Vcc  FE
 .
 I cQ 
R2 
 hFE 
Note that the bracketed quantity on the right hand side of this expression is the net current
flowing to ground, since the first term in the bracketed quantity is the transistor emitter current, while the second term is the current flowing through resistance R2. For Vcc = 3 volts, hFE
= 100, IcQ = 2 mA, and VbeQ = 720 mV, Pdis = 8 mW stipulates R2 = 1.11 K.

(3).

Equation (3-99) can now be exploited to determine the value of resistance R2. For Vcc = 3
volts, hFE = 100, R1 = 1.11 k, IcQ = 2 mA, and VbeQ = 720 mV, R2 = 355.1 . This required
value of resistance R2 means that the 8 mW power dissipation specification cannot be met in
light of all other stipulated operating requirements.
With Sbe = 1.8 mV/C, hFE = 100, R1 = 1.11K, and R2 = 355.1 , the temperature coefficient, i, of the Q-point collector current is found, using (3-102), to be i = 668.6 μA/C.
Given Tj = 30 C, the computed change in collector current is IcQ = iTj = 20.1 mA. In
truth, the collector current cannot be perturbed by this amount for at best, the transistor saturates as the collector circuit in Figure (3.25a) rises toward its maximum possible value of
Vcc/Rl = 3 mA. Even this maximum cannot be precisely achieved in that the collector-emitter
voltage can be reduced to only its small, but certainly nonzero, saturation value.
ENGINEERING COMMENTARY:
This simple, if not impractical, example teaches two lessons. The first is that all calculations
executed on an active network must be carefully scrutinized in light of the assumptions on
which these analyses are premised. In the case at hand, it is presumed at the outset that the
transistor operates in its linear domain. For linear operation of the circuit at hand, the collector current must remain smaller than about 3 mA, which is the collector current that theoretically flows if the collector-emitter voltage can be reduced to zero. Accordingly, the
calculated, temperature-induced change in quiescent collector current of better than 20 mA is
a meaningless computation, except to suggest that the transistor in question indeed saturates
merely because of an increase in the operating temperature of the emitter-base junction. And
mind you, this increased temperature most likely derives as a self-heating phenomena incurred by the very collector current we are attempting to stabilize in our biasing cell.
(4).
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Chapter 3
BJT Models & Biasing
The second lesson articulated by this example, is that the circuit in Figure (3.25a) is terrible.
Aside from the unpredictability of collector current due to the inherent uncertainty in the gain
parameter, hFE, it is difficult to sustain linear domain operation over routinely encountered
temperature excursions. For example, routine self-heating of a junction readily results in a 10
C increase in junction operating temperature. But even for this modest 10 C increment, the
calculated change in quiescent collector current is almost 6.7 mA, which is more than sufficiently large to incur device saturation.
3.4.1.2. Current Controlled Voltage Feedback Biasing
The shortfalls of the simple biasing circuit in Figure (3.25a) are circumvented if an
emitter degeneration resistance, Ree, is inserted into the emitter lead, as we show in Figure
(3.26a). As we suggested in the preceding section of material, this resistance inserts voltage
feedback, which takes the form of the indicated voltage, Vre, developed across resistance Ree, into
the base-emitter loop. This voltage is clearly proportional the quiescent collector current, thereby allowing is to adjust automatically the static base-emitter voltage when the quiescent collector current, IcQ, deviates from its design target. We note that this automatic adjustment capability
of base-emitter voltage VbeQ contrasts sharply with the previously considered biasing configuration, which effectively and deleteriously maintains constant base-emitter voltage.
+Vcc
+Vcc
R1
Rl
Rl
Zx
Zx
IcQ
Vbx

IcQ /hFE
VceQ

VbeQ
R2
IcQ


Vre
Vbx

R1||R2
Ree

( hh +1 ) I
(
FE
cQ

R2
V
R1+R2 cc

)

IcQ /hFE
VceQ

VbeQ


Vre

Ree

FE
( hh +1 ) I
FE
cQ
FE
(a).
(b).
Figure (3.26). (a). BJT biasing circuit incorporating emitter degeneration as a feedback vehicle for the
stabilization of the quiescent collector current. (b). Equivalent circuit of the network of (a).
We can glean a qualitative appreciation of the effectiveness of resistance Ree in Figure
(3.26a) by assuming at the outset that the quiescent collector current meets its design goal.
Subsequent to establishing the correct collector current in the steady state, let IcQ increase, perhaps because of self-heating. The increase in IcQ is sensed directly as an increase in the
aforementioned voltage, Vre. A study of the equivalent circuit in Figure (3.26b) allows us to
surmise that voltage Vbx at the base node of the transistor is approximately constant. In particular, a constant, current invariant Vbx materializes if the Thévenin resistance, (R1||R2), is small
and/or the minimum anticipated value of hFE is large. The latter parametric condition minimizes
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Chapter 3
BJT Models & Biasing
the quiescent base current conducted by the transistor, thereby ensuring a small voltage drop
across the aforementioned Thévenin resistance. Since Vbe = Vbx − Vre, an increase in voltage Vre,
is therefore met by a decrease in the base-emitter voltage, Vbe. In turn, diminished Vbe serves to
reduce the collector current, thereby offsetting, albeit partially, the parasitic increase in quiescent
collector current. Of course, this self-correction mechanism also functions for a decreasing IcQ in
that diminished IcQ produces a reduced Vre, which manifests an increased Vbe that mitigates the
original current reduction.
By inspection of the equivalent circuit in Figure (3.26b),
 I cQ 
 R2 
 hFE  1 
(3-103)
  VbeQ  

Vcc   R1 R2  
 Ree I cQ ,
h
h
 R1  R2 


FE
FE


whence



R1 
 Vcc   1 
VbeQ 
R2 
 R2  

.
I cQ  hFE 
(3-104)

R

R
R
R

h

1
R
 FE  ee 
2   1 2
 1




The dependence of IcQ on the poorly controlled gain parameter, hFE, is substantively reduced if
we choose resistances that ensure (R1||R2) << (hFE + 1)Ree. It is interesting that this inequality
conflates with our previous requirement that voltage Vbx, which energizes the base terminal of
our network, be a constant, independent of the transistor base current. The quantification of the
extent to which the parallel combination of resistances R1 and R2 is smaller than the net resistance, (hFE + 1)Ree depends on the desired degree of accuracy ascribed to setting the actual
collector current. For example, if IcQ is to be controlled to within ±10% of its design target,
(R1||R2) ≤ (hFE + 1)Ree/10 constitutes an appropriate design tack. If, on the other hand, ±1% control is required, (R1||R2) should be of the order of 100-times smaller than (hFE + 1)Ree. Degree of
accuracy notwithstanding, (R1||R2) << (hFE + 1)Ree collapses (3-104) to the approximate result,

 α  R2  

R1 
I cQ   FE 
(3-105)
 Vcc   1 
VbeQ  ,
R2 

 Ree   R1  R2  

where αFE is given by (3-48). Observe that as long as the minimum anticipated value of hFE is
much larger than one, as it inevitably is, αFE approaches unity, and IcQ in (3-105) is rendered virtually independent of hFE. Accordingly, the inclusion of an emitter degeneration resistance (Ree)
serves to mitigate the undesirably pronounced sensitivity of quiescent collector current on hFE.
Recall that this undesirable sensitivity is an implicit signature of the awful biasing circuit in Figure (3.25a).
Apart from virtually eliminating the problem of IcQ sensitivity to hFE, emitter degeneration proves effectual from the standpoint of relative temperature insensitivity. Using (3-101),
ΔI cQ
α S
(3-106)
σi 
 FE be ,
ΔT j
Ree
which demonstrates that unlike the collector current temperature sensitivity in the circuit of Figure (3.25a), the emitter-base junction temperature sensitivity coefficient, Sbe, is not amplified by
the large current gain parameter, hFE. To be sure, (3-100) offers an arguably best case sensitivity
measure because of numerous approximations invoked in the course of its development. To wit,
(3-106) presumes that the temperature coefficient of resistance Ree is small and can be ignored,
the dependence of Vcc on temperature is negligibly small, and the small temperature sensitivity of
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Chapter 3
BJT Models & Biasing
parameter αFE is inconsequential. Note that the temperature coefficients of resistances R1 and R2
are likely to prove immaterial in that these resistances appear only as resistive ratios in the quiescent collector current expression. In particular, let R1 and R2 be laid out proximately on the integrated circuit chip so that they experience essentially the same operating temperature. If, in
addition, these resistors comparably sized diffused or implanted geometric structures, and if their
individual values are within an order of magnitude or so of one another, the temperature dependence of either R1/R2 or R2/R1 is unlikely to prove significant to the observed performance of the
circuit. For a given sensitivity target, (3-106) establishes a convenient basis for selecting
appropriate emitter degeneration.
The degree to which current IcQ modulates in response to signals appearing at the
collector port of the network in Figure (3.26a) can be discerned through quantifying the indicated
output impedance, Zx, established at the collector terminal. To this end, the subject transistor can
be replaced by the transistor small signal model of Figure (3.15). At very high frequencies,
impedance Zx is invariably dominated by the shunting substrate capacitance, Cs, which is a relatively large parasitic capacitance in conventional monolithic processes. Accepting this contention without proof, attention presently focuses on uncovering only the low frequency component,
say Rx, of impedance Zx. Accordingly, all capacitances in the transistor model at hand are set to
zero. The ohmic resistances, re, and rc, are also ignored without fear of significant accuracy
impairment, so that the resultant low frequency, small signal equivalent circuit is the topology
delineated in Figure (3.27). In this representation, voltage Vcc is supplanted by a short circuit
since the small signal value of this ideally constant voltage is zero. An additional noteworthy
point is that output resistance Rx does not include the collector load resistance, Rl. Such neglect
does not reflect engineering oversight since the analytical objective here is to discern the
sensitivity of quiescent collector current to signal changes at the collector terminal for any load
resistance that sustains transistor operation in its forward active domain.
Rx
rbb
Vx
R1||R2
r
ac I
I
IxacI
Ree
ro
Ix
IIx
Figure (3.27). Approximate low frequency model used to evaluate
the small signal resistance, Rout, presented at the
collector port of the biasing network shown in Figure (3.26a).
The desired resistance, Rx, is the voltage to current ratio, Vx/Ix, of mathematical ohmmeter variables, Ix and Vx. For convenience, we have delineated the branch current responses to the
imposed ohmmeter excitation in Figure (3.27). By KVL,
0   R1 R2  rbb  rπ  Ree  I  Ree I x
(3-107)
.
Vx   ro  Ree  I x   βac ro  Ree  I
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Chapter 3
BJT Models & Biasing
Solving the first of these two equations for current I, thence inserting this solution into the
second expression, leads to


V
βac Ree
Rx  x  Ree  R1 R2  rbb  rπ    1 
 ro .

Ix
R

R
R

r

r
ee
1 2
bb
π

(3-108)


βac Ree
 1 
 ro .

Ree  R1 R2  rbb  rπ 

The indicated approximation is reflective of the fact that the Early resistance, ro, and the AC
beta, βac, are typically large numbers. While Rx, is assuredly finite, (3-108) confirms that an
emitter degeneration resistance can render resistance Rx significantly larger than the Early resistance. Because of a potentially very large Rx, the biasing network in question supplies a quiescent collector current that is relatively unaffected by signal voltage swings at the collector node.
In other words in the limit as Rx approaches infinity, the collector port functions as an ideal current that sinks to ground. We recall from our good old basic circuit days that an ideal current
source (or sink) establishes a current that is independent of the voltage appearing across the
source. Thus, with resistance Rx very large, any voltage appearing with respect to ground at the
transistor collector terminal has almost no stamina to perturb the collector current.
EXAMPLE #3.6:
The silicon NPN transistor in the biasing circuit of Figure (3.26a) has a minimum hFE of 100 and a nominal base-emitter voltage of 720 mV at room temperature. Using a 6-volt power supply, design the circuit to provide a quiescent
collector current of 2 mA at a collector-emitter voltage of 1.5 volts at room
temperature. The desired quiescent collector current is to be maintained to
within ±5% for junction temperature increases as large as 50 °C. An emitterbase junction temperature coefficient of Sbe = 2.0 mV/°C can be presumed Using the HSPICE NPN BJT model parameters given in Table (3.1), simulate the
design at 27°C, 50 °C, and 75 °C. Assuming rbb = 250 Ω, r = 1.4 KΩ, ro = 28
KΩ, and βac = 95 amps/amp at the quoted quiescent operating point, estimate the
change in the collector Q-point current for a 500 mV signal swing at the collector
terminal.
SOLUTION #3.6:
(1).
Since the maximum allowable increase in quiescent collector current is 5% of its room
temperature, 2 mA value, IcQ ≤ (0.05) (2 mA) = 100 A. Given Tj = 50 C, i = IcQ /Tj ≤
2 A/C. Using (3-106), in which Sbe = 2 mV/C and FE = hFE /(hFE + 1) = 100/101 =
0.9901,
α S
(E6-1)
σi  FE be  2 μA / C ,
Ree
which results in Ree ≥ 990.1 Ω. Rounding this resistance upward to Ree = 1 KΩ is prudent
engineering action in that a larger than minimally required emitter degeneration resistance
provides for a small amount of proverbial breathing room with regard to the target temperature sensitivity of the biasing network.
(2).
The circuit in Figure (3.26a) yields
- 249 -
Chapter 3
BJT Models & Biasing
SPICE
TEXT
SYMBOL SYMBOL
DESCRIPTION
OF PARAMETER
VALUE
UNITS
Forward Short Circuit Gain
120
amps/amp
BR
f
r
Reverse Short Circuit Gain
1.2
amps/amp
CJC
CJE
CJS
IKF
IRB
IS
MJC
MJE
MJS
NF
NR
RB
Cjco
Cjeo
Cso
Ikf
Irb
Is
mc
mje
ms
nf
nr
rbb
Zero Bias B-C Depletion Capacitance
fF
fF
fF
mA
mA
fA
–
–
–
–
–
Zero Bias Base Resistance
1.1
3.8
10.2
11.5
4.5
3.2
0.333
0.5
0.5
1.05
1.0
270
RBM
Rbm
Minimum Base Resistance
35

RC
rc
Series Collector Resistance
20

RE
re
Series Emitter Resistance
1.5
TF
fo
Zero Bias Minority Carrier Transit Time
4.2

pSEC
TNOM
Tj
Junction Reference Temperature
27
VAF
VJC
VJE
VJS
XTI
XTF
XTB
Vaf
Vjc
Vje
Vjs
Forward Early Voltage
30
780
920
690
3
0.025
0.02
BF
Zero Bias B-E Depletion Capacitance
Zero Bias Substrate Depletion Capacitance
Forward Knee Current
Base Resistance Corner Current
Saturation Current
B-C Junction Grading Coefficient
B-E Junction Grading Coefficient
Substrate Junction Grading Coefficient
B-E Junction Injection Coefficient
B-C Junction Injection Coefficient
B-C Junction Built-In Potential
B-E Junction Built-In Potential
Substrate-Collector Built-In Potential
Temperature Exponent For IS
Temperature Exponent For TF
Temperature Exponent For BF

C
volts
mvolts
mvolts
mvolts
−
−
−
Table (3.1). HSPICE model parameters for a representative NPN bipolar junction transistor having a maximum unity gain frequency of approximately 35 GHz. Although the transistor characterized herewith is fictitious, the parameters are typical of a moderately high speed BJT.
 h 1
(E6-2)
Vcc  Rl I cQ  VceQ   FE
 Re e I cQ .
 hFE 
With Vcc = 5 volts, VceQ = 1.5 volts, hFE = 100, IcQ = 2 mA, and Ree = 1 KΩ, the collector load
resistance, Rl, computes as Rl = 1.24 KΩ.
(3).
The parallel resistance combination, (R1||R2) must be selected so that it is substantially
smaller than the effective resistance, (hFE + 1)Ree. Since the target quiescent collector current
is to be sustained to within ±5% over the quoted temperature excursion, (R1||R2) must be at
most (hFE + 1)Ree/20, or 5.05 KΩ. It is therefore reasonable to stipulate (R1||R2) = 4 KΩ, the-
- 250 -
Chapter 3
BJT Models & Biasing
reby ensuring, with ostensibly adequate safety margin, the validity of tacitly neglecting the
parallel resistance, (R1||R2), in (3-104).
(4).
Equation (3-105) can be recast as

  R1 R2  

 R1 
hFE
(E6-3)
I cQ  

VbeQ  ,
 Vcc  

  R1 R2    hFE  1 Ree   R1  
 R1 R2 
in which the only unknown quantity is the resistance ratio, (R1||R2)/R1. Solving for this ratio
results in the arguably sloppy, but nonetheless tractable, relationship,
  R1 R2    hFE  1 Ree 

 I cQ  VbeQ
hFE
R1 R2



 0.470 .
(E6-4)
R1
Vcc
Since (R1||R2) = 4 KΩ, R1 = 8.51 KΩ, whence
 R1 R2  R1  7.55 KΩ .
(E6-5)
R2 
R1   R1 R2 
(5).
For the computed biasing resistances and the quoted small signal transistor parameters, (3108) gives Rx = 428.8 KΩ. This means that for a collector terminal signal of 500 mV, the signal-induced change, IcQ, in the quiescent collector current is IcQ = 500 mV/428.8 KΩ =
1.17 A, which is only 0.058% of the target quiescent current.
+6 V
8.51 K
1.24 K
3.52 V
2.74 V
2.02 V
1K
7.55 K
Figure (3.28). Biasing circuit designed in Example #3.6.
The voltages shown in the boxes correspond
to the estimated respective node voltages for a
properly functioning circuit.
The completed design is shown in Figure (3.28). To facilitate circuit debugging with measured or simulated results, we have explicitly indicated target node voltages in the diagram.
ENGINEERING COMMENTARY:
The simulated results track very favorably with design predictions. In particular, the simulated quiescent collector current at 27 C is 1.98 mA, which is only 1% lower than the 2 mA
design target. At 50 °C, the simulated Q-point collector current is 2.01 mA, while at 75 °C,
the simulated value of IcQ is 2.05 mA, which is larger than the 27 °C value of 1.98 mA by only
3.54%. Moreover, at 27 °C, the simulated collector, base, and emitter voltages, measured
with respect to ground, are 3.55 volts, 2.75 volts, and 2.00 volts, respectively.
The circuit at hand is an excellent biasing circuit from the admittedly limited perspective that
- 251 -
Chapter 3
BJT Models & Biasing
both manual and computer-aided analyses confirm the predictability of the quiescent
collector current. It also offers adequate desensitization of this collector current with respect
to temperature changes, uncertainties in parameter hFE, and collector signal change, which at
500 mV is actually reasonably large by conventional small signal standards. However, it does
exude at least a few shortcomings, whose respective significance is better appreciated in a
subsequent chapter that addresses the gain characteristics of the amplifier that derives from
the considered biasing topology. In particular, it turns out that the maximum voltage gain afforded by the circuit at hand is the ratio of the collector load to emitter degeneration resistances. In the present case, this ratio is barely above one, but mitigation procedures are fortunately available. Moreover, the relatively large emitter degeneration resistance required for
suitable temperature compensation increases the equivalent input noise voltage, which is the
minimum signal strength that the amplifier can detect and process reliably. In most applications, this dilemma can likewise be addressed satisfactorily.
3.4.2. ACTIVE BIASING NETWORKS
Although passive biasing structures can stabilize the quiescent operating points of bipolar junction transistors over temperature, their ultimate effectiveness in monolithic circuit realizations is limited. One commonly encountered problem is that the ratios of requisite resistance values in these compensation schemes are large. Three problems prevail with respect to resistance
pairs whose values are divergent. The first is that controlling the accuracy of a resistance ratio is
progressively more difficult with increasing ratio value. The second is non-uniformity of
temperature coefficients between diverse resistance values. Yet another quandary is that the
resistance values required for acceptable temperature stabilization are often not synergistic with
the desired small signal performance that the network undergoing biasing must ultimately deliver. To this end, for example, we have noted that the large emitter degeneration resistance in
the design example of Figure (3.28) limits the achievable voltage gain of the cell without additional design heroics or modifications.
The foregoing shortfalls can be circumvented in integrated circuits by using companion
transistors to control the operating point stability of a given transistor. The basic approach is to
develop topological structures for which the quiescent variable of interest in a particular transistor is constrained to be directly proportional to the corresponding, and presumably easily stabilized, variable of a second, subsidiary transistor. This design tack can prove effective if the
second transistor offers volt-ampere characteristics that match those of the transistor for which
the quiescent collector current is to be stabilized. One advantage of this active stabilization approach is that judiciously laid out transistor pairs in monolithic embodiments can be matched
remarkably well so that their thermal properties track accurately with one another over broad
temperature ranges. As a result, design attention can focus on the relatively simple subsidiary
subcircuit, with the confidence that the monolithic process is capable of delivering transistor responses that track exceptionally well with the variables proscribed in these ancillary cells.
Interestingly enough, the methods documented in the following subsections are far less effective
with discrete, off the shelf component, circuit realizations since the relevant electrical
characteristics of even ostensibly identical transistor types from the same device manufacturer
are rarely matched well.
3.4.2.1. Diode-Resistor Current Mirror
A popular of active biasing compensation structure is the circuit offered in Figure
(3.29). A comparison of this circuit with the network in Figure (3.26a) shows that a diode-con- 252 -
Chapter 3
BJT Models & Biasing
nected transistor, Q2, (base and collector terminals are electrically tied together) is inserted in
series with the resistance R2 in the hope of incurring excellent stabilization of the quiescent
collector current, IcQ, conducted by transistor Q1. It is extremely important that, save possibly
for differences in emitter-base junction areas, the two BJTs be matched devices that are implemented in a manner that ensures that they conduct nearly identical current densities. If Q1 and
Q2 are indeed matched and conduct identical densities of static current, their base-emitter voltages, VbeQ, are necessarily equal. An equality of these two terminal voltages over all reasonable
temperature excursions additionally requires that the subject two transistors be laid out closely to
one another so that both experience essentially the same temperatures. Accordingly, if the emitter current of transistor Q2 is designated as IdQ,
 h 1
(3-109)
VbeQ  R2 I dQ  VbeQ   FE
 Ree I cQ .
 hFE 
+Vcc
R1
Rl
Rx
IdQ + IcQ /hFE
IcQ /hFE

VbeQ
Q2

VbeQ
IcQ

Q1 VceQ



R2
Ree
( hh +1 ) I
IdQ
FE
cQ
FE
Figure (3.29). Active biasing compensation through the addition of diode-connected transistor Q2 to the basic
passive structure of Figure (3.26a).
Since hFE >> 1 (if hFE is not much larger than one, it is time to contract with an alternate
processing foundry),
 R 
 h
 R 
(3-110)
I cQ   FE  2  I dQ   2  I dQ ,
 hFE  1  Ree 
 Ree 
which contends that IcQ mirrors current IdQ through the scale factor, R2/Ree. The fact that the subject constant of proportionality is a simple resistance ratio is symphonic. In particular, if R2 and
Ree are of the same order of magnitude, ratio R2/Ree can be controlled precisely (to within a
couple of percent or less) over broad ranges of temperature. In a word, stabilizing IdQ automatically renders current IcQ stable over temperature, despite the ramifications of other environmental
factors.
The most straightforward design circumstance is R2 = Ree, whereupon, IcQ is literally
the mirror image of IdQ. But note that if R2 is larger than Ree by a factor of k (k >1), IdQ, is a fac- 253 -
Chapter 3
BJT Models & Biasing
tor of k smaller than IcQ, which is advantageous from a circuit power dissipation perspective. In
other words, the added ancillary branch that is inserted for compensation purposes burns less
power than does the branch whose collector current is to be controlled. In this case, the junction
injection area of transistor Q1 must be made k-times larger than that of transistor Q2 to ensure
that both transistors conduct the identical densities of current that serve to sustain equivalence
between their base-emitter terminal voltages. In actual design environments, integer or non-integer values of k that do not exceed roughly six to eight prove viable and effective.
KVL applied to the R1-R2-Q2 branch of the circuit in Figure (3.29) gives
I cQ 

(3-111)
Vcc  R1  I dQ 
  VbeQ  R2 I dQ .
h
FE


Appealing to (3-109) for the elimination of the current variable, IdQ, in (3-111) results in a quiescent collector current of

Vcc  VbeQ
 R2  
I cQ  hFE 
(3-112)
.

 R1  R2    R1 R2    hFE  1 Ree 
As in the case of the passive compensation topology of Figure (3.26a), ensuring (R1||R2) << (hFE
+ 1)Ree all but eliminates a collector current dependence on hFE in that (3-112) consequently reduces to
 R2   Vcc  VbeQ 
(3-113)
I cQ  αFE 
.

Ree
 R1  R2 

This result advances a collector current temperature coefficient of
ΔI cQ
α S  R2 
(3-114)
σi 
 FE be 
.
ΔT j
Ree  R1  R2 
Two interesting observations surface from the last expression. The first, and most clarion, of the two observations is that the active compensation scheme delivers a collector current
temperature coefficient that is smaller than the coefficient associated with the passive compensator by the voltage divider factor, [R2/(R1 + R2]. The second point is that (3-114) combines with
(3-113) to yield
ΔI cQ
I cQ Sbe
α S  R2 
(3-115)
σi 
 FE be 
;
 
ΔT j
Ree  R1  R2 
Vcc  VbeQ
that is, the temperature coefficient of interest is actually independent of circuit resistances in that
it is set by the desired Q-point collector current, IcQ, the power supply voltage, Vcc, and, of course
the transistor properties implied by VbeQ and its voltage sensitivity factor, Sbe. Whereas the passive compensation scheme establishes its current temperature coefficient, σi, via the emitter
degeneration resistance, Ree, the active scheme effectively preserves a design degree of freedom
by relying on the supply voltage to establish the temperature coefficient of the circuit.
The resistance, Rx, presented to the collector load resistance, Rl, in Figure (3.29) can be
evaluated straightforwardly by substituting the low frequency, small signal equivalent circuit of a
BJT for both of the devices in the network at hand. In the process of such substitution, care must
be exercised to account for the fact that the potentially different junction injection areas of the
two transistors foment different values among corresponding small signal parameters. But unless the reader is exhilarated by detailed circuit analyses, a strongly recommended alternative
analytical tack entails a direct exploitation of the resistance result in (3-108), which derives from
- 254 -
Chapter 3
BJT Models & Biasing
a small signal analysis of the biasing configuration in Figure (3.26a). Clearly, the only difference between the two circuits at hand is that a diode-connected transistor (Q2) is inserted in series with resistance R2 in Figure (3.26a). Since the diode-connected BJT is a two terminal element that is to be replaced by the low frequency version of the small signal model in Figure
(3.15), its small signal electrical properties mimic those of a simple two terminal resistance, say
Rd. The upshot of the matter is that (3-108) adapts to the circuit in Figure (3.29) merely by
replacing resistance R2 by the resistance sum, (R2 + Rd). It follows, essentially by inspection,
that Rx in Figure (3.29) is given by


βac1 Ree
Rx  Ree  R1  R2  Rd   rbb1  rπ1    1 
 ro1






R
R
R
R
r
r


ee
1
2
d
bb1
π1 

(3-116)


βac1 Ree
 1 
 ro1 .






R
R
R
R
r
r


ee
1
2
d
bb1
π1 

There are two outstanding issues about understanding the resistance result, Rx, in (3116). The first of these issues is to comprehend that under small signal conditions, the manner in
which transistor Q2 is connected into the biasing configuration leaves us little choice but to view
the Q2 subcircuit as a simple resistance. We can supply two justifications for this allegation.
First, with the base and collector terminals connected together, the normally three-terminal
transistor collapses to a two-terminal unit. This state of affairs means that Q2 cannot function as
an amplifier. Instead, and like any two terminal passive element, the Q2 subcircuit can be connected only as an element in series with a network branch or an element in shunt with a branch.
Second, upon modeling the two-terminal Q2 subcircuit with our conventional, low frequency
small signal transistor equivalent circuit, we are left with a two-terminal, memoryless (meaning
no capacitances or inductances prevail as elements in the model) linear subcircuit. This circumstance is transparent because the small signal transistor model is inherently a linear network. It
follows that the current flowing through the Q2 subcircuit is related linearly to the voltage developed across the subcircuit; that is, it behaves as a conventional George Ohm resistance.
The only remaining outstanding issue is the evaluation of resistance Rd in terms of
transistor model parameters. To this end, the equivalent circuit of the Q2 interconnection is
given in Figure (3.30), where as usual, the internal collector and emitter resistances of the
transistor are ignored. The use of our trusty mathematical ohmmeter we depict in the circuit before us delivers
Rd
Rd
rbb2
(ac2+1)I
r
Q2
ac2 I
ro2

Ix Vx

I
Figure (3.30). Low frequency, small signal equivalent circuit of the diode-connected transistor, Q2, in
the biasing network of Figure (3.29). The model is configured to determine the resistance,
Rd, established between the two terminals of the diode unit.
Ix 
Vx
ro2
  βac2  1 I 
Vx
ro2


Vx
  βac2  1 
,
 rbb2  rπ2 
whereupon
- 255 -
Chapter 3
BJT Models & Biasing
r
Vx
rbb2  rπ2
 rπ2 
.
(3-117)
 ro2  bb2
 
Ix
β

1
β

1
ac2
 ac2

Owing to a large AC beta, resistance Rd is small and very likely, it is significantly smaller than
the biasing resistance, R2, in the network of Figure (3.29). It therefore materializes that from a
practical perspective, the base to ground branch in Figure (3.29) is virtually electrically
indistinguishable from the base to ground branch in Figure (3.26a).
Rd 
EXAMPLE #3.7:
The silicon NPN transistors in the active biasing circuit of Figure (3.29) have a
minimum hFE of 100 and a nominal base-emitter voltage of 720 mV at room
temperature. Design the circuit so that with a collector load resistance to emitter
degeneration resistance ratio of 10, a quiescent collector current of 2 mA flows
through transistor Q1 when its collector-emitter voltage is 1.5 volts at room
temperature. The desired quiescent collector current is to be maintained to
within ±5% for junction temperature increases as large as 50 °C. An emitterbase junction temperature coefficient of Sbe = 2.0 mV/°C can be presumed Using the HSPICE BJT model parameters given in Table (3.1), simulate the design
at 27 °C, 50 °C, and 75 °C.
SOLUTION #3.7:
(1).
Since the maximum allowable increase in quiescent collector current is 5% of its room
temperature, 2 mA value, IcQ ≤ (0.05) (2 mA) = 100 A. Given Tj = 50 °C, σi = IcQ /Tj ≤
2 A/°C. Using (3-115), in which Sbe = 2 mV/°C and VbeQ = 720 mV,
I cQ Sbe
σi 
 2 μA / C ,
(E7-1)
Vcc  VbeQ
which offers Vcc ≥ 2.72 volts. A power supply voltage of Vcc = 3 volts is appropriate, particularly since 1.5 volt battery cells are readily available commercially.
(2).
The circuit in Figure (3.29) shows that
 h 1
Vcc  Rl I cQ  VceQ   FE
(E7-2)
 Re e I cQ .
 hFE 
With Vcc = 3 volts, VceQ = 1.5 volts, hFE = 100, IcQ = 2 mA, and Rl = 10Ree, we find that the
requisite emitter degeneration resistance is Ree = 68.1 Ω and hence, the collector load resistance follows as Rl = 10Ree = 681 Ω.
(3).
In Figure (3.29) and (3-109), let the emitter current conducted by transistor Q2 be equal to the
emitter current flowing in transistor Q1, which means that both are required to have the same
emitter-base junction injection areas. Accordingly, R2 = Ree = 68.1 Ω. Moreover, with R2 =
Ree, (R1||R2) < Ree. It follows that the requisite inequality, (R1||R2) << (hFE +1)Ree, is satisfied
by a factor of better than (hFE +1) = 101, which is more than sufficient in light of the ±5%
tolerance constraint imposed on the desired quiescent collector current.
(4).
Setting R2 = Ree in (3-112) now leads to R1 = 1,061 Ω. The completed design is shown in
Figure (3.31). As in the preceding design example, target node voltages are delineated in the
diagram.
- 256 -
Chapter 3
BJT Models & Biasing
+3 V
1,061
681
1.64 V
Q1
Q2
857.6 mV
137.6 mV
68.1
137.6 mV
68.1
Figure (3.31). Biasing circuit designed in Example #3.7. All resistances are in units of ohms. The voltages shown in
the boxes correspond to the estimated respective node
voltages for a properly functioning circuit.
ENGINEERING COMMENTARY:
Once again, the simulated results confirm the propriety of the design methodology. In
particular, the simulated quiescent collector current at 27 C is 1.97 mA, which is only 1.5%
lower than the 2 mA design target. At 50 °C, the simulated Q-point collector current is 2.00
mA, while at 75 °C, the simulated value of IcQ is 2.03 mA, which is larger than the 27 C value
of 1.97 mA by only 3.05%. Moreover, at 27 °C, the simulated collector, base, and emitter
voltages of transistor Q1, measured with respect to ground, are 1.66 volts, 884 mV, and 135.3
mV, respectively.
While the active compensator postured herewith is a superb biasing network, it, like virtually
all electronic networks, is imperfect. In the case at hand, the desired match between transistor base-emitter voltages over temperatures is non-exact because of the finite Early voltages
of the transistors and the fact that the collector-emitter voltages of the two transistors are unlikely to equate. In particular, observe that the collector-emitter voltage of transistor Q2 is
necessarily constrained to be its base-emitter terminal voltage, while the collector emitter voltage of Q1 is designed to be better than twice the base-emitter voltage. Accordingly, the
collector current correction for the Early effect differs for each transistor.
3.4.2.2. Diode Current Mirror
Discovering that the temperature coefficient of the collector current in the diode-resistor configuration of Figure (3.29) is independent of the emitter degeneration resistance, Ree, hints
at the possibility of designing a suitable biasing network that is divorced of emitter degeneration.
The advantages of dispelling emitter degeneration include decreased circuit power dissipation,
the potential for increased small signal gain, and reduced noise.
To the foregoing end, consider the diode mirror shown in Figure (3.32), which uses, in
addition to the collector load resistance, Rl, only one additional resistance, R, to bias the circuit.
KVL applied to the R-Q2 branches of the circuit gives
- 257 -
Chapter 3
BJT Models & Biasing
+Vcc
Rl
R
Rx
IdQ + IcQ /hFE
IcQ /hFE

Q2

VbeQ
VbeQ
IcQ

Q1 VceQ


( hh +1 ) I
 IdQ
FE
cQ
FE
Figure (3.32). Diode current mirror BJT biasing circuit. The
emitter-base junction area of transistor Q1 can be
k-times larger than that of transistor Q2, which is
otherwise identical to transistor Q1.
I cQ 

(3-118)
Vcc  R  I dQ 
  VbeQ .
h
FE 

Let us select transistor Q1 to be identical to transistor Q2, save for the fact that the emitter-base
junction area of Q1 is larger than that of Q2 by a factor of k. Assume further that base
conductivity modulation (or Early effect) in both transistors is insignificant and/or the collector
to emitter Q-point voltages of the two transistors are not widely divergent. Then, the fact that the
indicated circuit connections force an identity between the base-emitter terminal voltages of both
active devices means that the emitter current, IdQ, flowing through Q2 is k-times smaller than the
emitter current of Q1; that is,
 h  1  I cQ 
(3-119)
I dQ   FE
.

 hFE  k 
In accordance with the fundamental premise of an actively compensated biasing network, the desired Q-point collector current is merely proportional (by a factor of essentially k) to the current,
IdQ, conducted by the ancillary network, which in this case is comprised of resistance R and diode-connected transistor Q2. The substitution of (3-119) into (3-118) resultantly accrues
hFE Vcc  VbeQ
I cQ 
.
(3-120)
hFE  1 

R 1 

k


The virtual elimination of a collector current dependence on parameter hFE mandates satisfaction
of the inequality, (hFE + 1) >> k, which is easy enough to achieve. Thus,
αFE k Vcc  VbeQ
(3-121)
I cQ 
,
R
which infers a Q1 quiescent collector current that is about k-times the current conducted by the
R-Q2 subcircuit. It is a simple matter to confirm a collector current temperature coefficient of




- 258 -
Chapter 3
BJT Models & Biasing
σi 
ΔI cQ
ΔT j

I cQ Sbe
αFE kSbe

,
R
Vcc  VbeQ
(3-122)
whose final form is identical to the temperature factor associated with the diode-resistor mirror.
It is worthwhile repeating that the circuit at hand boasts numerous advantages over its resistordiode counterpart. However, one disadvantage is that the resistance, Rx, seen looking into the
collector node of transistor Q1 is simply the Early resistance, ro1. We see then that the lack of
emitter degeneration in transistor Q1 results in an output resistance that is significantly smaller
than the output resistances observed in the degenerated biasing cells.
3.4.2.3. Vbe Multiplier Biasing
In the bipolar biasing circuits considered to this point, the quiescent collector current is
consistently couched as a function of two transistor variables; namely, the DC beta, hFE, and the
quiescent base-emitter terminal voltage, VbeQ. The dependence of collector current on the
numerically unreliable metric, hFE, is reduced to insignificant levels through having sufficiently
large hFE. On the other hand, the current dependence on Vbe, which is tantamount to a quantification of the collector current sensitivity to junction operating temperature, is mitigated with prudently selected emitter degeneration resistance and/or the use of a large enough power line voltage, Vcc. If it were possible to eliminate, as opposed to merely reducing, the collector current
dependence on base-emitter voltage, practical circuits offering the engineer enhanced degrees of
design freedom and possessed of impressively small temperature sensitivities become reality.
+Vcc
R1
Rl
Rx
Im + IcQ /hFE
IcQ /hFE
Im
VbeQ
Q1 VceQ


FE



( hh +1 ) I
Ic
Ry
Rx

IcQ
cQ
FE
Q2 Vm
Vbe2


R2
Ree
Figure (3.33). BJT biasing circuit using active compensation in the form of the Vbe
multiplier comprised of transistor Q2 and resistances R3 and R4.
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Chapter 3
BJT Models & Biasing
To the foregoing end, we shall find that the circuit appearing in Figure (3.33) is
theoretically capable of realizing the ideal goal of a quiescent collector current, IcQ, which is
independent of base-emitter voltage VbeQ. The core subcircuit underpinning this alluring performance attribute is the so-called Vbe multiplier, which consists of transistor Q2 and the two
resistances, R3 and R4. An analytical inspection of the Vbe multiplier cell itself, which is redrawn
for convenience in Figure (3.34a), facilitates the task of evaluating the volt-ampere properties of
the biasing network in Figure (3.33). In Figure (3.34a), KCL relates the current, Im, conducted
by the voltage multiplier cell to the static collector current, Ic2, flowing in transistor Q2 as
Vbe2 /R4  Ic2 /hFE2 Im
Im

Ic2
R3
Ic2 /hFE2

R4
Vbe2
Q2

Rm
Vm


Vm

kmVbe2


Vbe2 /R4
(a).
(b).
Figure (3.34). (a). The Vbe multiplier used in the biasing scheme depicted in Figure
(3.31). (b). The Thévenin equivalent static circuit of the cell in (a).
h
V
I
Vbe2
1
(3-123)
I m  I c2  be2  c2   FE2
.
 I c2 
R4
hFE2
R4
 hFE2 
Moreover, the voltage, Vm, established across the cell in response to the flow of current, Im, is
V
I 
(3-124)
Vm  R3  be2  c2   Vbe2 .
hFE2 
 R4
If we solve (3-123) for the transistor current, Ic2, and substitute the result into the last expression,
we find that voltage Vm relates to base-emitter voltage Vbe2 and cell current Im as

 R3 
α
R 
(3-125)
Vm   1  FE2 3  Vbe2  
 I m  kmVbe2  Rm I m .

R
h
1
 FE2

4


This result fosters the multiplier equivalent circuit in Figure (3.34b), where model parameters km
and Rm are given by
α
R
km  1  FE2 3
R4
.
(3-126)
R3
Rm 
hFE2  1
The model in Figure (3.34b) suggests that the multiplier cell emulates a Thévenin voltage source whose voltage is Vbe2, multiplied by the larger than unity factor, km. The subject figure also suggests that the Thévenin resistance, Rm, associated with the foregoing voltage is relatively small. Note that if R3 = 0, km = 1 and Rm = 0, which renders voltage Vm identical to the
base-emitter voltage, Vbe2, of transistor Q2. This result reflects our expectations since R3 = 0
short circuits the base terminal of transistor Q2 to its collector terminal. In turn, this short circuit
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Chapter 3
BJT Models & Biasing
reduces the Q2-R3-R4 subcircuit to a simple diode-connected transistor whose terminal voltage is
Vbe2. For large hFE2, the multiplier approximates an ideal voltage source whose voltage value is


α
R 
R 
V
(3-127)
kmVbe2   1  FE2 3  Vbe2   1  3  Vbe2   R3  R4  be2 .
R4 
R4 
R4


The final form relationship in this expression is rendered transparent by mere inspection of the
network in Figure (3.34a). In particular, Vbe/R4 is clearly the current conducted by resistance R4.
If gain metric hFE is large, the base current conducted by transistor Q2 is nil. It therefore follows
that current Vbe/R4 also flows through resistance R3, which effectively places resistances R3 and
R4 in series with one another. Clearly, the last form of the expression in (3-127) represents the
product of this net series resistance and the current conducted by it.
+Vcc
+Vcc
R1
Rl
Im + IcQ /hFE
IcQ /hFE

Im
Ic
R3


R4
VbeQ
R1
IcQ
Rl
Im + IcQ /hFE

IcQ /hFE
Q1 VceQ



Im
( hh +1 ) I
FE
VbeQ
IcQ

Q1 VceQ


( hh +1 ) I
FE
cQ
FE
Rm
cQ
FE
Q2 Vm
Vbe2
R2



kmVbeQ

Ree
R2
Ree
Figure (3.35). Equivalent electrical representation of the biasing network in Figure (3.33), wherein the Vbe
multiplier comprised of Q2-R3-R4 is supplanted by its equivalent circuit, which we depict in of
Figure (3.34b). Transistors Q1 and Q2 are presumed to be identical devices that are biased to conduct the same collector current densities, thereby implying that Vbe2 = VbeQ and hFE2 = hFE.
Armed with the subcircuit model of Figure (3.34b), the biasing network in Figure
(3.33) can be represented electrically as the structure given in Figure (3.35). The latter
configuration exploits the presumption that transistors Q1 and Q2 are identical devices that conduct identical collector current densities. Accordingly, the voltages developed across both baseemitter terminals are identical, as are the DC beta values of the two transistors. It is instructive
to observe that the structure in Figure (3.35) collapses to the diode-resistor biasing scheme offered in Figure (3.29) for the special case of km = 1 (or R3 = 0). In Figure (3.35),
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Chapter 3
BJT Models & Biasing
I cQ 

Vcc  R1  I m 
   R2  Rm  I m  kmVbeQ
hFE 

(3-128)
and
 h  1
(3-129)
 kmVbeQ  Ree  FE
 I cQ  VbeQ .
 hFE 
Upon inserting the solution for current Im in (3-128) into (3-127), we find the solution for the
quiescent collector current, IcQ, conducted by transistor Q1 to be the somewhat involved expression,


 km  1 R1 V 
 Vcc  1 
 beQ 
R2  Rm 
 R2  Rm  


I cQ  hFE 
(3-130)
.

 R1  R2  Rm   R1  R2  Rm    hFE  1 Ree 




Our confidence in the accuracy of this messy disclosure is bolstered by the observation that for
km = 1 (thereby constraining resistance Rm to zero), (3-130) reduces to (3-112), which defines the
Q-point collector current appropriate to the diode-resistor compensation scheme of Figure (3.29).
Moreover, if km and resistance Rm are artificially set to zero, the network in Figure (3.35) collapses to the passive compensation scheme in Figure (3.26a) and reassuringly, (3-130) reduces to
(3-104), which is the applicable collector current expression.
 R2  Rm  I m
In order to desensitize the Q-point collector current in (3-129) to uncertainties in the
DC beta, hFE, it is necessary to enforce R1||(R2 + Rm) << (hFE + 1)Ree. As such, (3-129) can be
approximated as


 km  1 R1 V 
Vcc  1 
 beQ 
R2  Rm 
 R2  Rm  


I cQ  αFE 
(3-131)
.

Ree
 R1  R2  Rm  





But in addition, if the constant, km, is selected such that
R  Rm
km  1  2
,
(3-132)
R1
the resultant quiescent collector current, which becomes,
 R2  Rm  αFE Vcc 
(3-133)
I cQ  

,
 R1  R2  Rm  Ree 
is rendered independent of voltage VbeQ. Recall, however, that this conclusion rests on three
assumptions. First, transistors Q1 and Q2 must be physically identical BJTs, save possibly for
emitter injection area differences. Second, the two matched transistors must conduct identical
collector current densities. Third, either the collector-emitter voltages of the two matched BJTs
must be nominally the same or, as is seemingly more pragmatic, both devices must exude
negligible Early effect. At a minimum and despite the foregoing presumptions, (3-132) removes
the dominant source of collector current sensitivity with respect to operating temperature. We
observe further that for large hFE, which delivers αFE ≈ 1 and encourages R2 >> Rm = R3/(hFE +
1), (3-132), becomes, with the aid of (3-126),
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Chapter 3
BJT Models & Biasing
R3
R
 2 .
(3-134)
R4
R1
Two interesting sidebars accompany this simple design requirement. First, if R3/R4 < R2/R1, (km
− 1) < R2/R1, and the coefficient of voltage VbeQ in the numerator on the right hand side of (3130) is resultantly a positive number. Accordingly, IcQ displays its stereotypically positive
temperature coefficient in view of the fact that VbeQ must decrease with temperature to preserve
constant collector current. But if R3/R4 > R2/R1, (km − 1) > R2/R1, and the VbeQ coefficient in
question is a negative number, thereby implying that IcQ decreases with operating temperature;
that is, the circuit projects an atypical negative temperature coefficient. In effect, the nature of
the relationship of resistance ratio R3/R4 to ratio R2/R1 provides an additional design degree of
freedom that may prove fruitful with respect to the problem of minimizing the temperature
sensitivity of either the I/O gain or the I/O driving point impedances of the overall network.
As in the case of the diode-connected transistor, Q2, in the active biasing scheme of
Figure (3.29), the Vbe multiplier cell in the network of Figure (3.33) functions as a two terminal
linear resistance under small signal circumstances. As such, its net small signal terminal resistance, say Rv, as highlighted in the diagram of Figure (3.36a), derives as the voltage to current
ratio, Vx/Ix, in the low frequency equivalent circuit we provide in Figure (3.36b). The equivalent
circuit ignores the internal collector and emitter resistances of transistor Q2 and additionally, it
simplifies the analysis through the tacit neglect of the routinely large Early resistance, ro, of
transistor Q2. In an attempt to appease the analytical purists, the latter resistance appears in
shunt with the current controlled current source, βacI, which implies that the effective resistance
across the collector and emitter terminals of the Vbe multiplier cell is actually (ro||Rv). For the
indicated branch currents in Figure (3.36b),
RV
RV
Ixac2I

R3
R3
rbb2
Q2
ac2 I
r
R4
Ix V x
R4
Ix(ac2+1)I

I
(a).
(b).
Figure (3.36). (a). The Vbe multiplier cell embedded in the biasing network of Figure (3.33). (b). The
small signal equivalent circuit pertinent to the evaluation of resistance Rv across the collector-emitter terminals of transistor Q2 in (a). As usual, the model ignores the series resistances in the collector and the emitter leads of transistor Q2. Also ignored is the relatively
large Q2 Early resistance, which appears as a simple shunting branch element across the
terminals at which resistance Rv is established.
 rbb2  rπ2  I  R4  I x   βac2  1 I 
Vx  R3  I x  βac2 I    rbb2  rπ2  I
.
Substituting current I from the first of the foregoing equations into the second leads to
- 263 -
(3-135)
Chapter 3
BJT Models & Biasing
Rv 

Vx
α R 
  1  ac2 4  R3  R4 Rd2 ,
Ix
R4  Rd2 

(3-136)
where
βac2
(3-137)
βac2  1
is known as the small signal, short circuit, common base current gain (more about this definition
later in the text), and Rd2 is the diode-connected transistor resistance introduced in (3-117). In
particular, observe in Figure (3.36a) that R3 = 0 and infinitely large R4 combine to reduce the Vbe
multiplier cell to a conventional diode-connected transistor. Reassuringly, Rv in (3-136) collapses to Rd2 for this special case. Since βac2 >> 1, αac2 in (3-137) approaches one, thereby
simplifying the expression for resistance Rv to
 Rd2 

R3 
(3-138)
Rv  
  R3  R4    1 
 Rd2  km Rd2 .
R4 
 Rd2  R4 

which is only modestly larger than Rd2.
αac2 
Like the diode-connected transistor in the active biasing scheme of Figure (3.29), the
Vbe multiplier subcircuit in Figure (3.33) appears in series with biasing resistance R2. Otherwise,
the two subject circuits are identical. It follows by inspection (don’t you just love when professors say that) that the resistance, Rx, presented to the collector load in Figure (3.33) derives directly from (3-116), subject to the caveat that Rd in (3-116) is replaced by Rv; that is


βac Ree
Rx  Ree  R1  R2  Rv   rbb  rπ    1 
 ro


R
R
R
R
r
r






ee
1
2
v
bb
π


(3-139)


βac Ree
 1 
 ro .

Ree  R1  R2  Rv   rbb  rπ 

EXAMPLE #3.8:
The silicon NPN transistors in the active biasing circuit of Figure (3.33) have a minimum hFE of 100 and a nominal base-emitter voltage of 720 mV at room temperature.
Design the circuit so that with a power supply voltage of Vcc = 6 volts, transistor Q1
conducts a quiescent collector current of IcQ = 2 mA when the collector-emitter voltage,
VceQ, of Q1 is 1.5 volts. The circuit is to be designed so that IcQ is nominally invariant
with temperature and is maintained at its design target to within ±3%. Use the HSPICE
BJT model parameters itemized in Table (3.1) to simulate the finalized design at 27 °C,
50 °C, 75 °C, and 100 °C.
SOLUTION #3.8:
(1).
In order to optimize thermal tracking between the identical transistors, Q1 and Q2, the collector current densities of these two transistors must be the same. Additionally, implementing
nominally identical collector-emitter voltages between the two devices is recommended to
mitigate the effects of base conductivity modulation. Thus, voltage Vm in Figure (3.33)
should equate to voltage VceQ, which is stipulated to be 1.5 volts. Ignoring current Ic2/hFE2 in
comparison with current Vbe2/R4 in the companion subcircuit diagram of Figure (3.34a) results
in
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Chapter 3
BJT Models & Biasing

R 
(E8-1)
Vm   1  3  Vbe2  VceQ  1.5 volts .
R4 

Given identical collector current densities in the two transistors, Vbe2 = VbeQ ≈ 720 mV,
whence R3/R4 = 1.083. Recalling (3-133), which is the condition underpinning the nominal
thermal independence of current IcQ, R2/R1 = R3/R4 = 1.083.
(2).
In an attempt to conserve standby power, let Q2 conduct one-half the current of transistor Q1,
which is to say that the emitter-base junction area of Q1 must be twice that of Q2. Thus, Ic2
in Figure (3.34a) is 1 mA. Moreover, arbitrarily allow resistance R4 to conduct 1 mA of current, which makes current Im = IcQ ≈ 2 mA. It follows that Vbe2/R4 = VbeQ/R4 = 1 mA, or Rx =
720 Ω. Since R3/R4 = 1.083, R3 = 780 Ω. Note that Ic2/hFE2 = 1 mA/100 = 10 μA, which is
100-times smaller than the current, Vbe2/Rx, thereby supporting (E8-1) to within an error of
roughly 1%. We note that this error is safely within the 3% error tolerance assigned to current IcQ.
(3).
If IcQ/hFE << Im, which has been set to 2 mA, Figure (3.33) confirms
Vcc   R1  R2  I m  Vm .
(E8-2)
With Vcc = 6 volts, Im = 2 mA, and Vm = 1.5 volts, (R1 + R2) is 2,250 Ω. Recalling that R2/R1
= 1.083, this computation produces R1 = 1,080 Ω and R2 = 1,170 Ω. It is important to check
the propriety of the invoked approximations. In the present case, IcQ/hFE << Im has been presumed. Since IcQ/hFE = 2 mA/100 = 20 A and Im = 2 mA, Im exceeds IcQ/hFE by a factor of
100, which amounts to a computational error of around only 1%.
(4).
Equation (3-133) can now be used to determine the resistance, Ree. Although resistance Rm,
as defined by (3-126) can be ignored because of its inverse dependence on (hFE + 1), it is a
trivial matter to include Rm in the calculations. To this end Rm = 7.72 Ω in that R3 = 780 
and hFE = 100. Then, with R1 = 1,080 Ω, R2 = 1,170 Ω, αFE = 100/101, and Vcc = 6 volts, Ree
= 1,549 Ω. Since (3-133) presumes (R1||R2) = 561.6  << (hFE + 1)Ree = 156.5 KΩ, the
approximation is satisfied by a factor of 156.5 K/561.6  = 278.7, which is tantamount to a
computational error of 1/278.7 = 0.36%. This small error means that the computed value of
Ree is high by nominally (0.0036)(1,549 ) = 5.6 . Given the other approximations invoked in the course of this design procedure, it may be prudent to reduce Ree by very slightly
more than 5.6 . To this end, allow Ree = 1,540 Ω.
The only remaining resistance to be computed is the collector load resistance, Rl. From Figure (3.33),
 h 1
Vcc  Rl I cQ  VceQ   FE
(E8-3)
 Ree I cQ ,
 hFE 
which straightforwardly y Rl = 694.6 Ω. Figure (3.37) delineates the completed design. As
in the preceding design examples, the estimated static node voltages are underscored in the
schematic diagram.
ENGINEERING COMMENTARY:
The simulated results confirm the propriety of the design methodology. In particular, the
simulated quiescent collector current of transistor Q1 at 27 °C is 2.00 mA, while at 50 °C, 75
°C, and 100 °C the simulated Q-point collector current is 2.01 mA. Moreover, at 27 °C, the
simulated collector, base, and emitter voltages of transistor Q1, measured with respect to
ground, are 4.61 volts, 3.84 V, and 3.11 V, respectively. For Q2, the corresponding voltages
are 3.84 V, 3.05 V, and 2.32 V.
Design solutions are rarely unique. In the present example, different design results accrue if a
current other than 1 mA is arbitrarily chosen to flow through resistance R4. From a thermal
sensitivity perspective, the circuit at hand is clearly excellent. Nonetheless, it is assuredly not
(5).
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Chapter 3
BJT Models & Biasing
the panacea for BJT biasing for at least two reasons. First, a large number of resistances are
required. It is extremely important that the resistance pairs, R1-R2 and R3-R4, respectively
track one another well over temperature extremes. Second, we note as we did in the first
biasing example, that resistance Ree exceeds the value of the collector load resistance, thereby
limiting the available voltage gain in the absence of additional, design heroics.
+6 V
1080
694.6
x2
4.61 V
Q1
3.83 V
780
Q2
3.06 V
720
2.34 V
3.11 V
1540
1170
Figure (3.37). Biasing circuit designed in Example #3.8. All resistances are in units
of ohms. The voltages shown in the boxes correspond to the estimated
node voltages for a properly functioning circuit. Transistor Q2 has
twice the emitter-base junction area of transistor Q1.
3.4.2.4. N-Stage Current Mirror
The satisfactory performance of many analog networks depends on the reliable and
accurate implementation of current sources (or sinks) that act as high impedance active loads for
amplifier cells or provide static paths to ground for currents used in the biasing of other subcircuits used in these networks. Rather than implement a required multiplicity of current sinks
individually, expedience motivates us to derive these sources or sinks from a single reference
cell. To this end, a common embodiment is the network in Figure (3.38). The reference current,
Io, for the network derives from a subcircuit comprised of branch resistance R and diode-connected transistor Q0. The collectors of identical transistors Q1, Q2, · · · QN establish current
sinking paths. In these transistors, the emitter-base junction injection area of the nth transistor is
a factor of kn larger than the junction area of Q0. Assuming that the transistors are otherwise
identical, the obvious identity among the base-emitter terminal voltages of the (N+1) cascaded
transistors in Figure (3.38) produces, in general, an emitter current flowing in the nth transistor of
knIo. This current supports corresponding collector and base currents of αFEknIo and knIo/(hFE+1)
= (1−αFE)knIo, where parameter αFE is related to the DC beta, hFE, by (3-48). The validity of the
last statement rests on each transistor operating in its forward active regime. In turn, this requirement constrains the nth collector-emitter voltage, Vn, to be at least as large as the base-emitter terminal potential, Vbe, applied to all transistors. The tacit assumption of equal hFE among all
transistors stems from the presumption that either all forward Early voltages are large and/or that
no significant differences prevail among the individual collector-emitter voltages for all (N+1)
- 266 -
Chapter 3
BJT Models & Biasing
transistors.
Vcc
V1
R
V2
Rx1
Ir
Rx2
FEk1I o
Q1
VN
RxN
FEk2I o
FEkNIo
Q2
QN
Q0
x k1
Io
k1Io
x k2
x kN
k2Io
kNIo
Figure (3.38). An N-stage current mirror. All (N +1) transistors are fundamentally identical, save
for the fact that the nth transistor has an emitter-base junction injection area that is kntimes larger than that of the diode-connected transistor, Q0.
The reference current, Ir, conducted by resistance R in the circuit of Figure (3.38) must
accommodate the Q0 emitter current, Io, and each of the base currents conducted by transistors
Q1 through QN. Thus,
I r  I o   1  αFE  I o
N
 ki .
(3-140)
i 1
It follows that the current, Icn, conducted by the collector of the nth device is
αFE kn Vcc  Vbe 
αFE kn I r
(3-141)
I cn 
.

N
N


1   1  αFE  ki
1   1  αFE  ki  R

i 1
i 1 
Clearly, the relative insensitivity to the DC beta, hFE, of any collector current flowing in Q1
through QN demands that

N
 ki
i 1


1
  hFE  1 .
1  αFE
(3-142)
The satisfaction of this design requirement is progressively more challenging if one or more of
the area scaling factors is large and/or the number of current sinking stages is large. One caveat
to this stipulation is that (3-142) is easily satisfied when SiGe HBTs, which have very large hFE,
are deployed in the current sinking configuration.
In addition to potentially compromising collector current sensitivity to hFE, large area
scaling factors also limit the small signal output resistance, Rxn, achievable in the nth current sink.
It is clear that if the nth transistor in Figure (3.38) is replaced by its low frequency, small signal
BJT equivalent circuit, Rxn, as indicated in the circuit diagram, is merely the Early resistance, ron,
of the nth device. Ignoring internal collector and emitter series resistances, a reference to (3-56)
spawns, in terms of the forward Early voltage, Vaf,
Vn  Vaf
Vn  Vaf


Rxn  ron 
 
(3-143)
R ,
I cn
 αFE kn Vcc  Vbe  
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Chapter 3
BJT Models & Biasing
where the validity of (3-142) is presumed. We note in this last expression that the current sink
output resistance is inversely dependent on the area factor for the nth emitter-base junction.
Thus, too large of a scaling factor compromises the desired relative independence of the nth
collector current on the voltage impressed with respect to ground at the nth collector node.
An improved version to the N-stage current mirror, offered in Figure (3.39), addresses
the potential area scaling shortfall to which the circuit in Figure (3-48) may succumb. Transistor
Q0 in this revised circuit diagram no longer functions as a diode. Nonetheless, its emitter still
supports a static current of Io, as in the original diagram of Figure (3.38). The insertion of
transistor QP, as shown in Figure (3.39), forces its emitter to conduct the sum of the base currents flowing in transistors Q0 through QN, as well as the current, Ip, conducted by the resistance, Rp. In principle, resistance Rp, is not required. But without Rp, the emitter of transistor QP
conducts only the superposition of the small base currents in transistors Q0 through QN. This
small current risks a debilitating reduction in the DC beta of QP, as well as other undesirable effects. In contrast, if Rp is selected to conduct a current that is nominally equal to the emitter current of transistor Q0, hFE for transistor QP approximates the DC beta of transistor Q0 and hence,
the DC beta values of all other transistors.
Vcc
V1
R
V2
Rx1
QP
Q0
Rx2
FEk1I o
Ir
NP
Rp
Io
Q1
VN
RxN
FEk2I o
FEkNIo
Q2
QN
x k1
x k2
k1Io
x kN
k2Io
kNIo
Ip
Figure (3.39). An improved version of the N-stage current mirror in Figure (3.38). The insertion
of transistor QP diminishes the dependence of the nth collector current on both the
number of stages deployed and the area scaling factors of these individual stages.
At the risk of analytical prematurity, we offer an additional point −one whose significance will be clarified later when we talk about regulated cascode networks− before proceeding
with the analysis of the compensated topology in Figure (3.39). The foundational essence of this
point is that transistor Q0 implements active, global, shunt-shunt feedback around transistor QP,
which in turn functions as a voltage buffer. The important effect of this active feedback is that it
substantially reduces the net Thévenin resistance seen at the node, NP, to which the emitter of
transistor QP is incident. This is an important trademark of the modified current sink topology.
Its importance stems from the fact that the feedback compensation forces the impedance witnessed at node NP to converge toward zero. To be sure, the subject impedance is never going to
be zero, but it is indeed small, as we shall demonstrate in a subsequent chapter. If the impedance
at node NP is nearly zero, the corresponding NP node voltage, Vbe0, which is the quiescent baseemitter potential of transistor Q0, functions as virtually an ideal voltage source driving the baseemitter terminals of the other N transistors. Thus, the mirroring of voltage Vbe0 at the base-emitter terminals of Q1 through QN nears perfection in that Vbeo is rendered almost independent of
the net base current drained from node NP.
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Chapter 3
BJT Models & Biasing
An inspection of the circuit in Figure (3.39) reveals
N


(3-144)
I r  αFE I o   1  αFE   I p   1  αFE  I o  1  αFE  I o ki  .


i 1 
In addition to ensuring nominally identical hFE among all devices, it is convenient to set Ip = Io.
In this case,
N

2 
(3-145)
I r  I o   1  αFE  I o  1 
ki  ,


i 1 

which implies that the current flowing through resistance R in Figure (3.39) is essentially
invulnerable to the number of current sink stages and the area scaling factors invoked in the network. This observation conflates with our feedback argument above. Equally important is that it
contrasts sharply with (3-140), which depicts current Ir in the diagram of Figure (3.38) as potentially sensitive to loading effects. We note with additional interest that the subject current is
nominally equivalent to the emitter current conducted by transistor Q0. Since Ir ≈ Io, the collector current of the nth transistor becomes
α k V  2Vbe 
I cn  αFE kn I o  FE n cc
.
(3-146)
R
The result at hand suggests that the price paid for the incorporation of transistor QP into the cascaded current mirror is a doubling of the temperature sensitivity in that current Io is now dependent on twice the emitter-base junction potential, as opposed to the single Vbe dependence projected by (3-141). However, we can somewhat compensate for this doubling through a prudent
increase in the value of circuit resistance R.


3.4.2.5. Wilson Current Mirror
The Wilson current mirror mitigates the port resistance limitations pervasive of the Nstage mirror considered in the preceding subsection by exploiting shunt-series feedback to boost
output resistance Rx above the Early resistance of the output port transistor[10]. Its basic schematic diagram is shown in Figure (3.40). In this diagram, the grounded emitter transistor, Q2,
and the diode-connected device, Q3, implement feedback that is in shunt with the base terminal
of output transistor Q1 and in series with the emitter lead of Q1. The three transistors are similar, but transistors Q1 and Q3 have an emitter-base junction injection area that is k-times larger
than the corresponding injection area of transistor Q2. We can therefore presume that the DC
beta values and applied base-emitter terminal voltages of all active devices are nominally the
same, especially if each transistor has large forward Early voltages. All devices operate in their
forward active regimes. In the case of transistor Q1, such operation requires that the output port
voltage, V1, be at least as large as 2Vbe.
For transistor Q2, the reference current, Ir, flowing into the junction of resistance R
with the base of Q1 and the collector of Q2 satisfies
V  2Vbe
(3-147)
I r  cc
.
R
If the emitter of transistor Q2 conducts a current of I2, as indicated in Figure (3.40), the emitter
of transistor Q3 necessarily conducts kI2 in light of the area stipulations and since the base-emitter terminal voltages of these two devices are the same. Accordingly, the collector of Q2 con-
- 269 -
Chapter 3
BJT Models & Biasing
ducts current αFEI2, while a current of (1 −αFE)I2 flows into the base of Q2. In order to keep
Kirchhoff happy, the emitter of transistor Q1 necessarily supports a current, Ie1, of
V1
Vcc
Rx
R
Ir
I1
Ib1
Q1
xk
FE I2
(1FE )I2
Ie1
Q2
x1
Q3
xk
I2
kI2
Figure (3.40). Basic schematic diagram of the Wilson current mirror.
I e1  kI 2  1  αFE  I 2 ,
(3-148)
whence an output port current, I1, flowing into the collector of transistor Q1 of
1  αFE 

I1  αFE kI 2  1 
.
k


It follows that the current, Ib1, flowing into the base of transistor Q1 is
I
1  αFE 

Ib1  1   1  αFE  kI 2  1 
.
hFE
k


This current requires a resistance current, Ir, of

 k 
1  αFE  
I r  I b1  αFE I 2  αFE I 2 1  
 1 
 .
k

 hFE  

(3-149)
(3-150)
(3-151)
Observe that for large hFE, the output current mirrors, by a factor of k, the current conducted by
resistance R, which in turn closely approximates the collector current conducted by transistor Q2.
Specifically, (3-149), (3-151), and (3-147) combine to deliver
k Vcc  2Vbe 
I1  αFE kI 2  kI r 
.
(3-152)
R
Recalling that base-emitter terminal voltage Vbe decreases with junction temperature Tj
in accordance with the linear empirical expression, Vbe ≈ −SbeTj, the temperature-induced
perturbation, I1, in current I1 is seen to be
2k Sbe
2I1Sbe
ΔI1
(3-153)


,
ΔT j
R
Vcc  2Vbe
which in turn stipulates the temperature-induced per-unit change in current I1 as
- 270 -
Chapter 3
BJT Models & Biasing
2Sbe ΔT j
2ΔVbe
ΔI1
(3-154)

 
.
I1
Vcc  2Vbe
Vcc  2Vbe
As is generally the prerequisite implicit to a design scenario focused on realizing a high degree
of temperature stability, an appropriately large supply voltage (Vcc in this case) is recommended.
Rx
rbb1
Vx
r1
Ia
rbb2
Iaac2 Ib
ro2
R
ac2 Ib
r2
ac1 Ia
ro1
Ixac1 Ia
IxIa
Ix
Rd3
IxIaIb
Ib
(a).
Rx
rbb
Vx
r
Ia
krbb
Iaac Ib
Rss
ac Ib
kr
ac Ia
ro
Ixac Ia
IxIa
Ix
Rd3
IxIaIb
Ib
(b).
Figure (3.41). (a). Low frequency, small signal equivalent circuit of the Wilson current mirror
shown in Figure (3.40). The small signal output resistance, Rx, is the voltage to current ratio, Vx /Ix. (b). The model of (a) redrawn to account explicitly for the fact that
the emitter-base junction injection areas of transistors Q1 and Q3 in Figure (3.40)
are a factor of k larger than the junction injection area of transistor Q2.
Figure (3.41a) depicts the approximate low frequency, small signal model for evaluating the output resistance highlighted as Rx. In this equivalent circuit, resistance Rd3 derives from
(3-117) in that it represents the small signal terminal resistance of diode-connected transistor Q3.
The circuit analysis, which is sufficiently sloppy to warrant the company of a nice pinot noir, can
more easily be interpreted if the given junction injection ratios are judiciously exploited. To this
- 271 -
Chapter 3
BJT Models & Biasing
end, assume that all transistors are characterized by large forward knee currents and Early voltages. Then, since Q1 and Q3 conduct roughly the same collector current, their base, base-emitter diffusion, and Early resistances are nominally the same; that is
rbb1  rbb3  rbb
rπ1  rπ3  rπ
ro1  ro3  ro
.
(3-155)
In view of the fact that the small signal beta of each device is the product of forward
transconductance gm, which nominally varies directly with quiescent collector current, and baseemitter diffusion resistance rπ, which varies inversely with collector current,
βac1  βac2  βac3  βac .
(3-156)
From (3-117),
 r  rπ3 
 rbb  rπ 
rbb  rπ
Rd 3  ro3  bb3
.
(3-157)
  ro 
 
βac  1
 βac3  1 
 βac  1 
Since the emitter-base geometry of transistor Q2 is smaller than that of either Q1 or Q3 by a factor of k and since this device conducts a current that is correspondingly k-times smaller than the
current supported by transistors Q1 and Q3,
rbb2  krbb3  k rbb
(3-158)
rπ2  krπ3  k rπ
.
ro2  kro3  k ro
The immediate modeling ramification of (3-155) through (3-158) is the equivalent circuit given
in Figure (3.41b). In this diagram, resistance Rss is given by
Rss   kro  R  R ,
(3-159)
which invariably is adequately approximated by merely the circuit resistance, R.
A conventional analysis of the circuit in Figure (3.41b) generates the following equilibrium equations:
Rd3 I x  Rd3 I a   k  rbb  rπ   Rd3  I b
(3-160)
Rss  I a  βac I b    rbb  rπ  I a  k  rbb  rπ  I b  0 .
Vx  ro  I x  βac I a   k  rbb  rπ  I b
Unfortunately, these equations must be solved simultaneously to arrive at the desired expression
for the resistance, Rx = Vx/Ix, of interest. A second glass of the aforementioned pinot noir mitigates the inclination toward profanity and results in




r r 
r r 
αac βac Rss  k  bb π 
k  Rss  rbb  rπ   bb π 




V
 βac  1 
 βac  1 
r  
 ,
Rx  x  1 
o



Ix
r r 
 r  r 

  k  1 Rss  1  k  βac  2    bb π  
 k  1 Rss  1  k  βac  2   bb π  


 βac  1  
 βac  1  
(3-161)
where (3-156) is used and
βac
αac 
.
βac  1
(3-162)
- 272 -
Chapter 3
BJT Models & Biasing
For large AC beta and the invariably large Early resistance of a bipolar junction transistor, the
formidable resistance expression in (3-161) collapses to the simpler relationship,


βac Rss
Rx  1 
(3-163)
 ro ,
k

1
R

k
r

r




ss
bb
π 

which can, for reasonable area factor k, be substantially larger than the Early resistance, ro.
EXAMPLE #3.9:
The SPICE parameters of the silicon technology transistors utilized in the
Wilson current mirror of Figure (3.40) are itemized in Table (3.1). Design the
circuit so that the current, I1, conducted by transistor Q1 is 4 mA at room
temperature. Assuming an emitter-base voltage temperature sensitivity of Sbe =
2 mV/°C, this current is to be sustained to within ±5% up to operating temperatures of 75 °C. Use the HSPICE BJT model parameters in Table (3.1) to simulate the finalized design at 27 °C, 50 °C, and 75 °C. Additionally, simulate the
room temperature frequency response of the magnitude of the output impedance
presented at the collector of Q1.
SOLUTION #3.9:
(1).
Given that conventional silicon transistors are deployed in the subject Wilson mirror, the
base-emitter terminal potentials of all active devices are nominally 700 mV. But a slightly
more satisfying approach entails deducing these potentials by using the values of saturation
current Is, junction injection coefficient nf, and forward knee current Ikf given in Table (3.1).
To this end, Ikf = 11.5 mA yields a forward transport current, Icc, in (E3-1) of 1.342 mA for a
collector current of 1 mA. Then, with Icc = 1.342 mA, Is = AeJs = 3.2 fA, and nf = 1.05.
Additionally, the approximate room temperature value of base-emitter terminal potential, Vbe,
derives from (3-5) as Vbe ≈ 727.1 mV. It follows that voltage V1 must satisfy V1  2Vbe =
1.454 V.
(2).
In an attempt to conserve standby power, let transistor Q2 conduct one-fourth the current
flowing in both transistors Q1 and Q3, which effectively stipulates a requisite junction injection area factor of k = 4.
(3).
In (3-154), I1/I1 = 0.05, Tj = (75 °C − 27 °C) = 48 °C, Sbe = 2 mV/°C, and Vbe = 727.1 mV.
Then, since
2Sbe ΔT j
ΔI1

 0.05,
(E9-1)
I1
Vcc  2Vbe
Vcc  Vbe + (2SbeTj /0.05) = 5.294 V. Select Vcc = 6 V to provide at least token design headroom to compensate for parametric uncertainties and analytical approximations.
(4).
For a reference current, Ir, of about 1 mA, (3-151) produces R = 4.546 K. Once again,
engineering introspection of design-oriented modeling and circuit parameter uncertainties
motivates the selection of resistance R as the slightly smaller R = 4.3 K.
(5).
Figure (3.42) is the schematic diagram of the completed design, where we have selected voltage V1 as 2 V to satisfy the requirement of V1  1.454 V. As usual, estimated static node voltages are provided in the diagram.
- 273 -
Chapter 3
BJT Models & Biasing
6 V
2V
Zx
4.2 K
4 mA
Q1
1.45 V
x4
Q2
727 mV
x1
Q3
x4
Figure (3.42). Schematic portrayal of the Wilson circuit designed in
Example #3.9. The voltages shown in the boxes
correspond to the estimated respective node voltages
for a properly functioning circuit. The indicated 4 mA
collector current of transistor Q1 is a design target.
ENGINEERING COMMENTARY:
As in previous examples, the simulated results confirm the propriety of the design methodology. In particular, the simulated quiescent collector current, I1, of transistor Q1 at 27 °C is
4.008 mA, at 50 °C, the simulated Q-point current is 4.075 mA, and at 75 °C, the simulated
value of I1 is 4.149 mA. Thus, the largest temperature-induced change in current I1, which
unsurprisingly is observed at 75 °C, is only 3.52% larger (well within the target 5% deviation
target) than the subject current value at room temperature.
MODEL
IB
IC
VBE
VBC
VCE
BETADC
GM
RPI
RX
RMU
RO
CPI
CMU
CBX
CCS
BETAAC
FT
Q1
NT
3.5627E-05
4.0080E-03
0.7245
-0.5505
1.2750
112.5001
1.3667E-01
7.6227E+02
6.7224E+01
1.1718E+12
7.6179E+03
6.1650E-13
3.6996E-15
0.0000E+00
2.0741E-14
1.0418E+02
3.5071E+10
Q3
NT
3.6239E-05
3.9984E-03
0.7250
0.0000
0.7250
110.3338
1.3619E-01
7.4940E+02
6.7220E+01
6.0739E+11
7.4986E+03
6.1446E-13
4.4335E-15
0.0000E+00
2.8694E-14
1.0206E+02
3.5022E+10
Q2
NT
9.0492E-06
1.0226E-03
0.7250
-0.7245
1.4495
113.0081
3.4835E-02
3.0011E+03
2.6888E+02
1.1946E+12
3.0027E+04
1.5701E-13
8.8742E-16
0.0000E+00
5.8204E-15
1.0454E+02
3.5113E+10
The simulated output impedance magnitude is displayed in Figure (3.43). At the quiescent
operating point established by Vcc = 6 volts and V1 = 2 volts, SPICE discloses the operating
- 274 -
Chapter 3
BJT Models & Biasing
point parametric detail itemized above. Several noteworthy observations surface from this
itemization and the subject frequency response plot.
Magnitude Of Output Impedance, Zx, (K)
160
120
80
40
0
1
10
100
1000
Signal Frequency (MHz)
Figure (3.43). Simulated frequency response of the magnitude of the output impedance, Zx, for
the Wilson current mirror in Figure (3.42).
(a). The low frequency value of the simulated magnitude of the output impedance is 144.85
KΩ which is almost 19-times larger than the simulated value of the Early resistance
(shown above as RO = 7.6179 K) for transistor Q1. Accordingly, our prediction of an
output resistance that is substantially larger than the Early resistance is spot on.
(b). Despite the fact that SPICE predicts a unity current gain frequency (FT) of about 35 GHz
for all three transistors, the 144.85 KΩ resistance level attenuates by a factor of root two
at about 29 MHz.
(c). As presumed in the course of the analysis of the Wilson mirror, the DC beta (BETADC)
and the AC beta (BETAAC) are approximately the same for all transistors. Recall that the
smallest geometry transistor, which carries a collector current that is 4-times smaller than
the other two devices is Q3. Note that the values of rπ (RPI), ro, and rbb (RX) for transistor Q3 are nominally 4-times larger than the corresponding parameters for transistors Q1
and Q2. On the other hand, Cπ (CPI) for the higher current transistors is roughly 4-times
the Cπ value of Q3, which is as expected owing to the almost direct dependence of emitter-base diffusion capacitance on quiescent collector current. In contrast, the base-collector transition capacitances, Cμ (CMU) are relatively independent of current level, as expected, but they do scale with junction area. A similar statement prevails for the
substrate capacitances (CCS), which dominantly limit the attainable frequency response
of impedance Zx. Recall that substrate capacitances do not factor into the enumeration of
the unity gain frequency, fT, of the transistors.
3.5.0. SUPPLY INDEPENDENT BIASING
The last six biasing networks mitigate the deleterious effects of omnipresent junction
temperature rises in bipolar junction transistors. Unfortunately, these networks establish quies- 275 -
Chapter 3
BJT Models & Biasing
cent current responses that functionally depend on the supply voltage. A bias current dependence on applied power line voltage is hardly surprising. But in the portable electronic age in
which we live, such dependence comprises a serious problem. For example, degrading supply
voltages incurred by aging batteries is unavoidable. When these voltages decrease to very low
levels, quiescent collector currents certainly diminish. In the wake of these line voltage decreases, there is even the likelihood of disabling the forward active regime operation of certain
transistors. A more insidious problem that prevails in both portable and plug-in electronics is the
fluctuation in line voltage incurred by the interaction of nonzero Thévenin supply impedances
with large transient fluctuations in the currents delivered by these supplies. These fluctuations
can influence transient responses and distortion characteristics and in extreme cases, they can
even cause network instability. In high performance applications where these power line voltage
fluctuations are only minimally intolerable, biasing networks that deliver reduced sensitivity to
supply line variations are essential. Networks that function with maximal independence of the
power line voltage are said to deliver a high power supply rejection (PSR).
3.5.1. BASE-EMITTER VOLTAGE OFFSET BIASING
One practical example of a network capable of delivering biasing currents that are
nominally independent of supply line voltages is the voltage offset configuration appearing in
Figure (3.44). The NPN transistors, Q1 and Q2, are identical, save for the fact that the emitterbase junction injection area of Q2 is kn-times larger than that of Q1. If a quiescent current, IQ, is
presumed to flow in the emitter of transistor Q2, nominally the same current is conducted by the
emitter of the PNP diode-connected transistor, Q3. This statement reflects three assumptions.
First, it assumes that all transistors have large DC beta, which allows tacit neglect of all transistor
base currents. Second, all transistors have large forward knee currents, and third, they boast
large Early voltages. Since transistors Q3 and Q4 have identical junction injection areas and utilize identical emitter degeneration resistances, Ree, current IQ in Q3 is mirrored in the emitter of
transistor Q4 and hence, in the emitter of transistor Q1. From the Ebers-Moll model, it follows
that
V
n V
V
n V
IQ  I s  e be1 f T  1  kn I s  e be2 f T  1 .
(3-164)




If we neglect the unity terms within the bracketed quantities, this relationship produces
Vbe1  Vbe2  n f VT ln  kn  .
(3-165)
By using the Q3-Q4-Ree mirror to force equality between the emitter currents of transistors Q1
and Q2, the difference in emitter-base junction injection areas of Q1 and Q2 fosters an offset between the corresponding base-emitter terminal voltages, Vbe1 and Vbe2. This nonzero offset is
crucial in that it is the vehicle by which a nominal independence of current IQ on the supply line
voltage, Vcc, is established Specifically,
Vbe1  Vbe2  RIQ ,
(3-166)
whence, by (3-165),
n f VT ln  kn 
V  Vbe2

IQ  be1
,
R
R
which is independent of voltage Vcc.
(3-167)
Before celebrating the supply-independence advanced by (3-167), at least five important engineering caveats warrant careful engineering scrutiny. First, (3-165) and the current
mirroring action between transistors Q3 and Q4 require that transistors Q4 and Q2 function in
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Chapter 3
BJT Models & Biasing
their forward active regimes. On the other hand, transistors Q1 and Q3 automatically operate in
their linear domains because the diode connection of these transistors forces their base-collector
terminal voltages to zero. Such operation for transistor Q4 requires Vec4 ≥ Veb4, where Vec4 and
Veb4 respectively denote the emitter to collector and the emitter to base voltages developed on
Q4. In turn, Vec4 ≥ Veb4 demands a supply voltage that is large enough to satisfy,
Vcc
Ree
Ree
Ree /kp
Q3
Q4
x1
Q1
x1
Q5
x1
x kp
Rx
Q2
Vo
x kn
R
IQ
IQ
Rl
Io
Figure (3.44). Schematic diagram of a biasing network delivering quiescent currents IQ and Io that are, to first order, independent of
the applied supply voltage, Vcc.
Vcc  Ree IQ  Veb4  Vbe1 .
(3-168)
Assuming both Veb4 and Vbe1 are of the order of 700 mV, we see that Vcc in excess of approximately (1.4+ReeIQ) volts is commanded if the desired linear operation of the circuit is to be sustained. For transistor Q2, linear operation mandates
Vce2  Vcc  Veb3   Ree  R  IQ  Vbe2 ,
(3-169)
which in turn commands
Vcc  Ree IQ  Vbe2  Veb3  n f VT ln  kn 
(3-170)
You are encouraged to verify that the satisfaction of (3-170) actually guarantees the satisfaction
of (3-168). As might we might have expected intuitively, the bias circuit produces quiescent currents that are not precisely independent of supply voltage Vcc. Problems are manifested when, as
per (3-170), voltage Vcc drops too low.
The second important point relevant to (3-167) is that while the indicated current is
ostensibly independent of Vcc, subject to the constraint imposed by (3-170), it varies with
temperature. In fact and to the extent that resistance R boasts a small temperature coefficient, IQ
in (3-167) is PTAT in that Boltzmann voltage VT is directly proportional to absolute junction
temperature. But the subject temperature dependence is not entirely bad news in that the forward
transconductance of transistor Q2 (and also Q1, Q3, and Q4) is rendered relatively insensitive to
temperature. In particular, (3-54) confirms that the forward transconductance is inversely dependent on the effective Boltzmann voltage, nfVT. To wit, if the effects of knee current and Early
- 277 -
Chapter 3
BJT Models & Biasing
voltage continue to be ignored, (3-54) gives rise to
IQ
ln  kn 

g m2  g m1  g m3  g m4 
.
n f VT
R
(3-171)
This disclosure is notable for two reasons. First, if the circuit at hand is operated as an amplifier,
wherein the gain and particularly the open loop gain in a feedback configuration, is dependent on
forward transconductance, the overall I/O gain can be rendered temperature independent.
Second, the forward transconductances of transistors Q1 through Q4 are rendered independent of
relatively unpredictable transistor modeling parameters.
The third noteworthy point is that the current conducted by transistor Q5, whose emitter-base junction injection area is kp-times that of transistors Q3 and Q4, is, like current IQ, nominally invariant with supply voltage. This contention follows from the fact that the emitter
degeneration resistance of transistor Q5 is a factor of kp smaller than that of Q3, which implies
k p n f VT ln  kn 
I o  k p IQ 
,
(3-172)
R
since
R 
Ree IQ  Veb   ee  I o  Veb .
(3-173)
 kp 


If transistor Q5 operates in its linear regime where Vec5 ≥ Veb5, its transconductance is
k p IQ
k p ln  kn 
Io
gm5 
,


(3-174)
n f VT
n f VT
R
which, like the transconductances of the other transistors in the subject network, is essentially
independent of junction operating temperature. In practice, the area ratio, kp, should be kept under two or three in that large kp sustains proportionately larger collector current in Q5. In turn,
large current levels in Q5 diminish its Early resistance, which reduces the attainable output resistance, Rx, of the current source.
The fourth noteworthy point is that resistance Ree can be set to zero without affecting
the current solution in (3-167). In fact, it is desirable to set Ree to zero in some low voltage
applications, since zero emitter degeneration in the PNP devices in Figure (3.44) lowers the
minimum required value of the supply voltage, Vcc. It is included herewith to increase the output
resistance, Rx, so that the Q5-Ree/kp subcircuit emulates a current source whose output current is
rendered progressively more independent of supply voltage.
The final, and arguably most important, point to be offered is that while (3-167) defines
the desired current solution for the biasing network in Figure (3.44), it is unfortunately not the
only solution. In particular, Vbe1 = Vbe2 = 0 in (3-166) sets IQ = 0 as a plausible current solution.
In other words, it is possible that at power supply turn on, transistors Q1 and Q2, and thus all
other transistors in the network, lock to a nonconductive state. Slight parametric differences
among the current mirroring devices are likely to preclude this annoying null state, but given the
luck of the traditional circuit design engineer, gambling that the desired solution projected by (3167) is realized is imprudent. Accordingly, a startup circuit −preferably a subcircuit that consumes no, or at least negligible, steady state power− is a worthwhile investment in design time
and chip area.
To the foregoing end, a relatively simple startup addendum to the network of Figure
(3.44) is submitted in Figure (3.45). The startup cell comprised of capacitance C and transistors
- 278 -
Chapter 3
BJT Models & Biasing
Vp(t)
Ree
Ree
Ree /kp
Q3
Q4
x1
Q5
Vcap
+

x1
C
x kp
Q7
x kc
Q1
x1
t=
x kc
SW
Startup Cell
Q6
Rx
Q2

Vcc

Vo
x kn
R
IQ
0
IQ
Rl
Io
Figure (3.45). The supply-independent biasing network of Figure (3.44) with a startup cell incorporated to preclude a null current solution at network startup.
Q6 and Q7 conduct current only at circuit startup. Startup in the present sense corresponds to
closing switch SW. If switch closure occurs at time t = 0, the indicated line voltage, Vp(t),
approximates an ideal voltage step; that is, Vp(t) = Vccu(t). Assuming capacitance C is initially
uncharged, the idealized instantaneous jump in the power line voltage, Vp(t), is transmitted instantly to the base-emitter terminals of transistor Q7. With the emitter-base junction of Q7
strongly forward biased, a potentially large collector current is allowed to flow through Q7.
Clearly, this current can derive only from transistor Q3. The current conducted by Q3 is mirrored to transistor Q4. In response to this mirroring action, the current forced through transistor
Q1 by Q4 is accompanied by an increase in base-emitter terminal voltage Vbe1 from its zero value
that prevails prior to switch closure. Note, however, that voltage Vbe2 remains nominally zero as
Vbe1 commences its increase. We state this with confidence because the large forward bias imposed across the emitter-base junction of Q7 immediately after switch closure forces the current
conducted by Q3 to flow through Q7. In effect, Q7 bypasses the Q2-R subcircuit immediately
after time t = 0. But as voltage Vbe1 ultimately increases, transistor Q6 starts to conduct. Its
collector current can be supplied only by the capacitive branch of the startup cell. This branch
continues to supply current until capacitor C is fully charged, which occurs when the indicated
capacitor voltage, Vcap, is Vcc. As Vcap approaches Vcc to within an emitter-base junction threshold voltage, the base-emitter voltage of transistor Q7 drops below threshold potential, whereupon transistor Q7 enters cutoff. With C almost fully charged, Q6, like Q7, is starved of current,
and the startup cell is effectively disconnected from the rest of the circuit. As such, the biasing
network eventually achieves steady state in that the current induced to flow in Q3 necessarily
flows through Q2, as presumed in the analysis leading to (3-166). We should note that the re- 279 -
Chapter 3
BJT Models & Biasing
sponse time of the startup cell is effectively defined by the time required for capacitance C to
charge to within an emitter-base junction threshold potential of Vcc.
Vp(t)
Ree
Ree
Ree /kp
Q3
Q4
x1
Q5
Vcap
+

x1
C
x kp
Q7
x kc
Q1
x1
t=
x kc
SW
Startup Cell
Q6
Rx
Q2

Vcc

Vo
x kn
R
IQ
0
IQ
Rl
Io
Figure (3.45). The supply-independent biasing network of Figure (3.44) with a startup cell incorporated to preclude a null current solution at network startup.
EXAMPLE #3.10:
Design the supply-independent biasing network of Figure (3.45) so that at room
temperature operating conditions, current IQ is nominally 1 mA and current Io is
nominally 2 mA for supply voltages, Vcc, in the range of 1.8 volts to 6 volts. Verify the design through SPICE simulation. Specifically, simulate the static
describing functions from Vcc to both IQ and Io at operating temperatures of 27
°C, 50 °C, and 75 °C. Additionally, simulate the time domain responses of currents IQ and Io to a 6 volt battery switched into the circuit at time t = 0 with a 500
nSEC rise time. The HSPICE parameters for the NPN devices appear in Table
(3.1), while Table (3.2) lists the parameters for PNP transistors that are nominally complementary to the NPN units.
- 280 -
Chapter 3
BJT Models & Biasing
SPICE
TEXT
SYMBOL SYMBOL
FBF
f
DESCRIPTION
OF PARAMETER
VALUE
UNITS
Forward Short Circuit Gain
100
amps/amp
BR
r
Reverse Short Circuit Gain
1.3
amps/amp
CJC
CJE
CJS
IKF
IRB
IS
MJC
MJE
MJS
NF
NR
RB
Cjco
Cjeo
Cso
Ikf
Irb
Is
mc
mje
ms
nf
nr
rbb
Zero Bias B-C Depletion Capacitance
fF
fF
fF
mA
mA
fA
–
–
–
–
–
Zero Bias Base Resistance
2.4
4.7
12.2
9.4
3.3
4.3
0.333
0.5
0.5
1.02
1.0
310
RBM
Rbm
Minimum Base Resistance
43

RC
rc
Series Collector Resistance
22

RE
re
Series Emitter Resistance
1.7
TF
fo
Zero Bias Minority Carrier Transit Time
6.2

pSEC
TNOM
Tj
Junction Reference Temperature
27
VAF
VJC
VJE
VJS
XTI
XTF
XTB
Vaf
Vjc
Vje
Vjs
Forward Early Voltage
24
780
920
690
3
0.025
0.03
Zero Bias B-E Depletion Capacitance
Zero Bias Substrate Depletion Capacitance
Forward Knee Current
Base Resistance Corner Current
Saturation Current
B-C Junction Grading Coefficient
B-E Junction Grading Coefficient
Substrate Junction Grading Coefficient
B-E Junction Injection Coefficient
B-C Junction Injection Coefficient
B-C Junction Built-In Potential
B-E Junction Built-In Potential
Substrate-Collector Built-In Potential
Temperature Exponent For IS
Temperature Exponent For TF
Temperature Exponent For BF

C
volts
mvolts
mvolts
mvolts
−
−
−
Table (3.2). HSPICE model parameters for a PNP bipolar junction transistor. Although the transistor characterized herewith is fictitious, the parameters are typical of a moderately high speed BJT.
SOLUTION #3.10:
(1).
Current IQ is nonzero if and only if transistors Q1 and Q2 possess different emitter-base junction injection areas. Arbitrarily choose kn = 10, meaning that the emitter-base junction area
of transistor Q2 is ten-fold that of Q1. There is little point in making kn much larger than ten
or so since IQ in (3-166) is seen as dependent, not on kn, but on the natural logarithm of kn.
From Table (3.1) the junction injection coefficient, nf, of all NPN devices is nf = 1.05. Thus,
with a junction temperature of Tj = 27 °C = 300.16 °K, nf = 1.05, kn =10, and IQ = 1 mA,
resistance R in the network of Figure (3.45) derives from (3-167) as
- 281 -
Chapter 3
BJT Models & Biasing
R 
n f VT ln  kn 
IQ
 62.6 Ω .
(E10-1)
In view of the empirical nature of parameter nf, the unavoidable uncertainties associated with
the internal junction temperature, and the relatively small size of the requisite circuit resistance, we should allow for laser trimming resistance R to the precise value that yields the target IQ = 1 mA current.
(2).
The next step in the design process entails determining the appropriate emitter degeneration
resistance, Ree, which is commensurate with satisfactory circuit operation at the worst case
power line voltage of Vcc =1.8 volts. Equation (3-169) is the applicable relationship, but prior
to its use, the emitter-base voltage, Veb3, for transistor Q3 and the base-emitter voltage, Vbe2,
for transistor Q2 must be estimated. To this end, precise calculations are not required and
therefore, knee currents and Early voltages are ignored. These simplifications enable the use
of the simple Ebers-Moll relationship for the forward transport current in (3-5) for both NPN
and PNP transistors. In particular,

Vbe n f VT
I c  I cc  I s e

Vbe n f VT
 1  I se
,
(E10-2)
which applies to the NPN devices with minimal junction injection areas. In the case of PNP
transistors, the only required change entails the replacement of base-emitter terminal voltage
Vbe by emitter-base voltage Veb. For transistor Q2, Table (3.1) gives Is = 3.2 fA and, of
course, nf = 1.05. Since Q2 has a ten-fold increase in emitter-base junction area, Is in (E10-2)
must be multiplied by 10. Then with Ic ≈ IQ = 1 mA, Is = (10)(3.2 fA), nf = 1.05, and Tj =
300.16 °K, Vbe2 = 656.5 mV. On the other hand, the PNP transistor, Q3, which has minimal
junction area, offers from Table (3.2), Is = 4.3 fA and nf = 1.02. Accordingly, Ic = IQ = 1 mA
in (E10-2) produces Veb3 = 690.8 mV. Recalling (3-169), Vcc ≥ 1.8 volts leads to a resistance
Ree, that satisfies
1.8  Vbe2  Veb3  n f VT ln  kn 
(E10-3)
Ree 
 390.1 Ω .
IQ
Design conservatism is almost always judicious and thus, we select Ree = 300 Ω in an attempt
to win the praise of our boss.
(3).
If current Io is to be 2 mA, which is twice the value of current IQ, transistor Q5 in Figure
(3.45) should boast twice the emitter-base junction injection area of transistor Q3. Moreover,
the desired current mirroring between transistors Q3 and Q5 requires that the emitter
degeneration resistance for Q5 be one-half that of transistor Q3. Note in Figure (3.45) that
the emitter degeneration resistance of Q5 is delineated as Ree/kp, where kp, the junction area
factor of Q5, is kp = 2.
(4).
When the supply line battery switch is closed at time t = 0, capacitor C begins to charge, such
that the indicated capacitor voltage, Vcap, ultimately rises to 6 volts. To crude first order, the
rate of capacitor voltage charging matches the slew rate of the supply line, which is 6
volts/500 nSEC = 12 volts/μSEC. The current conducted by capacitance C in the circuit of
Figure (3.45) closely approximates the collector current, say Ic6(t) flowing in transistor Q6.
Accordingly,
dV (t)
ΔVc
 C
 C 12 volts/μSEC  .
I c6 (t)  C c
(E10-4)
dt
Δt
Most transistors in the subject network conduct steady state currents in the range of 1 mA. In
order to forestall excessive junction heating in the startup cell devices, it is prudent to limit
the maximum current in transistor Q6 to approximately 1 mA, especially if the indicated area
factor, kc, is set to one. With Ic6(t) =1 mA, (E10-4) delivers C = 83.3 pF. Set C = 85 pF,
which requires that this capacitance be realized as an off chip realization element.
- 282 -
Chapter 3
BJT Models & Biasing
In addition to concerns about junction heating, care must be exercised to preclude voltage
overstress in transistors Q6 and Q7. Specifically, note that if switch SW in Figure (3.45) were
ideal, the entire 6 volts of supply line energy is developed across the base-emitter terminals of
transistor Q7 immediately after switch closure. Depending on the process actually adopted
for the monolithic realization of the network we are addressing, high voltage options may
need to be exploited for transistors Q6 and Q7.
(5).
The variable awaiting computation is the resistance, Rl, in Figure (3.45). The current, Io,
flowing through resistance Rl is 2IQ, and this same current approximates the emitter current of
transistor Q5. Thus, the emitter-collector voltage, Vec5, supported by transistor Q5 is
Vec5  Vcc   Ree  2Rl  IQ .
(E10-5)
In order for Q5 to sustain operation in its forward active domain, Vec5 must be at least as large
as Veb5 even when Vcc lies at its minimum, 1.8-volt level. The Q5 emitter-base voltage, Veb5,
effectively matches Veb3, which has been calculated to be 690.8 mV. It follows that
Vcc  Ree IQ  Veb5
(E10-6)
Rl 
 404.6 Ω .
2IQ
Therefore, we choose Rl = 400 Ω.
6V
300
300
5.70 V
5.70 V
Q4
x1
0V
Q6
Q7
0

6

Q5
x1
85
SW
5.70 V
Q3
5.01 V
t=
150
x2
5.01 V
x1
x1
719 mV
800 mV
x 10
Q1
x1
Q2
Vo
65.0 mV
65
IQ
IQ
400
Io
Figure (3.46). The supply-independent biasing network, complete with startup
compensation, designed in Example #3.10. All resistances are in
units of ohms, the capacitor is in picofarads, and the voltage
supply is in units of volts. The indicated node voltage estimates
apply in the steady state subsequent to closure of switch SW.
(6). A schematic diagram of the completed design is shown in Figure (3.46). A first simulation of
the network reveals a simulated current, IQ, which is about 4.8% larger than its target factor.
Accordingly, resistance R in Figure (3.45) is increased from its originally computed 62.6 Ω to
(1.048)(62.6 Ω) = 65.6 Ω. Subsequent to running a few static simulations to confirm the pro- 283 -
Chapter 3
BJT Models & Biasing
priety of relevant design modifications, we resultantly converge to R = 65 Ω.
Several SPICE simulations were executed to confirm the propriety of the design submitted as
the schematic diagram in Figure (3.46)
(a). Figure (3.47) displays room temperature, simulated static sweeps of currents IQ and Io
versus the supply line voltage, Vcc. At Vcc = 6 volts, IQ = 1.01 mA, which is a scant 1%
larger than the target design value. Even if Vcc drops from its steady state 6-volt value
by 75% to 1.5 volts, IQ is maintained at 924.9 A, which represents a degradation of
only 8.04%. On the other hand, the output current, Io, is 1.97 mA at Vcc = 6 volts and
1.77 mA at Vcc = 1.5 volts, which computes as a 10.12% decrease with respect to the
current value at Vcc = 6 volts. The diminished rejection of the power supply voltage at
higher quiescent currents can be attributed to the fact that the Early resistances, and
particularly the Early resistances in the PNP devices are inversely proportional to bias
currents.
2.5
Io
Current (mA)
2.0
1.5
IQ
1.0
0.5
0
0
2
4
6
8
10
Power Supply Voltage, Vcc (volts)
Figure (3.47). Simulated static describing functions for the room temperature currents, IQ
and Io, in the circuit of Figure (3.46).
The observed nonzero slopes in the current curves can also be attributed to Early voltages that are always finite. Said slope is more pronounced in the output current
characteristic because the Early voltage of the PNP transistor, Q5, is smaller than those
of any of the NPN devices. Below nominally 1.5 volts, transistors that are not connected as diode elements operate in saturation to produce current characteristics that are
strongly sensitive to supply line voltage variations.
(b).
Figure (3.48) offers the foregoing static sweeps at 50 °C and 75 C, as well as at 27 °C.
Temperature sensitivity is apparent, which is as expected because of the current solutions in (3-167) and (3-172). In particular, if the junction injection coefficient, nf, and
circuit resistance, R, are temperature invariant, (3-167) and (3-7) deliver
ΔIQ
n f k ln  kn 

 3.21 μA / C .
(E10-7)
ΔT j
qR
Equation (3-172) demonstrates that the temperature sensitivity of the output current, Io,
is larger than the foregoing sensitivity by a factor of kp. Thus,
- 284 -
Chapter 3
BJT Models & Biasing
2.5
Current (mA)
2.0
Io
1.5
Tj = 75 °C
Tj = 50 °C
Tj = 27 °C
1.0
IQ
0.5
0
0
2
4
6
8
10
Power Supply Voltage, Vcc (volts)
Figure (3.48). Simulated static describing functions at 27 °C, 50 °C, and 75 °C for the currents, IQ and Io, in the circuit of Figure (3.46).
ΔIQ
k p n f k ln  kn 
ΔI o
 kp

 6.42 μA / C .
(E10-8)
ΔT j
ΔT j
qR
The increase in the temperature sensitivity of current Io by a factor of kp over the
sensitivity of current IQ rebukes the use of high junction injection area factors in the
PNP units deployed in the circuit at hand. An examination of the data from which Figure (3.48) derive indicates that IQ = 1.006 mA at T = 27 °C, while at 50 °C and 75 °C,
the subject current is 1.086 mA and 1.173 mA, respectively. Comparing the 75 °C
simulated result to its 27 °C counterpart,
ΔIQ
ΔT j

1.173  1.006  10 -3 
75  27
 3.48 μA / C ,
(E10-9)
which is 8.4% higher than the computed sensitivity in (E10-7). This error can be attributed to the temperature dependence of other transistor parameters (such as the Early
voltage). These second order temperature sensitivities are tacitly ignored in (3-167).
For output current Io, Io = 1.969 mA, 2.125 mA, and 2.295 mA at 27 °C, 50 °C, and 75
°C, respectively. It follows that
ΔI o

ΔT j
(c).
 2.295  1.969  10 -3 
75  27
 6.79 μA / C ,
(E10-10)
which is only 5.8% larger than the calculated temperature sensitivity in (E10-8).
Finally, Figure (3.49) offers the transient responses of currents IQ and Io to sudden
switch closure at time t = 0, subject to the condition that the switch in question induces
a supply line voltage rise time of 500 nSEC. Superimposed on this plot are the currents
conducted by the startup transistors, Q6 and Q7. The latter curves confirm that the
startup devices are effectively disconnected from the biasing network, well within 1
μSEC of switch closure. The currents, IQ and Io, are seen to settle to their respective
steady state values at 584.75 nSEC. Since the supply line does not reach 6 volts until
500 nSEC, the circuit can be said to settle within 84.75 nSEC after the time at which
the supply line is fully activated. Observe a significant amount of peaking in the cur-
- 285 -
Chapter 3
BJT Models & Biasing
Current (mA)
Line Voltage, Vp(t) (volts)
rent outputs, IQ and Io, while the currents conducted by the startup devices always remain below IQ and Io.
6
4
2
0
3
Io
2
IQ
1
Ic7
Ic6
0
0
0.2
0.4
0.6
0.8
1.0
1.2
Time (microseconds)
Figure (3.49). Transient current responses to switch closure in the circuit of Figure (3.46).
The current, Ic6, is the collector current conducted by transistor Q6, while Ic7
denotes the collector current of transistor Q7.
ENGINEERING COMMENTARY:
As suggested earlier, the circuit addressed in this example is one of several whose operation
is premised on formulating a subcircuit that establishes an offset between base-emitter terminal voltages of key transistors. All are capable of delivering static current responses that,
while not strictly independent of power supply voltage, are significantly desensitized with respect to supply line voltage variations. Most of these networks are able to limit current variations about nominal design targets to 5%-10%, despite upwards of 50% degradation in line
voltage. The prices paid for these supply-independent topologies are potentially significant
temperature sensitivities, a recurring need for resistor trimming, and the inevitable requirement of startup cells that preclude null current states. Three additional biasing circuits are reserved for your own edification in Problems #3.28 through #3.30.
3.5.2. BANDGAP REFERENCE CIRCUIT
Among the best of the supply-independent biasing schemes is the bandgap reference
circuit[11]. Aside from offering excellent output voltage desensitization to power line voltage
variations, the bandgap reference also delivers an outstandingly low temperature coefficient for
its reference output voltage over wide temperature ranges. This temperature insensitivity is a
critically important performance attribute in high performance analog integrated circuits, and
especially in circuits earmarked for high-speed data acquisition, data conversion, and information processing. Circuit performance vulnerability to temperature looms especially severe in
deep submicron technologies for which even modest current levels generate the high current
densities that can result in intrinsic temperature increases of as much as 50 °C or more. In the
subsections that follow, we shall see that the bandgap reference circuit produces a nominally
temperature invariant static output voltage by coalescing the electrical properties of two circuit
- 286 -
Chapter 3
BJT Models & Biasing
cells that we have already encountered; namely, the emitter-base junction voltage of a diode connected transistor and a PTAT voltage generator.
Conceptually, the bandgap reference cell is the system abstraction pictured in Figure
(3.50)[12]. An emitter-base junction forward bias is extracted as the base-emitter voltage, Vbe,
which is developed in response to a constant current, Ik, applied as shown, to a diode connected
BJT. The PTAT generator is formed as a conventional Boltzmann voltage that is scaled by a
constant amplifier gain of K. In turn, the output reference voltage, Vref, derives as the superposition of voltage Vbe and the amplified Boltzmann voltage, KVT; that is,
Vcc
Ik
Vbe

Vbe + KVT
Vref
Vaa
Boltzmann
Voltage
Generator
VT


K
KVT
Figure (3.50). System level abstraction of a bandgap reference circuit.
V
ref
 V
be
 KV .
(3-175)
T
To the extent that current source Ik emulates an ideal (infinite impedance) current source, voltage
Vbe is independent of the biasing voltage, Vcc. And if the amplifier shown in the diagram boasts
high power supply rejection, gain K is nominally independent of the amplifier biasing voltage,
Vaa. It follows that output voltage Vref is all but independent of applied biasing.
Equation (3-101) teaches that to first order, junction voltage Vbe decreases linearly with
increasing temperature. Obviously, the Boltzmann voltage, VT, increases linearly with temperature. It is therefore intuitively evident that in addition to achieving a reference response boasting
supply voltage independence, constant K, which we presume to be temperature invariant, can be
selected to achieve a reference voltage having zero temperature coefficient. In particular,
dV
V 
dV
ref
be  K  k    S  K  To  ,
(3-176)

 
be
 T 
dT
dT
q
 o 
where VTo is the Boltzmann voltage evaluated at a reference temperature of To. Generally, the
reference temperature is taken as 27 °C, or 300.16 °K. We see that the temperature coefficient,
dVref /dT, of reference output Vref is null if
S T
K  be o ,
(3-176)
V
To
which produces a corresponding, and somewhat idealized, reference output of
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Chapter 3
BJT Models & Biasing
V 
(3-177)
S T  T  V S T.
ref
be
T
be
be o  V 
be
be
 To 
This basic result confirms that voltage Vref indeed boasts zero temperature coefficient if junction
voltage Vbe decreases linearly with junction temperature at a constant rate of Sbe. Note that if Sbe
is indeed constant, the amplifier gain, K, in (3-176) is necessarily temperature invariant.
V
 V
 KV
V
3.5.2.1. Temperature Coefficient of Forward Biased Junction Voltage
At this point, we have taken the temperature sensitivities of a forward biased PN junction diode and the analogous forward biased emitter-base junction of a BJT to be the temperature
invariant constant, Sbe. We even suggested that in the absence of definitive other information, Sbe
≈ −2 mV/°C is an excellent approximation for the pre-CAD, mathematical analysis of a circuit or
system. For the vast majority of BJT circuits, these first order presumptions are acceptable and
even appropriate. But in bandgap reference cells, which can deliver temperature sensitivities as
small as a few tens to a few hundreds of parts per million/°C (ppm/°C), representing Sbe as a
mere constant is not sufficiently accurate even for first order investigations. We shall learn that
the forward biased, emitter-base junction voltage of a BJT does not decrease linearly with
temperature. In fact, it does decrease with increasing temperature, but we shall find that the
functional dependence, Vbe(T), on absolute junction temperature T embraces both linear and
natural logarithmic terms. It follows that Sbe, which we shall now more astutely designate as the
temperature function, Sbe(T), is at least weakly dependent absolute operating temperature.
The archival literature testifies that the functional dependence of the emitter-base voltage on junction temperature is expressible as[13],
  J 

T 
T 
 T 
Vbe (T)  Vgo  1    Vbeo    n f VT ln  c   m ln  o   .
(3-178)
To 
 T  

 To 
  J co 
In this expression To is the previously introduced junction reference temperature, Vgo is the reference temperature value of the bandgap potential (1.206 volts in silicon operated at To = 300.16
°K), and Vbeo is the reference temperature value of the emitter-base junction voltage; that is, Vbeo
≡ Vbe(To). Parameter m is an empirical constant that for silicon semiconductor lies in the range,
2.0 ≤ m ≤ 2.8. Unless explicitly stated otherwise, m is typically assigned a value of m = 2.3.
Parameter Jc symbolizes the density of collector current (collector current Ic divided by the emitter-base junction cross section area, Ae) at an absolute junction temperature of T. Finally, Jco is
the T = To value of collector current density Jc.
The temperature dependence of the current density ratio, Jc /Jco, is usually fixed by the
topology of the subcircuit in which the transistor undergoing scrutiny is embedded. In most of
the commonly utilized bandgap reference cells, this ratio is made proportional to absolute
temperature in accordance with the simple, linear relationship,
Jc
T

,
(3-179)
J co
To
assuming that any resistors utilized in the reference cell have negligibly small temperature
coefficients and assuming further that no significant temperature gradients prevail across the
bandgap reference chip. Accordingly, (3-178) becomes

T 
T 
T 
(3-180)
Vbe (T)  Vgo  1    Vbeo    n f  m  1 VT ln   .
To 

 To 
 To 
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Chapter 3
BJT Models & Biasing
Consequently, the temperature sensitivity, Sbe(T), is
Vgo  Vbeo
n f k  m  1 
 T 
dVbe (T)
Sbe (T) 
 

(3-181)
1  ln    .
dT
To
q
 To  

Since Vgo > Vbeo and m > 1, both terms on the right hand side of this relationship are negative,
thereby confirming a negative temperature coefficient for the emitter-base junction potential.
Figure (3.51) displays a plot of Sbe(T) versus temperature T for the case of Vbeo =730 mV, nf =
1.0, a reference temperature of To = 27 °C = 300.16 °K and three values of empirical parameter
m. The logarithmic temperature term is invariably inconsequential for practical operating
temperatures. Accordingly, and as is subliminally inferred by the curves in Figure (3.51), the
temperature sensitivity function, Sbe(T), is almost constant. For m = 2.4, we see that Sbe(T) varies
from −1.695 mV/°C at T = 0 °C to −1.733 mV/°C at T = 100 °C, which reflects a change of only
about 2.2% over this 100 °C interval. Note further that the slopes of these sensitivity plots
change modestly as a function of parameter m.
-25
0
25
50
75
100
125
Temperature Coefficient, Sbe(T), (mV/deg C)
-1.60
-1.65
-1.70
-1.75
-1.80
Junction Temperature (deg C)
Figure (3.51). The temperature sensitivity of the forward biased emitter-base junction
voltage of a BJT, depicted as a function of junction temperature. Note that
the sensitivity function, Sbe(T), is plotted in units of mV/°C.
The fundamental emitter-base potential relationships of (3-5) and (3-11) are interesting
from the standpoint of their implications for two identical transistors (save for different emitterbase junction areas) operated at different collector current densities. Denote these two collector
current densities as Jc2 and Jc1, which respectively correspond to base-emitter voltages of Vbe2(T)
and Vbe1(T). If the two transistors operate at the same junction temperatures, a simple algebraic
manipulation of the basic Ebers-Moll equations confirms that
 I   A  
J 
Vbe2 (T)  Vbe1(T)  n f VT ln  c2   n f VT ln  c2   e1   .
(3-182)
 J c1 
 I c1   Ae2  
If the circuit in which the two transistors are embedded is designed to ensure that the two collector currents, Ic1 and Ic2, are identical, we have
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Chapter 3
BJT Models & Biasing
n f kT  Ae1 
A 
(3-183)
Vbe2 (T)  Vbe1(T)  n f VT ln  e1  
ln 
,
q
 Ae2 
 Ae2 
which is directly proportional to absolute temperature. Equation (3-183) serves as a foundational
platform for constructing a PTAT generator. A slight shortfall of this contention is that identical
transistors carrying non-identical collector current densities are likely to be operating at slightly
different junction temperatures. Thus, care must be exercised to preclude widely divergent current densities or equivalently, significant temperature gradients between the two transistors.
3.5.2.2. Circuit Realization of the Bandgap Reference Cell
The topological foundation of most of the currently used bandgap reference cells is the
Brokaw circuit shown in Figure (3.52)[11]. The circuit at hand utilizes two monolithic, and therefore physically similar, BJTs, Q1 and Q2, whose emitter-base junction areas are Ae1 and Ae2,
respectively. If the operational amplifier (op-amp) in the schematic diagram3 is characterized by
a low frequency open loop voltage gain of Ao, the indicated differential input voltage, Vi, of the
op-amp is
Vcc
R
R
I1
Vi


Q1

I2

Op-Amp
 Loop
]
en
[Op in = A o
Ga
Q2

Vbe1

R2
I2
I1

Vbe2
Vref
R1
I1+I2
Figure (3.52). The basic schematic diagram of a bandgap reference circuit that
uses two bipolar junction transistors and a low bandwidth operational amplifier. The indicated currents flowing through resistances R1 and R2 and the emitter of transistor Q2 ignore the base
currents conducted by Q1 and Q2.
3
The op-amp need not be realized in bipolar junction transistor amplifier. In fact, since high response speed in the
op-amp is not an issue, but static power dissipation is an important design consideration, the op-amp is commonly
realized in metal-oxide-semiconductor field-effect transistor (MOSFET) technology.
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Chapter 3
BJT Models & Biasing
V 
i
V
ref
A
.
(3-184)
o
From the schematic diagram, we then see that
V
ref
V 
 R I I .
i
1
2
A
o


(3-185)
The last equation shows that for a finite reference output voltage, Vref, Vi ≈ 0 if gain Ao is very
large. Thus, a sufficiently large open loop gain in the utilized op-amp manifests identical currents, I1 and I2. This large open loop gain is but one of three preeminent requirements we shall
invoke on the bandgap reference cell. The second directive is very large input impedances at
both of the op-amp input ports. Large input impedances ensure that the op-amp draws negligible
input current, which means that current I1 approximates the collector current flowing in transistor
Q1. Similarly, current I2 is essentially the collector current conducted by transistor Q2. The final mandate is that transistors Q1 and Q2 be characterized by large static current gains, hFE. This
stipulation means that currents I1 and I2 respectively approximate the Q1 and Q2 emitter currents, as we depict in Figure (3.52). In light of the foregoing approximations, the reference output voltage, Vref, is given by
Vref  Vbe2   I1  I 2  R1  Vbe2  2I1R1 ,
(3-186)
while
Vbe2  Vbe1
(3-187)
,
R2
where it is understood that the base-emitter voltages, Vbe1 and Vbe2, are functions of junction
absolute temperature T. Recalling (3-183), (3-186) becomes
A 
 2R 
 2R 
(3-188)
 V
 1 V
V
 V
  1  n V ln  e1  ,
V
ref
be2  R  be2
be1
be2  R  f T  A 
 2 
 2 
 e2 
This result mirrors the generalized reference output response delineated in our system level schematic of Figure (3.50). In particular, the first term on the far right hand side of (3-188) is a baseemitter voltage that decreases with temperature at a near constant rate. On the other hand, the
second term on the right hand side of (3-188) is proportional to Boltzmann voltage VT. To the
extent that the resistance ratio, (R1/R2), projects negligibly small temperature coefficient, which it
does if both resistances have roughly the same temperature coefficient, this second term is
PTAT. In a word, the first term on the right hand side of (3-188) decreases with temperature,
while its superimposed second term companion increases with temperature. The irrefutable
conclusion is that the resistance ratio (R1/R2) can be selected to incur a zero temperature coefficient associated with the reference output voltage. Actually, a bit of thought serves to convince
that (R1/R2) can be selected to deliver a reference output voltage that has positive, zero, or negative temperature coefficient. As a relevant and important sidebar, we note that voltage Vref in (3188) is ostensibly independent of the supply voltage, Vcc. It is indeed, as long as Vcc is large
enough to ensure the linear operation of the op-amp, which is required to sustain equality between the two currents, I1 and I2.
I1 


Using (3-188), the temperature sensitivity, Sref(T), of the reference voltage is
- 291 -
Chapter 3
BJT Models & Biasing
 2R   n f VTo   A 
1 
 ln  e1  ,
(3-189)
ref



 A 
dT
dT
R
T
 2   o   e2 
where we recall VTo as the T = To value of Boltzmann voltage VT. Using (3-181), the last result
becomes

 T   
1 
 n  m  1 V 1  ln    
S (T)   V  V
ref
f
To 
T 
T  go beo2
o
 o   

(3-190)
 2R   n f VTo   A 
 ln  e1  .
  1 
 R  T
 A 
 2   o   e2 
This temperature coefficient is not constant because the temperature coefficient of the base-emitter terminal voltage of a BJT is slightly temperature variant, as is portrayed by Figure (3.51). At
the reference temperature To, (3-190) becomes
 2R   n f VTo   A 
1
 ln  e1  . (3-191)
 n  m  1 V    1  
S (T )   V  V
ref o
f
To 
 R  T
 A 
T  go beo2
o
 2   o   e2 
A straightforward algebraic consideration of this expression leads to the conclusion that the
reference temperature value, Sref(To), of the temperature coefficient of the reference output voltage is zero if resistance ratio (R1/R2) is chosen in accordance with
V V
go
beo2
 n  m  1
f
V
R
To
1 
 ρo .
(3-192)
R
A 
2
2n ln  e1 
f
A 
 e2 
If we combine this relationship with (3-191), we get
 n V   A    R

f To
 ln  e1    1    .
(3-193)
S (T )  2 
ref o
o
 To   Ae2    R2


 

The last result is interesting because it clearly conveys general design guidelines for particular
applications. Specifically, zero reference output temperature coefficient, Sref(To), at T = To requires (R1/R2) = ρo. On the other hand, Sref(To) > 0 (positive temperature coefficient) demands
(R1/R2) > ρo, while Sref(To) < 0 (negative temperature coefficient)commands (R1/R2) < ρo.
S
(T) 
dV
ref

dV
be2  
Upon substituting (3-192) and (3-180) into (3-188), we arrive at the reference output
voltage for the specific circumstance of Sref(To) = 0. We shall term this specific reference output
the optimized output response, Vropt. Then,

 T 
 V   m  1 n V 1  ln    ,

(3-194)
V
V
ropt
ref S (T )=0
go
f T
 T 
 o 
ref o

for which the corresponding optimal temperature coefficient, Sropt(T), is
dV
(T)
 m  1 n f k  To 
ropt
(3-195)
ln  
S
(T) 

ropt
dT
q
T
 
- 292 -
Chapter 3
BJT Models & Biasing
Figure (3.53) plots this optimal coefficient as a function of temperature. We note that the
temperature coefficient varies from +23 μV/°C at T = −25 °C to −34 μV/°C at T = 125 °C.
25
Temp. Coefficient, Sropt(T) (microvolt/deg C)
20
15
10
5
0
-25
-5
0
25
50
75
100
125
-10
-15
-20
-25
-30
-35
Temperature (deg C)
Figure (3.53). The optimized temperature coefficient of the reference output voltage generated
by the bandgap reference cell. The cell, which exploits BJTs having m = 2.4, is
designed to deliver zero temperature coefficient at 27 °C.
Uncertainties plague junction injection coefficient nf, parameter m, and the temperature
coefficient of resistance ratio (R1/R2). Additionally, inaccuracies materialize from the facts that
the op-amp open loop gain is not infinitely large, voltage Vbeo2 is not known precisely, and the
DC gain, hFE, of the BJTs, while doubtlessly large, is not infinitely large. Because of these
uncertainties and analytical approximations, laser trimming of at least one of the two circuit
resistances, R1 and R2, is essential to ensure the accurate satisfaction of the resistance ratio constraint in (3-192). When properly trimmed, reference output voltages boasting excellent power
line rejection and phenomenally small temperature coefficients are achieved.
Aside from displaying the temperature domain plot of the optimal temperature coefficient, as we have done in Figure (3.53), the impressive performance of the bandgap reference
generator is best examined numerically. To wit, in (3-194), take Vgo = 1.206 volts, m = 2.4, nf =
1, and To = 27 °C = 300.16 °K. Then, the optimum reference output voltage at T = 0 °C is, by
(3-194), 1.242093 volts, at T = 27 °C, Vropt = 1.242244 volts, and at T = 100 °C, Vropt =
1.241250 volts. Thus, the change, say Vropt, in reference voltage output for a temperature rise
of 100 °C from 0 °C is only 843.0 μV or 0.0679% of the reference temperature value of the optimum reference output voltage. This voltage change over a 100 °C increase in operating temperature amounts to a voltage perturbation of only 679 ppm/°C!
3.5.2.3. Large Bandgap Reference Output Voltage
As we noted in the preceding section, the bandgap reference in Figure (3.52) is capable
of a static response that is limited to the neighborhood of the bandgap potential of the
semiconductor technology exploited for transistors Q1 and Q2. For silicon devices, for example,
we witnessed a reference output voltage of the order of only 1.24 volts. Larger reference outputs, which are often required in analog applications, require a topological modification to the
- 293 -
Chapter 3
BJT Models & Biasing
reference cell along the lines of the network that appears in Figure (3.54). We observe that the
modification entails appending resistances R3, R4, and R5 to the basic cell of Figure (3.52). The
indicated interconnection of resistances R4 and R5 enables an increase in the original reference
output voltage, Vref, to a voltage, VA, that is larger than Vref by a factor of (1 + R4/R5). On the
other hand, we shall demonstrate that resistance R3 cancels the effects of base currents conducted
by transistors Q1 and Q2. If transistors Q1 and Q2 conduct the small base currents that materialize from given collector currents and large hFE, which is assured in SiGe technology, resistance
R3 can be supplanted by a short circuit.
In order to demonstrate the operation of the modified cell and how the effects of
transistor base currents can be cancelled through proper selection of resistance R3, the circuit in
Figure (3.54) is analyzed under the depicted condition of equal collector currents (I) flowing in
transistors Q1 and Q2. Clearly, we continue to assume that the op-amp has very large input
impedances at both of its input ports and essentially infinitely large open loop gain. To this end,
the loop comprised of the base-emitter junctions of Q1 and Q2 and resistances R2 and R3 yields
Vcc
R
R
I

Vi

Q1
R3

I

Op-Amp
 Loop
]
en
[Op in = A o
Ga
Q2
2Ib
VA
R4
Vref /R5+2Ib

 Vbe1
 Vbe2
R2
I + Ib
Vref /R5
R1
R5
Vref
I + Ib
2(I + Ib)
Figure (3.54). The bandgap reference supply of Figure (3.52), modified to enable a
larger than bandgap potential reference output voltage, VA. Resistance R3 is inserted to mitigate for the effects of base currents (Ib)
flowing into transistors Q1 and Q2.
Vbe2  Vbe1  R2 I   R2  R3  I b .
(3-196)
In addition, the reference voltage, Vref, is given by
Vref  Vbe2  2R1  I  Ib  .
(3-197)
If the solution for current I in (3-196) is substituted into (3-197), we learn that
2R1
2R
Vref  Vbe2 
(3-198)
Vbe2  Vbe1   1 R3 I b .
R2
R2
It is vital to recognize that by virtue of (3-188), the sum of the first two terms on the right hand
- 294 -
Chapter 3
BJT Models & Biasing
side of (3-198) is literally the approximate value of the reference voltage response of our original
bandgap cell. Indeed, we arrived at (3-188) by tacitly neglecting all bipolar transistor base currents. To this end, we are reassured that (3-198) is correct in that if we neglect base current Ib in
that expression, (3-198) collapses precisely to (3-188).
The enhanced reference voltage, VA, derives from

R 
(3-199)
V A  2R4 I b   1  4 Vref .
R5 

Substituting for Vref from (3-198), we find that




R 
2R1
R  R  
V A   1  4  Vbe2 
Vbe2  Vbe1   2  R4   1  4  1  R3  I b . (3-200)
R5  
R2
R5   R2  




The first product of terms on the right hand side of this relationship is the enhanced reference
voltage that materializes if the transistor base currents, Ib, are zero, which is approximated if the
static short circuit current gains of the two bipolar devices are large. It follows that the last term
in the subject equation is a voltage error precipitated by nonzero base currents. Fortuitously, this
error, which is proportional to Ib, vanishes if we select resistance R3 carefully. An inspection of
(3-200) reveals that the error contributed to VA by the transistor base currents is forced to zero if
R 
(3-201)
R3   2   R4 R5  .
 R1 
Resultantly, (3-200) reduces to



R 
R 
2R
(3-202)
VA   1  4  Vbe2  1 Vbe2  Vbe1     1  4  Vref ,
R5  
R2
R5 



which is to say the embellished reference output voltage, VA, approximates (because of the original tacit neglect of base currents) the original reference output, Vref, multiplied by the resistive
ratio factor, (1 + R4/R5).
Two design-oriented issues surface as an engineering consequence to (3-202). First, if
output voltage VA is to preserve the laudable thermal properties ascribed to voltage Vref, resistances R4 and R5 must possess identical temperature coefficients. This design requirement implicitly requires that the values of R4 and R5 must not be excessively divergent. In other words, the
modified bandgap reference cell works very well if VA is of the order of two to four times Vref,
but not as well if VA/Vref equals ten to twenty. Moreover, the two subject resistances must be laid
on chip out closely to ensure that both resistances experience the same operating temperature.
The second, and somewhat more subtle issue is that resistance R5 (and hence R4, as well), cannot
be too small. Two reasons underpin this directive. First, a small value of R5 yields a relatively
large current, Vref /R5, which must be supplied by the output port of the op-amp in Figure (3.54).
As it turns out, large output port currents make the task of achieving excellent power supply
rejection in the op-amp progressively more difficult. The second reason is that large Vref /R5,
which is conducted by both resistances R4 and R5, manifest potentially excessive power dissipation in these resistances. Because of this large dissipation, the thermal characteristics of the
resistances may differ markedly from those of the other passive and active circuit components,
thereby rendering the thermal compensation measures discussed in the preceding section problematic.
- 295 -
Chapter 3
BJT Models & Biasing
3.6.0. REFERENCES
[1]. J. J. Ebers and J. L. Moll, “Large-Signal Behavior of Junction Transistors,” Proceedings of the
IRE, vol. 42, pp. 1761-1772, Dec. 1954.
[2]. B. S. Meyerson, “High Speed Silicon Germanium Electronics,” Scientific American, vol. 270,
pp. 42-47, March 1994.
[3]. P. Ashburn, SiGe Heterojunction Bipolar Transistors. John Wiley & Sons, 2003, chap. 10.
[4]. H. K. Gummel and H. C. Poon, “An Integral Charge-Control Model of Bipolar Transistors,”
Bell System Technical Journal, vol. 49, pp. 115-120, May-June 1970.
[5]. C. T. Kirk, “A Theory of Transistor Cutoff Frequency (fT) at High Current Densities,” IEEE
Transactions on Electron Devices, vol. ED-9, pp. 164-174, Mar. 1962.
[6]. S. Dimitrijev, Understanding Semiconductor Devices. New York: Oxford University Press,
2000, pp. 334-341.
[7]. J. M. Early, “Effects of Space-Charge Layer Widening in Junction Transistors,” Proceedings of
the IRE, vol. 46, pp. 1141-1152, Nov. 1958.
[8]. H. N. Ghosh, “A Distributed Model of the Junction Transistor and its Application in the Prediction of the Emitter-Base Diode Characteristic, Base Impedance, and Pulse Response of the Device,” IEEE Transactions on Electron Devices, vol. ED-12, pp. 513-531, Oct. 1965.
[9]. J. R. Hauser, “The Effects of Distributed Base Potential on Emitter Current Injection Density
and Effective Base Resistance for Stripe Transistor Geometries,” IEEE Transactions on Electron Devices, vol. ED-11, pp. 238-242, May 1965.
[10]. G. R. Wilson, “A Monolithic Junction FET-NPN Operational Amplifier,” IEEE Journal of
Solid-State Circuits, vol. SC-3, pp. 341-348, Dec. 1968.
[11]. P. Brokaw, “A Simple Three-Terminal IC Bandgap Reference,” IEEE Journal of Solid-State
Circuits, vol. SC-9, pp. 388-393, Dec. 1974.
[12]. P. E. Allen and D. R. Holberg, CMOS Analog Circuit Design (2nd Edition). New York: Oxford
University Press, 2002, pp. 153-158.
[13]. Y. Tsividis, “Accurate Analysis of Temperature Effects in Ic-Vbe Characteristics with Application to Bandgap Reference Sources,” IEEE Journal of Solid-State Circuits, vol. SC-14, pp. 655657, June 1979.
EXERCISES
PROBLEM #3.1
Each of three bipolar junction transistor interconnections depicted in Figure (P3.1) can function as
an effective PN junction diode. For each of these connections, use the Ebers-Moll equations to derive an expression for the volt-ampere characteristic, Id -versus- Vd. Stipulate the effective saturation
current, say Io, associated with each diode emulation.
Id

Id
(a).


Vd
Vd

Id

(b).
Figure (P3.1)
- 296 -
Vd

(c).
Chapter 3
BJT Models & Biasing
PROBLEM #3.2
In the composite transistor structure of Figure (P3.2), both transistors are identical, and the indicated
supply voltage, Vcc, is large enough to reverse bias the base-collector junctions of both devices.
Moreover, the battery voltage, Vbb, is sufficiently large to forward bias the emitter-base junctions of
both transistors. Use the Ebers-Moll model to derive expressions for the current ratios, I2/I1, I3/I1,
and I4/I1.
I2
+Vcc
I3
I1

I4
Vbb

Figure (P3.2)
PROBLEM #3.3
Consider a bipolar junction transistor that is operated in a manner as to ensure a strongly reverse biased base-collector junction. Using the Ebers-Moll model, show that the emitter-base junction voltage, Ve, commensurate with zero emitter current, is negative and given by
Ve   n f VT ln β f  1 .


PROBLEM #3.4
A bipolar junction transistor is operated with forward bias applied to its emitter-base junction. Use the Ebers-Moll model to show that the resultant collector current -to- base current
transfer ratio, Ic /Ib, is smaller than the gain parameter, f, whenever the base-collector junction of the subject transistor is forward biased.
PROBLEM #3.5
Repeat Example #3.3 for the diode interconnections of Figures (3.6b) and (3.6c).
PROBLEM #3.6
Use the Ebers-Moll model of a bipolar junction transistor to show that the ratio, Ibe /Ibc, of forward to
reverse base current components can be cast in terms of the BJT emitter and collector currents, Ie
and Ic, respectively, in accordance with
 I  αr I c   1  1  α f βr 
Ibe
 .
  e

 α f I e  I c   1   1  αr  β f 
Ibc




(a).
(b).

Explain the engineering significance of a zero value to this ratio when Ie = rIc.
Explain the engineering significance of an infinite value to this ratio when Ic = fIe.
PROBLEM #3.7
Repeat Example #3.4 for the case in which the collector-emitter biasing voltage is only one
volt.
PROBLEM #3.8
Repeat Example #3.4 for the stipulated biasing point, but for a transistor having three-times the
emitter-base junction area.
- 297 -
Chapter 3
(a).
(b).
(c).
BJT Models & Biasing
Plot the unity gain frequency, fT, as a function of the quiescent collector current, IcQ.
Determine an approximate collector current commensurate with maximum achievable fT.
What is the maximum fT corresponding to the collector current found in Part (b)?
PROBLEM #3.9
An occasionally invoked metric for high frequency transistor performance is the transverse cutoff
frequency, say fb. This metric is the 3-dB frequency of the small signal, short circuit forward
transadmittance, yfe(jω), evidenced in a grounded emitter topology. The “short circuit” refers to a
collector-emitter port that supports zero signal. Use the model in Figure (3.14) to derive analytical
expressions for both yfe(jω) and fb. For simplicity, take rc = re = 0 in the BJT model.
PROBLEM #3.10
The BJT in the grounded base amplifier of Figure (P3.10) operates in its forward active regime because of the constant voltage supply, Vcc, and the static current sink, IQ. The input signal, whose
Thévenin resistance is Rs, is applied as the current, Is. The response to this input current is extracted
as the signal component, Ios, of the net current, Io, which flows through the collector load resistance,
Rl. Use the model of Figure (3.14), with resistances rc and re ignored, to respond to the following
directives.
+Vcc
Rl
Io
Is
Rs
IQ
Figure (P3.10)
(a).
(b).
(c).
(d).
Evaluate the current transfer function, Ai = Ios /Is, at low signal frequencies. Give an upper
limit to this current gain.
For a sinusoidal input signal, Is, infinitely large Rs and a short circuit load resistance, Rl, evaluate the current transfer function, Ai = Ios /Is, as a function of frequency ω.
Derive an expression for the 3-dB bandwidth of the current gain determined in Part (b). Suitable approximations can be invoked as long as these approximations are clearly rationalized.
Compare the gain-bandwidth product of the grounded base amplifier with the short circuit, unity gain frequency, fT, of the grounded emitter configuration.
PROBLEM #3.11
The BJT in the grounded collector amplifier of Figure (P3.11) operates in its forward active regime
because of the constant voltage supplies, Vcc and Vbb. The input signal, whose Thévenin resistance is
Rs, is applied as voltage Vs,. The response to this input signal voltage is extracted as the signal
component, Vos, of the net voltage, Vo, which is established across the indicated load resistance, Rl.
Use the model of Figure (3.14), with resistances rc and re ignored, to respond to the following directives.
(a). Explain why the configuration in Figure (P3.11) is commonly referenced as a grounded
collector amplifier, despite the fact that the collector is not actually grounded and instead is
incident with the positive terminal of a constant voltage source.
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Chapter 3
BJT Models & Biasing
+Vcc
Rs

Vbb

Vo

Rl
Vs

Figure (P3.11)
(b).
(c).
(d).
(e).
Evaluate the voltage transfer function, Av = Vos /Vs, at low signal frequencies. Give an upper
limit to this voltage gain.
For a sinusoidal input signal, Vs, zero Rs, and an infinitely large load resistance, Rl, evaluate
the voltage transfer function, Av = Vos /Vs, as a function of frequency ω.
Derive an expression for the 3-dB bandwidth of the voltage gain determined in Part (c). Suitable approximations can be invoked as long as these approximations are clearly rationalized.
Compare the gain-bandwidth product of the grounded collector amplifier with the short circuit, unity gain frequency, fT, of the grounded emitter configuration.
PROBLEM #3.12
Develop a design-oriented expression for the quiescent collector current, IcQ, in the PNP transistor
cell of Figure (P3.12). Assess the shortfalls of the biasing configuration.
+Vee
R2

VebQ


VecQ

Rl
R1
IcQ
Figure (P3.12)
PROBLEM #3.13
The silicon transistor in the biasing configuration of Figure (P3.13) operates in its linear regime
where it has a static beta of hFE, an emitter-base junction temperature coefficient of −Sbe, and a nominal base-emitter turn on voltage of Vbe.
(a). Derive a general expression for the quiescent collector current, IcQ.
(b). Stipulate the inequality that the circuit must satisfy if current IcQ is to be rendered relatively
insensitive to gain parameter hFE.
- 299 -
Chapter 3
BJT Models & Biasing
+Vcc
Rl
Rx
R1
IcQ
R2
Ree
Figure (P3.13)
(c).
(d).
For the condition deduced in Part (b), what is the temperature sensitivity, IcQ/Tj, of the circuit?
What small signal resistance, say Rx, is presented to load resistance Rl at the collector node of
the subject biasing network? Use reasonable approximations to simplify the expression for
this resistance. Comment on the ability of the circuit to sustain nominally constant Q-point
collector current in the face of signal changes presented at the collector node.
PROBLEM #3.14
Using a 6 volt power supply, design the circuit in Figure (P3.13) for IcQ = 1 mA and VceQ = 1 volt at
room temperature. The collector current is to be held constant to within ±2% over a temperature
range extending to 75 °C. Assume that the transistor is identical to that exploited in Example #3.6
and parameterized in Table (3.1). Use HSPICE or similar circuit simulation software to determine
the quiescent collector current at 27 °C, 50 °C, and 75 °C.
PROBLEM #3.15
The silicon transistor in the circuit of Figure (P3.15) operates in its linear regime where it has a
static beta of hFE, an emitter-base junction temperature coefficient of −Seb, and a nominal base-emitter turn on voltage of Veb. The inductors in this circuit are ideal; that is, they have infinitely large
quality factors or equivalently, zero series resistances.
IcQ
Ree
R1

Vee

R2
Figure (P3.15)
(a).
(b).
Derive a general expression for the quiescent collector current, IcQ.
Stipulate the inequality that the circuit must satisfy if current IcQ is to be rendered nominally
- 300 -
Chapter 3
(c).
BJT Models & Biasing
insensitive to current transfer parameter hFE.
Under the condition deduced in Part (b), what is the temperature sensitivity, IcQ/Tj, of the
circuit?
PROBLEM #3.16
The two identical silicon transistors in the low noise biasing circuit of Figure (P3.16) operate in their
linear regimes where they display a static beta of hFE, an emitter-base junction temperature coefficient of −Sbe, and a nominal base-emitter turn on voltage of Vbe. The inductor in this circuit is ideal;
that is, it has infinitely large quality factors or equivalently, zero series resistance.
R5
R2
Q2
Vcc
Q1
R4
R1
R3
Figure (P3.16)
(a).
(b).
(c).
(d).
Derive general expressions for the quiescent collector currents, IcQ1 and IcQ2, conducted by
transistors Q1 and Q2, respectively.
Stipulate the inequalities that the circuit must satisfy if the two quiescent collector currents are
to be nominally insensitive to gain parameter hFE.
Under the condition deduced in Part (b), what are the temperature sensitivity coefficients, i1
and i2, of the two quiescent collector currents?
The circuit exploits feedback, which is implemented by resistance R3 and the inductive coil.
Explain qualitatively the operation of this feedback loop.
PROBLEM #3.17
Vcc
R
Rl
Rl
Q1
R
Q2
Rk
Figure (P3.17)
The power supply voltage, Vcc, in the biasing network of Figure (P3.17) is 9 volts, and the two
transistors are identical to the devices addressed by Example #3.6 and Table (3.1). Design the circuit so that each bipolar junction transistor conducts a quiescent collector current of 2 mA ±5% at a
collector-emitter voltage of 1.5 volts over a temperature range extending from 27 °C to 75 °C. Use
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Chapter 3
BJT Models & Biasing
HSPICE or equivalent computer-aided analysis software to confirm the propriety of the finalized design at 27 °C, 50 °C, and 75 °C.
PROBLEM #3.18
Table (3.2) itemizes the HSPICE parameters for a representative, high-speed PNP bipolar junction
transistor. Use this device to address the following directives. Unless specified otherwise, assume a
junction operating temperature of 27 °C.
(a). Set up a computer-aided simulation that sweeps, for an emitter-collector voltage, Vec, of 1.5
volts, the static collector current as a function of the applied emitter-base voltage, Veb, for 0 ≤
Veb ≤ 800 mV. Plot these curves.
(b). Set up a computer-aided simulation that sweeps, for an emitter-collector voltage, Vec, of 1.5
volts, the static base current as a function of the applied emitter-base voltage, Veb, for 0 ≤ Veb ≤
800 mV. Plot these curves.
(c). Use the plots garnered from the execution of the preceding two parts of this problem to generate a plot of the DC beta, hFE, as a function of collector current for Vec = 1.5 volts.
(d). Use the foregoing simulated results to deduce the value of emitter-base junction voltage, Veb,
commensurate with a collector current of 2 mA. What is the DC beta of the transistor at this
collector current?
(e). Repeat Part (a) for a junction temperature of 75 C. Recall that the temperature induced
change, Veb, in static emitter-base terminal voltage can be represented as Veb ≈ −SebTj,
where Veb is the voltage change required to preserve constant collector current over the
temperature excursion, Tj. This temperature excursion can be taken as the temperature
difference between 27°C and 75 °C. Deduce the value of coefficient Seb, using the simulations
already executed.
PROBLEM #3.19
The power supply voltage, Vcc, in the biasing network of Figure (P3.19) is 3 volts, and the two
transistors are identical to the device parameterized in Table (3.2). Use the simulation results garnered in the preceding problem to design the circuit so that transistor Q1 conducts a quiescent
collector current of 2 mA ±5% at an emitter-collector voltage of 1.5 volts over a temperature range
extending from 27 °C to 75 °C. Use HSPICE or appropriate other computer-aided analysis software
to confirm the propriety of the finalized design at 27 °C, 50 °C, and 75 °C.
Q2
Q1
IcQ
R
Rl
Vcc
Figure (P3.19)
PROLEM #3.20
Redesign the circuit addressed in Example #3.8 under the constraint that the current conducted by
resistance R4 is to be nominally one-fourth of the collector current conducted by transistor Q2. Submit a finalized schematic diagram, and simulate the design at 27 °C, 50 °C, 75 °C, and 100 °C.
PROBLEM #3.21
Redesign the circuit addressed in Example #3.8 under the constraint that the collector load resistance
imposed on transistor Q1 is at least 5-times larger than the emitter degeneration resistance, Ree. In
this approach, it is likely impossible to achieve equal collector-emitter voltages between transistors
- 302 -
Chapter 3
BJT Models & Biasing
Q1 and Q2. Submit a finalized schematic diagram, and simulate the design at 27 °C, 50 °C, 75 °C,
and 100 °C. The SPICE parameters for the BJTs are itemized in Table (3.1)
PROBLEM #3.22
Derive a general expression for the static collector current, Ic, conducted by transistor Q2 in the network of Figure (3.33). Deduce the resultant temperature coefficient of this collector current.
PROBLEM #3.23
Using (3-116), use engineering rationale to develop an expression for the resistance, Rx, indicated as
the resistance presented at the collector terminal of transistor Q1. Compared to the biasing
configuration shown in Figure (3.29), what is the relative effectiveness of this circuit in the sense of
coping with signal fluctuations at the collector port?
PROBLEM #3.24
All four transistors in the cascode current mirror of Figure (P3.24) are identical, save for the fact that
transistors Q3 and Q4 each have emitter-base junction injection areas that are k-times larger than
those of transistors Q1 or Q2. All transistors operate in their forward active regimes where they
support identical base-emitter terminal potentials and identical values of the DC beta, hFE. Do not
tacitly ignore transistor base currents.
Vcc
V
R
Rx
Ir
IQ
Q1
x1
Q3
xk
Q2
x1
Q4
xk
Figure (P3.24)
(a).
(b).
(c).
(d).
(e).
(f).
(g).
What is the static collector-base operating voltage of transistor Q4?
What is the minimum allowable collector to ground potential, V, of transistor Q3?
Derive the equation that relates current IQ to the current, Ir, conducted by resistance R.
Derive the equation that relates current IQ to the supply voltage, Vcc, the circuit resistance, R,
and the base-emitter terminal potential, Vbe, of all transistors.
What condition must be satisfied to render current IQ nominally independent of hFE?
Derive an expression for the temperature sensitivity, IQ/Tj, of current IQ. Assume Vbe =
−SbeTj, where Sbe is an empirical constant and Tj represents the junction operating temperature of all devices.
Derive an expression for the indicated small signal resistance, Rx, established at the collector
of transistor Q3. As usual, assume low frequency conditions and ignore the internal series
resistances implicit to the collectors and emitters of all devices. Make sure to account for the
model parameter implications of the fact that while the current densities of transistors are
likely to be identical, the actual collector currents flowing in all devices are not necessarily the
same.
PROBLEM #3.25
In the two stage Wilson current sink of Figure (P3.25), the four transistors are identical except for
- 303 -
Chapter 3
BJT Models & Biasing
the fact that transistors Q1 and Q2 have twice the emitter-base junction injection area of transistors
Q3 and Q4.
V3
V4
Vcc
Rx3
Rx4
R
I3
I4
Q3
Ir
Q4
x1
x1
Q1
x2
Q2
x2
Figure (P3.25)
(a).
(b).
(c).
(d).
(e).
(f).
What are the minimum allowable collector to ground potentials, V3 and V4, which are applied
to the collectors of transistors Q3 and Q4, respectively?
Derive the equations that relate currents I3 and I4 to the reference current, Ir, which flows
through resistance R.
Derive the equations that relate currents I3 and I4 to the supply voltage, Vcc, the circuit resistance, R, and the base-emitter terminal potential, Vbe, of all transistors.
What condition renders currents I3 and I4 almost independent of hFE?
Derive an expression for the temperature sensitivities, I3 /Tj and I4 /Tj, of currents I3 and
I4. Assume Vbe = −SbeTj, where Sbe is an empirical constant and Tj represents junction
operating temperature of all devices.
Derive expressions for the indicated small signal resistances, Rx3 and Rx4, established at the
collectors of transistors Q3 and Q4. As usual, assume low frequency conditions and ignore
the internal series resistances in the collector and emitter leads of all devices. Make sure to
account for the model parameter implications of the fact that while the current densities of all
transistors are likely to be nearly identical, the actual collector currents flowing in all devices
are not necessarily the same.
PROBLEM #3.26
In the Wilson current mirror of Figure (P3.26), the transistors are identical except for the indicated
differences in emitter-base junction injection ratios.
(a). Derive an expression for the low frequency, small signal resistance, Ry, presented to the junction formed of resistance R, the base terminal of transistor Q1, and the collector terminal of
Q2. Ignore internal series resistances in the collectors and emitters of all transistors and
additionally, ignore all forward Early resistances. Provide an engineering justification for the
somewhat cavalier neglect of the Early resistance in each transistor.
(b). Reduce the expression gleaned in (a) for the case of very large βac. Discuss the size of the
resultant resistance in terms of the injection area ratio, k.
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Chapter 3
BJT Models & Biasing
V1
Vcc
Rx
R
I1
Ir
Q1
xk
Ry
Q2
x1
Q3
xk
Figure (P3.26)
PROBLEM #3.27
Using the low frequency, small signal model of a bipolar junction transistor, develop an analytical
expression for the small signal resistance, Rx, in the supply-independent biasing configuration of
Figure (3.45). Ignore the internal collector and emitter resistances of all transistors. Furthermore,
assume that the two startup transistors, Q6 and Q7, are large geometry devices so that the substrate
capacitance of transistor Q6 emulates a short circuit for the signal frequencies of interest.
PROBLEM #3.28
An advantage of the startup cell in the supply-independent biasing scheme of Figure (3.45) is that it
consumes no steady state power. A disadvantage is that it utilizes a capacitor whose value is
generally sufficiently large to cast aspersions on its practical on chip circuit implementation. The
circuit in Figure (P3.28) supplants the subject startup cell with a resistance, Rk, which is chosen large
enough to ensure that it minimally affects the overall standby power dissipation of the network.
Prove that a small current flowing through resistance Rk thwarts an initial null lockup state. What
engineering logic underpins deploying resistance Rk?
Vcc
Rk
Ree
Ree
Ree /kp
Q3
Q4
x1
Q1
x1
Q5
x1
x kp
Q2
Vo
x kn
R
IQ
IQ
Figure (P3.28)
- 305 -
Rl
Io
Chapter 3
BJT Models & Biasing
PROBLEM #3.29
The circuit in Figure (P3.29) provides an output current, Io, which is dependent on the supply
voltage, Vcc. Indeed, current Io is a natural logarithmic function of Vcc. It therefore shows a voltage
sensitivity that is smaller than the current sensitivities of biasing networks whose currents of interest
are linearly related to the supply line voltage. All three transistors in the circuit are identical and are
characterized by the parameters in Table (3.1).
Vcc
Io
R1
Rl
Vo
Q1
Q2
x1
x1
R2
Q3
x1
Figure (P3.29)
(a).
(b).
(c).
(d).
(e).
Assuming large static beta, hFE, in transistor Q1, show that the output current, Io, is given by
 V  2Vbe 
V
I o  T ln  cc
,
R2  R1 I s

where VT is the familiar Boltzmann voltage, and Is is the saturation current of the NPN transistors. It is to be understood that while Vbe is the base-emitter terminal voltage of each of the
transistors, it can be supplanted by the junction threshold voltage for most design purposes.
Is current Io PTAT? Explain briefly.
Using a nominal 5-volt supply, design the circuit so that Io = 5 mA at room temperature. Select resistance Rl so that the collector-emitter potential supported by transistor Q1 is 2Vbe, even
if the supply voltage degrades to 2.5 volts.
Use SPICE to simulate the room temperature, static describing function, Io versus Vcc, for the
design effected in the preceding part of this problem. The simulation should span a Vcc range
of 0 ≤ Vcc ≤ 10 volts. Does the simulated current sensitivity to supply voltage corroborate with
theoretical expectations.
Repeat Part (d) for junction operating temperatures of 50 °C and 75 °C. Does the simulated
current dependence on temperature track with analytical expectations? Explain briefly.
PROBLEM #3.30
Figure (P3.30) proposes a supply-independent biasing scheme that can function as either a current
source (via the current, Io1, conducted by PNP transistor Q6) or a current sink (via the current, Io2,
flowing in the collector of NPN transistor Q3) or both, simultaneously. Current Io1 derives from current IQ because of the current mirror formed of transistors Q5 and Q4, while current Io2 is likewise
directly dependent on IQ because of the effective mirror established by transistors Q1 and Q3. To
the extent that the Early voltages of all active devices are large, currents Io1 and Io2 show a first order
independence of the supply voltage, Vcc. All NPN transistors are identical, save for the indicated
junction area factors, and have the parameters itemized in Table (3.1). Similarly, all PNP devices
are identical, except for the delineated area factors, and have the parameters listed in Table (3.2).
- 306 -
Chapter 3
BJT Models & Biasing
Vcc
Q5
Q4
Q6
x1
x1
IQ
x kp
IQ
Io1
Vo1
Q2
Vo2
x1
Io2
Q1
Q3
x1
R
x kn
Figure (P3.30)
(a).
(b).
(c).
(d).
(e).
(f).
(g).
Assuming all transistor base currents can be ignored and all transistor Early voltages and knee
currents are large, derive expressions for the output currents, Io1 and Io2.
What is the minimum value, say Vmin, of supply voltage Vcc that is commensurate with linear
active operation of transistors Q1, Q2, Q4, and Q5?
Are null current states at startup an issue? If they are, mitigate the problem with a suitably
connected resistance. Show the modified circuit schematic diagram.
Using a 6-volt supply, design the circuit so that IQ = 1 mA, Io1 = 2 mA, and Io2 = 3 mA at room
temperature. Select voltage Vo1 as the maximum voltage permissible with linear active mode
operation of transistor Q6 when Vcc = Vmin. Additionally, select voltage Vo as the minimum
voltage that ensures linear active regime operation of transistor Q3 when Vcc = Vmin.
Use SPICE to simulate the room temperature, static describing functions, Io1 versus Vcc and Io2
versus Vcc, for the design executed in the preceding part of this problem, over a Vcc range
spanning 0 ≤ Vcc ≤ 10 volts. What perturbations in currents Io1 and Io2 are observed when Vcc
degrades by 50%?
Repeat Part (d) for junction operating temperatures of 50 °C and 75 °C. Do the simulated current dependencies on temperature track with analytical expectations? Explain briefly.
Assuming a startup resistance is required, select its value such that the increased power
dissipation of the entire circuit is less than 2% of the dissipation observed for the circuit with
no startup provision. Simulate the transient responses of currents Io1 and Io2 when Vcc is a 6volt pulse waveform characterized by 500 nSEC rise and fall times. What is the simulated settling time of the entire circuit? To the latter end, define the settling time as that time subsequent to step input excitation at which currents Io1 and Io2 are maintained at 90% of their
respective steady state values.
PROBLEM #3.31
In this problem, we should like to investigate, albeit semi-quantitatively, the input to output (I/O)
nonlinearity that pervades the emitter-base junction voltage (Ve) to collector current (Ic) response.
To this end, recall that for Ve > Veon, where Veon represents the emitter-base junction threshold voltage, and Vc ≤ 0, the collector current is given approximately by the Ebers-Moll relationship,

Ve n f VT
I c  β f Ib  I s e

1 .
Let Ve = VeQ + Vm cos(ωt), where VeQ is the quiescent emitter-base junction forward bias that corresponds to a quiescent current of IcQ. Plot the approximate current ratio, Ic /IcQ, as a function of the
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Chapter 3
BJT Models & Biasing
normalized time, (ωt), for normalized signal amplitudes, Xm = Vm /nfVT, of Xm = 0.005, 0.05, 0.5, 5,
and 50. Your plots should embrace at least four periods of the applied sinusoid. Comment on the
nonlinearities postured by your plots.
PROBLEM #3.32
The current mirror offered in Figure (P3.32) is driven by the off chip current source, Ii. The three
transistors are identical except that the emitter-base junction areas of transistors Q1 and Q2 are each
k-times larger than the emitter-base junction area of transistor Q3. Voltages Vcc and Vpp are sufficiently large to ensure that all three active devices operate in their saturation domains.
(a). Derive an analytical relationship that links output current Io to static input current Ii.
Vcc
Vpp
Ii
Io
Rout
Q3
Q1
Ree
Q2
Ree
R
Figure (P3.32)
(b).
(c).
(d).
What purpose is served by resistance R in the mirror.
Does accurately predictable and reproducible current mirroring between Io and Ii require that
the quiescent collector-emitter voltages of all three transistors be nominally the same? What
is the effect of Early voltage on this mirroring?
If voltage Vpp changes by the small (positive or negative) amount, Vpp, estimate the resultantly small change, Io, in static current Io. Note that this estimate requires that you determine an expression for the indicated output resistance, Rout. In the course of determining Rout,
series collector and emitter resistances can be ignored. If you make other approximations, be
sure to justify them.
PROBLEM #3.33
In the cascode current mirror of Figure (P3.33), all transistors are identical (inclusive of emitter-base
junction areas), all transistors are biased identically, and all devices operate in their linear regimes.
Vcc
Vpp
Ii
Io
Rout
Q2
Q3
Q1
Q4
Figure (P3.33)
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Chapter 3
(a).
(b).
(c).
(d).
BJT Models & Biasing
Deduce relationship that links output current Io to the constant static input current Ii.
Determine an expression for the indicated small signal output resistance, Rout. The internal series collector and emitter resistances can be tacitly ignored, as can the Early resistances of
transistors Q1 and Q2. However, do not ignore the Early resistances of transistors Q3 and Q4.
Why is it sensible to ignore the Early resistances of transistors Q1 and Q2, but unreasonable to
ignore the Early resistances in the other two transistors? Submit concise, but clear, engineering rationale for your answers.
If voltage Vpp changes by the small (positive or negative) amount, Vpp, estimate the resultantly small change, Io, in static current Io.
PROBLEM #3.34
In the current source shown in Figure (P3.34), the two PNP transistors are similar, as are the two
NPN devices. However and in general, the ith transistor has an emitter-base junction injection area
of Ai. All devices operate in their saturation domains, and care is taken to ensure that the two NPN
collector-emitter voltages are the about the same; ditto for the two static emitter-collector voltages of
the two PNP transistors.
Vcc
Q1
Q2
Q4
Q3
II
Iout
R1
R2
Figure (P3.34)
(a).
(b).
Determine an expression for the static current transfer ratio, Iout /Ii.
How critical are the NPN and PNP DC betas, hFEn and hFEp, respectively, to a reasonably accurate determination of this current ratio?
PROBLEM #3.35
In the controlled current source of Figure (P3.35), the two transistors operate in their linear active,
forward regimes, have negligible base conductivity modulation, and offer very large DC beta. Current Ii is a constant current sink.
Vcc
R1
Vcntr
Q1
Q2
Io
Ii
Figure (P3.35)
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R2
Chapter 3
(a).
(b).
BJT Models & Biasing
Derive an expression for the indicated output current, Io, in terms of the supply line voltage,
Vcc, the applied control voltage Vcntr, and pertinent circuit resistances.
If voltages Vcc and Vcntr are reasonably insensitive to circuit operating temperatures, and if the
temperature coefficients of both circuit resistances are very small, comment on the temperature sensitivity of current Io. In particular, is current Io independent of operating temperature,
a strong function of temperature, or a weak function of temperature? Provide clear, but concise engineering rationale for your position on this matter.
PROBLEM #3.36
In the current sink reference depicted in Figure (P3.36), the current response is the indicated current,
Iref. All transistors operate in their forward active domains. Transistors Q1 and Q2 have identically
sized emitter-base junction areas, while NPN transistor Q3 has an emitter-base junction area that is
kn-times that of Q2. The PNP BJT, Q5, has an emitter-base junction area that is larger than that of
transistor Q4 by a factor of kp.
+Vcc
Q4
x1
Q5
Rl
x kp
Rx
R
Iref
Q2
Q1
x1
Q3
x1
x kn
Figure (P3.36)
(a).
(b).
Derive an analytical expression for the reference output current, Iref.
What must the collector-emitter voltages of transistors Q1, Q2, and Q5 satisfy to ensure that
these devices operate in their linear regimes? What must the emitter-collector voltages of
transistors Q4 and Q5 satisfy so that these transistors similarly operate in their forward active
regimes. Attempt, as best as possible, to express your engineering disclosures in terms of current Iref power line voltage, Vcc, and area ratios, kn and kp.
PROBLEM #3.37
Return to the current reference topology of Figure (P3.36).
(a). Derive an expression for the small signal output resistance, Rx. Make reasonable assumptions
to simplify this result.
(b). If voltage Vcc changes by the small amount, Vcc, what is the resultant change Iref, in the
reference current?
PROBLEM #3.38
The op-amp in the bandgap reference cell of Figure (P3.38) can be presumed to have infinitely large
open loop gain and infinitely large input impedances at both of its input ports. All transistors are biased to ensure that they operate in their forward active regimes. Transistors Q1 through Q5 and
transistors Q6, Q8, and Q10 have the same emitter-base junction injection areas. On the other hand,
the emitter-base junction areas of transistors Q7 and Q9 are a factor of k larger than the areas of
transistors Q6 and Q8.
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Chapter 3
BJT Models & Biasing
x1
x1
x1
x1
Q1
Q5
Q3
Q2
Q4
I2
I1
x1
Vcc
x1
xk
xk
Q7
Q9
Vref
op-amp

x1
R
x1
Q10
Q6
Q8
Figure (P3.38)
(a).
(b).
(c).
Determine a general expression for the reference output voltage, Vref.
What design condition must be satisfied if Vref is to offer zero temperature coefficient at the
room temperature reference of To? Take To = 27 °C.
In view of the design condition developed in Part (b), what is the corresponding optimized
value, say Vropt, of output voltage Vref?
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