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Monolithic Amplifier Circuits:
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Introduction
Device Physics:
The Bipolar Transistor
• In analog design the transistors are not simply switches
• Second-order effects impact their performance
• With each generation these effects become more significant
Chapter 4
• The goal is to develop a circuit model for each device by
formulating its operation
Jón Tómas Guðmundsson
– to do this a good understanding of the underlying principles is
necessary
tumi@hi.is
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Bipolar Junction Transistors
Bipolar Junction Transistors
emitter
base
collector
p+−type
n−type
p−type
• The bipolar junction transistor (BJT) is a semiconductor device
containing three adjoining alternately doped regions
• pnp bipolar junction transistor
• The central region is known as base and the outer regions emitter
and collector
• The emitter is much more doped than the collector
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• circuit symbol, voltage and current polarities keep the device in
the active region
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• The pnp transistor is basically two very closely spaced
p-n-junctions
• npn bipolar junction transistor
• The base region is ∼ 1 µm thick, so the junctions are very closely
spaced and interact with each other
• circuit symbol, voltage and current polarities keep the device in
the active region
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Bipolar Junction Transistors
Bipolar Junction Transistors
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• Thus the transistor is capable of current and voltage gain
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Bipolar Junction Transistors
Bipolar Junction Transistors
• The bipolar junction transistor has four regions of operation
• The cutoff region
• The most common region of operation is the active region
– E-B-junctions are forward biased
– E-B-junctions are reverse biased
– C-B-junctions are reverse biased
– C-B-junctions are reverse biased
– gives the largest signal gain
– represents an “off” state for the transistor as a switch
– almost all linear amplifiers operate in the active region
• The inverted region is when
– E-B-junctions are reverse biased
• The saturation region is when
– C-B-junctions are forward biased
– E-B-junctions are forward biased
– C-B-junctions are forward biased
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• In circuit applications the transistor typically has a common
terminal between input and output
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Bipolar Junction Transistors
Bipolar Junction Transistors
• The four operating regions of the bipolar junction transistor
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Bipolar Junction Transistors
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Bipolar Junction Transistors
• Common emitter is the most popular
• The common emitter amplifier has
• This gives three amplifier types
– output variables vEC and iC
– common base (CB)
– input variables vEB and iB
– common emitter (CE)
– common collector (CC) (emitter follower)
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• if two of the voltages (or currents) are known the third is also
known (Kirchhoff’s laws)
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Bipolar Junction Transistors
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Bipolar Junction Transistors
• The holes injected from the emitter that reach the collector
junction are ICp
• The current components within the transistor
• The current due to thermally generated electrons near the
junction are ICn
IC = ICp + ICn
• The holes injected from the emitter to base are IEp
IE = IEp + IEn
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Bipolar Junction Transistors
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Bipolar Junction Transistors
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• The ratio IC /IE in the active region is defined as the dc alpha
αdc =
IC
ICp + ICn
=
IE
IEp + IEn
• If the E-B junction is forward biased ICp ICn and
• It is preferred that αT is unity, but due to recombination in base
αT is slightly less than unity
αdc =
• The emitter injection efficiency
IEp
IEp
=
γ=
IE
IEn + IEp
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• As the emitter is more heavily doped γ −→ 1 and IEn −→ 0
• The base transport factor is defined as the ratio of the hole
current diffusing into the collector to the hole current injected at
the E-B junction
ICp
αT =
IEp
and measures the injected hole current compared to the total
emitter current
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IB = IE − IC = IB1 + IB2 − IB3
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ICp
IEp + IEn
which can be written as
IEp
1
ICp
= αT
αdc =
IEp 1 + IEn /IEp
IEp + IEn
or
αdc = γαT
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Bipolar Junction Transistors
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Bipolar Junction Transistors
• Beta is defined as
βdc =
or
βdc =
IC
IC
=
IB
IE − IC
IC /IE
αdc
=
1 − IC /IE
1 − αdc
• The emitter current crossing the E-B depletion region is
evaluated as two minority carrier diffusion currents
• Note that as αdc −→ 1 then βdc −→ ∞
IE = IEp (0) + IEn (000 )
or
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Bipolar Junction Transistors
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IE = −qADB
d∆pB d∆nE −
qAD
E
dx x=0
dx00 x00 =0
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Bipolar Junction Transistors
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• For holes in the base region
DB
or
where LB =
• The collector current crossing the C-B junction is evaluated as
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∆pB (x)
d2 ∆pB (x)
=
2
dx
L2B
DB τB is the minority carrier diffusion length
• The solution is
IC = ICp (W ) + ICn (00 )
or
√
d2 ∆pB (x)
∆pB (x)
=
2
dx
τB
∆pB (x) = C1 exp(x/LB ) + C2 exp(−x/LB )
d∆pB IC = −qADB
dx x=W
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d∆nC + qADC
dx0 x0 =0
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• Similar equations can be found for ∆nE (x00 ) and ∆nC (x0 )
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Bipolar Junction Transistors
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Bipolar Junction Transistors
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• If there is no recombination of holes in the base τB = ∞ then
d2 ∆pB (x)
=0
dx2
which has solution of the form
∆pB (x) = C1 x + C2
• The two boundary conditions yield
∆pB (0) = pB0 (exp(qVEB /kT ) − 1) = C1 × 0 + C2
• The junction voltage VEB and VCE determine the minority carrier
concentrations at the edges of the depletion region
– give the boundary conditions
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Bipolar Junction Transistors
∆pB (W ) = pB0 (exp(qVCB /kT ) − 1) = C1 × W + C2
∆pB (0) − ∆pB (W )
x
∆pB (x) = ∆pB (0) −
W
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Bipolar Junction Transistors
• Similarly
∆nE (x00 ) = C1 exp(−x00 /LE ) + C2 exp(x00 /LE )
and with the boundary conditions
∆nE (x00 ) = nE0 (exp(qVEB /kT ) − 1) exp(−x00 /LE )
• Thus the electron current is
• Thus the slope of ∆pB (x) is
∆pB (0) − ∆pB (W )
−
W
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IEn (000 ) = −qADE
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d∆nE (x00 ) qADE
00 = LE nE0 (exp(qVEB /kT )−1)
dx
x =0
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Bipolar Junction Transistors
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which gives
IEp (0) =
d∆pB (x) qADE
[∆pB (0) − ∆pB (W )]
=
dx x=0
W
• The total collector current is the sum of the hole and electron
currents
DB pB0
DC nC0
(exp(qVCB /kT ) − 1)
+
IC = −qA
LC
W
qADB
pB0 [(exp(qVEB /kT ) − 1) − (exp(qVCB /kT ) − 1)]
W
DB pB0
(exp(qVEB /kT ) − 1)
W
• The base current is then
+qA
• The total emitter current is the sum of the hole and electron
currents
DB pB0
DE nE0
(exp(qVEB /kT ) − 1)
IE = qA
+
LE
W
−qA
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Bipolar Junction Transistors
• Simlarly the hole current is
IEp (0) = −qADB
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DB pB0
(exp(qVCB /kT ) − 1)
W
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IB = −qA
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Bipolar Junction Transistors – Ebers – Moll
DC nC0
DE nE0
(exp(qVEB /kT )−1)+qA
(exp(qVCB /kT )−1)
LE
LC
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Bipolar Junction Transistors – Ebers – Moll
IE = IF − αR IR
IF = IF0 (exp(qVEB /kT ) − 1)
DB pB0
DE nE0
+
= qA
×(exp(qVEB /kT ) − 1)
LE
W
|
{z
}
• Similarly the collector current can be split up into
IR = IR0 (exp(qVCB /kT ) − 1)
DB pB0
DC nC0
×(exp(qVCB /kT ) − 1)
+
= qA
LC
W
|
{z
}
IF0
DB pB0
(exp(qVCB /kT )−1)
αR IR = αR IR0 (exp(qVCB /kT )−1) = −qA
W
IR0
and
αF IF = αF IF0 (exp(qVEB /kT )−1) = qA
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• The emitter current can be written as
• The currents are now given different names
and
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DB pB0
(exp(qVEB /kT )−1)
W
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Bipolar Junction Transistors – Ebers – Moll
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Bipolar Junction Transistors – Ebers – Moll
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• The emitter current can be written as
IE = IF − αR IR
• The collector current is then written as
IC = αF IF − IR
• With
• The Ebers - Moll circuit for an ideal pnp bipolar junction
transistor
IB = IE − IC = (1 − αF )IF + (1 − αR )IR
• These equations are known as the Ebers - Moll equations for
an ideal pnp bipolar junction transistor
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Bipolar Junction Transistors – Ebers – Moll
• Note that
• If
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and
αF
1 − αF
αR
1 − αR
then only three numbers are necessary for the Ebers-Moll
equations to be completely specified, βF , βR , and IS (or αF , αR ,
and IS ) and all other parameters can be calculated
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BJT - Small-signal models
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• Small signal models are then derived to calculate the circuit gain
and terminal impedances
• We recall
βR =
• The direct connection between the doping densities, base width,
lifetime, etc. and the Ebers - Moll equations makes them
particulary useful in integrated circuit analysis
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• Analog circuits are often operated with signal levels that are
small compared to the bias voltages and currents
αF IF0 = αR IR0 = IS
βF =
• The complete set of input-output I − V characteristics can be
calculated from these equations
IC = IS exp
VBE
VT
• The transconductance is defined as
gm =
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dIC
dIS exp(VBE /VT )
IS
VBE
IC
=
=
exp
=
dVBE
dVBE
VT
VT
VT
• Note that gm = 38 mS for IC = 1 mA at 25 ◦ C.
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BJT - Small-signal models
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BJT - Small-signal models
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• The minority carriers are supplied by the base lead so the device
has input capacitance (npn)
Cb =
∆Qh
IC
= τF gm = τF
∆VBE
VT
where
WB2
2Dn
is the base transit time in the forward direction
τF =
• A change in the base-emitter voltage ∆VBE causes a change in
the minority carrier charge in the base ∆Qe
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• Typical values are
– τF = 50 − 500 ps for npn
– τF = 1 − 40 ps for pnp
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BJT - Small-signal models
• In the forward active region the base current is related to the
collector current
IC
IB =
βF
and
IC
d
∆IC
∆IB =
dIC βF
or
−1
∆IC
IC
d
β0 =
=
∆IB
dIC βF
BJT - Small-signal models
• Typical values of β0 are close to those of βF
• If βF is constant then βF = β0
• A single value of β is often assumed for a transistor and then
used in both ac and dc calculations
where β0 is the small signal current gain of the transistor
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BJT - Small-signal models
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BJT - Small-signal models
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• r0 is the small-signal output resistance of the transistor
• Small-signal input inpedance of the device
rπ =
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∆VBE
β0
∆VBE
=
β0 =
∆IB
∆IC
gm
r0 =
1 VA
gm VT
• The small-signal input shunt resistance for a bipolar transistor
depends on the current gain and is inversly proportional to IC
• Small changes ∆VCE in VCE lead to changes ∆IC in IC
∆IC =
∂IC
∆VCE
∂VCE
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• The large-signal characteristics of the transistor when the Early
effect is included is
VCE
VBE
IC = IS exp
1+
VT
VA
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or
∆VCE
VA
=
= r0
IC
IC
where VA is the Early voltage
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BJT - Small-signal models
BJT - Small-signal models
• Note that r0 is inversly proportional to IC and thus r0 can be
related to gm
1
r0 =
ηgm
where
kT
η=
qVA
• Combination of the above small-signal elements yields the
small-signal model often referred to as the hybrid-π model
• So if VA = 100 V then η = 2.6 × 10−4 at 25◦ C
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• This is the basic model that arises directly from essential
processes in the device
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BJT - Small-signal models
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BJT - Small-signal models
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• An increase ∆VCE in VCE leads to a decrease ∆IB in IB modelled
by
∆VCE
∆VCE ∆IC
∆IC
rµ =
=
= r0
∆IB1
∆IC ∆IB1
∆IB1
• Typically IB1 < 0.1IB so
rµ ≈ 2β0 r0 − 5β0 r0
• All pn junctions have voltage dependent capacitance
• Technological limitations in the fabrication of a transistor give
rise to a number of parasitic elements that have to be taken into
account as well
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BJT - Small-signal models
– Cje is for the base-emitter junction
– Cµ is for the base-collector junction
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BJT - Frequency Response
• The base-collector junction capacitance
Cµ = – Ccs is for the collector-substrate junction
Cµ0
1−
V
ψ0
n
where V is forward bias and n ∼ 0.2 − 0.5
• Furthermore there are resistive parasitics
– rb is series resistance in base
– rc is series resistance in collector
• The addition of the resistive and capacitive parasitics to the
basic small-signal circuit gives the complete small-signal
equivalent circuit
– rex is series resistance in emitter
• The capacitance Cπ contains the base-charging capacitance Cb
and the emitter-base depletion layer capacitance Cje
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Cπ = Cb + Cje
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=⇒ Example 4.1
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BJT - Frequency Response
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BJT - Frequency Response
• The parasitic circuit elements influence the high-frequency gain
of the transistor
• The capability of a transistor to handle high frequency is
specified by the frequency where the magnitude of the
shor-circuit sommon-emitter gain falls to unity
• This frequency is the transition frequency fT
• Then the small signal voltage v1 is
• To determine fT we neglect rex and rµ and redraw the complete
small signal equivalent circuit
v1 =
rπ
ii
1 + rπ (Cπ + Cµ )s
• If rc is assumed small then ro and Ccs have no influence
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BJT - Frequency Response
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• If the current fed forward through Cµ is neglected we can write
gm r π
io ≈
ii
1 + rπ (Cπ + Cµ )s
and the high frequency current gain
β0 (Cπ +Cµ )
jω
gm
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|β(jω)| = 1
we find
ω = ωT =
= β(jω)
• At high frequencies the imagniary part of the denominator
dominates and the equation simplifies to
gm
β(jω) ≈
jω(Cπ + Cµ )
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• When the gain is unity
and thus
β0
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BJT - Frequency Response
io ≈ gm v1
io
(jω) ≈
ii
1+
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and
fT =
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gm
Cπ + Cµ
1
gm
2π Cπ + Cµ
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BJT - Frequency Response
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BJT - Frequency Response
• The frequency ωβ is defined as the frequency where
β0
|β(jω)| = √
2
or has fallen 3 dB down from the low frequency value
• That is
• The frequency response of a bipolar junction transistor |β(jω)|
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τT =
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This discussion is based on sections 1.1, 1.4, 2.1 - 2.4 and 2.6 in Neudeck (1983).
It is worth looking at the original paper by Ebers and Moll (1954). A detailed
discussion on the small-signal models and frequency response is found in sections
1.3 and 1.4 of Gray et al. (2001).
• A time constant τT is associated with ωT
or
gm
1
β0 Cπ + Cµ
Further reading
BJT - Frequency Response
τT =
ωβ =
1
ωT
References
Cje + Cb + Cµ
Cje + Cµ
Cπ + Cµ
=
= τF +
gm
gm
gm
Ebers, J. J. and J. L. Moll (1954). Large-signal behavior of junction transistors. Proceedings of the
IRE 42 (12), 1761–1772.
Gray, P. R., P. J. Hurst, S. H. Lewis, and R. G. Meyer (2001). Analysis and Design of Analog
Integrated Circuits (4 ed.). New York: John Wiley & Sons.
=⇒ Example 4.2
Neudeck, G. W. (1983). The Bipolar Junction Transistor. Reading, Massachusets: Addison-Wesley.
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