VI. Transistor amplifiers: Biasing and Small Signal Model
6.1
Introduction
Transistor amplifiers utilizing BJT or FET are similar in design and analysis. Accordingly
we will discuss BJT amplifiers thoroughly. Then, similar FET circuits are briefly reviewed.
Consider the circuit below. The operating point of the BJT is shown in the iC vCE space.
iC
RB
+
iB
+
vBE
vCE
_ _
iE
VBB
RC
VCC
Let us add a sinusoidal source with an amplitude of ∆VBB in series with VBB . In response to
this additional source, the base current will become iB + ∆iB leading to the collector current
of iC + ∆iC and CE voltage of vCE + ∆vCE .
i C +∆ i C
RB
i B +∆ i B
+
+
vBE +∆ vBE _ _
+
− ∆ VBB
~
vCE +∆ vCE
RC
VCC
VBB
For example, assume without the sinusoidal source, the base current is 150 µA, iC = 22 mA,
and vCE = 7 V (the Q point). If the amplitude of ∆iB is 40 µA, then with the addition of
the sinusoidal source iB + ∆iB = 150 + 40 cos(ωt) and varies from 110 to 190 µA. The BJT
operating point should remain on the load line and collector current and CE voltage change
with changing base current while remaining on the load line. For example when base current
is 190 µA, the collector current is 28.6 mA and CE voltage is about 4.5 V. As can be seen
from the figure above, the collector current will approximately be iC +∆iC = 22+6.6 cos(ωt)
and CE voltage is vCE + ∆vCE = 7 − 2.5 cos(ωt).
The above example shows that the signal from the sinusoidal source ∆VBB is greatly amplified
and appears as signals in collector current and CE voltage. It is clear from the figure that
this happens as long as the BJT stays in the active-linear state. As the amplitude of ∆iB
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
158
is increased, the swings of BJT operating point along the load line become larger and larger
and, at some value of ∆iB , BJT will enter either the cut-off or saturation state and the
output signals will not be a sinusoidal function. Note: An important observation is that
one should locate the Q point in the middle of the load line if we want to have the largest
output signal.
The above circuit, however, has two major problems: 1) The input signal, ∆VBB , is in
series with the VBB DC voltage making design of previous two-port network difficult, and
2) The output signal is usually taken across RC as RC × iC . This output voltage has a DC
component which is of no interest and can cause problems in the design of the next-stage,
two-port network.
The DC voltage needed to “bias” the BJT (establish the Q point) and the AC signal of
interest can be added together or separated using capacitor coupling as dis iscussed below.
6.1.1
Capacitive Coupling
For DC voltages (ω = 0), the capacitor is an open circuit (infinite impedance). For AC
voltages, the impedance of a capacitor, Z = −j/(ωC), can be made sufficiently small by
choosing an appropriately large value for C (the higher the frequency, the lower the C value
that one needs). This property of capacitors can be used to add and separate AC and DC
signals. Example below highlights this effect.
+15 V
Consider the circuit below which includes a DC source of
15 V and an AC source of vi = Vi cos(ωt). We are interested to calculate voltages vA and vB . The best method
to solve this circuit is superposition. The circuit is broken into two circuits. In circuit 1, we “kill” the AC source
and keep the DC source. In circuit 2, we “kill” the DC
source and keep the AC source. Superposition principle
states that vA = vA1 + vA2 and vB = vB1 + vB2 .
R2
vA
+
−
vi
C1
+
−
v
B
+15 V
R2
vA1
C1
R1
+
R2
A
+
−
+15 V
vA2
v
R
1
+
B
R1
R2
−
B1
vi
C1
+
−
vi
C1
v
B2
R
1
Consider the first circuit. It is driven by a DC source and, therefore, the capacitor will act
as open circuit. The voltage vA1 = 0 as it is connected to ground and vB1 can be found by
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
159
voltage divider formula: vB1 = 15R1 /(R1 + R2 ). As can be seen both vA1 and vB1 are DC
voltages.
In the second circuit, resistors R1 and R2 are in parallel. Let RB = R1 k R2 . The circuit
is a high-pass filter: VA2 = Vi and VB2 = Vi (RB )/(RB + 1/jωC). If we operate the circuit
at frequency above the cut-off frequency of the filter, i.e., RB 1/ωC, we will have VB2 ≈
VA2 = Vi and vB2 ≈ vA2 = Vi cos(ωt). Therefore, for ω 1/RB C
vA = vA1 + vA2 = Vi cos(ωt)
vB = vB1 + vB2 =
R1
× 15 + Vi cos(ωt)
R1 + R 2
Obviously, the capacitor is preventing the DC voltage to appear at point A, while the voltage
at point B is the sum of DC signal from 15-V supply and the AC signal.
Using capacitive coupling, we can reconfigure our previous amplifier circuit as is shown in
the figure below. Capacitive coupling is used extensively in transistor amplifiers.
vCE +∆ vCE
∆ vCE
i C +∆ i C
∆ VBB
RB
i B +∆ i B
+
+
vBE +∆ vBE _ _
+
−
~
vCE +∆ vCE
RC
VCC
VBB
BJT amplifier circuits are analyzed using superposition, similar to the example above:
1) DC Biasing: The input AC signal is set to zero and capacitors act as open circuit. This
analysis establishes the Q point in the active-linear state.
2) AC Response: DC bias voltages are set to zero. The response of the circuit to an AC
input is calculated and the transfer function, input and output impedances, etc. are found.
The break up of the problem into these two parts have an additional advantage as the
requirement for accuracy are different in the two cases. For DC biasing, we are interested in
locating the Q point roughly in the middle of active-linear state. The exact location of the
Q point is not important. Thus, a simple model, such as large-signal model of page 114 is
quite adequate. We are, however, interested to compute the transfer function for AC signals
more accurately. We will develop a model which is more accurate for small AC signals in
this section.
FET-based amplifiers are similar. FET should be biased similar to BJT and the analysis
method is broken into the DC biasing and the AC response.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
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6.2
BJT Biasing
VCC
This simple bias circuit is usually referred to as “fixed bias” as a
fixed voltage is applied to the BJT base. As we like to have only one
power supply, the base circuit is also powered by VCC . (To avoid
confusion, we will use capital letters to denote DC bias values e.g.,
IC .) Assuming that BJT is in active-linear state, we have:
BE-KVL:
VCC = IB RB + VBE
→
IB =
RB
RC
iC
+
iB
+
vBE
VCC − VBE
RB
vCE
_ _
VCC − VBE
RB
= IC RC + VCE → VCE = VCC − IC RC
RC
= VCC − β
(VCC − VBE )
RB
IC = βIB = β
CE-KVL:
VCC
VCE
For a given circuit (known RC , RB , VCC , and BJT β) the above equations can be solved to
find the Q-point (IB , IC , and VCE ). Alternatively, one can use the above equations to design
a BJT circuit to operate at a certain Q point. (Note: Do not memorize the above equations
or use them as formulas, they can be easily derived from simple KVLs).
Example 1: Find values of RC , RB in the above circuit with β = 100 and VCC = 15 V so
that the Q-point is IC = 25 mA and VCE = 7.5 V.
Since the BJT is in the active-linear state (VCE = 7.5 > Vγ ), IB = IC /β = 0.25 mA. BE-KVL
and CE-KVL result in:
BE-KVL:
VCC + RB IB + VBE = 0
CE-KVL:
VCC = IC RC + VCE
→
15 − 0.7
= 57.2 kΩ
0.250
15 = 25 × 10−3 RC + 7.5 →
→
RB =
RC = 300 Ω
Example 2: Consider the circuit designed in example 1. What is the Q point if β = 200.
We have RB = 57.2 kΩ, RC = 300 Ω, and VCC = 15 V but IB , IC , and VCE are unknown.
Assuming that the BJT is in the active-linear state:
BE-KVL:
VCC + RB IB + VBE = 0
→
IB =
VCC − VBE
= 0.25 mA
RB
IC = β IB = 50 mA
CE-KVL:
VCC = IC RC + VCE
→
VCE = 15 − 300 × 50 × 10−3 = 0
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
161
As VCE < vγ the BJT is not in the active-linear state (since IC > 0, the BJT should be in
saturation).
The above examples show the problem with our simple fixed-bias circuit as the β of a
commercial BJT can depart by a factor of 2 from its average value given in the manufacturers’
spec sheet. More importantly, environmental conditions (mainly temperature) can play an
important role. In a given BJT, IC increases by 9% per ◦ C for a fixed VBE (because of the
change in β). Consider a circuit which is tested to operate perfectly at 25◦ C. At 35◦ C, β
and IC will be roughly doubled and the BJT can be in saturation! In fact, the circuit has a
build-in positive feedback. If the temperature rises slightly, the corresponding increase in β
makes IC larger. Since the power dissipation in the transistor is VCE IC , the transistor may
get hotter which increases transistor β and IC further and can cause a “thermal runaway.”
The problem is that our biasing circuit fixes the value of IB (independent of BJT parameters)
and, as a result, both IC and VCE are directly proportional to BJT β (see formulas in the
previous page). A biasing scheme should be found that make the Q-point (IC and VCE )
independent of transistor β and insensitive to the above problems → Use negative feedback!
6.2.1
Voltage-Divider Biasing
VCC
This biasing scheme can be best analyzed and understood if we replace R1 and R2 of the voltage divider with its Thevenin equivalent:
VBB
R2
VCC
=
R1 + R 2
R1
RC
iC
and RB = R1 k R2
+
vBE
Analysis below also shows that the Q point is independent of BJT
parameters:
IE ≈ IC = βIB
vCE
_ _
R2
RE
Thevenin
Equivalent
VCC
{
The emitter resistor, RE , provides the negative feedback. Suppose
IC becomes larger than the designed value (e.g., larger β due to an
increase in temperature). Then, VE = RE IE will increase. Since
VBB and RB do not change, KVL in the BE loop shows that IB
should decrease which will reduce IC back towards its design value.
If IC becomes smaller than its design value opposite happens, IB
has to increase which will increase and stabilize IC .
RC
iC
RB
+
−
+
iB
+
vBE
VBB
vCE
_ _
RE
VBB − VBE
RB + βRE
BE-KVL:
VBB = RB IB + VBE + IE RE
→
IB =
CE-KVL:
VCC = RC IC + VCE + IE RE
→
VCE = VCC − IC (RC + RE )
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
+
iB
162
Choose RB such that RB βRE (this is the condition for the feedback to be effective):
IC ≈ I E ≈
VBB − VBE
RE
VBB − VBE
βRE
RC + R E
(VBB − VBE )
−
RE
and IB ≈
VCE = VCC − IC (RC + RE ) ≈ VCC
Note that now both IC and VCE are independent of β.
Another way to see how the circuit works is to consider BE-KVL: VBB = RB IB +VBE +IE RE .
If we choose RB βRE ≈ (IE /IB )RE or RB IB IE RE (rhe feedback condition above),
the KVL reduces to VBB ≈ VBE + IE RE , forcing a constant IE independent of the BJT β.
As IC ≈ IE this will also fixes the Q point of BJT. If the BJT parameters change (different
β due to a change in temperature), the circuit forces IE to remain fixed and changes IB
accordingly. This biasing scheme is one of several methods which fix IC (and VCE ) and allow
the BJT to adjust IB (through negative feedback) to achieve the proper bias. This class of
biasing methods is usually called “self-bias” schemes.
Another important point follows from VBB ≈ VBE + IE RE . As VBE is not a constant and
can change slightly (can drop to 0.6 or increase to 0.8 V for a Si BJT), we need to ensure
that IE RE is much larger than possible changes in VBE . As changes in VBE = vγ is about
0.1 V, we need to ensure that VE = IE RE 0.1 or VE > 10 × 0.1 = 1 V.
Example: Design a stable bias circuit with a Q point of IC = 2.5 mA and VCE = 7.5 V.
Transistor β ranges from 50 to 200.
Step 1: Find VCC : As we like to have the Q-point to be located in the middle of the load
line, we set VCC = 2VCE = 2 × 7.5 = 15 V.
Step 2: Find RC and RE :
VCE = VCC − IC (RC + RE )
→
RC + RE =
7.5
= 3 kΩ
2.5 × 10−3
We are free to choose RC and RE (usually the AC response sets the values of RC and RE as is
discussed later). We have to ensure, however, that VE = IE RE > 1 V or RE > 1/IE = 400 Ω.
Let’s choose RE = 1 kΩ which gives RC = 3 − RE = 2 kΩ (both commercial values).
Step 3: Find RB and VBB : We need to set RB βRE . As any commercial BJT has a range
of β values and we want to ensure that the above inequality is always satisfied, we should
use the minimum β value:
RB βmin RE
→
RB = 0.1βmin RE = 0.1 ∗ 50 ∗ 1, 000 = 5 kΩ
VBB ≈ VBE + IE RE = 0.7 + 2.5 × 10−3 × 103 = 3.2 V
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
163
Step 4: Find R1 and R2
R1 R2
= 5 kΩ
R1 + R 2
R2
3.2
=
=
= 0.21
R1 + R 2
15
RB = R 1 k R 2 =
VBB
VCC
The above are two equations in two unknowns (R1 and R2 ). The easiest way to solve these
equations are to divide the two equations to find R1 and use that in the equation for VBB :
R1 =
5 kΩ
= 24 kΩ
0.21
R2
= 0.21
R1 + R 2
→
0.79R2 = 0.21R1
→
R2 = 6.4 kΩ
Reasonable commercial values for R1 and R2 are and 24 kΩ and 6.2 kΩ, respectively.
The voltage divider biasing scheme is used frequently in BJT amplifiers. There are two
drawbacks to this biasing scheme that may make it unsuitable for some applications:
1) Because VB > 0, a coupling capacitor is needed to attach the input signal to the amplifier
circuit. As a result, this biasing scheme leads to an “AC” amplifier (cannot amplify DC
signals). In some applications, we need “DC” amplifiers. Biasing with two voltage sources,
discussed below, can solve this problem.
2) The voltage divider biasing requires 3 resistors (R1 , R2 , and RE ), and a coupling capacitor.
In ICs, resistors and large capacitors take too much space compared to transistors. It is
preferable to reduce their numbers as much as possible. For IC applications, “currentmirrors” are usually used to bias BJT amplifiers as is discussed below.
6.2.2
Biasing with 2 Voltage Sources
VCC
This biasing scheme is also a self-bias method and is similar
to the voltage-divider biasing. Basically, we have assigned a
voltage of −VEE to the ground (reference voltage) and chosen
VEE = VBB . As such, all of the currents and voltages in the
circuit should be identical to the voltage-divider biasing. We
should find that this is a stable bias point as long as RB βRE .
BE-KVL:
RB IB + VBE + RE IE − VEE = 0
IE
+ RE IE = VEE − VBE →
RB
β
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
IE =
VEE − VBE
RE + RB /β
RC
iC
RB
+
iB
+
vBE
vCE
_ _
RE
−VEE
164
Similar to the bias with one power supply, if we choose RB such that, RB βRE , we get:
VEE − VBE
= const
RE
= RC IC + VCE + RE IE − VEE
IC ≈ I E ≈
CE-KVL:
VCC
VCE = VCC + VEE − IC (RC + RE ) = const
Therefore, IC , and VCE are independent β and bias point is stable. Similar to the voltagedivider bias, we need to ensure that RE IE ≥ 1 V to account for possible variation in VBE .
Bias with two power supplies has certain advantages over biasing with one power supply, it
has two resistors, RB and RE (as opposed to three), and in fact, in most applications, we can
remove RB altogether and directly couple the input signal (without a coupling capacitor) to
the BJT). As such, such a configuration can also amplify “DC” signals.
6.2.3
Biasing in ICs: Current Mirrors
VCC
The self-bias schemes above, voltage-divider and bias with 2 voltage
sources, essentially operate the same way: They force IE to have
a given value independent of the BJT parameters. In principle, the
same objective can be achieved if we could bias the BJT with a current
source as is shown. In this case, no bias resistor is needed and we only
need to include resistors necessary for AC operation. As such, biasing
with a current source is the preferred way in most integrated circuits.
Such a biasing can be achieved with a current mirror circuit.
Consider the circuit shown with two identical transistors, Q1
and Q2 . Because both bases and emitters of the transistors
are connected together, KVL leads to vBE1 = vBE2 . As BJT’s
are identical, they should have similar iB (iB1 = iB2 = iB )
and, therefore, similar iE = iE1 = iE2 and iC = iC1 = iC2
KCL:
iE
β+1
βiE
β+1
2iE
iE
2iE
β+2
Iref = iC +
=
+
=
iE
β+1
β+1 β+1
β+1
Io
β
1
=
=
Iref
β+2
1 + 2/β
iB =
Io = i C =
RC
iC
RB
+
iB
+
vBE
vCE
_ _
RE
I
−VEE
I ref
2iE
Io
β +1
iC
iC
Q1
_
iE
+
+
vBE2 _
vBE1
−VEE
Q2
iE
We have explicitly used iC = βiB and iE = (β + 1)iB to illustrate the impact of β.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
165
For β 1, Io ≈ Iref (with an accuracy of 2/β). This circuit is called a “current mirror”
as the two transistors work in tandem to ensure that current Io remains the same as Iref
no matter what circuit is attached to the collector of Q2 . As such, the circuit behaves as
a current source and can be used to bias BJT circuits, i.e., Q2 collector is attached to the
emitter circuit of the BJT amplifier to be biased.
VCC
Value of Iref can be set in many ways. The simplest is by using
a resistor Rc as is shown. By KVL, we have:
RC
Io
iC
VCC = RC Iref + vBE1 − VEE
Iref
I ref
Q1
VCC + VEE − vBE1
=
= const
RC
_
iE
+
+
vBE2 _
vBE1
−VEE
Q2
iE
Current mirror circuits are widely used for biasing BJTs. In the simple current mirror circuit
above, Io = Iref with a relative accuracy of 2/β and Iref is constant with an accuracy of
small changes in vBE1 . Variations of the above simple current mirror, such as Wilson current
mirror and Widlar current mirror, have Io = Iref even with a higher accuracy and also
compensate for the small changes in vBE . Wilson mirror is especially popular because it
replace Rc with a transistor.
The right hand part of the current mirror circuit can be duplicated such that one current
mirror circuit can bias several BJT circuits as is shown. In fact, by coupling output of two
or more of the right hand BJTs, integer multiples of Iref can be made for biasing circuits
which require a higher bias current.
VCC
RC
I ref
Io
Io
2Io
−VEE
A large family of BJT circuit, including current mirrors, differential amplifiers, and emittercoupled logic circuits include identical BJT pairs. These circuits are rarely made of discrete
transistors because if one chooses two commercial BJTs, e.g., two 2N3904, there is no guaranty that β1 = β2 . However, if two identical BJTs are manufactured together on one chip
next to each other, β1 ≈ β2 within a couple of percent.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
166
6.3
Biasing FETs
Field-effect transistors can also be used in amplifier circuits by operating the FET in the
active state. Similar to BJT amplifiers, we need to apply a DC bias (in addition to the input
AC signal) so that the FET remains in the active state for the entire period of the AC signal.
VGG
The fixed-bias scheme for FETs is shown. Note that RG is not necessary
for biasing but is necessary for AC operation as without RG the input
AC signal will be grounded through VGG .
GS-KVL:
VDD
RD
RG
VGG = VGS
ID = K(VGS − Vt )2 = K(VGG − Vt )2
DS-KVL:
VDD = ID RD + VDS
→
VDS = VDD − KRD (VGG − Vt )2
Similar to the BJT β, both Vt and K vary due to the manufacturing and environmental
conditions. For example, as temperture is increased, both Vt and K decrease: decreasing
K decreases ID while decreasing Vt raises ID . The net effect (usually) is that ID decreases.
While the “thermal runaway” is not a problem in FETs, the bias point is not stable.
Similar to the BJT bias circuits, addition of a resistor RS provides the negative feedback
necessary to stabilize the bias point. For the voltage divider self bias, VG is set by R1 and R2 .
Since VGS = VG − RS ID , any decrease in ID would increase VGS and increases ID . Similarly,
any increase in ID would decrease VGS and decreases ID . As a result, ID will stay nearly
constant (because ID = K(VGS − Vt )2 , ID does not remain constant like IC in a BJT, rather
it variation become much smaller by the negative feedback). Another difference between
voltage-divider self-bias for FET with that of BJT si that in the case of BJT, we have to
ensure that RB βRE for negative feedback to be effective. THis generally limits the value
of R1 and R2 . In a FET, IG = 0 and no such limitaion exists. Therefore, R1 and R2 can be
taken to be large (MΩ) which is important in the AC response as is discussed later.
Self bias with 2 power supplies and FET current mirror bias are also shown below.
R2
RD
RD
iD
iD
R1
RS
VDD
VDD
VDD
R1
R
Io
I ref
RS
−VSS
Voltage-divider (Self Bias)
Bias with 2 power supplies
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
FET Current Mirror
167
6.4
BJT Small Signal Model
We calculated the DC behavior
of the BJT (DC biasing) with a
simple large-signal model. In the
active-linear state, this model is
simply: vBE = 0.7 V, iC = βiB .
This model is sufficient for calculating the Q point as we are only
interested in ensuring sufficient design space for the amplifier, i.e.,
Q point should be in the middle
of the load line in the active-linear
state. In fact, for our good biasing scheme with negative feedback,
the Q point location is independent
of BJT parameters (and, therefore,
independent of model used!).
iB
iC
vγ
vBE
vsat
vCE
A comparison of the simple largesignal model with the iv characteristics of the BJT shows that our
simple large-signal model is crude.
For example, the input AC signal results in small changes in vBE around 0.7 V (Q point) and
corresponding changes in iB . The simple model cannot be used to calculate these changes
(It assumes vBE is constant!). Also for a fixed iB , iC is not exactly constant as is assumed
in the simple model (see iC vs vCE graphs). As a whole, the simple large signal model is not
sufficient to describe the AC behavior of BJT amplifiers where more accurate representations
of the amplifier gain, input and output resistance, etc. are needed.
A more accurate, but still linear, model can be developed by assuming that the changes in
transistor voltages and currents due to the AC signal are small compared to corresponding
Q-point values and using a Taylor series expansion. Consider function f (x). Suppose we
know the value of the function and all of its derivative at some known point x0 . Then,
the value of the function in the neighborhood of x0 can be found from the Taylor Series
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
168
expansion as:
df (∆x)2 d2 f f (x0 + ∆x) = f (x0 ) + ∆x
+ ...
+
dx x=x0
2 dx2 x=x0
Close to our original point of x0 , ∆x is small and the high order terms of this expansion
(terms with (∆x)n , n = 2, 3, ...) usually become very small. Typically, we consider only the
first order term, i.e.,
df f (x0 + ∆x) ≈ f (x0 ) + ∆x
dx x=x0
The Taylor series expansion can be similarly applied to function of two or more variables
such as f (x, y):
∂f ∂f f (x0 + ∆x, y0 + ∆y) ≈ f (x0 , y0 ) + ∆x
+ ∆y
∂x x0 ,y0
∂y x0 ,y0
In a BJT, there are four parameters of interest: iB , iC , vBE , and vCE . The BJT iv characteristics plots, specify two of the above parameters, vBE and iC in terms of the other two,
iB and vCE , i.e., vBE is a function of iB and vCE (written as vBE (iB , vCE ) similar to f (x, y))
and iC is a function of iB and vCE , iC (iB , vCE ).
Let’s assume that BJT is biased and the Q point parameters are IB , IC , VBE and VCE . We
now apply a small AC signal to the BJT. This small AC signal changes vCE and iB by small
values around the Q point:
iB = IB + ∆iB
vCE = VCE + ∆vCE
The AC changes, ∆iB and ∆vCE results in AC changes in vBE and iC that can be found
from Taylor series expansion in the neighborhood of the Q point, similar to expansion of
f (x0 + ∆x, y0 + ∆y) above:
vBE (IB + ∆iB , VCE + ∆vCE ) = VBE
iC (IB + ∆iB , VCE
∂vBE ∂vBE +
∆iB +
∆vCE
∂iB Q
∂vCE Q
∂iC ∂iC + ∆vCE ) = IC +
∆iB +
∆vCE
∂iB Q
∂vCE Q
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
169
where all partial derivatives are calculated at the Q point and we have noted that at the Q
point, vBE (IB , VCE ) = VBE and iC (IB , VCE ) = IC . We denote the AC changes in vBE and iC
as ∆vBE and ∆iC , respectively:
vBE (IB + ∆iB , VCE + ∆vCE ) = VBE + ∆vBE
iC (IB + ∆iB , VCE + ∆vCE ) = IC + ∆iC
So, by applying a small AC signal, we have changed iB and vCE by small amounts, ∆iB and
∆vCE , and BJT has responded by changing , vBE and iC by small AC amounts, ∆vBE and
∆iC . From the above two sets of equations we can find the BJT response to AC signals:
∆vBE =
∂vBE
∂vBE
∆iB +
∆vCE ,
∂iB
∂vCE
∆iC =
∂iC
∂iC
∆iB +
∆vCE
∂iB
∂vCE
where the partial derivatives are the slope of the iv curves near the Q point. We define
hie ≡
∂vBE
,
∂iB
hre ≡
∂vBE
,
∂vCE
hf e ≡
∂iC
,
∂iB
hoe ≡
∂iC
∂vCE
Thus, response of BJT to small signals can be written as:
∆vBE = hie ∆iB + hre ∆vCE
∆iC = hf e ∆iB + hoe ∆vCE
which is our small-signal model for BJT.
We now need to relate the above analytical model to circuit elements so that we can solve
BJT circuits. Consider the expression for ∆vBE
∆vBE = hie ∆iB + hre ∆vCE
Each term on the right hand side should have units of Volts. Thus, hie should have units of
resistance and hre should have no units (these are consistent with the definitions of hie and
hre ). Furthermore, the above equation is like a KVL: the voltage drop between the base and
emitter (∆vBE ) is equal to the sum of voltage drops across two elements. The voltage drop
across the first element is hie ∆iB . So, it is a resistor with a value of hie . The voltage drop
across the second element is hre ∆vCE . Thus, it is a dependent voltage source.
B
∆i
Β +
+
∆v
E
V1 = hie ∆ iB
−
ΒΕ
−
+
V2 = hre ∆ v
CE
B
Β
E
h ie
+
∆v
−
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
∆i
ΒΕ
hre ∆v CE
+
−
−
170
Now consider the expression for ∆iC :
∆iC = hf e ∆iB + hoe ∆vCE
Each term on the right hand side should have units of Amperes. Thus, hf e should have no
units and hoe should have units of conductance (these are consistent with the definitions of
hoe and hf e .) Furthermore, the above equation is like a KCL: the collector current (∆iC )
is equal to the sum of two currents. The current in first element is hf e ∆iB . So, it is a
dependent current source. The current in the second element is proportional to hoe /∆vCE .
So it is a resistor with the value of 1/hoe .
∆i
C
+
i1 = h fe ∆ iB
∆v
i = h oe ∆v
2
CE
∆i
C
C
1/hoe
CE
−
C
+
hfe ∆ iB
∆v
CE
−
E
E
Now, if put the models for BE and CE terminals together we arrive at the small signal
“hybrid” model for BJT. It is similar to the hybrid model for a two-port network.
∆i
B
B
+
∆v
E
BE
_
∆i
h ie
hre ∆v CE
C
+
-
C
+
hfe ∆ iB
1/hoe
∆v
CE
-
E
The small-signal model is mathematically valid only for signals with small amplitudes. But
this model is so useful that is often used for signals with amplitudes approaching those of
Q-point parameters by using average values of “h” parameters. “h” parameters are given in
the manufacturer’s spec sheets for each BJT. It should not be surprising to note that even in
a given BJT, “h” parameter can vary substantially depending on manufacturing statistics,
operating temperature, etc. Manufacturer’s’ spec sheets list these “h” parameters and give
the minimum and maximum values. Traditionally, the geometric mean of the minimum and
maximum values are used as the average value in design (see the table below).
Since hf e = ∂iC /∂iB and BJT β = iC /iB , β is sometimes called hF E in manufacturers’ spec
sheets and has a value quite close to hf e . In most electronic text books, β, hF E and hf e are
used interchangeably.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
171
Typical hybrid parameters of a general-purpose 2N3904 NPN BJT
Minimum
1
0.5 × 10−4
100
1
25
10
rπ = hie (kΩ)
hre
β ≈ hf e
hoe (µS)
ro = 1/hoe (kΩ)
re = hie /hf e (Ω)
Maximum
10
8 × 10−4
400
40
1,000
25
Average*
3
2 × 10−4
200
6
150
15
* Geometric mean.
As hre is small, it is usually ignored in analytical calculations as it makes analysis much
simpler. This model, called the hybrid-π model, is most often used in analyzing BJT circuits.
In order to distinguish this model from the hybrid model, most electronic text books use a
different notation for various elements of the hybrid-π model:
rπ = hie
B
∆i
∆i
B
C
+
∆v
BE
ro =
C
1
hoe
β = hf e
B
∆i
∆i
B
C
+
hfe ∆ iB
h ie
1/hoe
_
∆v
BE
=⇒
β∆ i
rπ
B
C
ro
_
E
E
The above hybrid-π model includes a current-controlled current source. A variant of the
hybrid-π model can be developed which includes a voltage-controlled current source by noting
(∆vBE = rπ ∆iB :
β∆iB = β
∆vBE
= gm ∆vBE
rπ
β
Transfer conductance
rπ
rπ
1
=
Emitter resistance
re ≡
gm
β
gm ≡
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
B
∆i
∆i
B
C
+
∆v
BE
gm ∆ vBE
rπ
C
ro
_
E
172
6.5
FET Small Signal Model
Similar to the BJT, the simple large-signal model of FET (page 127) is sufficient for finding
the bias point; but we need to develop a more accurate model for analysis of AC signals.
The main issue is that the FET large signal model indicates that iD only depends on vGS
and is independent of vDS in the active state. In reality, iD increases slightly with vDS in
the active state.
We can develop a small signal model for FET in a manner similar to the procedure described
in detail for the BJT. The FET characteristics equations specify two of the FET parameters,
iG and iD , in terms of the other two, vGS and vDS . (Actually FET is simpler than BJT as
iG = 0 at all times.) As before, we write the FET parameters as a sum of DC bias value
and a small AC signal, e.g., iD = ID + ∆iD . Performing a Taylor series expansion, similar
to pages 169 and 170, we get:
iG (VGS + ∆vGS , VDS + ∆vDS ) = 0
iD (VGS
∂iD ∂iD ∆vGS +
∆vDS
+ ∆vGS , VDS + ∆vDS ) = iD (VGS , VDS ) +
∂vGS Q
∂vDS Q
Since iG (VGS +∆vGS , VDS +∆vDS ) = IG +∆iG and iD (VGS +∆vGS , VDS +∆vDS ) = ID +∆iD ,
we find the AC components to be:
∆iG = 0
and
Defining
gm ≡
∂iD
∂vGS
and
∂iD ∂iD ∆iD =
∆vGS +
∆vDS
∂vGS Q
∂vDS Q
ro ≡
∂iD
∂vDS
We get:
∆iG = 0
and
∆iD = gm ∆vGS + ro ∆vDS
This results in the hybrid-π model for
the FET as is shown. Note that the
FET hybrid-π model is similar to the BJT
hybrid-π model with rπ → ∞.
G
∆i = 0
∆i
G
D
D
gm ∆ vGS
+
∆v
GS
_
ro
S
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
173
6.6
BJT Amplifier Circuits
As we have developed different models for DC signals (simple large-signal model) and AC
signals (small-signal model), analysis of BJT circuits follows these steps:
DC biasing analysis: Assume all capacitors are open circuit. Analyze the transistor circuit
using the simple large signal mode as described in page 114.
AC analysis:
1) Kill all DC sources
2) Assume coupling capacitors are short circuit. The effect of these capacitors is to set a
lower cut-off frequency for the circuit. This is analyzed in the last step.
3) Inspect the circuit. If you identify the circuit as a prototype circuit, you can directly use
the formulas for that circuit. Otherwise go to step 4.
4) Replace the BJT with its small signal model.
5) Solve for voltage and current transfer functions and input and output impedances (nodevoltage method is the best).
6) Compute the cut-off frequency of the amplifier circuit.
Several standard BJT amplifier configurations are discussed below and are analyzed. For
completeness, circuits include standard bias resistors R1 and R2 . For bias configurations
that do not utilize these resistors (e.g., current mirror), simply set RB = R1 k R2 → ∞.
6.6.1
Common Collector Amplifier (Emitter Follower)
VCC
DC analysis: With the capacitors open circuit, this circuit is the
same as our good biasing circuit of page 162 with RC = 0. The
bias point currents and voltages can be found using procedure
of pages 162-164.
R1
vi
Cc
vo
AC analysis: To start the analysis, we kill all DC sources:
R2
RE
VCC = 0
R1
vi
Cc
vi
Cc
vo
R2
RE
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
C
B
vo
E
R1
R2
RE
174
We can combine R1 and R2 into RB (same resistance that we encountered in the biasing
analysis) and replace the BJT with its small signal model:
vi
Cc
B
∆i
∆i
B
C
+
∆v
RB
BE
β ∆i
rπ
B
Cc
rπ
B
E
vo
∆i
ro
RB
vo
E
_
vi
C
B
RE
ro
β ∆i
B
C
RE
The figure above shows why this is a common collector configuration: the collector is common
between the input and output AC signals. We can now proceed with the analysis. Node
voltage method is usually the best approach to solve these circuits. For example, the above
circuit has only one node equation for node at point E with a voltage vo :
vo − v i vo − 0
vo − 0
+
− β∆iB +
=0
rπ
ro
RE
Because of the controlled source, we need to write an “auxiliary” equation relating the control
current (∆iB ) to node voltages:
∆iB =
vi − v o
rπ
Substituting the expression for ∆iB in our node equation, multiplying both sides by rπ , and
collecting terms, we get:
vi (1 + β) = vo 1 + β + rπ
1
1
+
ro RE
= vo
"
rπ
1+β+
ro k R E
#
Amplifier Gain can now be directly calculated:
Av ≡
vo
=
vi
1+
1
rπ
(1 + β)(ro k RE )
Unless RE is very small (tens of Ω), the fraction in the denominator is quite small compared
to 1 and Av ≈ 1.
To find the input impedance, we calculate ii by KCL:
ii = i1 + ∆iB =
vi − v o
vi
+
RB
rπ
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
175
Since vo ≈ vi , we have ii = vi /RB or
Ri ≡
vi
= RB
ii
Note that RB is the combination of our biasing resistors R1 and R2 . With alternative biasing
schemes which do not require R1 and R2 (and, therefore, RB → ∞), the input resistance of
the emitter follower circuit will become large. In this case, we cannot use vo ≈ vi . Using the
full expression for vo from above, the input resistance of the emitter follower circuit becomes:
Ri ≡
vi
= RB k [rπ + (RE k ro )(1 + β)]
ii
which is quite large (hundreds of kΩ to several MΩ) for RB → ∞. Such a circuit is in fact
the first stage of the 741 OpAmp.
The output resistance of the common collector amplifier (in fact for all transistor amplifiers)
is somewhat complicated because the load can be configured in two ways (see figure): First,
RE , itself, is the load. This is the case when the common collector is used as a “current
amplifier” to raise the power level and to drive the load. The output resistance of the circuit
is Ro as is shown in the circuit model. This is usually the case when values of Ro and Ai
(current gain) is quoted in electronic text books.
VCC
VCC
R1
R1
Cc
vi
vi
Cc
vo
R2
vo
RE = RL
R2
RE is the Load
vi
Cc
rπ
B
B
vo
vi
Cc
rπ
B
E
∆i
β∆ i
B
ro
RL
Separate Load
E
∆i
RB
RE
RE
C
RB
B
vo
β∆ i
B
ro
RE
RL
C
Ro
R’o
Alternatively, the load can be placed in parallel to RE . This is done when the common
collector amplifier is used as a buffer (Av ≈ 1, Ri large). In this case, the output resistance
is denoted by Ro0 (see figure). For this circuit, BJT sees a resistance of RE k RL . Obviously,
if we want the load not to affect the emitter follower circuit, we should use RL to be much
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
176
larger than RE . In this case, little current flows in RL which is fine because we are using
this configuration as a buffer and not to amplify the current and power. As such, value of
Ro0 or Ai does not have much use.
Cc
vi
When RE is the load, the output resistance can
be found by killing the source (short vi ) and finding the Thevenin resistance of the two-terminal
network (using a test voltage source).
rπ
B
∆i
β∆ iB
B
RB
iT
E
+
−
ro
vT
C
Ro
vT
KCL:
iT = −∆iB +
− β∆iB
ro
KVL (outside loop):
− rπ ∆iB = vT
Substituting for ∆iB from the 2nd equation in the first and rearranging terms we get:
Ro ≡
(ro ) rπ
vT
=
iT
(1 + β)(ro ) + rπ
Since, (1 + β)(ro ) rπ , the expression for Ro simplifies to
Ro ≈
rπ
rπ
(ro ) rπ
=
≈
= re
(1 + β)(ro )
(1 + β)
β
As mentioned above, when RE is the load the common collector is used as a “current amplifier” to raise the current and power levels . This can be seen by checking the current gain
in this amplifier: io = vo /RE , ii ≈ vi /RB and
Ai ≡
io
RB
=
ii
RE
We can calculate Ro0 , the output resistance
when an additional load is attached to the circuit (i.e., RE is not the load) with a similar
procedure: we need to find the Thevenin resistance of the two-terminal network (using a
test voltage source).
We can use our previous results by noting that
we can replace ro and RE with ro0 = ro k RE
which results in a circuit similar to the case
with no RL . Therefore, Ro0 has a similar expression as Ro if we replace ro with ro0 :
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
vi
Cc
rπ
B
∆i
β∆ iB
B
RB
iT
E
ro
+
−
RE
vT
C
R’o
vi
Cc
rπ
B
∆i
RB
B
iT
E
β∆ iB
+
−
r’o
vT
C
R’o
177
Ro0
vT
(ro0 ) rπ
≡
=
iT
(1 + β)(ro0 ) + rπ
In most circuits, (1 + β)(ro0 ) rπ (unless we choose a small value for RE ) and Ro0 ≈ re
In summary, the general properties of the common collector amplifier (emitter follower)
include a voltage gain of unity (Av ≈ 1), a very large input resistance Ri ≈ RB (and can
be made much larger with alternate biasing schemes). This circuit can be used as buffer for
matching impedance, at the first stage of an amplifier to provide very large input resistance
(such in 741 OpAmp). The common collector amplifier can be also used as the last stage
of some amplifier system to amplify the current (and thus, power) and drive a load. In this
case, RE is the load, Ro is small: Ro = re and current gain can be substantial: Ai = RB /RE .
Impact of Coupling Capacitor:
Up to now, we have neglected the impact of the coupling capacitor in the circuit (assumed
it was a short circuit). This is not a correct assumption at low frequencies. The coupling
capacitor results in a lower cut-off frequency for the transistor amplifiers. In order to find the
cut-off frequency, we need to repeat the above analysis and include the coupling capacitor
impedance in the calculation. In most cases, however, the impact of the coupling capacitor
and the lower cut-off frequency can be deduced be examining the amplifier circuit model.
Consider our general model for any
amplifier circuit. If we assume that
coupling capacitor is short circuit
(similar to our AC analysis of BJT
amplifier), vi0 = vi .
Vi
Ro
Cc
+
−
+
V’i
−
Ri
+
−
Io
+
AVi
Vo
ZL
−
Voltage Amplifier Model
When we account for impedance of the capacitor, we have set up a high pass filter in the
input part of the circuit (combination of the coupling capacitor and the input resistance of
the amplifier). This combination introduces a lower cut-off frequency for our amplifier which
is the same as the cut-off frequency of the high-pass filter:
ωl = 2π fl =
1
Ri Cc
Lastly, our small signal model is a low-frequency model. As such, our analysis indicates
that the amplifier has no upper cut-off frequency (which is not true). At high frequencies,
the capacitance between BE , BC, CE layers become important and a high-frequency smallsignal model for BJT should be used for analysis. You will see these models in upper division
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
178
courses. Basically, these capacitances results in amplifier gain to drop at high frequencies.
PSpice includes a high-frequency model for BJT, so your simulation should show the upper
cut-off frequency for BJT amplifiers.
6.6.2
Common Emitter Amplifier
DC analysis: Recall that an emitter resistor is necessary to provide stability of the
bias point. As such, the circuit configuration as is shown has as a poor bias. We
need to include RE for good biasing (DC
signals) and eliminate it for the AC signals. The solution is to include an emitter
resistance and use a “bypass” capacitor to
short it out for AC signals as is shown.
VCC
RC
R1
vi
VCC
vo
Cc
RC
R1
vi
vo
Cc
R2
R2
Cb
RE
Good Bias using a
by−pass capacitor
Poor Bias
For this new circuit and with the capacitors open circuit, this circuit is the same as our
good biasing circuit of page 162. The bias point currents and voltages can be found using
procedure of pages 162-164.
AC analysis: To start the analysis, we kill all DC sources, short out Cb (which shorts out
RE ), combine R1 and R2 into RB , and replace the BJT with its small signal model. We
see that the emitter is now common between the input and output AC signals (thus, the
common emitter amplifier). Examination of the circuit shows that:
vi
vi = rπ ∆iB
vo = −(RC k ro ) β∆iB
vo
β
β
RC
= − (RC k ro ) ≈ − RC = −
vi
rπ
rπ
re
Ri = R B k r π
Cc
B
C
∆i
RB
B
rπ
β∆ iB
vo
RC
ro
Av ≡
E
Ro
R’o
The negative sign in Av indicates a 180◦ phase shift between the input and output signals.
This circuit has a large voltage gain but has a medium value for the input resistance.
As with the emitter follower circuit, the load can be configured in two ways: 1) RC is the
load; or 2) the load is placed in parallel to RC . The output resistance can be found by killing
the source (short vi ) and finding the Thevenin resistance of the two-terminal network. For
this circuit, we see that if vi = 0 (killing the source), ∆iB = 0. In this case, the strength of
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
179
the dependent current source would be zero and this element would become an open circuit.
Therefore,
Ro = r o
Ro0 = RC k ro
Lower cut-off frequency: Both the coupling and bypass capacitors contribute to setting
the lower cut-off frequency for this amplifier, both act as a high-pass filter with:
1
Ri Cc
1
ωl (bypass) = 2π fl = 0
RE Cb
ωl (coupling) = 2π fl =
0
where RE
≡ RE k re
0
Note that usually RE re and, therefore, RE
≈ re .
In the case when these two frequencies are far apart, the cut-off frequency of the amplifier
is set by the “larger” cut-off frequency. i.e.,
ωl (bypass) ωl (coupling)
→
ωl (coupling) ωl (bypass)
→
1
Ri Cc
1
ωl = 2π fl = 0
RE Cb
ωl = 2π fl =
When the two frequencies are close to each other, there is no exact analytical formulas, the
cut-off frequency should be found from simulations. An approximate formula for the cut-off
frequency (accurate within a factor of two and exact at the limits) is:
ωl = 2π fl ≈
1
1
+ 0
Ri Cc RE Cb
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
180
6.6.3
Common Emitter Amplifier with Emitter resistance
VCC
A problem with the common emitter amplifier is that its gain
depend on BJT parameters: Av ≈ (β/rπ )RC . Some form of
feedback is necessary to ensure stable gain for this amplifier.
One way to achieve this is to add an emitter resistance. Recall
impact of negative feedback on OpAmp circuits: we traded gain
for stability of the output. Same principles apply here.
R1
RC
Cc
vo
vi
DC analysis: With the capacitors open circuit, this circuit is the
RE
R2
same as our good biasing circuit of page 162. The bias point
currents and voltages can be found using procedure of pages
162-164.
AC analysis: To start the analysis, we kill all DC sources, combine R1 and R2 into RB and
replace the BJT with its small signal model. Analysis is straight forward using node-voltage
method.
vi
vE − v i
vE
vE − v o
+
− β∆iB +
=0
rπ
RE
ro
vo
vo − v E
+
+ β∆iB = 0
RC
ro
vi − v E
∆iB =
(Controlled source aux. Eq.)
rπ
C1
B
∆i
∆i
B
C
+
∆v
RB
BE
β∆ iB
rπ
_
C
vo
ro
E
RE
RC
Substituting for ∆iB in the node equations and noting 1 + β ≈ β, we get :
vE − v i vE − v o
vE
+β
+
=0
RE
rπ
ro
vo
vo − v E
vE − v i
+
−β
=0
RC
ro
rπ
Above are two equations in two unknowns (vE and vo ). Adding the two equation together
we get vE = −(RE /RC )vo and substituting that in either equations we can find vo . Using
rπ /β = re , we get:
Av =
vo
RC
RC
=
≈
vi
re (1 + RC /ro ) + RE (1 + re /ro )
re (1 + RC /ro ) + RE
where we have simplified the equation noting re ro . For most circuits, RC ro and
re RE . In this case, the voltage gain is simply Av = −RC /RE .
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
181
The input resistance of the circuit can be found from (prove it!)
Ri = R B k
vi
∆iB
Noting that ∆iB = (vi − vE )/rπ and vE = −(RE /RC )vo = −(RE /RC )Av vi , we get:
Ri = R B k
rπ
1 + Av RC /RE
Substituting for Av from above (complete expression for Av with re /ro 1), we get:
"
RE
+ re
Ri = R B k β
1 + RC /ro
!#
For most circuits, RC ro and re RE . In this case, the input resistance is simply
Ri = RB k (βRE ).
As before the minus sign in Av indicates a 180◦ phase shift between input and output
signals. Note the impact of negative feedback introduced by the emitter resistance: The
voltage gain is independent of BJT parameters and is set by RC and RE (recall OpAmp
inverting amplifier!). The input resistance is also increased dramatically.
B
As with the emitter follower circuit, the load can
be configured in two ways: 1) RC is the load. 2)
Load is placed in parallel to RC . The output resistance can be found by killing the source (short
vi ) and finding the Thevenin resistance of the
two-terminal network (by attaching a test voltage
source to the circuit).
iT = −∆iB − i1 = −∆iB 1 +
rπ
RE
vT = −∆iB rπ − i2 ro = −∆iB ro
iT
C
B
β∆ iB
rπ
i2
E
ro
+
−
Ro
B
∆i
iT
C
B
β∆ iB
rπ
i2
E
ro
+
−
RC
rπ
β+1+
RE
vT
RE
i1
Resistor RB drops out of the circuit because it is
shorted out. Resistors rπ and RE are in parallel.
Therefore, i1 = (rπ /RE )∆iB and by KCL, i2 =
(β + 1 + rπ /RE )∆iB . Then:
∆i
i1
+ rπ
vT
RE
R’o
Then:
Ro =
1 + ro /re
vT
= ro + RE ×
iT
1 + RE /rπ
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
182
where we have used rπ /β = re . Generally ro re (first approximation below) and for most
circuit, RE rπ (second approximation) leading to
RE /re
RE ro
RE
Ro ≈ r o + r o ×
≈ ro +
= ro
+1
1 + RE /rπ
re
re
Value of Ro0 can be found by a similar procedure. Alternatively, examination of the circuit
shows that
Ro0 = RC k Ro ≈ RC
Lower cut-off frequency: The coupling capacitor together with the input resistance of
the amplifier lead to a lower cut-off frequency for this amplifier (similar to emitter follower).
The lower cut-off frequency is given by:
ωl = 2π fl =
1
Ri Cc
A Possible Biasing Problem: The gain of the common
emitter amplifier with the emitter resistance is approximately
RC /RE . For cases when a high gain (gains larger than 5-10) is
needed, RE may be become so small that the necessary good
biasing condition, VE = RE IE > 1 V cannot be fulfilled. The
solution is to use a by-pass capacitor as is shown. The AC signal
sees an emitter resistance of RE1 while for DC signal the emitter
resistance is the larger value of RE = RE1 + RE2 . Obviously formulas for common emitter amplifier with emitter resistance can
be applied here by replacing RE with RE1 as in deriving the amplifier gain, and input and output impedances, we “short” the
bypass capacitor so RE2 is effectively removed from the circuit.
VCC
R1
RC
Cc
vo
vi
R2
R E1
R E2
Cb
The addition of by-pass capacitor, however, modifies the lower cut-off frequency of the circuit.
Similar to a regular common emitter amplifier with no emitter resistance, both the coupling
and bypass capacitors contribute to setting the lower cut-off frequency for this amplifier.
Similarly we find that an approximate formula for the cut-off frequency (accurate within a
factor of two and exact at the limits) is:
ωl = 2π fl =
1
1
+ 0
Ri Cc RE Cb
0
where RE
≡ RE2 k (RE1 + re )
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
183
6.6.4
Common Base Amplifier
VCC
By setting the signal ground at the base of the BJT, one arrives
at the common base amplifier (the input sginal is still applied
between the base and the emitter). While it is possible to bias
this configuration with a voltage divider self-bias, the preferred
method is to bias this amplifier with two power supplies (or a
current mirror). The bias point currents and voltages can be
found using procedure of pages 164-165.
RC
vo
vi
Cc
AC analysis: To start the analysis, we kill all DC sources and
replace the BJT with its small signal model. We see that base
is now common between the input and output AC signals (thus,
the common base amplifier).
∆i
B
B
β ∆ iB
rπ
vi
ro
=⇒
RC
vi
E
ii
E
∆i
RE
C
Cc
B
Cc
−VEE
β ∆ iB
vo
C
RE
rπ
RE
ro
vo
RC
B
Using node voltage method and noting ∆iB = −vi /rπ :
vo
vo − v i
+ β∆iB +
=0
RC
ro
1
β
1
1
+ vi − −
+
RC ro
rπ ro
vo
vo
1
β
≈ vi
RC k r o
rπ
Av ≡
!
=0
vo
β
β
RC
= (RC k ro ) ≈ RC =
vi
rπ
rπ
re
which is exactly the gain of the common emitter amplifier (with no emitter resistor) except
for the positive sign. This should not be surprising as compared to a common emitter, we
have switched the terminals of the input signal (leading to the change in the sign of Av ) and
the output voltage is vCB = vCE − vBE ≈ vCE because of the high gain of the amplifier.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
184
The input resistance of the circuit can be found by finding ii from the circuit above and
computing vi /ii to be
Ri =
rπ (ro + RC )
rπ
rπ
rπ (ro + RC )
≈
≈
≈
= re
rπ + RC + ro (1 + β)
ro (1 + β)
1+β
β
In the approximation, we first used the fact that rπ + RC ro (1 + β) and then RC ro .
Note that the input resistance is quite small.
As before, the load can be configured in two ways: 1) RC is the load; or 2) load is placed
in parallel to RC . The output resistance can be found by killing the source (short vi ) and
finding the Thevenin resistance of the two-terminal network. For this circuit, we see that
if vi = 0 (killing the source), ∆iB = 0. In this case, the strength of the dependent current
source would be zero and this element would become an open circuit. In addition, emitter
would be effectively grounded and resistors RE and rπ are effectively shorted out of the
circuit. Therefore,
Ro = r o
Ro0 = RC k ro ≈ RC
which are similar to the common amplifier with no emitter resistor.
As a whole, this circuit is similar to common emitter amplifier with no resistor (large voltage
gain, medium output resistance) but has a very low input resistance (re ). As such, it is
rarely used as a voltage amplifier (except for very specialized cases).
Following the formula in page 13, the short circuit current-gain of this amplifier is:
Ai =
ZI
re RC
Av =
=1
ZL + Z o
0 + R c re
Therefore, this circuit has a low input resistance, a medium output resistance and currentgain of unity and, therefore, is a “current buffer”: It accepts an input signal current with
a low input resistance and deliver nearly equal current to a much higher output resistance.
Common-base amplifiers are mostly used as a current buffer, typically forming circuits including two BJTs (cascode amplifier) which are utilized specially in integrated circuits.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
185
6.7
FET Amplifier Circuits
As expected, FET amplifiers are very similar to the BJT amplifiers. There are four basic
FET amplifiers: 1) common-drain or source follower (similar to common collector or emitter
follower), 2) common-source (similar to common emitter), 3) common source with a source
resistor (similar to common emitter with an emitter resistor) and common gate (similar to
common base).
The analysis technique are exactly the same: 1) DC-biasing analysis, and 2) AC analysis in
which we replace FET with its small signal model. In fact, by comparing the small signal
model for an FET that that of a BJT, we should be able to find the answer immediately by
replacing β/rπ = gm in the formulas of the equivalent BJT circuits and then let rπ → ∞ (and
of course, replace RC → RD , RE → RS , and RB = R1 k R2 → RG = R1 k R2 ). Therefore,
we will only solve the common-source amplifier in detail and summarize the results for the
other configurations.
6.7.1
Common Source Amplifier
DC analysis: Recall that a source resistor
is necessary to provide stability for the bias
point. As such, the circuit configuration as
is shown has a poor bias. We need to include RS for good biasing (DC signals) and
eliminate it for AC signals. The solution
is to include a source resistance and use a
“bypass” capacitor to short it out for AC
signals similar to the BJT common-emitter
amplifier.
VDD
R1
vi
VDD
R1
RD
vo
Cc
vi
RD
vo
Cc
R2
R2
Cb
RS
Good Bias using a
by−pass capacitor
Poor Bias
AC analysis: To start the analysis, we kill all DC sources, short out Cb (which shorts out
RS ), combine R1 and R2 into RG , and replace the FET with its small signal model. We see
that the source is now common between the input and output AC signals (thus, the common
source amplifier). Examination of the circuit shows that:
vi = ∆vGS
vo = −(RD k ro ) gm ∆vGS
vo
Av ≡
= −gm (RD k ro ) ≈ −gm RD
vi
Ri = R G
Ro = r o
Ro0 = RD k ro
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
vi
Cc
RG
G
∆i = 0
+
∆v
GS
_
vo
D
G
gm ∆ vGS
ro
RD
S
Ro
R’o
186
which are exactly the same as formulas for a BJT common emitter amplifier if we let β/rπ =
gm and rπ → ∞. Note that as an FET can be biased with large (MΩ) R1 and R2 (see
page 167), the input resistance of this amplifier is considerably larger than that of a common
emitter amplifier and can even be made to be infinitely large (resistance of the Gate insulator)
by removing RG and biasing the circuit with two voltage supplies or a current mirror.
Lower cut-off frequency: As Ri is very large, the lower cut-off frequency is set by the
bypass capacitor (unless Cc is chosen to be very small) .
ωl = ωl (bypass) = 2π fl =
where RS0 ≡ RS k
1
RS0 Cb
1
gm
Note that usually RS 1/gm and, therefore, RS0 ≈ 1/gm .
6.7.2
Common Source Amplifier with Source resistance
Similar to common-emitter amplifier, the common source amplifier gain depends on the FET parameters (gm ). Addition of
a source resistance will remove this dependency (similar to the
common emitter amplifier with an emitter resistor). Details of
the AC analysis is left as an exercise. The parameters of this
amplifier are:
RD
gm RD
≈−
Av = −
1 + g m RS
RS
Ri = R G
Ro = 1/gm k ro
1
ωl = 2π fl =
Ri Cc
VDD
R1
vi
RD
vo
Cc
R2
RS
Ro0 = RD k Ro ≈ RD
Similar to the common-emitter amplifier, the gain is set by RD and RS and is independent of
the FET parameters. The input resistance of the circuit is large (much larger than common
emitter amplifier because R1 and R2 can be large).
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
187
6.7.3
Common Drain Amplifier
VDD
This circuit is similar to the common-collector amplifier (or the
emitter follower). Details of the AC analysis is left as an exercise.
The parameters of this amplifier are:
Av =
gm ro RS
≈1
ro + (1 + gm ro )RS
R1
Cc
vi
vo
Ri = R G
Ro = 1/gm k ro
1
ωl = 2π fl =
Ri Cc
R2
Ro0 = RS k Ro ≈ RS
RS
Similar to the emitter follower, the source follower is a voltage buffer. It is superior to the
emitter follower because of its very large input resistance.
6.7.4
Common Gate Amplifier
VDD
This circuit is similar to the BJT common-base amplifier. Details of the AC analysis is left as an exercise. The parameters of
this amplifier are:
RD
vo
Av = −gm (RD k ro ) ≈ −gm RD
Ri =
RS (ro + RD )
RS (ro + RD )
ro
1
≈
≈
=
ro + RD + (1 + gm ro )RS
(1 + gm ro )RS
gm ro
gm
Ro0 = RD k ro ≈ RD
1
ωl = 2π fl =
Ri Cc
Ro = r o
vi
Cc
RS
−VSS
Note that in the approximation for Ri , we first used the fact that ro + RD (1 + gm ro )RS
and then RD ro .
Similar to the common-base amplifier, this is a poor voltage amplifier because of its low input
resistance but has a short-circuit current gain of unity, low input impedance, and medium
output impedance and can be used as a current buffer.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
188
Summary of Transistor Amplifiers?
VCC
Common Collector (Emitter Follower):
R1
(RE k ro )(1 + β)
≈1
Av =
rπ + (RE k ro )(1 + β)
Cc
vi
vo
Ri = RB k [rπ + (RE k ro )(1 + β)] ≈ RB
(ro ) rπ
rπ
Ro =
≈
= re
(1 + β)(ro ) + rπ
β
Ro0 =
rπ
(ro0 ) rπ
≈
(1 + β)(ro0 ) + rπ
β
R2
1
2π fl =
Ri Cc
where ro = ro k RC
RE
VCC
Common Emitter:
RC
R1
β
RC
β
(RC k ro ) ≈ − RC = −
rπ
rπ
re
Ri = R B k r π
Av = −
vo
Cc
vi
R2
Cb
RE
Ro0 = RC k ro ≈ RC
1
1
0
+ 0
where RE
≡ RE k re
2π fl =
Ri Cc RE Cb
Ro = r o
VCC
Common Emitter with Emitter Resistance:
Av = −
R1
RC
RC
RC
≈−
≈−
re (1 + RC /ro ) + RE
re + R E
RE
"
RE
+ re
Ri = R B k β
1 + RC /ro
!#
≈ RB k βRE ≈ RB
RE
RE /re
Ro ≈ r o + r o ×
≈ ro
+1
1 + RE /rπ
re
Ro0 = RC k Ro ≈ RC
and 2π fl =
RC
Cc
vo
vi
R2
RE
1
Ri Cc
VCC
RC
Common Base Amplifer:
β
RC
(RC k ro ) ≈
rπ
re
rπ (ro + RC )
≈ re
Ri =
rπ + RC + ro (1 + β)
vo
Av =
Ro = r o
Ro0 = RC k ro ≈ RC
vi
Cc
1
and 2π fl =
Ri Cc
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
RE
−VEE
189
VDD
Common Drain (Source Follower):
Av =
R1
gm ro RS
≈1
ro + (1 + gm ro )RS
Cc
vi
vo
Ri = R G
R2
Ro = 1/gm k ro
1
ωl = 2π fl =
Ri Cc
Ro0
RS
= RS k Ro ≈ RS
VDD
Common Source:
R1
Av = −gm (RD k ro ) ≈ −gm RD
RD
vo
Cc
vi
Ri = R G
Ro = r o
Ro0 = RD k ro
ωl = ωl (bypass) = 2π fl =
R2
1
RS0 Cb
where RS0 ≡ RS k
Cb
RS
1
gm
VDD
Common Source with Source Resistance:
RD
gm RD
≈−
Av = −
1 + g m RS
RS
Ri = R G
Ro = 1/gm k ro
1
ωl = 2π fl =
Ri Cc
R1
vi
R2
Ro0 = RD k Ro ≈ RD
Ro0 = RD k ro ≈ RD
1
ωl = 2π fl =
Ri Cc
Ro = r o
RS
VDD
RD
Av = −gm (RD k ro ) ≈ −gm RD
1
RS (ro + RD )
≈
ro + RD + (1 + gm ro )RS
gm
vo
Cc
Common Gate Amplifer:
Ri =
RD
vo
vi
Cc
RS
−VSS
?
If bias resistors are not present (e.g., bias with current mirror), let RB or RG → ∞ in the
“full” expression for Ri .
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
190
6.8
Exercise Problems
In circuit design, use 5% commercial resistor and capacitor values (1, 1.1, 1.2, 1.3, 1.5, 1.6,
1.8, 2, 2.2, 2.4, 2.7, 3., 3.3, 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1 × 10 n where n is
an integer). Use Si BJTs, with β = 200, βmin = 100, rπ = 5 kΩ, ro = 100 kΩ.
Problem 1. Show that this circuit is a stable biasing scheme.
Problem 2 to 5. Compute Io assuming identical transistors.
Problem 6 to 8: Find the bias point and AC amplifier parameters of these circuits (Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
V CC
I ref
Io
I ref
VCC
Io
Q3
Io
I ref
Q3
RC
RB
IC
Q1
Q2
Q1
Q2
Q2
−VSS
−VEE
Problem 1
Q1
Problem 2
−VEE
Problem 3
Problem 4
15 V
9V
34 k
Io
I ref
Q3
Q2
30k
4.7 µ F
vi
vo
0.47 µ F
Q1
vo
vi
18k
16 V
1k
22k
1k
vi
270
510nF
1.5k
vo
5.9 k
240
47 µ F
6.2k
510
−VSS
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9: Design a BJT amplifier with a gain of 4 and a lower cut-off frequency of 100 Hz.
The Q point parameters should be IC = 3 mA and VCE = 7.5 V.
Problem 10: Design a BJT amplifier with a gain of 10 and a lower cut-off frequency of
100 Hz. The Q point parameters should be IC = 3 mA and VCE = 7.5 V. A power supply of
15 V is available.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
191
Problem 11. Design a BJT amplifier with a gain of 5 and a lower cut-off frequency of 10 Hz,
powered by a 16 V supply. Set the Q-point parameters to be VCE = 10 V and Ic = 5 mA.
Problem 12. Consider the BJT circuit below with R1 = 47 kΩ, R2 = 39 kΩ, RE =
1.5 kΩ, RL = 50 kΩ, C1 = 100 nF, C2 = 0.47 µF, and VCC = 15 V. An input signal with
vi = cos(5000t) is applied to the circuit. Calculate expressions for voltages vB , vE , and vo
(include both AC and DC parts in the expression for each voltage). Manufacturers’ spec
sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ.
Problems 13 to 16: Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
Problems 17. Find the bias point and AC amplifier parameters of these circuits (Manufacturers’ spec sheets give K = 0.25 mA/V2 and Vt = 1 V, gm = 0.25 mA/V, and ro = 100 kΩ).
Problems 18. Find the bias point and AC amplifier parameters of these circuits (Manufacturers’ spec sheets give K = 0.20 mA/V2 and Vt = 3 V, gm = 0.2 mA/V, and ro = 100 kΩ).
Problem 19. Find the bias point and AC amplifier parameters of these circuits (Manufacturers’ spec sheets give K = 0.20 mA/V2 and Vt = 4 V, gm = 0.2 mA/V, and ro = 100 kΩ).
15 V
VCC
R
v
C
i
vB
vE
C2
RE
−9 V
2k
vo
vi
1
1
R2
39 k
18k
vo
Problem 12
510
4V
vi
110k
vo
1k
vo
vi
Cc
51k
18k
Problem 15
20 V
2k
1M
1k
vi
18 V
500k
vo
vi
Cc
1k
vo
1k
Problem 14
12 V
vo
Cc
1M
1k
0.47 µ F
22k
510
Problem 13
vi
vo
0.47 µ F
47 µ F
6.2 k
22k
vi
0.33 µ F
RL
9V
1k
Cb
1.3M
10k
−5 V
Problem 16
Problem 17
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
Problem 18
Problem 19
192
Problems 20 to 22: Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
15 V
33k
vi
2k
18k
Q1
0.47 µF
33k
Q2
vo
4.7 µF
6.2k
500
22k
18 V
15 V
1k
15k
2k
Q2
vi
4.7 µF
6.2k
Problem 20
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
vo
Q1
500
vi
1k
Problem 21
3.6k
1.5k
Q1
4.7 µF
2.7k
510
vo
Q2
510
Problem 22
193
6.9
Solution to Selected Exercise Problems
Problem 1. Show that this is a stable biasing scheme.
This is another stable biasing scheme which eliminates RB thereby, greatly reducing the
input resistance and increasing the value of the coupling capacitor (or lowering the cut-off
frequency). This scheme uses Rc as the feedback resistor.
We assume that the BJT is in the active-linear state. Since IB IC ,
by KCL I1 = IC + IC ≈ IC . Then:
BE-KVL:
VCC = RC IC + RB IB + VBE = (RC + RB /β) IC + VBE
IC =
VCC − VBE
RC + RB /β
VCC
RC
RB
I1
IC
If, RB /β RC or RB βRC , we will have (setting VBE = Vγ ):
IC =
VCC − Vγ
RC
Since IC is independent of β, the bias point is stable. We still need to prove that the BJT
is in the active-linear state. We write a KVL through BE and CE terminals:
VCE = RB IB + VBE = RB IB + Vγ > Vγ
Since VCE > Vγ , BJT is indeed in the active state.
To see the negative feedback effect, rewrite BE-KVL as:
IB =
VCC − Vγ − RC IC
RB
Suppose that the circuit is operating and BJT β is increased (e.g., an increase in the temperature). In this case IC will increase which raises the voltage across resistor RC (RC IC ).
From the above equation, this will lead to a reduction in IB which, in turn, will decrease
IC = βIB and compensate for any increase in β. If BJT β is decreased (e.g., a decrease
in the temperature), IC will decrease which reduces the voltage across resistor RC (RC IC ).
From the above equation, this will lead to an increase in IB which, in turn, will increase
IC = βIB and compensate for any decrease in β.
Note: The drawback of this bias scheme is that the allowable AC signal on VCE is small.
Since VCE ± ∆VCE > Vγ in order for the BJT to remain in active state, we find the amplitude
of AC signal, ∆VCE < RB IB = (RB /β)IC . Since, RB /β RC for bias stability thus,
∆VCE RC IC . This is in contrast with the standard biasing with emitter resistor in which
∆VCE is comparable to RC IC . Also, there is a feedback for the AC signals.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
194
Problem 2. Compute Io assuming identical transistors.
Because both bases and emitters of the transistors Q1 and Q2
are connected together, KVL leads to vBE1 = vBE2 . As BJT’s
are identical, they should have similar iB (iB1 = iB2 = iB )
and, therefore, similar iE = iE1 = iE2 and iC = iC1 = iC2 .
Using iC = βiB and iE = (β + 1)iB to illustrate the impact
of β:
iE
iB =
β+1
KCL:
iE3
iB3
KCL:
βiE
Io = i C =
β+1
I ref
Io
Q3
i B3
iC
Q1
iB
2i B
iB
iC
Q2
−VEE
2iE
= 2iB =
β+1
iE3
2iE
=
=
β+1
(β + 1)2
Iref = iC + iB3 =
V CC
2iE
βiE
+
β + 1 (β + 1)2
1
1
β
Io
=
≈
=
Iref
β + 2/(β + 1)
1 + 2/β(β + 1)
1 + 2/β 2
As can be seen, this is a better current mirror than our simple version as Io ≈ Iref with an
accuracy of 2/β 2 . Similar to our simple current-mirror circuit, Iref can be set by using a
resistor Rc .
Problem 3. Compute Io assuming identical transistors.
This is the MOS version of our simple current mirror. Because both gates and sources of the transistors Q1 and Q2
are connected together, KVL leads to vGS1 = vGS2 . The
drain of Q1 is connected to its gate: vDS1 = vGS1 . Therefore,
vDS1 = vGS1 > vGS1 − Vt , Q1 will be in the active state with
Iref = iD1 = K(vGS1 − Vt )2 . If Q2 is also in active state, then
Io = iD2 = K(vGS2 − Vt )2 . Since, vGS1 = vGS2 , then Io = Iref .
Io
I ref
Q1
Q2
−VSS
Note that as opposed to the BJT version, there is no 2/β effect here. However, a sufficient
voltage should be applied to Q2 to ensure that it is in the active state.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
195
Problem 4. Compute Io assuming identical transistors.
I ref
Because both bases and emitters of the transistors Q1 and Q2
are connected together, KVL leads to vBE1 = vBE2 . As BJT’s
are identical, they should have similar iB (iB1 = iB2 = iB )
and, therefore, similar iE = iE1 = iE2 and iC = iC1 = iC2 .
Using iC = βiB and iE = (β + 1)iB to illustrate the impact
of β:
iB =
KCL:
i B3
Q3
2i B
iC
iB
iB
Q1
i E3
iC
Q2
iE
β+1
−VEE
βiE
β+2
2iE
+
=
iE
β+1 β+1
β+1
iE3
β+2
=
=
iE
β+1
(β + 1)2
iE3 = 2iB + ic =
iB3
KCL:
Io
Iref = iC + iB3 =
Io = iC3 =
β+2
β(β + 1) + β + 2
βiE
+
iE =
iE
2
β + 1 (β + 1)
(β + 1)2
β(β + 2)
β
iE3 =
iE
β+1
(β + 1)2
Io
β(β + 2)
β(β + 2)
=
=
==
Iref
β(β + 1) + β + 2
β(β + 2) + 2
1
2
1+
β(β + 2)
≈
1
1 + 2/β 2
This circuit is called the Wilson current mirror after its inventor. It has a reduced β dependence compared to our simple current mirror and has a greater output impedance compared
to the current mirror of problem 2.
Problem 5. Compute Io assuming identical transistors.
Io
I ref
This is the MOS version of the Wilson current mirror. Solution is similar to those of Problems 3 and 4. The advantage of
this current mirror over the simple current mirror of Problem
3 is its much larger output resistance.
Q3
Q1
Q2
−VSS
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
196
Problem 6. Find the bias point and AC amplifier parameters of this circuit
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
9V
DC analysis:
Replace R1 and R2 with their Thevenin equivalent and
proceed with DC analysis (all DC current and voltages
are denoted by capital letters):
18k
vi
RB = 18 k k 22 k = 9.9 kΩ
22
VBB =
9 = 4.95 V
18 + 22
22k
KVL: VBB = RB IB + VBE + 103 IE
4.95 − 0.7 = IE
IE = 4 mA ≈ IC ,
IB =
9.9 × 103
+ 103
201
IB =
vo
0.47 µ F
IE
IE
=
1+β
201
1k
VCC
!
RB
VBB
IC
= 20 µA
β
RE
KVL: VCC = VCE + 103 IE
VCE = 9 − 103 × 4 × 10−3 = 5 V
DC Bias summary: IE ≈ IC = 4 mA,
IB = 20 µA,
VCE = 5 V
AC analysis: The circuit is a common collector amplifier. Using the formulas in page 189,
Av ≈ 1
Ri ≈ RB = 9.9 kΩ
rπ
Ro ≈
= 25 Ω
β
ωl
1
1
fl =
=
=
= 36 Hz
3
2π
2πRi Cc
2π × 9.9 × 10 × 0.47 × 10−6
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
197
Problem 7. Find the bias point and AC amplifier parameters of this circuit
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
DC analysis: Replace R1 and R2 with their Thevenin equivalent
and proceed with DC analysis (all DC current and voltages are
denoted by capital letters). Since all capacitors are replaced with
open circuit, the emitter resistance for DC analysis is 270+240 =
510 Ω.
15 V
34 k
vo
vi
4.7 µ F
RB = 5.9 k k 34 k = 5.0 kΩ
5.9
15 = 2.22 V
VBB =
5.9 + 34
270
5.9 k
5.0 × 103
+ 510
201
!
VCE = 15 − 1, 510 × 3 × 10
DC Bias: IE ≈ IC = 3 mA,
RC
RB
VBB
= 10.5 V
IB = 15 µA,
47 µ F
VCC
IC
IE = 3 mA ≈ IC ,
IB =
= 15 µA
β
KVL: VCC = 1000IC + VCE + 510IE
−3
240
IE
IE
=
IB =
1+β
201
KVL: VBB = RB IB + VBE + 510IE
2.22 − 0.7 = IE
1k
R E=
270 + 240 =
510
VCE = 10.5 V
AC analysis: The circuit is a common collector amplifier with an emitter resistance. Note
that the 240 Ω resistor is shorted out with the by-pass capacitor. It only enters the formula
for the lower cut-off frequency. Using the formulas in page 189 (with RE = 270 Ω) and
noting re = rπ /beta = 25 Ω:
Av =
1, 000
RC
=
= 3.70
RE
270
Ri ≈ RB = 5.0 kΩ
R o ≈ ro
RE
+ 1 = 1.2 M Ω
re
The lower cut-off frequency can be found from formula on page 183:
0
RE
= RE2 k (RE1 + re ) = 240 k (270 + 25) = 132 Ω
1
1
ωl
=
+
fl =
=
0
2π
2πRi Cc 2πRE
Cb
1
1
+
= 31.5 Hz
2π × 5, 000 × 4.7 × 10−6 2π × 132 × 47 × 10−6
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
198
Problem 8. Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
16 V
Because the forward bias voltage for BE junction, VBE = vγ , changes
as the temperature changes, the bias point changes slightly even in
the presence of the RE . Although this change is small, in some cases
a diode is added to the the voltage divider self-bias to compensate
for this small changes. Assuming that the BJT is in active state, the
base voltage has to be large enough to forward bias the BE junction
and, therefore, the diode would also be forward biased.
30k
vi
1.5k
vo
510nF
6.2k
510
We can find the Thevenin equivalent of our bias circuit
(see circuit) by noting:
R2
VBB = Voc =
(VCC − vγ ) + vγ = 2.74 + 0.83vγ (V)
R1 + R 2
RB = RT = R1 k R2 = 5.14 kΩ
R1
V CC
+
+
−
−
VBB = RB IB + VBE + 510IE
IE
+ vγ + 510IE
2.74 + 0.83vγ = 5.14 × 103
201
2.74 − 0.17vγ
IC
IE =
= 4.9 mA ≈ IC ,
IB =
= 24 µA
536
β
VCC
RC
RB
VBB
Note that the dependence of IE to vγ is reduced by a factor of 6 ı.e.,
IE now scales as 2.74 − 0.17vγ instead of 2.74 − vγ (the case with no
diode).
CE-KVL:
γ
R2
DC analysis:
BE-KVL:
V
RE
VCC = 1, 500IC + VCE + 103 IE
VCE = 16 − 2, 100 × 4.9 × 10−3 = 5.7 V
AC analysis: Since the diode is forward biased and can be represented by an independent
voltage source, it does not enter the AC analysis (because we short out the DC voltage
sources). As such, this is a common emitter amplifier with an emitter resistor. Using the
formulas in page 189:
Av =
1, 500
RC
=
= 2.94
RE
510
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
199
Ri ≈ RB = 5.1 kΩ
fl =
R o ≈ ro
RE
+ 1 = 2.14 M Ω
re
ωl
1
= 60.7 Hz
=
2π
2πRi Cc
Problem 9: Design a BJT amplifier with a gain of 4 and a lower cut-off frequency
of 100 Hz. The Q point parameters should be IC = 3 mA and VCE = 7.5 V.
VCC
The prototype of this circuit is a common emitter amplifier with an
emitter resistance. Using formulas of page 189
R1
RC
|Av | ≈
=4
RE
Cc
The lower cut-off frequency will set the value of Cc .
vo
vi
R2
We start with the DC bias: As VCC is not given, we need to
choose it. To set the Q-point in the middle of load line, set
VCC = 2VCE = 15 V. Then, noting IC ≈ IE ,:
15 − 7.5 = 3 × 10−3 (RC + RE )
→
→
4RE + RE = 2.5 kΩ
RC
RC + RE = 2.5 kΩ
Values of RC and RE can be found from the above equation
together with the AC gain of the amplifier, AV = RC /RE = 4:
→
RE
VCC
VCC = RC IC + VCE + RE IE
RC
=4
RE
RC
RB
VBB
RE
RE = 500 Ω, RC = 2. kΩ
Commercial values are RE = 510 Ω and RC = 2 kΩ. Use these commercial values for the
rest of analysis.
We need to check if VE > 1 V, the condition for good biasing. VE = RE IE = 510×3×10−3 =
1.5 > 1, it is OK (See next example for the case when VE is smaller than 1 V).
We now proceed to find RB and VBB . RB is found from good bias condition (and trying to
have RB as large as possible) and VBB from a KVL in BE loop:
RB (β + 1)RE
BE-KVL:
→
RB = 0.1(βmin + 1)RE = 0.1 × 101 × 510 = 5.1 kΩ
VBB = RB IB + VBE + RE IE
VBB = 5.1 × 103
3 × 10−3
+ 0.7 + 510 × 3 × 10−3 = 2.28 V
201
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
200
Bias resistors R1 and R2 are now found from RB and VBB :
R1 R2
= 5 kΩ
R1 + R 2
R2
2.28
=
=
= 0.152
R1 + R 2
15
RB = R 1 k R 2 =
VBB
VCC
R1 can be found by dividing the two equations: R1 = 33 kΩ. R2 is found from the equation
for VBB to be R2 = 5.9 kΩ. Commercial values are R1 = 33 kΩ and R2 = 6.2 kΩ.
Lastly, we have to find the value of the coupling capacitor:
ωl =
1
= 2π × 100
Ri Cc
Using Ri ≈ RB = 5.1 kΩ, we find Cc = 3 × 10−7 F or a commercial values of Cc = 300 nF.
So, are design values are: R1 = 33 kΩ, R2 = 6.2 kΩ, RE = 510 Ω, RC = 2 kΩ. and
Cc = 300 nF.
Problem 10: Design a BJT amplifier with a gain of 10 and a lower cut-off frequency of 100 Hz. The Q point parameters should be IC = 3 mA and VCE = 7.5 V.
A power supply of 15 V is available.
VCC
The prototype of this circuit is a common emitter amplifier with an
emitter resistance. Using formulas of page 184:
|Av | ≈
RC
= 10
RE
R1
RC
Cc
vo
vi
The lower cut-off frequency will set the value of Cc .
R2
RE
We start with the DC bias: As the power supply voltage is given,
we set VCC = 15 V. Then, noting IC ≈ IE ,:
VCC = RC IC + VCE + RE IE
15 − 7.5 = 3 × 10−3 (RC + RE )
→
RC + RE = 2.5 kΩ
Values of RC and RE can be found from the above equation together with the AC gain of
the amplifier AV = RC /RE = 10:
RC
= 10
RE
→
10RE + RE = 2.5 kΩ
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
→
RE = 227 Ω,
and RC = 2.27 kΩ
201
We need to check if VE > 1 V which is the condition for good biasing: VE = RE IE =
227 × 3 × 10−3 = 0.69 < 1. Therefore, we need to use a bypass capacitor and modify our
circuits as is shown.
VCC
For DC analysis, the emitter resistance is RE1 + RE2 while for
AC analysis, the emitter resistance will be RE1 . Therefore:
DC Bias:
AC gain:
RC + RE1 + RE2 = 2.5 kΩ
RC
Av =
= 10
RE1
R1
Cc
vo
vi
R E1
R2
R E2
Above are two equations in three unknowns. A third equation is
derived by setting VE = 1 V to minimize the value of RE1 + RE2 .
Cb
VCC
VE = (RE1 + RE2 )IE
1
RE1 + RE2 =
= 333
3 × 10−3
Now, solving for RC , RE1 , and RE2 , we find RC = 2.2 kΩ,
RE1 = 220 Ω, and RE2 = 110 Ω (All commercial values).
RC
RC
RB
VBB
R E1+ R E2
We can now proceed to find RB and VBB :
RB (β + 1)(RE1 + RE2 )
RB = 0.1(βmin + 1)(RE1 + RE2 ) = 0.1 × 101 × 330 = 3.3 kΩ
KVL:
VBB = RB IB + VBE + RE IE
VBB = 3.3 × 10
33
× 10−3
+ 0.7 + 330 × 3 × 10−3 = 1.7 V
201
Bias resistors R1 and R2 are now found from RB and VB B:
R1 R2
= 3.3 kΩ
R1 + R 2
R2
1
=
=
= 0.066
R1 + R 2
15
RB = R 1 k R 2 =
VBB
VCC
R1 can be found by dividing the two equations: R1 = 50 kΩ and R2 is found from the
equation for VBB to be R2 = 3.6k Ω. Commercial values are R1 = 51 kΩ and R2 = 3.6k Ω
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
202
Lastly, we have to find the value of the coupling and bypass capacitors:
0
RE
= RE2 k (RE1 + re ) = 110 k (220 + 25) = 76 Ω
Ri ≈ RB = 3.3 kΩ
1
1
ωl =
+ 0
= 2π × 100
Ri Cc RE Cb
This is one equation in two unknown (Cc and CB ) so one can be chosen freely. Typically
0
Cb Cc as Ri ≈ RB RE RE
. This means that unless we choose Cc to be very small,
the cut-off frequency is set by the bypass capacitor. The usual approach is the choose C b
based on the cut-off frequency of the amplifier and choose Cc such that cut-off frequency of
the Ri Cc filter is at least a factor of ten lower than that of the bypass capacitor. Note that
in this case, our formula for the cut-off frequency is quite accurate (see discussion in page
179) and is
ωl ≈
1
0
RE
Cb
= 2π × 100
This gives Cb = 20 µF. Then, setting
1
1
0
Ri Cc
RE Cb
1
1
= 0.1 0
Ri Cc
RE Cb
0
Ri Cc = 10RE
Cb
→
Cc = 4.7 × 10−6 = 4.7 µF
So, are design values are: R1 = 50 kΩ, R2 = 3.6 kΩ, RE1 = 220 Ω, RE2 = 110 Ω, RC =
2.2 kΩ, Cb = 20 µF, and Cc = 4.7 µF.
An alternative approach is to choose Cb (or Cc ) and compute the value of the other from
the formula for the cut-off frequency. For example, if we choose Cb = 47 µF, we find
Cc = 0.86 µF.
Problem 11. Design a BJT amplifier with a gain of 5 and a lower cut-off frequency
of 10 Hz, powered by a 16 V supply. Set the Q-point parameters to be VCE = 10 V
and Ic = 5 mA.
Answer: A common-emitter amplifier with R1 = 18 kΩ, R2 = 2.2 kΩ, RE = 200 Ω,
RC = 1.0 kΩ. and Cc = 10 µF.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
203
Problem 12. Consider the BJT circuit below with R1 = 47 kΩ, R2 = 39 kΩ,
RE = 1.5 kΩ, RL = 50 kΩ, C1 = 100 nF, C2 = 0.47 µF, and VCC = 15 V. An input
signal with vi = cos(5000t) is applied to the circuit. Calculate the expressions for
voltages vB , vE , and vo . (Include both AC and DC parts in the expression for
each voltage.)
The voltages at Base and Emitter will be the sum of DC
and AC signals, e.g., vB = VB + ∆vB . First we calculate
the DC voltages, VB and VE . Replacing R1 and R2 with
their Thevenin equivalent, we have:
47 × 103 × 39 × 103
RB = R 1 k R 2 =
= 21.3 kΩ
47 × 103 + 39 × 103
R2
VCC = 6.80 V
VBB =
R1 + R 2
KVL: VBB = RB IB + VBE + RE IE
VBB − VBE = [RB + RE (β + 1)]IB
6.80 − 0.7
IB =
= 18.9 µA
21.3 × 103 + 1.5 × 103 × 201
IC ≈ IE = βIB = 3.78 mA
VE = RE IE = 1, 500 × 3.78 × 10
−3
VCC
R
v
C
i
1
1
vB
R2
vE
RE
C2
vo
RL
VCC
RB
VBB
RE
= 5.67 V
VB = VE + VBE = 5.67 + 0.7 = 6.37 V
AC voltages: The circuit is a voltage follower. But, we have to check to see if capacitors
affect the signal. The frequency of the input signal is 5000/(2π) = 796 Hz. The impact of
C1 coupling capacitor is to set a lower cut-off frequency for the amplifier.
Ri ≈ RB = 21.3 kΩ
1
→ fl = 75 Hz 796 Hz
2π fl =
Ri Cc
Thus: ∆vB = vi → vB = VB + ∆vB = 6.37 + cos(5000t)
Av ≈ 1
→
∆vE = ∆vB = vi
→
vE = VE + ∆vE = 5.67 + cos(5000t)
Capacitor C2 and resistor RL act as a high-pass filter. They separate the DC voltage. To
consider their impact on the AC signal, note:
1
→ fl = 6.8 Hz 796 Hz
RL C2
Thus: ∆vo = ∆vE = ∆vB = vi → vo = 0 + ∆vo = cos(5000t)
2π fl =
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
204
Problem 13. Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
15 V
39 k
15 V
2k
39 k
vo
vi
2k
39 k
vo
vi
0.33 µ F
0.33 µ F
15 V
2k
vo
vi
0.33 µ F
47 µ F
6.2 k
510
510
6.2 k
510
510 || 510
= 255
6.2 k
DC Response
AC signals
For DC signals, capacitors are open circuit and the emitter resistor is only 510 Ω. For AC
signals, capacitors are short circuit and the emitter resistor is 510 k 510 = 255 Ω.
DC analysis:
6.2 k
× 15 = 2.06 V
6.2 k + 39 k
IC
+ 0.7 + 510IC
VBB = RB IB + VBE + IE RE = 5.35 × 103
200
IC
IC ≈ IE = 2.53 mA → IB =
= 12.6 µA
β
RB = 6.2 k 39 = 5.35 kΩ
VCC = RC IC + VCE + RE IE
VBB =
→ VCE = 15 − 2.53 × 10−3 (2, 510) = 8.65 V
So Q point values are: IC ≈ IE = 2.53 mA, IB = 12.6 µA, and VCE = 8.65 V.
AC analysis: This is common emitter amplifier with emitter resistance:
RC
2, 000
=
= 7.8
RE
255
Ri ≈ RB = 4.8 k
RE
+ 1 = 1.1 MΩ
Ro ≈ r o
re
Av ≈
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
205
Problems 14 & 15. Find the bias point and AC amplifier parameters of these
circuits (Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
Both of these circuits are PNP versions of problem 6. For DC bias we should get the
same value for currents and voltages would be negative: IC = IE = 4 mA, IB = 20 µA,
VCE = −5 V, and VBE = −0.7 V. The amplifier parameters are exactly identical to those of
problem 6.
Problem 16. Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
This circuit is also similar to the circuit of problem 6.
Here BJT is biased with two voltage sources (Note that
in problem 6, VBB ≈ 5 V and here −VEE = −5 and
VCC = 9 − 5 = 4 V. As such, the Q-point parameters
should be the same.
DC Analysis: We short vi and,therefore, the BJT base would be grounded:
BE-KVL:
CE-KVL:
0 = VBE + 103 IE − 5
IC
IB =
= 20 µA
β
→
4V
vi
vo
1k
−5 V
4V
vo
IC ≈ IE = 4.3 mA
4 = VCE + 103 IE − 5
1k
−5 V
VCE = 9 − 103 × 4.3 × 10−3 = 4.7 V
DC Bias summary: IE ≈ IC = 4 mA, IB = 20 µA, and VCE = 5 V.
AC analysis: The circuit is a common collector amplifier. Using the formulas in page 189,
Av ≈ 1
Ri = rπ + (RE k ro )(1 + β) ≈ rπ + ro (1 + β) = 20 MΩ
rπ
Ro ≈
= 25 Ω
β
1
1
ωl
=
=
fl =
= 0.034 Hz
7
2π
2πRi Cc
2π × 2 × 10 × 0.47 × 10−6
Note that because there are no bias resistors (RB → ∞), we have used the full formulas for
Ri and the amplifier has a large input resistance.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
206
Problem 17. Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give K = 0.25 mA/V2 , Vt = 1 V, gm = 0.25 mA/V,
and ro = 100 kΩ).
DC Bias: Replacing the bias circuit with its Thevenin equivalent, we get:
VGG =
12 V
51, 000
× 12 = 3.80 V
51, 000 + 110, 000
110k
vo
vi
RG = 51 k k 110 k = 34.8 kΩ
Cc
1k
51k
Since iG = 0,
GS-KVL:
2k
3.8 = 34, 800iG + VGS + 1, 000iD = VGS + 1, 000iD
12 V
2k
Assume NMOS is in active state,
vo
34.8k
iD = K(VGS − Vt )2 = 0.25 × 10−3 (VGS − 1)2
+
−
Substituting for iD in GS-KVL, we get:
3.8 V
1k
2
3.8 = VGS + 0.25(VGS − 1)2 = 0.25VGS
+ 0.5VGS + 0.25
VGS = 2.9 V and VGS = −4.9 V
Negative root is unphysical, so VGS = 2.9 and iD = 0.9 mA. Then,
DS-KVL:
12 = 2, 000iD + VDS + 1, 000iD = VDS + 2.7
→
VDS = 9.3 V
As VDS = 9.3 > VGS − Vt = 2.9 − 1 = 1.95, our assumption of NMOS in active state is
correct. Therefore, Bias Summary: VGS = 2.9 V, VDS = 9.3 V, and iD = 0.9 mA.
AC Analysis: This is a common source amplifier with a source resistor. Using formulas in
page 190:
gm RD
0.25 × 10−3 × 2 × 103
0.5
=−
=−
= −0.4
−3
3
1 + g m RS
1 + 0.25 × 10 × 10
1.25
Ri = RG = 34.8 kΩ
1
k 100 × 103 ≈ 4 kΩ
Ro = 1/gm k ro =
0.25 × 10−3
1
ωl = 2π fl =
Ri Cc
Av ≈ −
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
207
Problem 18. Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give K = 0.20 mA/V2 and Vt = 3 V, gm = 0.2 mA/V,
and ro = 100 kΩ).
DC Bias:
20 V
Replacing the bias circuit with its Thevenin equivalent,
we have RG = 500 kΩ and VGG = 10V:
GS-KVL:
DS-KVL:
1M
Cc
3
20 = vDS + 10 iD
1M
Assume NMOS in active state: iD = K(vGS − Vt )2 and
vDS > vGS − Vt . Substituting for iD in GS-KVL, we get:
GS-KVL:
−3
10 = vGS + 10 × 0.2 × 10 (vGS − 3)
1k
Cb
20 V
2
1k
2
10 = vGS + 0.2vGS
− 1.2vGS + 1.8
2
vGS
− vGS − 41 = 0
→
vo
vi
10 = vGS + 103 iD
3
1k
vGS = −5.92 V and vGS = 6.92 V
vo
500k
+
−
10 V
1k
Negative root is unphysical so vGS = 6.92 V.
GS-KVL give iD = 3.08 mA. DS-KVL gives vDS = 20 − 6.16 = 13.8 V Since vDS = 13.8 >
vGS − Vt = 6.92 − 3 = 3.92 V, our assumption of NMOS in active state is justified. Bias
summary: vGS = 6.92 V, vDS = 13.8 V, and iD = 3.08 mA
AC Analysis: This is a common source amplifier with NO source resistor. Using formulas in
page 190:
Av ≈ −gm RD = −0.2 × 10−3 × 103 = −0.2
Ri = RG = 500 kΩ
Ro = ro = 100 kΩ
Note: Problems 17 & 18 show some fundamental differences between FET and BJT amplifiers. BJTs have a much larger gain compared to FET (compare gm = 20 − 40 mA/V for a
typical BJT with gm = 0.2 − 05mA/V for an NMOS). Therefore, typically RD and RS are a
factor of 10 or more larger than typical RC and RE values.
In addition, BJTs are more “linear” as iC = βiB and β does not vary considerably, while in
a MOSFET, iD = (vGS − Vt )2 so FET response is quadratic instead of linear. Because of
the more linear behavior and the higher gain, BJTs are used most often in amplifier circuits.
A first-stage FET source follower is also used to increase the input resistance of the overall
circuit considerably.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
208
Problem 19. Find the bias point and AC amplifier parameters of these circuits
(Manufacturers’ spec sheets give K = 0.20 mA/V2 and Vt = 4 V, gm = 0.2 mA/V,
and ro = 100 kΩ).
18 V
DC Bias:
Replacing the bias circuit with its Thevenin equivalent,
we have RG = 360 kΩ and VGG = 5V:
GS-KVL:
13 = vGS + 104 iD
DS-KVL:
18 = vDS + 104 iD
500k
vi
3
−3
5 = vGS + 50 × 10 × 0.2 × 10 (vGS − 4)
5 = vGS +
2
10vGS
10k
1.3M
Assume NMOS in active state: iD = K(vGS − Vt )2 and
vDS > vGS − Vt . Substituting for iD in GS-KVL, we get:
GS-KVL:
vo
Cc
− 80vGS + 160
2
18 V
361k
+
−
13 V
vo
10k
2
10vGS
− 81vGS + 155 = 0
→
vGS = 3.1 V and vGS = 5 V
Since VGS = 3.1 < Vt = 4 V required for NMOS On, this root is unphysical so vGS = 5 V.
GS-KVL give iD = 0.2 mA. DS-KVL gives vDS = 18 − 10 = 8 V Since vDS = 8 > vGS − Vt =
5 − 4 = 1 V, our assumption of NMOS in active state is justified. Bias summary: vGS = 5 V,
vDS = 8 V, and iD = 0.2 mA
AC Analysis: This is a common drain amplifier (or source follower). Using formulas in page
190:
Av =
gm ro RS
0.2 × 10−3 × 100 × 103 × 50 × 103
=
= 0.87
ro + (1 + gm ro )RS
100 × 103 + (1 + 0.2−3 × 100 × 103 ) × 50 × 103
Ri = RG = 360 kΩ
Ro = 1/gm k ro ≈ 5 kΩ
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
209
Problem 20: Find the bias point and AC amplifier parameters of this circuit
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
15 V
This is a two-stage amplifier. The first stage (Q1)
is a common emitter amplifier and the second
stage (Q2) is an emitter follower. The two stages
are coupled by a coupling capacitor (0.47 µF).
33k
vi
2k
18k
Q1
0.47 µF
Q2
vo
4.7 µF
DC analysis:
6.2k
When we replace the coupling capacitors with
open circuits, we see the that bias circuits for
the two transistors are independent of each other.
Each bias circuit can be solved separately.
500
22k
1k
For Q1, we replace the bias resistors (6.2k and 33k) with their Thevenin equivalent and
proceed with DC analysis:
RB1 = 6.2 k k 33 k = 5.22 kΩ
and
VBB1 =
BE-KVL: VBB1 = RB1 IB1 + VBE1 + 103 IE1
2.37 − 0.7 = IE1
IE1 = 3.17 mA ≈ IC1 ,
5.22 × 103
+ 500
201
IB1 =
IB1
6.2
15 = 2.37 V
6.2 + 33
IE1
IE1
=
=
1+β
201
!
IC1
= 16 µA
β
CE-KVL: VCC = 2 × 103 IC1 + VCE1 + 500IE1
VCE1 = 15 − 2.5 × 103 × 3.17 × 10−3 = 7.1 V
DC Bias summary for Q1: IE1 ≈ IC1 = 3.17 mA,
IB1 = 16 µA,
VCE1 = 7.1 V
Following similar procedure for Q2, we get:
RB2 = 18 k k 22 k = 9.9 kΩ
and
BE-KVL: VBB2 = RB2 IB2 + VBE2 + 103 IE2
8.25 − 0.7 = IE2
IE2 = 7.2 mA ≈ IC2 ,
9.9 × 103
+ 103
201
IB2 =
22
15 = 8.25 V
18 + 22
IE2
IE2
IB2 =
=
1+β
201
VBB2 =
!
IC2
= 36 µA
β
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
210
CE-KVL: VCC = VCE2 + 103 IE2
VCE2 = 15 − 103 × 7.2 × 10−3 = 7.8 V
DC Bias summary for Q2: IE2 ≈ IC2 = 7.2 mA,
IB2 = 36 µA,
VCE2 = 7.8 V
AC analysis:
We start with the emitter follower circuit (Q2) as the input resistance of this circuit will
appear as the load for the common emitter amplifier (Q1). Using the formulas in page 189:
Av2 ≈ 1
Ri2 ≈ RB2 = 9.9 kΩ
1
ωl2
1
=
fl2 =
=
= 34 Hz
3
2π
2πRB2 Cc2
2π × 9.9 × 10 × 0.47 × 10−6
Since Ri2 = 9.9 kΩ is NOT much larger than the collector resistor of common emitter
amplifier (Q1), it will affect the first circuit. Following discussion in pages 176 and 177, the
effect of this load can be taken into by replacing RC in common emitter amplifiers formulas
with RC0 = RC k RL = RC1 k Ri2 = 2 k k 9.9 kΩ = 1.66 kΩ.
1.66k
RC0
=
= 3.3
|Av1 | ≈
RE
500
Ri1 ≈ RB1 = 5.22 kΩ
1
1
ωl1
=
=
= 6.5 Hz
fl1 =
2π
2πRB1 Cc1
2π × 5.22 × 103 × 4.7 × 10−6
The overall gain of the two-stage amplifier is then Av = Av1 ×Av2 = 3.3. The input resistance
of the two-stage amplifier is the input resistance of the first-stage (Q1), Ri = 9.9 kΩ. To
find the lower cut-off frequency of the two-stage amplifier, we note that:
Av1 (jω) =
Av1
1 − jωl1 /ω
and
Av (jω) = Av1 (jω) × Av2 (jω) =
Av2 (jω) =
Av2
1 − jωl2 /ω
Av1 Av2
(1 − jωl1 /ω)(1 − jωl2 /ω)
From above, it is clear that the maximum value of Av (jω) is Av1 Av2 and the cut-off frequency,
√
ωl can be found from |Av (jω = ωl )| = Av1 Av2 / 2 (similar to procedure we used for filters).
For the circuit above, since ωl2 ωl1 the lower cut-off frequency would be very close to ωl2 .
So, the lower-cut-off frequency of this amplifier is 34 Hz.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
211
Problem 21: Find the bias point and AC amplifier parameters of this circuit
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
15 V
This is a two-stage amplifier. The first stage (Q1) is
a common emitter amplifier and the second stage (Q2)
is an emitter follower. The circuit is similar to the twostage amplifier of Problem 20. The only difference is that
Q2 is directly biased from Q1 and there is no coupling
capacitor between the two stages. This approach has its
own advantages and disadvantages that are discussed at
the end of this problem.
33k
I1
2k
vi
4.7 µF
I B2
I C1
Q1
6.2k
500
Q2
V B2
vo
1k
DC analysis:
Since the base current in BJTs are typically much smaller that the collector current, we start
by assuming IC1 IB2 . In this case, I1 = IC1 + IB2 ≈ IC1 ≈ IE1 (the bias current IB2 has
no effect on bias parameters of Q1). This assumption simplifies the analysis considerably
and we will check the validity of this assumption later.
For Q1, we replace the bias resistors (6.2k and 33k) with their Thevenin equivalent and
proceed with DC analysis:
RB1 = 6.2 k k 33 k = 5.22 kΩ
and
VBB1 =
BE-KVL: VBB1 = RB1 IB1 + VBE1 + 103 IE1
2.37 − 0.7 = IE1
IE1 = 3.17 mA ≈ IC1 ,
5.22 × 103
+ 500
201
IB1 =
IB1
6.2
15 = 2.37 V
6.2 + 33
IE1
IE1
=
=
1+β
201
!
IC1
= 16 µA
β
CE-KVL: VCC = 2 × 103 IC1 + VCE1 + 500IE1
VCE1 = 15 − 2.5 × 103 × 3.17 × 10−3 = 7.1 V
DC Bias summary for Q1: IE1 ≈ IC1 = 3.17 mA,
IB1 = 16 µA,
VCE1 = 7.1 V
To find the bias point of Q2, we note:
VB2 = VCE1 + 500 × IE1 = 7.1 + 500 × 3.17 × 10−3 = 8.68 V
BE-KVL: VB2 = VBE2 + 103 IE2
8.68 − 0.7 = 103 IE2
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
212
IE2 = 8.0 mA ≈ IC2 ,
IB2 =
IC2
= 40 µA
β
KVL: VCC = VCE2 + 103 IE2
VCE2 = 15 − 103 × 8.0 × 10−3 = 7.0 V
DC Bias summary for Q2: IE2 ≈ IC2 = 8.0 mA,
IB2 = 40 µA,
VCE2 = 7.0 V
We now check our assumption of IC1 IB2 . We find IC1 = 3.17 mA IB2 = 41 µA. So,
our assumption was justified.
It should be noted that this bias arrangement is also stable to variation in transistor β. The
bias resistors in the first stage will ensure that IC1 (≈ IE1 ) and VCE1 is stable to variation
of Q1 β. Since VB2 = VCE1 + RE1 × IE1 , VB2 will also be stable to variation in transistor β.
Finally, VB2 = VBE2 + RE2 IE2 . Thus, IC2 (≈ IE2 ) will also be stable (and VCE2 because of
CE-KVL).
AC analysis:
As in problem 20, we start with the emitter follower circuit (Q2) as the input resistance
of this circuit will appear as the load for the common emitter amplifier (Q1). Using the
formulas in page 189 and noting that this amplifier does not have bias resistors (RB1 → ∞):
Av2 ≈ 1
Ri2 = rπ + (RE k ro )(1 + β) = 5 × 103 + 201 × 103 = 201 kΩ
Note that because of the absence of the bias resistors, the input resistance of the circuit is
very large, and because of the absence of the coupling capacitors, there is no lower cut-off
frequency for this stage.
Since Ri2 = 201 kΩ is much larger than the collector resistor of common emitter amplifier
(Q1), it will NOT affect the first circuit. The parameters of the first-stage common emitter
amplifier can be found using formulas of page 189.
RC
2, 000
=
=4
RE
500
Ri1 ≈ RB1 = 5.22 kΩ
ωl1
1
1
fl1 =
=
=
= 6.5 Hz
2π
2πRB1 Cc1
2π × 5.22 × 103 × 4.7 × 10−6
|Av1 | ≈
The overall gain of the two-stage amplifier is then Av = Av1 × Av2 = 4. The input resistance
of the two-stage amplifier is the input resistance of the first-stage (Q1), Ri = 9.9 kΩ. The
lower cut-off frequency of the two-stage amplifier is 6.5 Hz.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
213
This two-stage amplifier has many advantages over that of problem 20. It has three less
elements. Because of the absence of bias resistors, the second-stage does not load the first
stage and the overall gain is higher. Also because of the absence of a coupling capacitor
between the two-stages, the overall cut-off frequency of the circuit is lower. Some of these
issues can be resolved by design, e.g., use a large capacitor for coupling the two stages, use
a large RE2 , etc.. The drawback of this circuit is that the bias circuit is more complicated
and harder to design.
Problem 22: Find the bias point and AC amplifier parameters of this circuit
(Manufacturers’ spec sheets give: β = 200, rπ = 5 kΩ, ro = 100 kΩ).
We start with replacing 2.7 k and 15 kΩ voltage divider with its Thevenin equivalent (as seen in circuit
below)
RB = 2.7 k 15 = 2.29 kΩ
VBB =
18 V
15k
vi
3.6k
Q1
4.7 µF
2.7 k
× 18 = 2.75 V
2.7 k + 15 k
2.7k
510
I1
I C1
I B2
V B2
1.5k
vo
Q2
510
Writing a KVL through BE terminals of Q1 and assuming that
Q1 is in the active-linear state (IC1 ≈ IE1 = βIB1 ), we get:
IC1
+ 0.7 + 510IC1
100
= IC1 /β = 38.5 µA
VBB = RB IB1 + VBE1 + 510IE1 = 2.29 × 103
IC1 ≈ IE1 = 3.85 mA
CE1-KVL:
KCL:
→
IB1
18 = 3.6 × 103 I1 + VCE1 + 510IE1
I1 = IC1 + IB2
We assume IB2 IC1 . Then, from KCL above, I1 ≈ IC1 = 3.85 mA. Substituting for I1
and IE1 in CE1-KVL, we find VCE1 = 2.18 V. Since VCE1 > Vγ = 0.7 V, our assumption of
Q1 being in the active-linear state is justified.
To find the Q-point of Q2, we first calculate the voltage at the collector of Q1 through a
KVL its CE terminals:
VC1 = VB2 = VCE1 + 510IE1 = 2.18 + 1.96 = 4.14 V
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
214
We assume that Q2 is in the active-linear state. We can calculate IC2 ≈ IE2 from a KVL:
VC1 = VB2 = VBE2 + 510IE2
IE2 ≈ IC2 = 6.75 mA
→
→
IB2
4.14 = 0.7 + 510IE2
IC2
=
= 67.5 µA
β
Since IB2 = 67.5 µA IC1 = 3.85 mA, our assumption of IB2 IC1 is justified. Lastly, we
can find VCE2 from a KVL through CE terminals of Q2:
18 = 1.5 × 103 IC2 + VCE2 + 510IE2
→
VCE2 = 4.43 V
Ans since VCE2 = 4.43 V > Vγ = 0.7 V, our assumption of Q2 being in the active-linear
state is justified.
Therefore, the operating points of BJTs are: IE1 ≈ IC1 = 3.85 mA, IB1 = 38.5 µA, VCE2 =
2.18 V and IE2 ≈ IC2 = 6.75 mA, IB2 = 67.5 µA, VCE2 = 4.43 V
AC analysis:
As in problems 20 & 21, we start with the Q2 circuit as the input resistance of this circuit
will appear as the load for the Q1 circuit. Q2 is configured as a common emitter amplifer
with an emitter resistor. Using the formulas in page 189 and setting RB1 → ∞:
Av2 ≈ −
1, 500
RC
=−
≈ −3
RE
510
Ri2 = rπ + (RE k ro )(1 + β) = 5 × 103 + 201 × 510 = 108 kΩ
Note that because of the absence of the bias resistors, the input resistance of the circuit is
very large, and because of the absence of any coupling capacitors, there is no lower cut-off
frequency for this stage.
Since Ri2 = 108 kΩ is much larger than the collector resistor of common emitter amplifier
(Q1), it will NOT affect the first circuit. The parameters of the first-stage common emitter
amplifier can be found using formulas of page 189.
3, 600
RC
=−
= −7.06
RE
510
Ri1 ≈ RB1 = 2.29 kΩ
ωl1
1
1
fl1 =
=
=
= 14.8 Hz
2π
2πRB1 Cc1
2π × 2.29 × 103 × 4.7 × 10−6
Av1 ≈ −
The overall gain of the two-stage amplifier is then Av = Av1 ×Av2 = 21. The input resistance
of the two-stage amplifier is the input resistance of the first-stage (Q1), Ri = 2.3 kΩ. The
lower cut-off frequency of the two-stage amplifier is 14.8 Hz.
ECE65 Lecture Notes (F. Najmabadi), Winter 2007
215