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H IGH-F REQUENCY R ESPONSE OF SIMPLE BJT A MPLIFIERS
At high frequencies, the amplifier response is characterized by midband and high-frequency poles. Single BJT
amplifiers are analyzed.
Common-emitter amplifier high-frequency response
- AC model of a simple BJT common-emitter amplifier is shown in Figure 1.
R
B
R
C
Q
1
R
S
v
v o
s
− −
R
B
R
C
R
S
v
o
v
s
− −
r
o
g
m
v
π
r
π
r
b
v
π
−
C
π
C
μ
(a) (b)
Figure 1. Common-Emitter Equivalent Circuits
(a) Midband AC Equivalent, (b) High-Frequency Equivalent
- The input portion of the high-frequency equivalent circuit is simplified for analysis as shown in Figure 2.
R
C
R'
S
v
o
v
i
− −
r
o
g
m
v
π
r
π
r
b
v
π
−
C
π
C
μ
R
S
R
B
=
v
S
R
S
( R )
S
R
= B
Figure 2. Simplified Input Portion of High-Frequency Equivalent Circuit
- Miller's Theorem is used to simplify the high-frequency equivalent circuit made complex by the presence of C
μ
which interconnects the input and output sections of the circuit.
- The two-port network accentuated by the shaded area has a midband voltage gain of
o
m o C m C
v
A g r R g R
v
π
which is used in Miller's Theorem,
1 2
Z A Z
Z Z
A A
. (10.5-2)
General Linear
Network
V = A V
1 2 1 2
V V
Z
− −
Ι
Ι
Ι
Ι
1i 2i
in out
General Linear
Network
V = A V
1 2 1 2
V V
−
−
Z Z
1 2
Ι Ι Ι Ι
1s in out 2s
with
with
(a) (b)
Figure 3. Miller's Equivalent Circuits : (a) Interconnecting Impedance, (b) Port-Shunting Impedance
- The use of Miller's Theorem results in the following equivalent circuit:
R
C v
o
−
o
g
m
v
π
r
π
r
b
v
π
−
C
i C
'
R'
S
v
i
−
R
S
R
B
=
v
S
R
S
( R )
S
R
= B
Figure 4. Miller's Theorem Applied to a Common-Emitter Amplifier
where
( )
( )
'
'
i m C
i
m C
j C
C C g R
j C A
j C g R
μ
μ
μ
and
( )
( )
'
'
' '
m C
m C
o
o m C m C
A
g R j C g R
C C
j C A g R j C g R
μ
μ
μ
ω
ω ω
- The voltage gain of the circuit is therefore:
'
'
o o i
V
S i s
i S B
m C
o S
S b
i
v v v v
A
v v v v
r
j C R R
g R
j C R
R r r
j C
π
π
π
π
ω
ω
ω
Simplifying this expression to yield an expression for the gain which clearly shows the poles of the amplifier:
( ) ( )
( )
'
'
'
m C B
V
S B b B b o C
i S b
g R R r
A
R R r r R r r j C R
j C r R r
π
π π
π
ω
ω
- The high frequency poles for the common-emitter amplifier as shown in Figure 1 are:
1 '
P
o C
j C R
and
( )
2
'
P
i S b
j C r R r
π
.
- Simply put, the voltage gain characteristics of the amplifier at high frequencies is composed of the midband
voltage and the lowpass transfer characteristics of the input and output portions of the high-frequency equivalent
circuit: [ ]
( )
'
'
V Vm
o C
i S b
A A
j C R
j C r R r
π
.
Common-collector amplifier high-frequency response
- AC model of a simple BJT common-collector amplifier is shown in Figure 5.
- The pole introduced by the C
i
is of primary interest since C
o
is less than the input capacitance.
Therefore, assume that the pole introduced by C
o
is sufficiently high so that C
o
can be replaced by an
open circuit.
- The voltage gain of the common-collector amplifier is:
o m E S
V
s S b m E i S b S
v g r R R
A
v R r g R r j C r R r R
π
π π
- The dominant high-frequency pole is
1
P
S b S b
i m C E
m E m E
R r R r
C r C C g R R r
g R g R
π π μ π
.
- Common-emitter amplifiers with emitter resistors have the same basic topology as the common-collector
amplifier. The resultant pole location remains identical.
Common-base amplifier high-frequency response
- AC model of a simple BJT common-base amplifier is shown in Figure 8.
R
E
R
C
Q
1
R
S
v
o
v
s
− −
R
E
R
C
R
S
v
o
v
i
− −
g
m
v
π
r
π
r
b
v
π
−
C
π
C
μ
(a) (b)
R
B
C
B
v
s
R
E
R
S
R
S
=
R '
S
=
Figure 8. Common-Base Equivalent Circuits
(a) Midband AC Equivalent, (b) High-Frequency Equivalent
- Note that there are no capacitors bridging the input and output terminals: therefore, there do not exist Miller
effect high valued capacitors. Therefore, the poles are at very high frequencies.
- To simplify the analysis, assume that the base current is small so the voltage across r
b
is also small. Then r
b
can
be ignored: that is, let r
b
= 0 and v
e
≈ −v
π
.
1
m S E
F
P T
S E
r g r R R
C r R R C r
π π
π π π π
= ≈ = and
2
P
C
C R
μ
- Both poles are at very high frequencies. Therefore, common-base amplifiers are not usually the frequency
limiting elements in a multistage amplifier.
