EEE 194RF
Simple BJT Amps: High Frequency
1
HIGH-FREQUENCY RESPONSE OF SIMPLE BJT AMPLIFIERS
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.
C
R
+
S
Q
+
1
s
R
r
S
v
C
+
+
g
v
o
s
R
B
−
−
r
B
π
−
v
C
π
R
o
v
C
−
(b)
Cµ
RB
r
b
+
vi
v S (R
= R
S
S
−
+
+
g m vπ
rπ
RB )
vπ Cπ
RC
ro
vo
−
−
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 Cµ
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
vo
= − g m ( ro RC ) = − g m RC ' ,
vπ
Z
which is used in Miller's Theorem, Z1 =
1− A
A=
Z
Ι 1i
+
V
1
−
Ι 2i
General Linear
Network
Ι in
with
V =AV
2
1
(a)
•
o
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.
= RS
•
r
−
R'S
•
v
m π
π
(a)
•
µ
b
+
R
v
R
Ιout
+
Z2 =
AZ
A−1
Ι 1s
Ι in
+
V
V
1
−
−
2
Z1
.
(10.5-2)
Ιout
General Linear
Network
with
V =AV
2
1
Ι 2s
+
Z 2 V2
−
(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:
EEE 194RF
Simple BJT Amps: High Frequency
2
R'
S
= R
R
S
B
r
b
+
v
v
=
R
S
S
where
and
•
+
i
r
(R
−
+
g m vπ
S
R )
π
B
v
π
C
C
i
R'
C
v
o
−
Figure 4. Miller's Theorem Applied to a Common-Emitter Amplifier
o
−
1
jω Cµ
1
1
=
=
⇒ Ci = Cµ (1 + g m RC' )
'
jω Ci 1 − A
jω Cµ 1 − ( − g m RC ) 
A
1 + g m RC' )
jω Cµ
(
1
− g m RC'
=
=
⇒ Co = Cµ
jω Co
A −1
g m RC'
jω Cµ ( − g m RC' − 1)
The voltage gain of the circuit is therefore:
 v   v  v   v 
AV =  o  =  o  π   i 
 vS   vπ  vi   vs 


=  − g m  RC'




1
rπ


jω Ci
1  
  RS RB 



jω Co    '

1    RS 
 RS + rb +  rπ

ω
j
C

i

 
Simplifying this expression to yield an expression for the gain which clearly shows the poles of the amplifier:
•





− g m RC' RB rπ
1
1

AV = 


' 
'
 RS ( RB + rb + rπ ) + RB ( rb + rπ )  1 + jω Co RC  1 + jω Ci  rπ ( RS + rb )  


The high frequency poles for the common-emitter amplifier as shown in Figure 1 are:
ω P1 =
•
1
1 + jω Co RC'
and
ωP2 =
1
1 + jω Ci  rπ

(R
'
S
+ rb ) 

.
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:




1
1

AV = [ AVm ] 
 .
' 
'
1 + jω Co RC  1 + jω Ci  rπ ( RS + rb )  



Common-collector amplifier high-frequency response
• AC model of a simple BJT common-collector amplifier is shown in Figure 5.
EEE 194RF
Simple BJT Amps: High Frequency
3
C
R
R
S
+
r
+
R
R
B
+
1
R
s
b
+
Q
v
r
S
v
B
−
v
π
C
v
µ
R
s
π
C
g
π
v
m π
r
o
R
−
B
o
R
−
v
B
−
o
−
(a)
•
(b)
Figure 5. Common-Collector 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 6. For
simplicity, ro is considered to be very large compared to RC and RE.
C
R'S
= R
S
R
r
B
+
=
(RS
S
RS
v
rπ
vi
v
µ
b
+
π
g v
m π
Cπ
R
−
RB )
C
+
v
RB
−
o
−
Figure 6. Simplified Input Portion of High-Frequency Equivalent Circuit
•
The gain of the shaded two-port network is
Z1 =
•
C
+
Z
1− A
Z2 =
A=
vec
= − g m ( RC + RE ) which is used in Miller's Theorem,
vπ
AZ
.
A−1
(10.5-2)
When Miller's Theorem is applied to the common-collector amplifier,
1 + g m ( RC + RE ) 
Ci = Cπ + Cµ 1 + g m ( RC + RE )  and CO = Cµ 
 ≈ Cµ
 g m ( RC + RE ) 
with the equivalent circuit shown in Figure 7:
R'
S
= R
R
S
E
r
b
+
+
vi
v S (R
= R
S
rπ
R )
v
π
g v
m π
Cι
Cο
R
−
C
+
E
S
R
−
B
vo
−
Figure 7. Miller's Theorem Applied to a Common-Collector Amplifier
EEE 194RF
•
•
Simple BJT Amps: High Frequency
The pole introduced by the Ci is of primary interest since Co is less than the input capacitance.
Therefore, assume that the pole introduced by Co is sufficiently high so that Co can be replaced by an
open circuit.
The voltage gain of the common-collector amplifier is:
AV =
•
vo
g m rπ RE
RS '
≈
vs RS '+ rb + (1 + g m RE ) rπ + jω Ci rπ ( RS + rb ) RS
The dominant high-frequency pole is
ω P1 ≈
•
4
1
  R +r
Ci  rπ  S b
  1 + g m RE


 
=
1
  R +r
Cπ + Cµ 1 + g m ( RC + RE )   rπ  S b
  1 + g m RE
{
}


 
.
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 '
S
= R
R
Q
g
E
+
R
E
R
−
C
B
B
R
v
C
o
r
=
v
π
v
π
+
C
C
π
µ
R
i
R
s
v
m π
−
1
v
s
R
+
S
+
v
S
S
R
R
E
+
C
−
r
b
−
•
•
−
(b)
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 rb is also small. Then rb can
be ignored: that is, let rb = 0 and ve ≈ −vπ.
The poles are located at
ω P1 =
•
o
S
(a)
•
v
rπ + (1 + g m rπ ) ( RS ' RE )
Cπ rπ ( RS ' RE )
≈
βF
= ωT
Cπ rπ
and
ωP2 =
1
.
Cµ RC
Both poles are at very high frequencies. Therefore, common-base amplifiers are not usually the frequency
limiting elements in a multistage amplifier.