Frequency Response of
Transistor Amplifiers
 Frequency response of CS amplifier,
qualitative analyses
 Quantitative analyses of frequency response
for CS and CE amplifiers - optional
 Cascode amplifier - optional
1/22
CS amplifier
medium
frequency
Full
circuit:
Medium frequency
small-signal
equivalent circuit
Av   g m RD
Ri  RG
RO  RD || ro  RD
2/22
Frequency response of transistor amplifiers
 mid-frequency :
– coupling capacitors → short-circuits
– internal parasitic capacitances → open-circuits
 low-frequency:
- coupling capacitors → equivalent impedances
- internal parasitic capacitances → open-circuits
 high-frequency :
- coupling capacitors → short-circuits
- internal parasitic capacitances → equivalent impedances
• must be taken into consideration:
- output resistance of the signal source
- load resistance
3/22
• No capacitors in the
real
signal
source
load
small- signal equivalent
circuit
• Frequency independent
behavior
Av ( j )  cst
PS
Mid-frequency
against frequency
variations
R
4/22
• Coupling capacitors in the
real
signal
source
load
small-signal equivalent circuit
• Frequency dependent
behavior
• High pass type
f , Av ( j ) 
PS
Low-frequency
Sets the
lower cutoff
frequency fL
5/22
• Parasitic capacitors in the
real
signal
source
load
small-signal equivalent circuit
• Frequency dependent
behavior
• Low pass type
f , Av ( j ) 
PS
High-frequency
Sets the
upper cutoff
frequency
fH
6/22
Frequency response
Due to coupling
capacitances
Due to parasitic
capacitances
7/22
Optional
CS amplifier
real
signal
source
load
vo  j 
Av ( j ) 
vi  j 
vo(j)=Fo(j)id(j);
id(j)=Fs(j)vg(j);
vg(j)=Fi(j)vi(j);
PS
Av  Fi  j   Fs  j   Fo  j 
 Analysis in the low-frequency range
8/22
Analysis in
low-frequency
range
Optional
Fi  j  
vg  j 
vi  j 
Fi  j  
jRG CCi
1  j R  RG CCi
HPF, introduces a pole at the frequency
1
f Li 
2 R  RG CCi
Frequency of the pole: CCi multiplied by the total
resistance seen by CCi
Generalization:
1
f Li 
2 R  Ri CCi
9/22
Analysis in
low-frequency
range
Optional
id  j 
Fs  j  
v g  j 
id  j   g mvgs  j 
f Ls 
1
 1

2  || RS CS
 gm

Generalization:

1
vg  j   vgs  j   g m vgs  j  RS ||
jCS




Frequency of the pole: CS multiplied by
the total resistance seen by CS
1
f Ls 
2 R ' || RS CS


10/22
Analysis in
low-frequency
range
Optional
vo  j 
Fo  j  
id  j 
f Lo 
1
2 RD  RL CCo
Generalization:
Frequency of the pole: CCo multiplied by
the total resistance seen by CCo
1
f Lo 
2 Ro  RL CCo
• Dominant pole: the greater break frequency between fLi, fLs, fLo
if the nearest pole or zero will be at least a decade away.
• usually given by fLs, for equal coupling capacitances
11/22
 Analysis in the high-frequency range
Cds is not shown
here because it
generates a pole
at a much higher
frequency that
the one generated
by Cgs and Cgd
Cgd is reflected to the input according to Miller’s theorem
Cgd ,ech  1  av Cgd
av   g m RL'   g m RL || RD || ro 
Cgd ,ech  1  g m R'L Cgd
Optional
12/22
 Analysis in the high-frequency range
Optional
Ci  Cgs  Cgd ,ech  Cgs  1  g m R'L Cgd
vo  j 
RG
1
Av  j  

g m R' L
vi  j 
R  RG
1  j R || RG Ci
RG
Avo  
g m R' L
R  RG
1
fH 
2 R || RG Ci
1
fH 
2 R || Ri Ci
13/22
Numerical example
Optional
CCi=CCo=Cs=10F, R=20K, RG=2M, RD=10k, RL=20k,
Rs=10K, I=400µA.
K=100A/V2, (W/L)=18, VA=100V. Cgs=Cgd= 1pF @ I=400A
g m  1.2mS
Solution:
f Li
ro  250KΩ
1
1


 8mHz
3
6
2 ( R  RG )CCi 2 20  200010 10 10
f Ls 
f Lo
1
 1

2 
|| RS CS
 gm


1
 21Hz
 1

2 
|| 10 103 10 10 6
 1,2

1
1


 0,5Hz
3
6
2 RD  RD  2 10  2010 10 10
fL=21Hz
The output resistance of the signal source
R does not affect fLi but affects fH
14/22
RG
2
'
Avo  
g m RL  
1.2  6.5  7.7
R  RG
0.02  2
Avo
dB
Optional
 20 log( 7,7)  17,7
Ci  Cgs  1  g m ro || RD || RL Cgd  1  1  1,2250|| 10 || 201  9,8pF
1
1
fH 

 820KHz
3
12
2 R || RG Ci 2 20 || 2000 10  9,8 10
15/22
CE amplifier
Optional
 Analysis in the
low-frequency range
The effect of each
capacitor is analyzed
considering the other
two capacitors with infinite capacity (zero equivalent impedance)
f Li 
1
1

2π( R  Ri )CCi 2π( R  RB || rbe )CCi
1
1
f Le 

;
'
2π R || RE CE 2πRE CE

f Lo

RB  RB1 || RB 2
RE  RE || R '  RE ||
1
1


2π( RO  RL )CCo 2π( RC  RL )CCo
rbe  RB || R
 1
16/22
 Analysis in the high-frequency range
Optional
RB || rbe
1
Av ( j ) 
g m ( RC || RL )
R  RB || rbe
1  jRiCi
Ci  Cbe  (1  g m RL )Cbc
RB || rbe
Avo 
g m ( RC || RL )
R  RB || rbe
Ri  rbe || RB || R
1
fH 
2π(rbe || RB || R)Ci
17/22
Cascode amplifiers
Optional
 for the CS and CE amplifiers the magnitude of the gain and
the bandwidth are in inverse ratio due to Miller’s effect
 when the gain increases, the parasitic capacitance reflected
to the input also increases, so the high breaking frequency
decreases
RG
Avo  
g m R' L
R  RG
1
fH 
2πR || Ri Cgs  1  g m R'L Cgd 
to reduce the multiplication effect of the parasitic capacitance
it is often use the cascode configuration:
- connection of a CS (CE) amplifier followed by a
CG (CB) amplifier
 a technique to build wideband amplifiers
18/22
The cascode configuration with MOSFET
Optional
medium
frequency
CS
Av 2 
Av   gm RD
same current through T1, T2
gm  gm1  gm2
Av1  1
vo
 g m 2 RD  g m RD
vo1
Av  Av1  Av 2

vo1
1 

Av1 
  g m1  ro1 ||
vi
gm2 

1
 ro1
gm2
CG
Ri  RG1 || RG 2  RG
Ro  RD || (ro1  ro 2  gm2ro1ro 2 )  RD
19/22
High frequency
Optional
Ci  Cgs1  (1  Av1 )Cgd1  Cgs1  2Cgd1
The multiplication factor of Cgd1 is 2, considerable smaller that the one
in CS configuration (1  gm R2 ). The value for Ci results much more
reduced, that leads to a much higher value of the high cutoff frequency:
fH 
1
2π( R || RG )Ci
 the other two poles due to Cgs2 and Cgd2 results to considerable higher
frequency than fH
 Ci introduces the dominant pole at high frequency and determines the
20/22
pass-band of the amplifier
Numerical illustration
RD  10K
RG1  RG 2  4MΩ
R  20KΩ
I  400μA
RL  20KΩ
(W / L)  18
VA  100V
K  100
Optional
μA
V2
Cgs  Cgd  1pF
g m  2K
RG  RG1 || RG 2
Av 
W
I  2 100  18  400  1200 μS  1.2 mS
L
 4 || 4  2 M
R
 g m ( RD || RL )   20 1,2  (10 || 20)  7.9
R  RG
20  2000
Ci  Cgs  2Cgd  1  2  1  3 pF
1
1
fH 

 2.7 MHz
2 ( R || RG )Ci 2 (20 || 2000) 103  3 10 12
21/22
The cascode
configuration with BJT
Optional
For identical T1 and
T2 transistors
RB  RB1 || RB 2
rbe  rbe1  rbe2
g m  g m1  g m2
vo
RB || rbe1 
1 
  g m1
 g m 2 ( RC || RL )
Av  
vi R  RB || rbe1 
gm2 
RB || rbe
Av  
g m ( RC || RL )
R  RB || rbe
1
fH 
2 (rbe || RB || R)(Cbe  2Cbc )
22/22