# Finding the Open Circuit Time Constants

I. Finding the Open Circuit Time Constants τj’s
A. Example: CE Amplifier
V+
iSUP
iOUT
RS
+
+
V
− s
RL
vOUT
−
+
V
− BIAS
V−
• Small signal model
C&micro;
RS
+
+
Vs
+
−
rπ
Vπ
gmV
π
Cπ
−
ro roc
RL
Vout
−
(a)
EECS 6.012 Spring 1998
Lecture 22
B. Procedure
• Eliminate all independent sources (e.g., Vs --&gt; 0)
• Open-circuit all capacitors
• Find Thevenin resistance by applying it and measuring vt.
C. Time Constant for Cπ
RS
C&micro; --&gt; 0
rπ
+
vt
↓
Cπ -&gt; 0
it
gmVπ
ro || roc
RL
_
• Result: (by inspection)
R Tπ = R s r π
τ
C
πo
= R
C
Tπ π
EECS 6.012 Spring 1998
Lecture 22
D. Time Constant for C&micro;
• Same procedure:
+
vt
↓
_
RS
io
Vo
↓
it
+
C&micro; --&gt; 0
rπ
Vπ
gmVπ
Cπ -&gt; 0
ro || roc
RL
_
• Let R’in = Rs || rπ and R’out = ro || roc || RL
− vt
+
R'in = RS r π
+
it
gmv π
vπ
R'out = ro roc RL
−
vt + vπ
i t = ---------------- + g m v π
R′ out
v
π
---------- = – i eliminate vπ
t
R′
in
v
t
---- = R
= R′
+ R′  1 + g R′ 
T&micro;
out
in 
m out 
i
t
τ
C
&micro;o
= R
C =
T&micro; &micro;
R′
out
+ R′  1 + g R′  C
in 
m out  &micro;
EECS 6.012 Spring 1998
Lecture 22
E. Dominant Pole for CE Amplifier
• Sum Individual time constants
b 1 = ( R Tπ C π + R T&micro; C &micro; )
b
1
= R′ C + R′  1 + g R′  C + R′
C
in π
in 
m out  &micro;
out &micro;
• Assume τ1 &gt;&gt; τ2
b1 = τ1 + τ2 ≈ τ1
ω
1
1
≈ ------ = ------------------------------------------------------------------------------------------------------------
3dB b
R′ C + R′
1 + g R′  C + R′
C
1
in π
in 
m out  &micro;
out &micro;
• Very similar result to the Miller effect calculation
• Addtional term R’outC&micro; is taken into account
EECS 6.012 Spring 1998
Lecture 22
II. Common Collector Frequency Response
V+
+
RS
RL
+
V
− s
vOUT
−
iSUP
+
V
− BIAS
V−
A. Small Signal Model
• Add C&micro; and Cπ to the two-port model from Chapter 8
Cπ
RS
Vs
+
−
E
B
Rout
+
+
Vin
Rin
C&micro;
−
+
−
Vin
RL
Vout
−
C
EECS 6.012 Spring 1998
Lecture 22
B. Low Frequency Voltage Gain
R
v
 R in 


L
out
----------- =  ----------------------  ( 1 )  -------------------------- 
v
 R S + R in 
 R L + R out 
s
• Substituting values for input and output resistance
R
v
 rπ + βo R L 


L
out

----------- =  ----------------------------------------  ( 1 )  ----------------------------------------------------------------R +r +B R 
R + 1 ⁄ g  +  R ⁄ β 
v


S π
o L
L 
m  S o
s
C. Use Miller Approximation to Find Dominant Pole
• Voltage gain from B to E across Cπ
A
R
R
L
L
= -------------------------- = ---------------------------vCπ
R
+R
1⁄g +R
out
L
m
L
• Total Capacitance seen at the input CT = (1-AvCπ)Cπ + C&micro;
C
1⁄g


m
 +C
= C  ---------------------------------T
π  1 ⁄ g  + R 
&micro;
m
L


• Thevenin resistance seen by CT
RT = RS || Rin
EECS 6.012 Spring 1998
Lecture 22
D. Common Collector Dominant Pole
• The dominant time constant for a CC amplifier is
C
π
τ =  R || R  ------------------------ + C
in  1 + g R
&micro;
 S
m L
• Substitute for Rin and look at ω3dB
ω
3dB
1
= ---------------------------------------------------------------------------------------------------------C


π
 R ||  r + β R    -----------------------+C 
o L   1 + g R
&micro;
 S  π
m L
• Effect of Cπ is small
• In general RS &lt; Rin---&gt; frequency response is dominated by RSC&micro;
EECS 6.012 Spring 1998
Lecture 22
III. Common Base Amplifier Frequency Response
V+
iSUP
iOUT
RL
Is
IBIAS
RS
V−
A. Small Signal Model
E
C
Iout
Is
RS
Rin
−Iin
π
C
Rout
C&micro;
RL
B
• DC Gain
R
i

 R out 
S 
out
---------- =  ----------------------  ( 1 )  -------------------------- 
i
 R in + R S 
 R L + R out 
s
EECS 6.012 Spring 1998
Lecture 22
B. Use Method of Open Circuit Time Constants
• Thevenin resistance across Cπ
R
Tπ
= R || R = R ||  1 ⁄ g  ≈ 1 ⁄ g
S
in
S 
m
m
• Thevenin resistance across C&micro;
R
T&micro;
= R
out
|| R ≈ β r || R ≈ R
L
o o
L
L
• Summing the open circuit time constants and taking reciprocal
ω
3dB
1
≈ ----------------------------------------------C ⁄g  + R C L &micro;
 π m
IV. Summary of Frequency Response of Single-Stages
CE/CS: with voltage output - suffers from Miller effect
CE/CS: with current output - “wideband”
CB/CG: “wideband”
CC/CD: “wideband”
• “Wideband” means that the stage can operate near the frequency
limit of the device ... fT
• Frequency limitation is set by external circuit RS and RL
EECS 6.012 Spring 1998
Lecture 22