Open-Circuit Time Constant Analysis
1  a1s  a2 s 2 
H (s )  K
1  b1s  b2 s 2 
General
Form
 amsm
 bns n
When the poles and zeros are easily found, then it is relatively
easy to determine a dominant pole, if one exists. But sometimes
it is not easy to determine the dominant pole.
The coefficient b1 in the transfer function is especially important
b1 
1
p 1

1
p 2

1
p 3
 ...
1
pn
  p 1   p 2   p 3  . . .  pn
How do we determine the pi or pi values? We next examine all
of the capacitors in the overall circuit individually.
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Open-Circuit Time Constant Analysis
We consider each capacitor in the overall circuit one at a
time by setting every other small capacitor to an open
circuit and letting independent voltage sources be short
circuits.
The value of b1 is computed by summing the individual time
constants, called the “sum of the open-circuit time
constants.”
n
b1   RioCi
RC is a time constant
i 1
And the pole frequency H is given by
H 
1

b1
1
n
R C
i 1
io
i
2
Open-Circuit Time Constant (OCTC) Description
The method of open-circuit time constants provides a
simple and powerful way to obtain a reasonably good
estimate of the upper 3-dB frequency, fH. The capacitors
that contribute to the high-frequency response are
considered one at a time, with independent source VS
turned off (set to zero), and all other capacitances set to
zero (that is, open-circuited). The Thévenin resistance
presented to each capacitance is then determined, and
the time constants (pi) are summed to find the overall
cutoff frequency fH is found from 1/(2ppi).
3
Open-Circuit Time Constant (OCTC) Computation Rules
• For each “small” capacitor Cj in the circuit:
–
–
–
–
Open-circuit all other “small” capacitors
Short circuit all “big” capacitors (e.g., coupling capacitors)
Turn off all independent sources (but not dependent sources)
Replace the capacitor under consideration (Cj) with a current
or voltage source for resistance calculation (or determine by
inspection)
– Find the Thévenin equivalent input resistance Rj as seen by
the capacitor Cj
– RjCj is the open-circuit time constant for the jth capacitor
• Procedure is best illustrated with an example . . .
4
Open-Circuit Time Constant (OCTC) Example 1
Doing the full analysis gives
VO
1
1


VS 1  s[ R1C1  ( R1  R2 )C 2 ]  s 2 ( R1 R2C1C 2 ) 1  b1s  b2 s 2
Standard format
Remember:
b1   1   2
and b2   1 2
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Open-Circuit Time Constant (OCTC) Example 1
Circuit
Determining 2
Determining 1
Set C1 to open, & replace
capacitor C2 with voltage
source Vx & determine
Thévenin resistance
through which current
Ix flows.
Set C3 to open, & replace
capacitor C1 with voltage
source Vx & determine
Thévenin resistance
through which current
Ix flows.
Vx = Ix (R1 + R2), so
Then 2 = (R1 + R2)C2
Vx = Ix (R1), so
Then 1 = R1C2
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Open-Circuit Time Constant (OCTC) Example 1
Recalling the expression for the transfer function,
VO
1
1


2
VS 1  s[ R1C1  ( R1  R2 )C 2 ]  s ( R1 R2C1C 2 ) 1  b1s  b2 s 2
1
2
Let’s put in some numbers:
Suppose R1 = R2 = 10 k and C1 = C2 = 100 pF.
What are the pole frequencies?
 1  R1C1  (10 4 )100  1012 sec  1  sec  p1  1 MHz
 2  ( R1  R2 )C2  (10 4  10 4 )100  1012 sec  2  sec  p 2  0.5 MHz
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Open-Circuit Time Constant (OCTC)
Why does the Open-Circuit Time Constant method work?
Answer:
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Common-gate MOSFET Amplifier
C in  C gs
C L'  C gd  C L
Vout
C ds  0
RL'  rO RL
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Common-gate MOSFET Amplifier – Focus on Cin
Set CL to open
Rin 
rO  RL
1  gm rO

 rO  RL  
 1  Cin  Rsig 

1

g
r
m O 



+
Vx
rO  RL
1  gm rO
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Common-gate MOSFET Amplifier – Focus on C’L
Rout  rO  1  gmrO 
 2  CL'  RL' rO  1  gm Rsig 


Set Cin to open
rO  1  gmrO 
+
Vx
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Common-gate MOSFET Amplifier – Conclusion
The midband voltage gain of the CG stage is
(rO  RL' )
'
AV  

[
g
(
r
R
m O
L )]
'
(rO  RL )  gmrO Rsig
The two time constants are

 rO  RL  
 1  Cin  Rsig 

1

g
r
m O 



 2  CL'  RL' rO  1  gm Rsig 


1
1
fH 

2p b1 2p ( 1   2 )
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Miller’s Theorem vs. Miller’s Approximation
For Miller Theorem to work, ratio of V2/V1 (amplifier gain) must be
calculated in the presence of the impedance Z being transformed.
Most books use the mid-band gain of the amplifier and ignore
changes in the gain due to the feedback capacitor, Cgd. This is called
“Miller’s Approximation.”
The amplifier gain in the presence of Cgd is smaller than the midband gain (i.e., high-frequency portion of the Bode gain plot), so
Miller’s approximation overestimates the Cgd ,input term and it
underestimates the capacitor Cgd,output .
Note: But the OCTC method using b1 and fH does better.
Also, Miller’s Approximation “misses” the zero introduced by the
feedback capacitor Cgd or C (important for analyzing stability of
feedback amplifiers as it affects both gain and phase margins).
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The origin of the zero in the CS MOSFET amplifier
1)
2)
3)
4)
Definition of a zero: Vo(s = sz) = 0
Because Vout = 0, zero current will flow in ro, CL and RL
Using KCL, a current of gmvgs flows in Cgd.
Ohm’s law for Cgd gives:
sC gd vgs  gm vgs
gm
sZ 
C gd
and
gm
fH 
2p C gd
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Comparison of CS and CG MOSFET Amplifiers
1) Both CS and CG amplifiers have high gain gm  rO RL 
2) CS amplifier has an  infinite input resistance whereas CG
amplifier has a low input resistance ( 1/gm).
 CG amplifier has a much better high-frequency response.
 CS amplifier has a large capacitor at the input due to the
Miller’s effect: Cin  C gs  C gd [1  gm (rO RL )] compared to that
of a CG amplifier: Cin  C gs
 In addition, a CS amplifier has a zero.
Note: The Cascode amplifier combines the desirable properties
of high input impedance with a reasonably high-frequency
response. (It has a better high-frequency response than a twostage CS amplifier.)
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Caution: Miller’s Approximation
The main value of Miller’s Theorem is to demonstrate that a large
capacitance will appear at the input of a CS amplifier (Miller’s
capacitor).
Whereas, Miller’s Approximation gives a reasonable
approximation to fH, it fails to provide accurate values for each
pole and misses the zero in the transfer function.
 Miller’s approximation should be used only as a first
guess in analysis. Simulation can be used to more accurately
find the amplifier response.
 Stability analysis (i.e., gain and phase margins) should
utilize simulations unless a dominant pole exists in the
expression for fH.
Miller’s approximation breaks down when the gain is close to unity.
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Common-Drain (Source Follower) Stage Example
AV < 1
Time constant 1 from Cgs
 1  RsigC gd
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Common-Drain (Source Follower) Stage (2)
Time constant 2 from CL
1/gm
CL
 1 '
2  
RL  CL
 gm

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Common-Drain (Source Follower) Stage (3)
Time constant 3 from Cgs
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Common-Drain (Source Follower) Stage (4)
Note: vx  vgs , using KVL gives
vx  ix Rsig  (rO RL' )(ix  gm v gs )
vx  ix Rsig  (rO RL' )ix  gm (rO RL' )vx
vx 1  gm (rO RL' )  [ Rsig  (rO RL' )]ix
Rsig  (rO RL' )
vx
 Rgs 

ix 1  gm (rO RL' )


 Rsig  (rO RL' ) 
 C gs
3  
'
 1  gm (rO RL ) 


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Common-Drain (Source Follower) Stage (5)
1
 b1   1   2   3
2p f H
'
R

(
r
R
)
 1

1
sig
O
L
'
 RsigC gd  
RL  C L 
C gs
'
2p f H
1  gm (rO RL )
 gm



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Selected comments on high-frequency
response in MOSFET amplifiers
 Include internal-capacitances of MOSFETs and simplify
the circuit as much as possible.
 Use Miller’s approximation for Miller capacitance in
configurations with a large (and negative) voltage gain AV .
 Use the open-circuit time constant method to find fH .
 Do not neglect zeros in the CS and CD configurations.
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Common-source stage with active load example
 1  RsigCin
 2   rO 1 Ri 2  C 1
 3   rO 1 RL  C L'
Ro2
Three poles
1
 b1   1   2   3
2p f H
Ri2
rO1
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Dominant Pole Compensation
 Sometimes we must purposely introduce an additional “pole” in a circuit
(such as to control gain or phase margin in feedback amplifiers for stability).
This is called “dominant pole compensation. “
 This pole must be a “dominant pole” (that is, several octaves below any
zero or other pole).
 In this case, we can ignore transistor internal capacitances in the analysis
because the poles introduced by these capacitances are at higher frequencies
and do not significantly impact the “dominant pole.”
1. Dominant pole is introduce by capacitor between output & ground
2. Capacitor between input and output of a stage (i.e., uses Miller
Effect).
Example:
Dominant pole created by adding
large capacitance CL at the output .
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