1

Open-Circuit Time Constant Analysis

21 2

21 2

1( )

1

mm

nn

a s a s a sH s K

b s b s b s

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

1 1 2 3

1 2 3

1 1 1 1. . . . . .p p p pn

p p p pn

b

How do we determine the pi or pi values? We next examine all of the capacitors in the overall circuit individually.

General Form

2

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.”

And the pole frequency H is given by

11

n

io ii

b R C

1

1

1 1H n

io ii

bR C

RC is a time constant

3

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/(2ppi).

Open-Circuit Time Constant (OCTC) Description

4

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 . . .

5

Open-Circuit Time Constant (OCTC) Example 1

Doing the full analysis gives

2 21 1 1 2 2 1 2 1 2 1 2

1 1

1 [ ( ) ] ( ) 1O

S

V

V s R C R R C s R R C C b s b s

1 1 2 2 1 2andb b Remember:

Standard format

6

Open-Circuit Time Constant (OCTC) Example 1

Circuit Determining 2

Set C1 to open, & replace capacitor C2 with voltage source Vx & determine Thévenin resistance through which current Ix flows.

Vx = Ix (R1 + R2), so Then 2 = (R1 + R2)C2

Determining 1

Set C3 to open, & replace capacitor C1 with voltage source Vx & determine Thévenin resistance through which current Ix flows.

Vx = Ix (R1), so Then 1 = R1C2

7

Open-Circuit Time Constant (OCTC) Example 1

2 21 1 1 2 2 1 2 1 2 1 2

1 1

1 [ ( ) ] ( ) 1O

S

V

V s R C R R C s R R C C b s b s

Recalling the expression for the transfer function,

1 2

Suppose R1 = R2 = 10 k and C1 = C2 = 100 pF. What are the pole frequencies?

4 121 1 1

4 4 121 2 2 2

1

2

(10 )100 10 sec 1 sec 1 MHz

( ) (10 10 )100 10 sec 2 sec 0.5 MHz

p

p

R C

R R C

Let’s put in some numbers:

8

Open-Circuit Time Constant (OCTC)

Why does the Open-Circuit Time Constant method work? Answer:

9

Common-gate MOSFET Amplifier

'

0

in gs

L gd L

ds

C C

C C C

C

'L O LR r R

Vout

10

Common-gate MOSFET Amplifier – Focus on Cin

1O L

in

m O

r RR

g r

Set to openLC

1O L

m O

r R

g r

Vx

+

11O L

in sig

m O

r RC R

g r

11

Common-gate MOSFET Amplifier – Focus on C’L

1out O m OR r g r

1O m Or g r Vx

+

' '2 1L L O m sigC R r g R

Set to openinC

12

Common-gate MOSFET Amplifier – Conclusion

' '2 1L L O m sigC R r g R

11O L

in sig

m O

r RC R

g r

The midband voltage gain of the CG stage is

''

'

( )[ ( )]

( )O L

V m O L

O L m O sig

r RA g r R

r R g r R

The two time constants are

1 1 2

1 1

2 2 ( )Hf

bp p

13

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 mid-band 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).

Miller’s Theorem vs. Miller’s Approximation

14

The origin of the zero in the CS MOSFET amplifier

1) Definition of a zero: Vo(s = sz) = 0 2) Because Vout = 0, zero current will flow in ro, CL and RL 3) Using KCL, a current of gmvgs flows in Cgd. 4) Ohm’s law for Cgd gives:

and2

gd gs m gs

m mZ H

gd gd

sC v g v

g gs f

C Cp

15

Comparison of CS and CG MOSFET Amplifiers

1) Both CS and CG amplifiers have high gain 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: compared to that of a CG amplifier: 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 two-stage CS amplifier.)

m O Lg r R

[1 ( )]in gs gd m O LC C C g r R

in gsC C

16

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.

Caution: Miller’s Approximation

17

Common-Drain (Source Follower) Stage Example

AV < 1

Time constant 1 from Cgs

1 sig gdR C

18

Common-Drain (Source Follower) Stage (2)

Time constant 2 from CL

1/gm

'2

1L L

m

R Cg

CL

19

Common-Drain (Source Follower) Stage (3)

Time constant 3 from Cgs

20

Common-Drain (Source Follower) Stage (4)

'

' '

' '

'

'

Note: , using KVL gives

( )( )

( ) ( )

1 ( ) [ ( )]

( )

1 ( )

x gs

x x sig O L x m gs

x x sig O L x m O L x

x m O L sig O L x

sig O Lxgs

x m O L

v v

v i R r R i g v

v i R r R i g r R v

v g r R R r R i

R r RvR

i g r R

'

3 '

( )

1 ( )

sig O L

gs

m O L

R r RC

g r R

21

Common-Drain (Source Follower) Stage (5)

1 1 2 3

'

'

'

1

2

( )1 1

2 1 ( )

H

sig O L

sig gd L L gs

H m m O L

bf

R r RR C R C C

f g g r R

p

p

22

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.

Selected comments on high-frequency response in MOSFET amplifiers

23

Common-source stage with active load example

Ro2

Ri2

rO1

1

2 1 2 1

'3 1

sig in

O i

O L L

R C

r R C

r R C

1 1 2 3

1

2 H

bf

p

Three poles

24

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).

Dominant Pole Compensation

Dominant pole created by adding large capacitance CL at the output .

Example: