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M.H. Perrott Analysis and Design of Analog Integrated Circuits Lecture 9 Open Circuit Time Constant Technique Michael H. Perrott February 26, 2012 Copyright © 2012 by Michael H. Perrott All rights reserved.

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M.H. Perrott

Analysis and Design of Analog Integrated CircuitsLecture 9

Open Circuit Time Constant Technique

Michael H. PerrottFebruary 26, 2012

Copyright © 2012 by Michael H. PerrottAll rights reserved.

M.H. Perrott

Review of Our Analysis Techniques

Two port analysis allows us to quickly calculate small signal gain from cascaded network stages- So far, only purely resistive impedances have been considered 2

No IndependentSources

Block 2

No IndependentSources

Block 3

ZLVb

No IndependentSources

Block 1

Vin Va Vc

Linear NetworkLinear NetworkLinear Network

ZLZoutGmVbZinVbZoutGmVaZinVaZoutGmVinZinVin Vc

Zout,effective

Vth,effective Zin,effectiveVb

M.H. Perrott

The Problem with Complex Impedances

When complex impedances are considered (i.e., capacitors, inductors, and resistors), things get much more messy- Complex impedance calculations are time consuming- Capacitance between drain and gate of transistors

complicates calculation effort further

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No IndependentSources

Block 2

No IndependentSources

Block 3

ZLVb

No IndependentSources

Block 1

Vin Va Vc

Linear NetworkLinear NetworkLinear Network

ZLZoutGmVbZinVbZoutGmVaZinVaZoutGmVinZinVin Vc

Can we determine a faster analysis path to gain intuition?

M.H. Perrott

No IndependentSources

Block 2

No IndependentSources

Block 3

ZLVb

No IndependentSources

Block 1

Vin Va Vout

Linear NetworkLinear NetworkLinear Network

20log(Vout/Vin)(dB)

w (rad/s)w0 w1 w2wac0wac1wac2

MidBand Gain f (Hz) =w (rad/s)

Note:

General Frequency Response for Amplifiers

Midband gain can be calculated by assuming purely resistive impedances (as we have done so far)- Large valued capacitors used for AC coupling will be shorts in

this analysis For DC coupled circuits, typically DC gain = Midband Gain

- Small valued capacitors will be opens in this analysis 4

M.H. Perrott

No IndependentSources

Block 2

No IndependentSources

Block 3

ZLVb

No IndependentSources

Block 1

Vin Va Vout

Linear NetworkLinear NetworkLinear Network

20log(Vout/Vin)(dB)

w (rad/s)w0 w1 w2wac0wac1wac2

MidBand Gain f (Hz) =w (rad/s)

Note:

Our Focus Will Be on High Frequency Poles

We are particularly interested in knowing the bandwidth of our amplifier circuit- Bandwidth is primarily set by the lowest frequency pole, w0- Additional attenuation occurs at frequencies beyond the

amplifier bandwidth by higher frequency poles w1, w2, etc.5

M.H. Perrott

20log(Vout/Vin)(dB)

w (rad/s)w0 w1 w2wac0wac1wac2

MidBand Gain f (Hz) =w (rad/s)

Note:

Open Circuit Time Constant Technique

The Open Circuit Time Constant (OCT) technique allows us to quickly estimate the bandwidth of an amplifier circuit- We will see that it is most accurate when there is one dominant

pole, w0

This means that w1, w2, and higher poles are not close in frequency to w0

This will hold for opamps and other circuits that operate in feedback

There is still considerable value to the OCT method in providing design intuition even when there is not just one dominant pole

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M.H. Perrott

20log(Vout/Vin)(dB)

w (rad/s)w0 w1 w2wac0wac1wac2

MidBand Gain f (Hz) =w (rad/s)

Note:

Short Circuit Time Constant Technique

The Short Circuit Time Constant (SCT) technique allows us to quickly estimate the AC-coupled cutoff frequency, wac0- This has many similarities to the OCT method, but we will not

discuss in this class since AC coupling is not used very often in integrated circuits due to

the high cost of large valued capacitors When AC coupling is applied in integrated circuits, it is often

quite easy to estimate the AC-coupled cutoff frequency since there are relatively few poles in the circuit related to AC-coupling

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M.H. Perrott

20log(Vout/Vin)(dB)

w (rad/s)w0 w1 w2

MidBand Gain f (Hz) =w (rad/s)

Note:

Key Assumptions for the OCT Technique

Let us assume that the transfer function from Vin to Vout is

- Note that we are ignoring any AC-coupling poles/zeros This implies that are approximating DC gain = Midband gain The OCT method does not require this assumption – it just

simplifies the analysis to follow- Note also that DC gain equals K in the above transfer function

We see this by setting s = 08

Vout(s)

Vin(s)=

K

(τ0s+ 1)(τ1s+ 1) · · · (τn−1s+ 1)

M.H. Perrott

20log(Vout/Vin)(dB)

w (rad/s)w0 w1 w2

MidBand Gain f (Hz) =w (rad/s)

Note:

Key Idea of the OCT Technique

Assuming the transfer function from Vin to Vout is:

We can achieve a reasonable approximation of the bandwidth of the system by instead considering:

- Here i are the “time constants” corresponding to the poles of the circuit network

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Vout(s)

Vin(s)=

K

(τ0s+ 1)(τ1s+ 1) · · · (τn−1s+ 1)

Vout(s)

Vin(s)=

K³Pn−1i=0 τi

´s+ 1

M.H. Perrott

20log(Vout/Vin)(dB)

w (rad/s)w0 w1 w2

MidBand Gain f (Hz) =w (rad/s)

Note:

Bandwidth Estimate from OCT Technique

The OCT technique approximates the transfer function as:

The estimated bandwidth is found by substituting s = jw0 and solving for w0 such that the magnitude is /

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Vout(s)

Vin(s)=

K³Pn−1i=0 τi

´s+ 1

w0 =1Pn−1i=0 τi

⇒¯̄̄̄Vout(w0)

Vin(w0)

¯̄̄̄=

¯̄̄̄K

j1 + 1

¯̄̄̄=K√2

Bandwidth estimate found by inversing the sum of time constants!

M.H. Perrott

Why Is This Approximation Reasonable?

Consider a second order example:

Expanding:

- But notice (since the time constant values are > 0):

In fact: The worse case of 01o

2 = 0.25 occurs when 0 = 1:

The approximation will be better for o ≠ 111

Vout(w0)

Vin(w0)=

K

(jτ0w0 + 1)(jτ1w0 + 1)

Vout(w0)

Vin(w0)=

K

−τ0τ1w20 + j(τ0 + τ1)w0 + 1

j(τ0 + τ1)w0 = j1 ⇒ τ0w0 < 1, τ1w0 < 1

τ0τ1w20 ≤ 0.25¯̄̄̄

Vout (w0)

Vin(w0)

¯̄̄̄=

¯̄̄̄K

j1 + 1− 0.25

¯̄̄̄=

K√1.56

≈ K√2

M.H. Perrott

Key Issues For the OCT Approximation

For the higher order transfer function

The OCT approximation for bandwidth is

As hinted at by our second order example:- The OCT approximation will have much better accuracy if the

time constants are different, and particularly if there is one dominant time constant

- The bandwidth estimate by the OCT method is typically conservative (i.e., actual bandwidth > OCT estimate) Complex poles can lead to actual bandwidth < OCT estimate

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Vout(s)

Vin(s)=

K

(τ0s+ 1)(τ1s+ 1) · · · (τn−1s+ 1)

BW ≈ 1Pn−1i=0 τi

rad/s

But how do we compute ∑ ?

M.H. Perrott

OCT Method of Calculating the Sum of Time Constants

OCT method calculates by the following steps:- Compute the effective resistance Rthj seen by each

capacitor, Cj, with other caps as open circuits AC coupling caps are not included – considered as shorts

- Form the “open circuit” time constant Tj = RthjCj for each capacitor Cj- Sum all of the “open circuit” time constants

As proved by Richard Adler at MIT

- This implies that the sum of the transfer function pole time constants is the same as the sum of the open circuit time constants

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⇒ BW ≈ 1Pmj=1RthjCj

rad/s

n−1Xi=0

τi =

mXj=1

RthjCj

M.H. Perrott

How Do You Tell if a Cap is for AC coupling or OCT?

In general, capacitors associated with AC coupling have the property that the amplifier gain increases as the capacitor goes from open to short- These capacitors are simply assumed to be shorts for

the OCT analysis In general, capacitors used in the OCT calculation

have the property that the amplifier gain decreases as the capacitor goes from open to short- These capacitors must all be considered in the OCT

analysis

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M.H. Perrott

Example: Second Order RC Network

Transfer function of the above network:

The sum of the time constants from the poles of the above network are obtained by inspection of the first order coefficient in the above transfer function

For more complex networks, the direct approach of explicitly calculating the transfer function is quite tedious

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C1 C2

R1 R2

VoutVin

Vout(s)

Vin(s)=

1

R1R2C1C2s2 + (R1C1 + R1C2 +R2C2)s+ 1

⇒ BW ≈ 1Pn−1i=0 τi

=1

R1C1 +R1C2 +R2C2rad/s

M.H. Perrott

OCT Method Applied to Second Order RC Network

Obtain the Thevenin resistance values seen by each capacitor with other capacitors as opens

Bandwidth estimate from OCT method:

- Note that OCT method agrees with estimate based on direct calculation of the transfer function, but is much faster!

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Rth1 Rth2

R1 R2

C1 C2

R1 R2

VoutVin

Rth1 = R1 ⇒ Rth1C1 = R1C1

Rth2 = R1 +R2 ⇒ Rth2C2 = (R1 +R2)C2

⇒ BW ≈ 1Pmj=1RthjCj

=1

R1C1 + (R1 +R2)C2rad/s

M.H. Perrott

Example: Common Source Amplifier

Estimate the bandwidth of the above amplifier using the OCT method- What capacitances should be considered?- What Thevenin resistances must be calculated?

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Rin

RLID

Vin

Vout

M.H. Perrott

Key Capacitances for CMOS Devices

S D

GVGS

VD>ΔV

ID

LDLD

overlap cap: Cov = WLDCox + WCfringe

B

CgcCcb

Cov

CjdbCjsb

Cov

Side View

gate to channel cap: Cgc = CoxW(L-2LD)

channel to bulk cap: Ccb - ignore in this class

S D

Top View

W

E

L

E

E

source to bulk cap: Cjsb = 1 + VSB ΦB

Cj(0)

1 + VSB ΦB

Cjsw(0)WE + (W + 2E)

junction bottom wall cap (per area)

junction sidewall cap (per length)

drain to bulk cap: Cjsd = 1 + VDB ΦB

Cj(0)

1 + VDB ΦB

Cjsw(0)WE + (W + 2E)

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(make 2W for "4 sided" perimeter in some cases)

L

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M.H. Perrott

CMOS Hybrid- Model with Caps (Device in Saturation)

RS

RG

RD

RD

RS

RG

-gmbvsvgs

vs

rogmvgs

ID

Csb

Cgs

CgdCdb

Cgs = Cgc + Cov = CoxW(L-2LD) + Cov23

Cgd = Cov

Csb = Cjsb (area + perimeter junction capacitance)

Cdb = Cjdb (area + perimeter junction capacitance)

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M.H. Perrott

Back to Common Source Amplifier

Of the above capacitors, only Cgs, Cgd, and Cdb must be considered- Csb is grounded on both sides

Thevenin resistance calculations- Cdb: Rthd || Rd- Cgs and Cgd: these involve new Thevenin resistance

calculations20

Rin

RLID

Vin

VoutCgd

Cgs Csb

Cdb

M.H. Perrott

OCT Thevenin Resistance Calculations

Cgs: Thevenin resistance between gate and source

Cgd: Thevenin resistance between gate and drain

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Rthgs

Rthgd

Cgs

Cgd

Rthgs

Rthgd

RS

RG

RD RD

RS

RG

-gmbvsvgs

vs

rogmvgs

ID

Rthgs =RS(1 +RD/ro) +RG(1 + (gmb + 1/ro)RS +RD/ro)

1 + (gm + gmb)RS + (RS +RD)/ro

Rthgd = (RD +RG)(1− rods/ro) + rodsgmRGwhere rods = ro||

RD1 + (gm + gmb)RS

M.H. Perrott

OCT Calculations for Common Source Amplifier

Estimated bandwidth from OCT method:

- The above calculations are straightforward given the Thevenin resistance formulas for Rthd, Rthgd, and Rthgs

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Rin

RLID

Vin

VoutCgd

Cgs Csb

Cdb

BW ≈ 1Pmj=1RthjCj

=1

(Rthd ||Rd)Cdb +RthgdCgd +RthgsCgsrad/s