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)
2π
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)
2π
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)
2π
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)
2π
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)
2π
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)
2π
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)
2π
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
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