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High-Frequency Analysis of Circuits
Asst. Prof. MONTREE SIRIPRUCHYANUN, D. Eng.
Dept. of Teacher Training in Electrical Engineering,
Faculty of Technical Education
King Mongkut’s Institute of Technology North Bangkok
http://www.te.kmitnb.ac.th/msn
mts@kmitnb.ac.th
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Overview
•
Background
– So far, our treatment of small-signal analysis of amplifiers has been
for low frequencies where internal capacitances do not affect
operation. However, we will see that internal capacitances do exist.
We will see how these capacitances affect the frequency response
of amplifiers and why we modeled real op amps to have a
frequency response with a single dominant pole.
To fully understand and model the frequency response of
amplifiers, we utilize Bode plots again. We will look at two methods
to facilitate the high-frequency circuit analysis. One technique
called open-circuit time constant (OCT) method approximates the
high-frequency response by calculating the time constant
associated with each capacitor in a circuit. Miller’s theorem
provides a way to deal with bridging capacitors (between input and
output).
– In this lecture, we will focus on understanding and analyzing the
frequency response of MOSFET-based amplifiers. High-frequency
analysis of BJT-based amplifiers are similar.
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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High-Frequency MOSFET Model
•
Let’s review the MOSFET high-frequency model
complete
simplified
– when source is connected to the body, model is simplified
(remove Csb)
– in saturation, Cgb ≅ 0
– further simplify model by removing Cdb for hand calculations
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224510 Advanced coomunication circuit design
High-Frequency BJT Model
Cμ
rb
B
C
gmvπ
Cπ
rπ
rο
vπ
E
•
In BJTs, the PN junctions (EBJ and CBJ) also have capacitances associated with them
– Cμ is the reverse-biased CB junction
•
Where mjc is between 0.2 and 0.5, V0 is between 0.5V and 1V
–
Cπ represents the capacitance of the forward-biased EBJ which exhibits both the
junction cap and diffusion cap
–
At high frequencies, base resistance (rb) also plays an important role in device
operation
•
Where Cje is the junction cap and τF is the base-transit time
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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Amplifier Transfer Function
|A| (dB)
|A| (dB)
A0
ωH
•
•
•
•
AM
-3dB
ω
-3dB
ωL
ωH
ω
Voltage-gain frequency response of amplifiers seen so far take one of two forms
– Direct-Coupled (DC) amplifiers exhibit low-pass characteristics
Æ flat gain from DC to ωH
– Capacitively-coupled amplifiers exhibit band-pass characteristics – attenuation at low
frequency due to impedance from coupling capacitances increasing for low frequencies
We will focus on the high-frequency portion of the response (ωH)
– Gain drops due to effects of internal capacitances of the device
Bandwidth is the frequency range over which gain is flat
– BW = ωH or ωH-ωL ≈ ωH (ωH >> ωL)
Gain-Bandwidth Product (GB) – Amplifier figure of merit
– GB ≡ AMωH
where AM is the midband gain
– We will see later that it is possible to trade off gain for bandwidth
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Gain Function A(s)
•
We can represent the frequency dependence of gain with the following
expression:
– Where FL(s) and FH(s) are the functions that account for the frequency
dependence of gain on frequency at the lower and upper frequency ranges
• We will use open-circuit time constant method to find FH(s)
• Assume FL(s) = 1 for now
– We can solve for AM by assuming that large coupling capacitors are short
circuits and internal device capacitances are open circuits (what we have
done so far for low-frequency small-signal analysis)
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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High-Frequency Response
•
We can express function FH(s) with the general form:
•
•
Where ωP and ωZ represent the frequencies of high-frequency poles and zeros
The zeros are usually at infinity or sufficiently high frequencies such that the numerator Æ 1
and assuming there is one dominant pole (other poles at much higher frequencies), we can
approximate the function as…
•
– This simplifies the determination of the BW or ωH
If a dominant pole does not exist, the upper -3dB frequency ωH can be found from a plot of
|FH(jω)|. Alternatively, we can approximate with the following formula
–
Note: if ωP1 is a dominant pole, then reduces to ωH=ωP1
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Open-Circuit Time Constant Method
•
It may be difficult to find the poles and zeros of the system (which is usually the
case). We can find approximate values of ωH using the following method.
– We can multiply out factors and represent FH(s) in an alternative form:
– Where a and b are coefficients related to the zero and pole frequencies
– We can show that
and b1 can be obtained by considering the various capacitances in the highfrequency equivalent circuit one at a time while reducing all other capacitors
to zero (or open circuits); and calculating (summing) the RC time constants
due to the circuit associated with each capacitor.
– This is called the open-circuit time constant (OCT) method
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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Calculating OCTs
The approach:
• For each capacitor:
– set input signal to zero
– replace all other capacitors with open circuits
– find the effective resistance (Rio) seen by the capacitor Ci
• Sum the individual time constants (RCs or also called the open-circuit time
constants)
•
•
This method for determining b1 is exact. The approximation comes from using
this result to determine ωH.
– This equation yields good results even if there is no single dominant pole
but when all poles are real
We will see an example of this method when we analyze the high-frequency
response of different amplifier topologies
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224510 Advanced coomunication circuit design
Miller’s Theorem
•
Let’s review Miller’s Thoeorem
Consider the circuit network below (left) with two nodes, 1 and 2. An admittance Y (Y=1/Z)
is connected between the two nodes and these nodes are also connected to other nodes in
the network. Miller’s theorem provides a way for replacing the “bridging” admittance Y with
two admittances Y1 and Y2 between node 1 and gnd, and node 2 and gnd.
1
V1
–
–
I1
Y
I2
2
1 I1
V2
V1
I2 2
Y1
The relationship between V2 and V1 is given by K=V2/V1
To find Y1 and Y2
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
Y2
V2
Caveat:
The Miller equivalent
circuit is valid only as
long as the conditions
that existed in the
network when K was
determined are not
changed.
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High-Frequency Response of CS Amp
•
Let’s now find the high-frequency response of a common-source amplifier
– First, redraw using a high-frequency small-signal model for the nMOS
•
There are three ways to find the upper 3-dB frequency ωH
– Use open-circuit time constant method
– Use Miller’s theorem
– Brute force calculations to find vout/vin
Let’s investigate them all
•
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Using OCT on CS Amplifier
•
Find the RC time constants associated with Cgd and Cgs in the following circuit
C gd
Rs
g mvgs
vi
•
vgs
C gs
R L||r o
vo
Replace Cgd with an open circuit and find the resistance seen by Cgs
Rs
gmvgs
vtst
•
RL||ro
Itst
vo
Replace Cgs with an open-ckt and find the resistance seen by Cgd
itst
Rs
vtst
vgs
gmvgs
RL||ro
vo
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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Using OCTs Cont’d
•
Summing the two time constants yields ωH
– From the above equation, it is not difficult to imagine that Cgd has a more
significant effect on reducing BW
– The resulting frequency dependence of gain is…
•
Let’s compare this result with what we get using Miller’s theorem
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224510 Advanced coomunication circuit design
Using Miller’s Theorem on CS Amplifier
•
Redraw the high-frequency small-signal model using Miller’s theorem
Rs
Cgd(1+gmRL')
vi
vgs
gmvgs
RL'
Cgs
vo
Cgd[1+1/(gmRL')]
~= Cgd
CT
–
•
•
Assuming a dominant pole introduced by Cgd in parallel with Cgs
– Miller multiplication of Cgd results in a large input capacitance
Notice that this approximation for ωH is close to the approximation found using OCT
assuming that RsCgd(1+gmRL’) dominates
Let’s verify these two approaches by deriving the exact high-frequency transfer function of
the CS amplifier
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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High-Frequency Response of CS Amplifier
•
Replace the input source and series resistance with a Norton equivalent
sCgd(vgs-vo)
Cgd
vi/Rs
gmvgs
Rs
vgs
Cgs
RL'
vo
Notice zero is negative. This
has implications on stability
(we will see later)
•
– The exact solution gives a zero (at a high frequency) and two poles
– Notice that the s term (in the denominator) is the same as the solution using
the OCT method
Unfortunately, the denominator is too complicated to extract useful info as is.
So, assuming the two poles are widely separated (greater than an order of
magnitude), we can rewrite the expression for the denominator as…
224510 Advanced coomunication circuit design
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HF Response of CS Amplifier
•
Rewrite the denominator as:
– And from the solution on the previous slide we can write…
•
So the second pole is usually at a much higher frequency and we can assume a
dominant pole (ωP1)
•
Using either Miller’s theorem or OCTs enables a way to quickly find
approximations of the amplifier’s high-frequency response
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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Summary of 3 Approaches
•
To summarize, here are the dominant pole frequencies that we
found using the three approaches
– OCT method
– Miller Theorem
– Brute Force
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224510 Advanced coomunication circuit design
Frequency Response of CG Amplifier
•
One way to avoid the frequency limitations of Miller multiplication of Cgd is to utilize
a CG amplifier configuration
Cgd
Rs
vs
•
-(gm+gmb)vx
Cgs
vx
Cdb
-(gm+gmb)vx
RD
Csb
CD=Cgd+Cdb
Rs
vs
vx
RD
CS=Cgs+Csb
Using OCT method, we find two time constants
– At the input (source node)
– At the output (drain node)
•
•
The output drives additional load capacitance such that the output pole is dominant
The frequency response of CG amplifiers is then combined with a CS stage to
build a cascode circuit
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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Cascode Amplifier
•
Cascoding enables high bandwidth by suppressing Miller multiplication of Cgd. Let’s
investigate how with the following high-frequency model of a cascode stage.
RL
vOUT
Cgd2
-gm2+gmb2vx
Cdb2+CL
Cdb2+CL
Vb
Cgd1
Cgs2
Rs
Rs
vIN
Cdb1+Csb2
Cgd1
gmvgs1
vin
vx
Cgs1
Cgs1
vout
RL
Cdb1+Csb2+Cgs2
– Use OCT method to find the time constants associated with each capacitor.
The time constant associated with Cgd1 is…
NOTICE:
Miller multiplication is ~2
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224510 Advanced coomunication circuit design
Frequency Response of Source Followers
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Start with a high-frequency small-signal model of the source follower circuit
Rs
Rs
vOUT
vIN
vin
Cgd
vgs
Cgs
CL
gmvgs
vout
CL
•
Directly solving for vout/vin yields:
•
– The zero is due to Cgs that directly couples the signal from the input to the output
If poles are far apart, then the s term represents the dominant pole
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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More on Source Follower
•
Other important aspects of a source follower are its input and output
impedances (since they are often used as buffers)
Let’s calculate the input impedance using the high-freq small-signal
models
•
Zin
Cgs
vgs
gmvgs
vout
1/gmb
CL
Now calculate the output impedance (ignoring gmb for simplicity)
•
Rs
–
–
–
At low frequency, Zout ≈ 1/gm
At high frequency, Zout ≈ Rs
Shape of the response depends on the relative size of Rs and 1/gm
|Zout|
Cgs
vgs
gmvgs
vout
|Zout|
1/gm
Rs
Rs
1/gm
Zout
ω
Zout can look inductive
or capacitive depending
on Rs and 1/gm
ω
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224510 Advanced coomunication circuit design
Differential Pair
•
We have seen that a symmetric differential amplifier can be analyzed with a
differential half circuit. This still holds true for high-frequency small-signal
analysis.
RD
+vd/2
-vd/2
Rs
RD
vout
Rs
Cgd
gmvgs
Rs
vd/2
Cgs
Cdb
RD
vout
I
– The response is identical to that of a common-source stage
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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High-Frequency CMRR
•
The CMRR of a differential pair degrades at high frequency due to a
number of factors. The most important is the increase in CM gain with
frequency due to capacitance on the tail node.
Use the common-mode equivalent half circuit to understand how CM
gain increases with frequency
– Draw the small-signal equivalent model and see the effect of CTAIL
on the vout/vin transfer function
•
RD
vout,CM
vin,CM
2ro
vin,CM
CTAIL
I/2
gmvgs
vgs
Zero at ωZ = 1/roCTAIL (since ro is big, ωZ occurs at a low frequency)
There are additional poles at higher frequencies due to CTAIL and
other internal capacitances (that we have ignored)
The zero causes the CM gain to increase with frequency until the higher
frequency poles kick in Æ CMRR degrades due to the zero
–
–
•
vout,CM
RD
2ro
CTAIL
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224510 Advanced coomunication circuit design
HF CMRR plots
•
The impact of the zero in the CM gain on CMRR can
be illustrated as shown
– Remember CMRR = Ad/Acm
|Acm| (dB)
ωZ
ω (log scale)
|Ad| (dB)
•
There is a trade off between CMRR and voltage
headroom
– Wider current source devices enable lower vds
– Wider current source device means larger CTAIL
ωP
ω (log scale)
-20dB/dec
CMRR (dB)
-40dB/dec
ωZ
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
ωP
ω (log scale)
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Example: Diff Pair with Active Load
Q3
Q4
y
z
Q1
+vd /2
• diff half circuit Æ node x is virtual gnd
Vo
• Cy = Cgd1 + Cdb1 + Cgs3 + Cdb3 + Cgs4 + Cpar
Cload
Q2
• Cz = Cgd2 + Cdb2 + Cgd4 + Cdb4 + Cload
x
-vd /2
• Ro = ro4 // ro2
Q5
Vy =
gm1 Vid
1
2
Io =
(gm3 + sCy)
A = gm1 Ro
gm2 Vid
2
+ gm3vy
Vo = RoIo
(1 + sCz/2gm)
1
(1 + sCzRo) (1 + sCz/gm)
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224510 Advanced coomunication circuit design
Example: 2 Stage CMOS Op Amp
Q3
+vd /2
Q4
Q1
Cx
Q6
Gm1 R1
Cc
Q2
C1
Gm2 R2
C2
Cload
-vd /2
Q5
Q7
• C1= Cgd2 + Cdb2+ Cgd4 + Cdb4+ Cgs6
• C2= Cgd7 + Cdb7 + Cdb6+ Cload
• Cx= Cc + Cgd6
A = Gm1R1 Gm2R2
Using OCT:
next pole is at C2/Gm2
• Gm2=gm6
• R1= ro2 // ro4
• R2= ro6 // ro7
• R1= ro2 // ro4
dominant pole is at (Gm2R2)CxR1
unity gain freq is at Gm1/Cx
• Gm1=gm1 =gm2
How do you determine the phase margin?
224510 Advanced coomunication circuit design
Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN
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