Frequency response
Transcript of Frequency response
Chapter 11 Frequency Response
11.1 Fundamental Concepts 11.2 High-Frequency Models of Transistors 11.3 Analysis Procedure 11.4 Frequency Response of CE and CS Stages 11.5 Frequency Response of CB and CG Stages 11.6 Frequency Response of Followers 11.7 Frequency Response of Cascode Stage 11.8 Frequency Response of Differential Pairs 11.9 Additional Examples
1
Chapter Outline
CH 11 Frequency Response 2
CH 11 Frequency Response 3
High Frequency Roll-off of Amplifier
As frequency of operation increases, the gain of amplifier decreases. This chapter analyzes this problem.
Example: Human Voice I
Natural human voice spans a frequency range from 20Hz to 20KHz, however conventional telephone system passes frequencies from 400Hz to 3.5KHz. Therefore phone conversation differs from face-to-face conversation.
CH 11 Frequency Response 4
Natural Voice Telephone System
Example: Human Voice II
CH 11 Frequency Response 5
Mouth RecorderAir
Mouth EarAir
Skull
Path traveled by the human voice to the voice recorder
Path traveled by the human voice to the human ear
Since the paths are different, the results will also be different.
Example: Video Signal
Video signals without sufficient bandwidth become fuzzy as they fail to abruptly change the contrast of pictures from complete white into complete black.
CH 11 Frequency Response 6
High Bandwidth Low Bandwidth
Gain Roll-off: Simple Low-pass Filter
In this simple example, as frequency increases the impedance of C1 decreases and the voltage divider consists of C1 and R1 attenuates Vin to a greater extent at the output.
CH 11 Frequency Response 7
CH 11 Frequency Response 8
Gain Roll-off: Common Source
The capacitive load, CL, is the culprit for gain roll-off since at high frequency, it will “steal” away some signal current and shunt it to ground.
1||out m in D
L
V g V RC s
CH 11 Frequency Response 9
Frequency Response of the CS Stage
At low frequency, the capacitor is effectively open and the gain is flat. As frequency increases, the capacitor tends to a short and the gain starts to decrease. A special frequency is ω=1/(RDCL), where the gain drops by 3dB.
1222
LD
Dm
in
out
CR
Rg
V
V
CH 11 Frequency Response 10
Example: Figure of Merit
This metric quantifies a circuit’s gain, bandwidth, and power dissipation. In the bipolar case, low temperature, supply, and load capacitance mark a superior figure of merit.
LCCT CVVMOF
1...
Example: Relationship between Frequency Response and Step Response
CH 11 Frequency Response 11
2 2 21 1
1
1H s j
R C
0
1 1
1 expoutt
V t V u tR C
The relationship is such that as R1C1 increases, the bandwidth drops and the step response becomes slower.
CH 11 Frequency Response 12
Bode Plot
When we hit a zero, ωzj, the Bode magnitude rises with a slope of +20dB/dec.
When we hit a pole, ωpj, the Bode magnitude falls with a slope of -20dB/dec
21
210
11
11
)(
pp
zz
ss
ss
AsH
CH 11 Frequency Response 13
Example: Bode Plot
The circuit only has one pole (no zero) at 1/(RDCL), so the slope drops from 0 to -20dB/dec as we pass ωp1.
LDp CR
11
CH 11 Frequency Response 14
Pole Identification Example I
inSp CR
11
LDp CR
12
22
221
2 11 pp
Dm
in
out Rg
V
V
CH 11 Frequency Response 15
Pole Identification Example II
inm
S
p
Cg
R
1||
11
LDp CR
12
CH 11 Frequency Response 16
Circuit with Floating Capacitor
The pole of a circuit is computed by finding the effective resistance and capacitance from a node to GROUND.
The circuit above creates a problem since neither terminal of CF is grounded.
CH 11 Frequency Response 17
Miller’s Theorem
If Av is the gain from node 1 to 2, then a floating impedance ZF can be converted to two grounded impedances Z1 and Z2.
v
F
A
ZZ
11v
F
A
ZZ
/112
CH 11 Frequency Response 18
Miller Multiplication
With Miller’s theorem, we can separate the floating capacitor. However, the input capacitor is larger than the original floating capacitor. We call this Miller multiplication.
CH 11 Frequency Response 19
Example: Miller Theorem
FDmSin CRgR
1
1F
DmD
out
CRg
R
1
1
1
High-Pass Filter Response
121
21
21
11
CR
CR
V
V
in
out
The voltage division between a resistor and a capacitor can be configured such that the gain at low frequency is reduced.
CH 11 Frequency Response 20
Example: Audio Amplifier
nFCi 6.79 nFCL 8.39
In order to successfully pass audio band frequencies (20 Hz-20 KHz), large input and output capacitances are needed.
200/1
100
m
i
g
KR
CH 11 Frequency Response 21
Capacitive Coupling vs. Direct Coupling
Capacitive coupling, also known as AC coupling, passes AC signals from Y to X while blocking DC contents.
This technique allows independent bias conditions between stages. Direct coupling does not.
Capacitive Coupling Direct Coupling
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Typical Frequency Response
Lower Corner Upper Corner
CH 11 Frequency Response 23
CH 11 Frequency Response 24
High-Frequency Bipolar Model
At high frequency, capacitive effects come into play. Cb
represents the base charge, whereas C and Cje are the junction capacitances.
b jeC C C
CH 11 Frequency Response 25
High-Frequency Model of Integrated Bipolar Transistor
Since an integrated bipolar circuit is fabricated on top of a substrate, another junction capacitance exists between the collector and substrate, namely CCS.
CH 11 Frequency Response 26
Example: Capacitance Identification
CH 11 Frequency Response 27
MOS Intrinsic Capacitances
For a MOS, there exist oxide capacitance from gate to channel, junction capacitances from source/drain to substrate, and overlap capacitance from gate to source/drain.
CH 11 Frequency Response 28
Gate Oxide Capacitance Partition and Full Model
The gate oxide capacitance is often partitioned between source and drain. In saturation, C2 ~ Cgate, and C1 ~ 0. They are in parallel with the overlap capacitance to form CGS and CGD.
CH 11 Frequency Response 29
Example: Capacitance Identification
CH 11 Frequency Response 30
Transit Frequency
Transit frequency, fT, is defined as the frequency where the current gain from input to output drops to 1.
C
gf mT 2
GS
mT C
gf 2
Example: Transit Frequency Calculation
THGSn
T VVL
f 22
32
GHzf
sVcm
mVVV
nmL
T
n
THGS
226
)./(400
100
65
2
CH 11 Frequency Response 31
Analysis Summary
The frequency response refers to the magnitude of the transfer function.
Bode’s approximation simplifies the plotting of the frequency response if poles and zeros are known.
In general, it is possible to associate a pole with each node in the signal path.
Miller’s theorem helps to decompose floating capacitors into grounded elements.
Bipolar and MOS devices exhibit various capacitances that limit the speed of circuits.
CH 11 Frequency Response 32
High Frequency Circuit Analysis Procedure
Determine which capacitor impact the low-frequency region of the response and calculate the low-frequency pole (neglect transistor capacitance).
Calculate the midband gain by replacing the capacitors with short circuits (neglect transistor capacitance).
Include transistor capacitances. Merge capacitors connected to AC grounds and omit those
that play no role in the circuit. Determine the high-frequency poles and zeros. Plot the frequency response using Bode’s rules or exact
analysis.
CH 11 Frequency Response 33
Frequency Response of CS Stagewith Bypassed Degeneration
1
1
SmbS
bSDm
X
out
RgsCR
sCRRgs
V
V
In order to increase the midband gain, a capacitor Cb is placed in parallel with Rs.
The pole frequency must be well below the lowest signal frequency to avoid the effect of degeneration.
CH 11 Frequency Response 34
CH 11 Frequency Response 35
Unified Model for CE and CS Stages
CH 11 Frequency Response 36
Unified Model Using Miller’s Theorem
Example: CE Stage
fFC
fFC
fFC
mAI
R
CS
C
S
30
20
100
100
1
200
GHz
MHz
outp
inp
59.12
5162
,
,
The input pole is the bottleneck for speed.
CH 11 Frequency Response 37
Example: Half Width CS Stage
XW 2
22
12
1
221
2
1
,
,
XY
Lm
outL
outp
XYLminS
inp
CRg
CR
CRgCR
CH 11 Frequency Response 38
CH 11 Frequency Response 39
Direct Analysis of CE and CS Stages
Direct analysis yields different pole locations and an extra zero.
outinXYoutXYinLThev
outXYLinThevThevXYLmp
outXYLinThevThevXYLmp
XY
mz
CCCCCCRR
CCRCRRCRg
CCRCRRCRg
C
g
1||
1
1||
||
2
1
CH 11 Frequency Response 40
Example: CE and CS Direct Analysis
outinXYoutXYinOOS
outXYOOinSSXYOOmp
outXYOOinSSXYOOmp
CCCCCCrrR
CCrrCRRCrrg
CCrrCRRCrrg
21
212112
212111
||
)(||||1
)(||||1
1
Example: Comparison Between Different Methods
MHz
MHz
outp
inp
4282
5712
,
,
GHz
MHz
outp
inp
53.42
2642
,
,
GHz
MHz
outp
inp
79.42
2492
,
,
KR
g
fFC
fFC
fFC
R
L
m
DB
GD
GS
S
2
0
150
100
80
250
200
1
Miller’s Exact Dominant Pole
CH 11 Frequency Response 41
CH 11 Frequency Response 42
Input Impedance of CE and CS Stages
rsCRgC
ZCm
in ||1
1
sCRgC
ZGDDmGS
in
1
1
Low Frequency Response of CB and CG Stages
miSm
iCm
in
out
gsCRg
sCRgs
V
V
1
As with CE and CS stages, the use of capacitive coupling leads to low-frequency roll-off in CB and CG stages (although a CB stage is shown above, a CG stage is similar).
CH 11 Frequency Response 43
CH 11 Frequency Response 44
Frequency Response of CB Stage
Xm
S
Xp
Cg
R
1||
1,
CCX
YLYp CR
1,
CSY CCC Or
CH 11 Frequency Response 45
Frequency Response of CG Stage
OrX
mS
Xp
Cg
R
1||
1,
SBGSX CCC
YLYp CR
1,
DBGDY CCC
Similar to a CB stage, the input pole is on the order of fT, so rarely a speed bottleneck.
Or
CH 11 Frequency Response 46
Example: CG Stage Pole Identification
111
,1
||
1
GDSBm
S
Xp
CCg
R
22112
, 11
DBGSGDDBm
Yp
CCCCg
Example: Frequency Response of CG Stage
KR
g
fFC
fFC
fFC
R
d
m
DB
GD
GS
S
2
0
150
100
80
250
200
1
MHz
GHz
Yp
Xp
4422
31.52
,
,
CH 11 Frequency Response 47
CH 11 Frequency Response 48
Emitter and Source Followers
The following will discuss the frequency response of emitter and source followers using direct analysis.
Emitter follower is treated first and source follower is derived easily by allowing r to go to infinity.
CH 11 Frequency Response 49
Direct Analysis of Emitter Follower
1
1
2
bsas
sg
C
V
V m
in
out
m
LS
mS
LLm
S
g
C
r
R
g
CCRb
CCCCCCg
Ra
1
CH 11 Frequency Response 50
Direct Analysis of Source Follower Stage
1
1
2
bsas
sg
C
V
V m
GS
in
out
m
SBGDGDS
SBGSSBGDGSGDm
S
g
CCCRb
CCCCCCg
Ra
Example: Frequency Response of Source Follower
0
150
100
80
250
100
200
1
m
DB
GD
GS
L
S
g
fFC
fFC
fFC
fFC
R
GHzjGHz
GHzjGHz
p
p
57.279.12
57.279.12
2
1
CH 11 Frequency Response 51
CH 11 Frequency Response 52
Example: Source Follower
1
1
2
bsas
sg
C
V
V m
GS
in
out
1
22111
22111111
))((
m
DBGDSBGDGDS
DBGDSBGSGDGSGDm
S
g
CCCCCRb
CCCCCCCg
Ra
CH 11 Frequency Response 53
Input Capacitance of Emitter/Source Follower
Lm
GSGDin Rg
CCCCC
1
//
Or
CH 11 Frequency Response 54
Example: Source Follower Input Capacitance
1211
1 ||1
1GS
OOmGDin C
rrgCC
CH 11 Frequency Response 55
Output Impedance of Emitter Follower
1
sCr
RrsCrR
I
V SS
X
X
CH 11 Frequency Response 56
Output Impedance of Source Follower
mGS
GSS
X
X
gsC
sCR
I
V
1
CH 11 Frequency Response 57
Active Inductor
The plot above shows the output impedance of emitter and source followers. Since a follower’s primary duty is to lower the driving impedance (RS>1/gm), the “active inductor” characteristic on the right is usually observed.
CH 11 Frequency Response 58
Example: Output Impedance
33
321 1||
mGS
GSOO
X
X
gsC
sCrr
I
V
Or
CH 11 Frequency Response 59
Frequency Response of Cascode Stage
For cascode stages, there are three poles and Miller multiplication is smaller than in the CE/CS stage.
12
1,
m
mXYv g
gA
XYx CC 2
CH 11 Frequency Response 60
Poles of Bipolar Cascode
111, 2||
1
CCrRS
Xp 121
2
,
21
1
CCC
g CSm
Yp
22,
1
CCR CSL
outp
CH 11 Frequency Response 61
Poles of MOS Cascode
12
11
,
1
1
GDm
mGSS
Xp
Cg
gCR
11
221
2
,
11
1
GDm
mGSDB
m
Yp
Cg
gCC
g
22,
1
GDDBLoutp CCR
Example: Frequency Response of Cascode
KR
g
fFC
fFC
fFC
R
L
m
DB
GD
GS
S
2
0
150
100
80
250
200
1
MHz
GHz
GHz
outp
Yp
Xp
4422
73.12
95.12
,
,
,
CH 11 Frequency Response 62
CH 11 Frequency Response 63
MOS Cascode Example
12
11
,
1
1
GDm
mGSS
Xp
Cg
gCR
3311
221
2
,
11
1
DBGDGDm
mGSDB
m
Yp
CCCg
gCC
g
22,
1
GDDBLoutp CCR
CH 11 Frequency Response 64
I/O Impedance of Bipolar Cascode
sCCrZ in
111 2
1||
sCCRZ
CSLout
22
1||
CH 11 Frequency Response 65
I/O Impedance of MOS Cascode
sCg
gC
Z
GDm
mGS
in
12
11 1
1
sCCRZ
DBGDLout
22
1||
CH 11 Frequency Response 66
Bipolar Differential Pair Frequency Response
Since bipolar differential pair can be analyzed using half-circuit, its transfer function, I/O impedances, locations of poles/zeros are the same as that of the half circuit’s.
Half Circuit
CH 11 Frequency Response 67
MOS Differential Pair Frequency Response
Since MOS differential pair can be analyzed using half-circuit, its transfer function, I/O impedances, locations of poles/zeros are the same as that of the half circuit’s.
Half Circuit
CH 11 Frequency Response 68
Example: MOS Differential Pair
33,
11
331
3
,
1311,
1
11
1
])/1([
1
GDDBLoutp
GDm
mGSDB
m
Yp
GDmmGSSXp
CCR
Cg
gCC
g
CggCR
Common Mode Frequency Response
12
1
SSmSSSS
SSSSDm
CM
out
RgsCR
CRRg
V
V
Css will lower the total impedance between point P to ground at high frequency, leading to higher CM gain which degrades the CM rejection ratio.
CH 11 Frequency Response 69
Tail Node Capacitance Contribution
Source-Body Capacitance of M1, M2 and M3
Gate-Drain Capacitance of M3
CH 11 Frequency Response 70
Example: Capacitive Coupling
EBin RrRR 1|| 222
HzCRr B
L 5422||
1
1111
HzCRR inC
L 9.221
222
CH 11 Frequency Response 71
MHzCRR inD
L 92.621
2212
MHzCR
Rg
S
SmL 4.422
1
11
111
22 1 v
Fin A
RR
Example: IC Amplifier – Low Frequency Design
CH 11 Frequency Response 72
77.3|| 211 inDmin
X RRgv
v
Example: IC Amplifier – Midband Design
CH 11 Frequency Response 73
Example: IC Amplifier – High Frequency Design
)21.1(2
)15.1(
1
)15.2(2
)308(2
2223
2
1
GHz
CCR
GHz
MHz
DBGDLp
p
p
CH 11 Frequency Response 74