Hisham Lopez Engh Burns Rusciano Mallek Simirov Rooney Ri ...
Transcript of Hisham Lopez Engh Burns Rusciano Mallek Simirov Rooney Ri ...
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1 Zechariah
Pettit Daniel
Borgerding Liuchang
Li Andrew
Mun Brian
Crist Difeng
Liu Aimee
Salt Julien Di
Tria
2 Erik Lee
Nick Robbins
Bijan Choobineh
Wing Yi Lwe
Pangzhou Li
Travis Cook
Wentai Wang
Hisham Abbas
3 Ryan Wade
Jiayu Hong
Jean-Francois Burnier
Morgan Hardy
Bodhisatta Pramanik
Nagulapally Spurthi
Alfonso Raymundo
Corey Wright
4 Mohamad Samusdin
Aqila-Sarah Zulkifli
Honghao Liu
Clayton Hawken
Christopher Little
Antonio Montoya
Jaehyuk Han
Logan Heinen
5 Nicholas Riesen
Satvik Shah
Alex McCullough
Wei Shen Theh
Minh Nguyen
Trevor Brown
Zhong Zhang
Mingda Yang
6 Abdussamad
Hisham Brenda Lopez
Benjamin Engh
Blake Burns
Mark Rusciano
Daniel Mallek
Ilya Simirov
Bryce Rooney
EE 330 Class Seating
EE 330
Lecture 21
• Small Signal Analysis
• Small Signal Modelling
Small-Signal Operation
VIN
Q-point
VINQ
VOUTQ
VOUT
Input
Range
Output
Range
• If slope is steep, output range can be much larger than input range
• This can be viewed as voltage gain in the circuit
Review from Last Lecture
Small signal operation of nonlinear circuits
VIN=VMsinωt
VM is small
Nonlinear CircuitVIN VOUT = ?
VIN
tVM
-VM
• Small signal concepts often apply when building amplifiers
• If small signal concepts do not apply, usually the amplifier will not perform
well
• Small signal operation is usually synonymous with “locally linear”
• Small signal operation is relative to an “operating point”
Review from Last Lecture
Amplification with Transistors
An amplifier, electronic amplifier or (informally) amp is an
electronic device that increases the power of a signal.
From Wikipedia: (Oct. 2014)
•It is difficult to increase the voltage or current very much with passive RC circuits
•Voltage and current levels can be increased a lot with transformers but not practical
in integrated circuits
•Power levels can not be increased with passive elements (R, L, C, and Transformers)
• Often an amplifier is defined to be a circuit that can increase power levels (be careful
with Wikipedia and WWW even when some of the most basic concepts are discussed)
• Transistors can be used to increase not only signal levels but power levels to a load
• In transistor circuits, power that is delivered in the signal path is supplied by a
biasing network
Review from Last Lecture
Consider the following MOSFET and BJT Circuits
R1
Q1
VIN(t)
VOUT
VCC
VEE
BJT MOSFET
R1
VIN(t)
VOUT
VDD
VSS
M1
• MOS and BJT Architectures often Identical
• Circuit are Highly Nonlinear
• Nonlinear Analysis Methods Must be used to analyze these and almost any
other nonlinear circuit
Review from Last Lecture
Small signal analysis using nonlinear models
RVVtV2L
WμCVV TSSM
OXDDOUT
2sin
VDD
R
M1
VIN
VOUT
VSS
Assume M1 operating in saturation region
By selecting appropriate value of VSS, M1
will operate in the saturation region
VIN=VMsinωt
VM is small
VIN
tVM
-VM
2
OX
D IN SS T
μC WI V -V -V
2L
OUT DD DV =V -I R
2
OX
OUT DD IN SS T
μC WV V V -V -V R
2L
2TSSOX
DQ VV2L
WC μI
Termed Load Line
Review from Last Lecture
Small signal analysis example
RVVtV2L
WμCVV TSSM
OXDDOUT
2sin
VDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
VM is small
RVV
tV1-VV
2L
WμCVV
TSS
MTSS
OXDDOUT
2
2 sin
Recall that if x is small 2
1+x 1+2x
2 sinOX MOUT DD SS T
SS T
μC W 2V tV V V V 1- R
2L V V
2 2OX OX M
OUT DD SS T SS T
SS T
μC W μC W 2V sin tV V V V R V V R
2L 2L V V
2
OX OX
OUT DD SS T SS T M
μC W μC WV V V V R V V R V sin t
2L L
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
2
OX OX
OUT DD SS T SS T M
μC W μC WV V V V R V V R V sin t
2L L
By selecting appropriate value of VSS, M1
will operate in the saturation region
Assume M1 operating in saturation region
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
2
OX OX
OUT DD SS T SS T M
μC W μC WV V V V R V V R V sin t
2L L
Quiescent Output ss Voltage Gain
OX
v SS T
μC WA V V R
L
Assume M1 operating in saturation region
2
OX
OUTQ DD SS T
μC WV V V V R
2L
OUT OUTQ V MV V A V sin t
Note the ss voltage gain is negative since VSS+VT<0!
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
OX
v SS T
μC WA V V R
L
But – this expression gives little insight into how large the gain is !
Assume M1 operating in saturation region
And the analysis for even this very simple circuit was messy!
2
OX
OUTQ DD SS T
μC WV V V V R
2L
OUT OUTQ V MV V A V sin t
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωtVIN
tVM
-VM
VM=0t
VDD
VSS
VOQ
OUT OUTQ V MV V A V sin t
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt OX
v SS T
μC WA V V R
L
VIN
tVM
-VM
VM t
VDD
VSS
VOQ
VOUT
OUT OUTQ V MV V A V sin t
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt OX
v SS T
μC WA V V R
L
VIN
tVM
-VM
VM t
VDD
VSS
VOQ
VOUT
OUT OUTQ V MV V A V sin t
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
OX
v SS T
μC WA V V R
L
VIN
tVM
-VM
VM t
VDD
VSS
VOQ
VOUT
Serious Distortion occurs if signal is too large or Q-point non-optimal
Here “clipping” occurs for high VOUT
OUT OUTQ V MV V A V sin t
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt OX
v SS T
μC WA V V R
L
VIN
tVM
-VM
VM t
VDD
VSS
VOQ
VOUT
Serious Distortion occurs if signal is too large or Q-point non-optimal
Here “clipping” occurs for low VOUT
OUT OUTQ V MV V A V sin t
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
OX
v SS T
μC WA V V R
L
DQ
v
SS T
2I RA
V V
Thus, substituting from the expression for IDQ we obtain
2TSSOX
DQ VV2L
WC μI But recall:
Note this is negative since VSS+VT < 0
OUT OUTQ V MV V A V sin t
Small signal analysis example
VDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
DQ
v
SS T
2I RA
V V
Observe the small signal voltage gain is twice the
Quiescent voltage across R divided by VSS+VT
Can make |AV |large by making |VSS+VT |small
• This analysis which required linearization of a nonlinear output voltage is quite
tedious.
• This approach becomes unwieldy for even slightly more complicated circuits
• A much easier approach based upon the development of small signal models
will provide the same results, provide more insight into both analysis and
design, and result in a dramatic reduction in computational requirements
Small signal analysis example
VDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
OUT OUTQ V MV V A V sin t
However, there are invariably small errors in this analsis
OUT OUTQ V M
V V A V sin t + ε t
RVVtV2L
WμCVV TSSM
OXDDOUT
2sin
To see the effects of the approximations consider again
222 sin 2 sinOXOUT DD SS T M SS T
μC RWV V t V V V t+ V V
2LMV
21 cos 2
2 sin2
2OXOUT DD M SS T M SS T
μC RW tV V V V V V t+ V V
2L
2
sin cos 22
22OX OX OXM
OUT DD SS T SS T M M
μC RW μC W μC RWVV V + V V V V R V t V t
2L L 4L
Note presence of second harmonic distortion term !
(Consider what was neglected in the previous analysis)
recall
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωtOUT OUTQ V M
V V A V sin t
OUT OUTQ V M
V V A V sin t + ε t
2
sin cos 22
22OX OX OXM
OUT DD SS T SS T M M
μC RW μC W μC RWVV V + V V V V R V t V t
2L L 4L
2
2
2
OX MOUTQ DD SS T
μC RW VV V + V V
2L
OXV SS T
μC WA V V R
L
OX2 M
μC RWA V
4L
Nonlinear distortion term
sin cos 2OUT OUTQ V M 2 MV V A V t A V t
Small signal analysis exampleVDD
R
M1
VIN
VOUT
VSS
VIN=VMsinωt
2
2
2
OX MOUTQ DD SS T
μC RW VV V + V V
2L
OXV SS T
μC WA V V R
L
OX2 M
μC RWA V
4L
Nonlinear distortion term
sin cos 2OUT OUTQ V M 2 MV V A V t A V t
Total Harmonic Distortion:
Recall, if then 0
sink kb kωT+k
x t
2
1
2
kb
THDb
k
Thus, for this amplifier, as long as M1 stays in the saturation region
2 OX
M2 M 2 M
OXV M V SS TSS T
μC WRVA V A V4LTHD
μC WA V A 4 V +VR V +V
L
Distortion will be much worse (larger and more harmonic terms) if M1 leaves saturation region.
Distortion will be small for VM<<|VSS+VT|
Consider the following MOSFET and BJT Circuits
R1
Q1
VIN(t)
VOUT
VCC
VEE
BJT MOSFET
R1
VIN(t)
VOUT
VDD
VSS
M1
One of the most widely used amplifier architectures
• Analysis was very time consuming
• Issue of operation of circuit was
obscured in the details of the analysis
Consider the following MOSFET and BJT Circuits
R1
Q1
VIN(t)
VOUT
VCC
VEE
BJT MOSFET
R1
VIN(t)
VOUT
VDD
VSS
M1
One of the most widely used amplifier architectures
Small signal analysis using nonlinear models
R1
Q1
VOUT
VCC
VEE
VIN(t)
EE
t
-V
V
CQ S EI J A e
Assume Q1 operating in forward active region
By selecting appropriate value of VSS, M1
will operate in the forward active region
VIN=VMsinωt
VM is small
VIN
tVM
-VM
IN EE
t
V -V
V
C S EI J A e
OUT CC C 1V =V -I R
IN EE
t
V -V
V
OUT CC S E 1V V J A R e
M EE
t
V sin t -V
V
OUT CC S E 1V V J A R e
Small signal analysis using nonlinear models
R1
Q1
VOUT
VCC
VEE
VIN(t)
EE
t
-V
V
CQ S EI J A e
VIN=VMsinωt
VM is small
M EE
t
V sin t -V
V
OUT CC S E 1V V J A R e
MEE
t t
V sin t-V
V V
OUT CC S E 1V V J A R e e
EE
t
-V
V M
OUT CC S E 1
t
V sin tV V J A R e 1+
V
EE EE
t t
-V -V
V V M
OUT CC S E 1 S E 1
t
V sin tV V J A R e J A R e
V
Recall that if x is small xe 1+x (truncated Taylor’s series)
\
Small signal analysis using nonlinear models
R1
Q1
VOUT
VCC
VEE
VIN(t)EE
t
-V
V
CQ S EI J A e
VIN=VMsinωt
VM is small
EE EE
t t
-V -V
V V M
OUT CC S E 1 S E 1
t
V sin tV V J A e R J A e R
V
1CQ
OUT CC 1 M
IV V R V sin t
CQ
t
RI
V
Quiescent Output
ss Voltage Gain
Comparison of Gains for MOSFET and BJT Circuits
R
Q1
VIN(t)
VOUT
VCC
VEE
BJT MOSFET
R1
VIN(t)
VOUT
VDD
VSS
M1
DQ
VM
SS T
2I RA
V V
CQ 1
VB
t
I RA
V
If IDQR =ICQR1=2V, VSS+VT= -1V, Vt=25mV
0CQ 1
VB
t
I R 2VA - =-8
V 25mV
DQ
VM
SS T
2I R 4VA = - 4
V V -1V
Observe AVB>>AVM
Due to exponential-law rather than square-law model
Operation with Small-Signal Inputs
• Analysis procedure for these simple circuits was very tedious
• This approach will be unmanageable for even modestly more complicated
circuits
• Faster analysis method is needed !
End of Lecture 21
Small-Signal Analysis
IOUTSS
VOUTSS
VINSS or IINSS
Biasing (voltage or current)
Nonlinear
Circuit
IOUTSS
VOUTSS
VINSS or IINSSLinear Small
Signal Circuit
• Will commit next several lectures to developing this approach
• Analysis will be MUCH simpler, faster, and provide significantly more insight
• Applicable to many fields of engineering
Simple dc Model
Small
Signal
Frequency
Dependent Small
Signal
Better Analytical
dc Model
Sophisticated Model
for Computer
Simulations
Simpler dc Model
Square-Law Model
Square-Law Model (with extensions for λ,γ effects)
Short-Channel α-law Model
BSIM Model
Switch-Level Models
• Ideal switches
• RSW and CGS
Small-Signal Analysis
Operation with Small-Signal Inputs
Why was this analysis so tedious?
What was the key technique in the analysis that was used to obtain a simple
expression for the output (and that related linearly to the input)?
Because of the nonlinearity in the device models
ME E
t t
V sin t-V
V V
O U T C C S E 1V V J A R e e
1CQ
OUT CC 1 M
IV V R V sin t
CQ
t
RI
V
Linearization of the nonlinear output expression at the operating point
Operation with Small-Signal Inputs
1CQ
OUT CC 1 M
IV V R V sin t
CQ
t
RI
V
Quiescent Output
ss Voltage Gain
E E
t
-V
V
C Q S EI J A e
Small-signal analysis strategy
1. Obtain Quiescent Output (Q-point)
2. Linearize circuit at Q-point instead of linearize the nonlinear solution
3. Analyze linear “small-signal” circuit
4. Add quiescent and small-signal outputs to obtain good approximation
to actual output
Small-Signal Principley
x
Q-point
XQ
YQ
Nonlinear function
y=f(x)
Small-Signal Principley
x
Q-point
XQ
YQ
Region around
Q-Point
y=f(x)
Small-Signal Principley
x
Q-point
XQ
YQ
Region around
Q-Point
Relationship is nearly linear in a small enough region around Q-point
Region of linearity is often quite large
Linear relationship may be different for different Q-points
y=f(x)
Small-Signal Principley
x
Q-point
XQ
YQ
Region around
Q-Point
Relationship is nearly linear in a small enough region around Q-point
Region of linearity is often quite large
Linear relationship may be different for different Q-points
y=f(x)
Small-Signal Principle y
x
Q-point
XQ
YQxss
yss
Device Behaves Linearly in Neighborhood of Q-Point
Can be characterized in terms of a small-signal coordinate system
y=f(x)
Small-Signal Principle
QQ X
Q
X=
f=
x
y-y
x-x
Qx
Q Q
=x
f
xy - y x - x
y
x
Q-point
XQ
YQ
XQINT
y=f(x)
(x,y)
(xQ,yQ) or (xQ,yQ)
y=mx+b
Q Q
Q Q
x=x x=x
y yf f
xx x
x
Qx=x
mf
x
Qx
Q Q
=x
f
xy - y x - x
Small-Signal Principle y
x
yQ
xQ
xSS
ySS
Q-point y=f(x)
Changing coordinate systems:
ySS=y-yQ
xSS=x-xQ
Q
Q Q
x=x
fy - y x - x
x
Q
SS SS
x=x
fy x
x
Small-Signal Principle y
x
yQ
xQ
xSS
ySS
Q-point y=f(x)
Q
SS SS
x=x
fy x
x
• Linearized model for the nonlinear function y=f(x)
• Valid in the region of the Q-point
• Will show the small signal model is simply Taylor’s series expansion
of f(x) at the Q-point truncated after first-order terms
Small-Signal Model:
Small-Signal Principle
y
x
yQ
xQ
xSS
ySS
Q-point y=f(x)
Q
SS SS
x=x
fy x
x
Mathematically, linearized model is simply Taylor’s series expansion of the nonlinear
function f at the Q-point truncated after first-order terms with notation xQ=x0
Small-Signal Model:
Q
Q Q
x=x
fy - y x - x
x
Q
Q Q
x=x
fy f x x - x
x
Q
Q Q
x=x
fy f x x - x
x
Observe:
Recall Taylors Series Expansion of nonlinear function f at expansion point x0
0
k0 0
x=xk=1
1 dfy=f x + x-x
k! dx
Q Q
y f x
Truncating after first-order terms (and defining “o” as “Q”):
Q
SS SS
x=x
fy x
x
Small-Signal Principle y
x
yQ
xQ
xSS
ySS
Q-point y=f(x)
Q
Q SS
x=x
fy f x x
x
Quiescent Output
ss Gain
How can a circuit be linearized at an operating point as an alternative to
linearizing a nonlinear function at an operating point?
V Nonlinear
One-Port
IConsider arbitrary nonlinear one-port network
Arbitrary Nonlinear One-Port
yi VQV=V
defn I
Vy
d f
SS
e
= i i
d
SS
ef
= v V
V
IQ
I
Q-point
VQ
iSS
vSS
Q
SS S
V=
S
V
I
Vi v
Linear model of the nonlinear
device at the Q-point
V Nonlinear
One-Port
I
I = f V
yV
iA Small Signal Equivalent Circuit:
I
V
2-Terminal
Nonlinear
Device
f(v)
Arbitrary Nonlinear One-Port
QV=V
Iy
V
yi V
• The small-signal model of this 2-terminal electrical network is a resistor of value 1/y
or a conductor of value y
• One small-signal parameter characterizes this one-port but it is dependent on Q-
point
• This applies to ANY nonlinear one-port that is differentiable at a Q-point (e.g. a diode)
V Nonlinear
One-Port
I
Linear small-signal model:
Small-Signal Principle
Goal with small signal model is to predict
performance of circuit or device in the
vicinity of an operating point (Q-point)
Will be extended to functions of two and
three variables (e.g. BJTs and MOSFETs)
Solution for the example of the previous lecture was based upon solving the
nonlinear circuit for VOUT and then linearizing the solution by doing a Taylor’s
series expansion
• Solution of nonlinear equations very involved with two or more
nonlinear devices
• Taylor’s series linearization can get very tedious if multiple nonlinear
devices are present
Standard Approach to small-signal
analysis of nonlinear networks
1. Solve nonlinear network
2. Linearize solution
1.Linearize nonlinear devices (all)
2. Obtain small-signal model from
linearized device models
3. Replace all devices with small-signal
equivalent
4 .Solve linear small-signal network
Alternative Approach to small-signal
analysis of nonlinear networks
Alternative Approach to small-signal
analysis of nonlinear networks
• Must only develop linearized model once for any
nonlinear device (steps 1. and 2.)
e.g. once for a MOSFET, once for a JFET, and once for a BJT
Linearized model for nonlinear device termed “small-signal model”
derivation of small-signal model for most nonlinear devices is less complicated than
solving even one simple nonlinear circuit
• Solution of linear network much easier than solution of
nonlinear network
1.Linearize nonlinear devices
2. Obtain small-signal model from
linearized device models
3. Replace all devices with small-signal
equivalent
4 .Solve linear small-signal network
Alternative Approach to small-signal analysis of
nonlinear networks
1.Linearize nonlinear devices
2. Obtain small-signal model from
linearized device models
3. Replace all devices with small-signal
equivalent
4 .Solve linear small-signal network
The “Alternative” approach is used almost exclusively
for the small-signal analysis of nonlinear networks
“Alternative” Approach to small-signal
analysis of nonlinear networks
Nonlinear
Network
dc Equivalent
Network
Q-point
Values for small-signal parameters
Small-signal (linear)
equivalent network
Small-signal output
Total output
(good approximation)
Nonlinear
Device
Linearized
Small-signal
Device
Linearized nonlinear devices
This terminology will be used in THIS course to emphasize difference
between nonlinear model and linearized small signal model
VDD
R
M1
VIN
VOUT
VSS
RM1
VIN
VOUT
Nonlinear network
Linearized small-
signal network
Example:
It will be shown that the nonlinear circuit has the linearized small-signal network given
Linearized Circuit Elements
Must obtain the linearized circuit element for ALL linear and nonlinear
circuit elements
(Will also give models that are usually used for Q-point calculations : Simplified dc models)
VDC
IDC
RC
Large
C
SmallL
LargeL
Small
IAC
VAC
Small-signal and simplified dc equivalent elements
Element ss equivalentSimplified dc
equivalent
VDCVDC
IDC IDC
R RR
dc Voltage Source
dc Current Source
Resistor
VACVACac Voltage Source
ac Current Source IAC IAC
Small-signal and simplified dc equivalent elements
C
Large
C
Small
L
Large
L
Small
C
L
Simplified
Simplified
Capacitors
Inductors
MOS
transistors
Diodes
Simplified
Element ss equivalentSimplified dc
equivalent
(MOSFET (enhancement or
depletion), JFET)
Element ss equivalent
Dependent
Sources
(Linear)
Simplified
Simplified
Bipolar
Transistors
Simplified dc
equivalent
VO=AVVIN IO=AIIIN VO=RTIIN IO=GTVIN
Small-signal and simplified dc equivalent elements
Example: Obtain the small-signal equivalent circuit
VDD
VINSS
C
R3
R1
R2
VOUT
C is large
VIN
R3
R1
R2
VOUTVIN
R1
R2//R3
VOUT
Example: Obtain the small-signal equivalent circuit
VDD
R
M1
VIN
VOUT
VSS
R
M1
VIN
VOUT RM1
VIN
VOUT
Example: Obtain the small-signal equivalent circuit
VINSS
VDD
M1
VOUT
VSS
C1
R1
R2
R4 R5
R3 R6
RL
C2
C4
C3
R7
C1,C2, C3 large
C4 small
IDD
Q1
VIN
M1
VOUT
R1
R2
R4 R5
R3 R6
RL
C4
R7
Q1M1
VOUT
R1//R2
R4
R5
R6
RL
C4
R7
Q1
VIN
How is the small-signal equivalent circuit
obtained from the nonlinear circuit?
What is the small-signal equivalent of the
MOSFET, BJT, and diode ?
yV
i
A Small Signal Equivalent Circuit
I
V
2-Terminal
Nonlinearl
Device
f(x)
Small-Signal Diode Model
QV=V
Iy
V
yi V
-1D
dD Q
IR
V
Thus, for the diode
Small-Signal Diode Model
-1D
dD Q
IR
V
For the diode
D
t
VV
D SI =I e
D
t
VV DQD
St tQ
II 1= I e
V VDQ
V
td
DQ
VR =
I
Example of diode circuit where small-signal
diode model is useful
Voltage Reference
D1D2
VREF
R0
VD2
I1 I2
ID1 ID2VD1
R1 R2
VX
RD1 RD2
VREF
R0
R1 R2
VX
Small-signal model of
Voltage Reference (useful for compensation
when parasitic Cs included)
End of Lecture 21