Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15
Lecture 15:
Small Signal Modeling
Prof. Niknejad

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Lecture Outline
 Review: Diffusion Revisited
 BJT Small-Signal Model
 Circuits!!!
 Small Signal Modeling
 Example: Simple MOS Amplifier

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Notation Review
 Since we’re introducing a new (confusing) subject, let’s adopt some
consistent notation
 Please point out any mistakes (that I will surely make!)
 Once you get a feel for small-signal analysis, we can drop the notation
and things will be clear by context (yeah right! … good excuse)
( , )
C BE CE
i f v v

Large signal
( , )
C C BE BE CE CE
I i f V v V v
      
small signal
DC (bias)
( , )
C c BE be CE ce
I i f V v V v
   
small signal
(less messy!)
c be ce
BE CE
Q Q
f f
i v v
v v
 
 
 
transconductance Output conductance
( , )
BE CE
Q V V

Quiescent Point
(bias)

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Diffusion Revisited
 Why is minority current profile a linear function?
 Recall that the path through the Si crystal is a zig-zag series
of acceleration and deceleration (due to collisions)
 Note that diffusion current density is controlled by width of
region (base width for BJT):
 Decreasing width increases current!
Density here fixed by potential (injection of carriers)
Physical interpretation: How many electrons (holes) have
enough energy to cross barrier? Boltzmann distribution give
this number.
Wp
Density fixed by
metal contact
Half go left,
half go right

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Diffusion Capacitance
 The total minority carrier charge for a one-sided
junction is (area of triangle)
 For a one-sided junction, the current is dominated
by these minority carriers:
2 , 0 0
1 1
( )( )
2 2
D
qV
kT
n dep p p p
Q qA bh qA W x n e n
     
0 0
,
( )
D
qV
n kT
D p p
p dep p
qAD
I n e n
W x
 

 
2
,
n
D
n p dep p
D
I
Q W x


Constant!

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Diffusion Capacitance (cont)
 The proportionality constant has units of time
 The physical interpretation is that this is the transit
time for the minority carriers to cross the p-type
region. Since the capacitance is related to charge:
 
2
,
p dep p
n
T
D n
W x
Q
I D


 
n T D
Q I

 n
d T d T
Q I
C g
V V
 
 
  
 
Diffusion Coefficient
Distance across
P-type base
 
2
,
p dep p
T
n
W x
q
kT



 Mobility
Temperature

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
BJT Transconductance gm
 The transconductance is analogous to diode
conductance

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Transconductance (cont)
 Forward-active large-signal current:
/
(1 )
BE th
v V
C S CE A
i I e v V
 
• Differentiating and evaluating at Q = (VBE, VCE )
/
(1 )
BE
qV kT
C
S CE A
BE Q
i q
I e V V
v kT

 

C C
m
BE Q
i qI
g
v kT

 


EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
BJT Base Currents
Unlike MOSFET, there is a DC current into the
base terminal of a bipolar transistor:
  /
(1 )
BE
qV kT
B C F S F CE A
I I I e V V
 
  
To find the change in base current due to change
in base-emitter voltage:
1
B B C
m
BE C BE F
Q Q
Q
i i i
g
v i v 
  
 
  
B
b be
BE Q
i
i v
v



m
b be
F
g
i v



EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Small Signal Current Gain
0
C
F
B
i
i
 

 

 Since currents are linearly related, the derivative is a
constant (small signal = large signal)
0
C B
i i

  
0
c b
i i



EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Input Resistance rπ
 
1 1 C m
B
BE F BE F
Q Q
i g
i
r
v v

 
 

  
 
 In practice, the DC current gain F and the small-signal
current gain o are both highly variable (+/- 25%)
 Typical bias point: DC collector current = 100 A
F
m
r
g



25mV
100 25k
.1mA
r   
i
R   MOSFET

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Output Resistance ro
Why does current increase slightly with increasing vCE?
Answer: Base width modulation (similar to CLM for MOS)
Model: Math is a mess, so introduce the Early voltage
)
1
(
/
A
CE
V
v
S
C V
v
e
I
i th
BE 

Base (p)
Emitter (n+)
Collector (n)
B
W

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Graphical Interpretation of ro
slope~1/ro
slope

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
BJT Small-Signal Model
b be
i r v


1
c m be ce
o
i g v v
r
 

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
BJT Capacitors
 Emitter-base is a forward biased junction 
depletion capacitance:
 Collector-base is a reverse biased junction 
depletion capacitance
 Due to minority charge injection into base, we have
to account for the diffusion capacitance as well
, , 0
1.4
j BE j BE
C C

b F m
C g



EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
BJT Cross Section
 Core transistor is the vertical region under the
emitter contact
 Everything else is “parasitic” or unwanted
 Lateral BJT structure is also possible
Core Transistor
External Parasitic

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Core BJT Model
 Given an ideal BJT structure, we can model most of the
action with the above circuit
 For low frequencies, we can forget the capacitors
 Capacitors are non-linear! MOS gate & overlap caps are
linear
m
g v
Base Collector
Emitter
Reverse biased junction
Reverse biased junction &
Diffusion Capacitance
Fictional Resistance
(no noise)

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Complete Small-Signal Model
Reverse biased junctions
“core” BJT
External Parasitics
Real Resistance
(has noise)

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Circuits!
 When the inventors of the bipolar transistor first
got a working device, the first thing they did was to
build an audio amplifier to prove that the transistor
was actually working!

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Modern ICs
 First IC (TI, Jack Kilby, 1958): A couple of transistors
 Modern IC: Intel Pentium 4 (55 million transistors, 3 GHz)
Source: Texas Instruments
Used without permission
Source: Intel Corporation
Used without permission

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
A Simple Circuit: An MOS Amplifier
DS
I
GS
V
s
v
D
R DD
V
GS GS s
v V v
 
o
v
Input signal
Output signal
Supply “Rail”

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Selecting the Output Bias Point
 The bias voltage VGS is selected so that the output is
mid-rail (between VDD and ground)
 For gain, the transistor is biased in saturation
 Constraint on the DC drain current:
 All the resistor current flows into transistor:
 Must ensure that this gives a self-consistent
solution (transistor is biased in saturation)
DD o DD DS
R
D D
V V V V
I
R R
 
 
,
R DS sat
I I

DS GS T
V V V
 

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Finding the Input Bias Voltage
 Ignoring the output impedance
 Typical numbers: W = 40 m, L = 2 m,
RD = 25k, nCox = 100 A/V2, VTn = 1 V,
VDD = 5 V
2
,
1
( )
2
DS sat n ox GS Tn
W
I C V V
L

 
2
,
1
( )
2 2
D
DD
R DS sat n ox GS Tn
D
V W
I I C V V
R L

   
2
2
5V μA 1
100μA 20 100 ( 1)
50k V 2
GS
V
    

2
.1 ( 1)
GS
V
  1.32
GS
V  .32 2.5
GS T DS
V V V
    

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Applying the Small-Signal Voltage
Approach 1. Just use vGS in the equation for the total
drain current iD and find vo
GS GS s
v V v
 
ˆ cos
s s
v v t


2
1
( )
2
O DD D DS DD D n ox GS s T
W
v V R i V R C V v V
L

     
Note: Neglecting charge storage effects. Ignoring
device output impedance.

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Solving for the Output Voltage vO
2
1
( )
2
O DD D DS DD D n ox GS s T
W
v V R i V R C V v V
L

     
2
2
1
( ) 1
2
s
O DD D DS DD D n ox GS T
GS T
v
W
v V R i V R C V V
L V V

 
     
 

 
DS
I
2
1 s
O DD D DS
GS T
v
v V R I
V V
 
  
 

 
2
DD
V

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Small-Signal Case
 Linearize the output voltage for the s.s. case
 Expand (1 + x)2 = 1 + 2x + x2 … last term can be
dropped when x << 1
1
vs
VGS VTn
–
-------------------------
-
+
 
 
 
2
1
2vs
VGS VTn
–
-------------------------
-
vs
VGS VTn
–
-------------------------
-
 
 
 2
+ +
=
Neglect
2
1 s
O DD D DS
GS T
v
v V R I
V V
 
  
 

 

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Linearized Output Voltage
For this case, the total output voltage is:
The small-signal output voltage:
2
1
2
s
DD
O DD
GS T
v
V
v V
V V
 
  
 

 
2
s DD
DD
O
GS T
v V
V
v
V V
 

“DC”
Small-signal output
s DD
o v s
GS T
v V
v A v
V V
  

Voltage gain

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
Plot of Output Waveform (Gain!)
Numbers: VDD / (VGS – VT) = 5/ 0.32 = 16 output
input
mV

EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Department of EECS University of California, Berkeley
There is a Better Way!
 What’s missing: didn’t include device output
impedance or charge storage effects (must solve
non-linear differential equations…)
 Approach 2. Do problem in two steps.
 DC voltages and currents (ignore small signals
sources): set bias point of the MOSFET ... we had
to do this to pick VGS already
 Substitute the small-signal model of the MOSFET
and the small-signal models of the other circuit
elements …
 This constitutes small-signal analysis