lecture 15: small signal modeling
Post on 14-Jan-2016
67 Views
Preview:
DESCRIPTION
TRANSCRIPT
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15
Lecture 15:
Small Signal Modeling
Prof. Niknejad
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Lecture Outline
Review: Diffusion Revisited BJT Small-Signal Model Circuits!!! Small Signal Modeling Example: Simple MOS Amplifier
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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 CEi f v vLarge signal
( , )C C BE BE CE CEI i f V v V v small signal
DC (bias)( , )C c BE be CE ceI i f V v V v
small signal(less messy!)
c be ceBE CEQ Q
f fi v v
v v
transconductance Output conductance
( , )BE CEQ V V
Quiescent Point(bias)
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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 givethis number.
Wp
Density fixed by metal contact
Half go left,half go right
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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
DqV
kTn dep p p pQ qA bh qA W x n e n
0 0,
( )DqV
n kTD p p
p dep p
qADI n e n
W x
2
,
nD
n p dep p
DI
Q W x
Constant!
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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 pnT
D n
W xQ
I D
n T DQ In
d T d T
Q IC g
V V
Diffusion Coefficient
Distance acrossP-type base
2
,p dep p
Tn
W xq
kT
Mobility
Temperature
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
BJT Transconductance gm
The transconductance is analogous to diode conductance
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Transconductance (cont)
Forward-active large-signal current:
/ (1 )BE thv VC S CE Ai I e v V
• Differentiating and evaluating at Q = (VBE, VCE )
/ (1 )BEqV kTCS CE A
BE Q
i qI e V V
v kT
C Cm
BE Q
i qIg
v kT
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
BJT Base Currents
Unlike MOSFET, there is a DC current into thebase terminal of a bipolar transistor:
/ (1 )BEqV kTB C F S F CE AI I I e V V
To find the change in base current due to change in base-emitter voltage:
1B B Cm
BE C BE FQ QQ
i i ig
v i v
Bb be
BE Q
ii v
v
mb be
F
gi v
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Small Signal Current Gain
0C
FB
i
i
Since currents are linearly related, the derivative is a constant (small signal = large signal)
0C Bi i
0c bi i
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Input Resistance rπ
1 1 C mB
BE F BE FQ Q
i gir
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
rg
25mV100 25k
.1mAr
iR MOSFET
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Output Resistance roWhy 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(/ACE
VvSC VveIi thBE
Base (p)
Emitter (n+)
Collector (n)
BW
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Graphical Interpretation of ro
slope~1/ro
slope
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
BJT Small-Signal Model
b bei r v1
c m be ceo
i g v vr
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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
, , 01.4j BE j BEC C
b F mC g
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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
mg v
Base Collector
Emitter
Reverse biased junction
Reverse biased junction &Diffusion Capacitance
Fictional Resistance(no noise)
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Complete Small-Signal Model
Reverse biased junctions“core” BJT
External Parasitics
Real Resistance(has noise)
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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!
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Modern ICs
First IC (TI, Jack Kilby, 1958): A couple of transistors Modern IC: Intel Pentium 4 (55 million transistors, 3 GHz)
Source: Texas InstrumentsUsed without permission
Source: Intel CorporationUsed without permission
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
A Simple Circuit: An MOS Amplifier
DSI
GSV
sv
DR DDV
GS GS sv V v
ov
Input signal
Output signal
Supply “Rail”
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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 DSR
D D
V V V VI
R R
,R DS satI I
DS GS TV V V
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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( )
2DS sat n ox GS Tn
WI C V V
L
2,
1( )
2 2D
DDR DS sat n ox GS Tn
D
V WI I C V V
R L
22
5V μA 1100μA 20 100 ( 1)
50k V 2 GSV
2.1 ( 1)GSV 1.32GSV .32 2.5GS T DSV V V
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Applying the Small-Signal Voltage
Approach 1. Just use vGS in the equation for the total
drain current iD and find vo
GS GS sv V v
ˆ coss sv v t
21( )
2O DD D DS DD D n ox GS s T
Wv V R i V R C V v V
L
Note: Neglecting charge storage effects. Ignoring device output impedance.
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Solving for the Output Voltage vO
21( )
2O DD D DS DD D n ox GS s T
Wv V R i V R C V v V
L
2
21( ) 1
2s
O DD D DS DD D n ox GS TGS T
vWv V R i V R C V V
L V V
DSI2
1 sO DD D DS
GS T
vv V R I
V V
2DDV
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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
1vs
VGS VTn–--------------------------+
2
12vs
VGS VTn–--------------------------
vs
VGS VTn–--------------------------
2
+ +=
Neglect
21 s
O DD D DSGS T
vv V R I
V V
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Linearized Output Voltage
For this case, the total output voltage is:
The small-signal output voltage:
21
2sDD
O DDGS T
vVv V
V V
2s DDDD
OGS T
v VVv
V V
“DC”
Small-signal output
s DDo v s
GS T
v Vv A v
V V
Voltage gain
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
Plot of Output Waveform (Gain!)
Numbers: VDD / (VGS – VT) = 5/ 0.32 = 16 output
input
mV
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 15 Prof. A. Niknejad
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
top related