Chapter 13
Small

Signal Modeling and Linear
Amplification
Chapter Goals
Understanding of concepts related to:
•
Transistors as linear amplifiers
•
dc and ac equivalent circuits
•
Use of coupling and bypass capacitors to modify dc and ac equivalent
circuits
•
Small

signal voltages and currents
•
Small

signal models for diodes and transistors
•
Identification of common

emitter amplifiers
•
Amplifier characteristics such as voltage gain, input and output
resistances and linear signal range
•
Rule

of

thumb estimates for voltage gain of common

emitter
amplifiers.
Introduction to Amplifiers
•
The BJT is an an excellent amplifier when biased in the forward

active
region.
•
The FET can be used as an amplifier if operated in the saturation
region.
•
In these regions, the transistors can provide high voltage, current and
power gains.
•
DC bias is provided to stabilize the operating point in the desired
operation region.
•
The DC Q

point also determines
–
The small

signal parameters of the transistor
–
The voltage gain, input resistance, and output resistance
–
The maximum input and output signal amplitudes
–
The overall power consumption of the amplifier
A Simple BJT Amplifier
The BJT is biased in the forward active region by dc voltage sources
V
BE
and
V
CC
= 10 V. The DC Q

point is set at, (
V
CE
,
I
C
) = (5 V, 1.5 mA) with
I
B
= 15
m
A.
Total base

emitter voltage is:
Collector

emitter voltage is: This produces a load line.
BJT Amplifier (continued)
An 8 mV peak change in
v
BE
gives a 5
m
A change in
i
B
and a 0.5 mA change in
i
C
.
The 0.5 mA change in
i
C
gives a 1.65 V
change in
v
CE
.
If changes in operating currents and
voltages are small enough, then
I
C
and
V
CE
waveforms are undistorted
replicas of the input signal.
A small voltage change at the base
causes a large voltage change at the
collector. The voltage gain is given
by:
The minus sign indicates a 180
0
phase shift between input and
output signals.
A Simple MOSFET Amplifier
The MOSFET is biased in the saturation region by dc voltage sources
V
GS
and
V
DS
= 10 V. The DC Q

point is set at (
V
DS
,
I
DS
) = (4.8 V, 1.56 mA) with
V
GS
=
3.5 V.
Total gate

source voltage is:
A 1 V
p

p
change in
v
GS
gives a 1.25 mA
p

p
change in
i
DS
and a 4 V
p

p
change
in
v
DS
. Notice the characteristic non

linear I/O relationship compared to the BJT.
A Practical BJT Amplifier using
Coupling and Bypass Capacitors
•
AC coupling through capacitors is
used to inject an ac input signal and
extract the ac output signal without
disturbing the DC Q

point
•
Capacitors provide negligible
impedance at frequencies of interest
and provide open circuits at dc.
In a practical amplifier design,
C
1
and
C
3
are large coupling capacitors or dc
blocking capacitors, their reactance (X
C
= Z
C
 = 1/
w
C
) at signal frequency is
negligible. They are effective open
circuits for the circuit when DC bias is
considered.
C
2
is a bypass capacitor. It provides a
low impedance path for ac current from
emitter to ground. It effectively
removes
R
E
(required for good Q

point
stability) from the circuit when ac
signals are considered.
DC and AC Analysis

Application of
Superposition
•
DC analysis:
–
Find the DC equivalent circuit by replacing all capacitors by open
circuits and inductors (if any) by short circuits.
–
Find the DC Q

point from the equivalent circuit by using the
appropriate large

signal transistor model.
•
AC analysis:
–
Find the AC equivalent circuit by replacing all capacitors by short
circuits, inductors (if any) by open circuits, dc voltage sources by
ground connections and dc current sources by open circuits.
–
Replace the transistor by its small

signal model (to be developed).
–
Use this equivalent circuit to analyze the AC characteristics of the
amplifier.
–
Combine the results of dc and ac analysis (superposition) to yield the
total voltages and currents in the circuit.
DC Equivalent for the BJT Amplifier
•
All capacitors in the original amplifier circuit are replaced by open
circuits, disconnecting
v
I
,
R
I
, and
R
3
from the circuit and leaving
R
E
intact. The the transistor
Q
will be replaced by its DC model.
DC Equivalent Circuit
AC Equivalent for the BJT Amplifier
•
The coupling and bypass capacitors are replaced by short circuits. The DC
voltage supplies are replaced with short circuits, which in this case connect
to ground.
AC Equivalent for the BJT Amplifier
(continued)
•
By combining parallel resistors into equivalent
R
B
and
R
, the equivalent AC
circuit above is constructed. Here, the transistor will be replaced by its
equivalent small

signal AC model (to be developed).
Hybrid

Pi Small

signal AC Model for
the BJT
•
The hybrid

pi small

signal
model is the intrinsic low

frequency representation of the
BJT.
•
The small

signal parameters are
controlled by the Q

point and
are independent of the geometry
of the BJT.
Transconductance:
Input resistance:
Output resistance:
Small

signal Current Gain and
Amplification Factor of the BJT
o
>
F
for
i
C
<
I
M
, and
o
<
F
for
i
C
>
I
M
,
however
,
o
and
F
are usually assumed to be about
equal
.
The amplification factor is given by:
For
V
CE
<<
V
A
,
m
F
represents the
maximum
voltage
gain an individual BJT can provide,
independent of the operating point.
Example
o
Calculation for 2N2222A
Choose the Q

point at about (5 V, 5 mA) for this analysis. Notice the slope of the
DC current gain characteristic in this region. Ideally, the slope would be zero.
at about
I
C
= 5 mA and 25
°
C
for
F
= 180
Given the tolerances usually encountered in forward current gain, the
assumption of
F
=
o
seems reasonable for preliminary analysis and
initial designs.
From Figure 3 for the 2N2222A BJT at the chosen Q

point…
Equivalent Forms of the Small

signal
Model for the BJT
•
The voltage

controlled current source
g
m
v
be
can be transformed into a
current

controlled current source,
•
The basic relationship i
c
=
i
b
is useful in both dc and ac analysis when
the BJT is biased in the forward

active region.
Small Signal Operation of BJT
For linearity,
i
c
should be directly proportional to
v
be
.
If we limit
v
be
to 5 mV, the relative change in
i
c
compared to
I
C
that
corresponds to small

signal operation is:
for
Small

Signal Analysis of the Complete
C

E Amplifier: AC Equivalent
•
The AC equivalent circuit is
constructed by assuming that all
capacitances have zero
impedance at signal frequency
and the AC voltage source is at
ground.
•
Assume that the DC Q

point has
already been calculated.
Small

Signal Analysis of Complete C

E
Amplifier: Small

Signal Equivalent
Overall voltage gain from source
v
i
to output voltage
v
o
across
R
3
is:
Capacitor Selection for the CE Amplifier
The key objective in design is to make the capacitive reactance
much smaller at the operating frequency
f
than the associated
resistance that must be coupled or bypassed.
C

E Amplifier Input Resistance
•
The input resistance, the total
resistance looking into the amplifier
at coupling capacitor
C
1
, represents
the total resistance presented to the
AC source.
C

E Amplifier Output Resistance
•
The output resistance is the total
equivalent resistance looking into the
output of the amplifier at coupling
capacitor
C
3.
The input source is set to 0
and a test source is applied at the output.
But v
be
=0
.
since
r
o
is usually
>> R
C
.
CE Amplifier Design Example
Using LabVIEW Virtual Instruments
Amplifier Power Dissipation
•
Static power dissipation in amplifiers is determined from their DC
equivalent circuits.
Total power dissipated in C

B
and E

B junctions is:
where
Total power supplied is:
The difference is the power dissipated by the bias resistors.
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