Chapter 13 Small-Signal Modeling and Linear Amplification

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Nov 2, 2013 (3 years and 8 months ago)

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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.