Jaeger/Blalock

1
Chapter 13Small-Signal Modeling and Linear
Amplification

• Microelectronic Circuit Design
• Richard C. Jaeger
• Travis N. Blalock

Chap13 - 1
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Chapter Goals

• Understanding of concepts related to
• Transistors as linear amplifiers
• dc and ac equivalent circuits
• Use of coupling and bypass capacitors and
  inductors to modify dc and ac equivalent circuits
• Small-signal voltages and currents
• Small-signal models for diodes and transistors
• Identification of common-source and
  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 and common-source amplifiers.

Chap13 - 2
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Introduction to Amplifiers

• BJT is an excellent amplifier when biased in
  forward-active region
• FET can be used as amplifier if operated in
  pinch-off or saturation region.
• In these regions, transistors can provide high
  voltage, current and power gains.
• Bias is provided to stabilize the operating point
  in desired operation region.
• Q-point also determines
• Small-signal parameters of transistor
• Voltage gain, input resistance, output resistance
• Maximum input and output signal amplitudes
• Power consumption

Chap13 - 3
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BJT Amplifier
BJT is biased in active region by dc voltage
source VBE. Q-point is set at (IC, VCE)(1.5 mA,
5 V) with IB 15 mA. Total base-emitter voltage
is
Collector-emitter voltage is
This gives the load line
Chap13 - 4
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BJT Amplifier (contd.)
If changes in operating currents and voltages are
small enough, then IC and VCE waveforms are
undistorted replicas of input signal. Small
voltage change at base causes large voltage
change at collector. Voltage gain is given
by Minus sign indicates 1800 phase shift
between input and output signals.
8 mV peak change in vBE gives 5 mA change in iB
and 0.5 mA change in iC. 0.5 mA change in iC
gives 1.65 V change in vCE .
Chap13 - 5
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MOSFET Amplifier
MOSFET is biased in active region by dc voltage
source VGS. Q-point is set at (ID, VDS)(1.56 mA,
4.8 V) with VGS 3.5 V. Total gate-source voltage
is
1 V p-p change in vGS gives 1.25 mA p-p change in
iD and 4 V p-p change in vDS.
Chap13 - 6
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Coupling and Bypass Capacitors
C1 and C3 are large coupling capacitors or dc
blocking capacitors, their reactance at signal
frequency is negligible. C2 is bypass capacitor,
provides low impedance path for ac current from
emitter to ground, removing RE (required for
good Q-point stability) from circuit when ac
signals are considered.

• AC coupling through capacitors is used to inject
  ac input signal and extract output signal without
  disturbing Q-point
• Capacitors provide negligible impedance at
  frequencies of interest and provide open circuits
  at dc.

Chap13 - 7
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DC and AC Analysis

• DC analysis
• Find dc equivalent circuit by replacing all
  capacitors by open circuits and inductors by
  short circuits.
• Find Q-point from dc equivalent circuit by using
  appropriate large-signal transistor model.
• AC analysis
• Find ac equivalent circuit by replacing all
  capacitors by short circuits, inductors by open
  circuits, dc voltage sources by ground
  connections and dc current sources by open
  circuits.
• Replace transistor by small-signal model
• Use small-signal ac equivalent to analyze ac
  characteristics of amplifier.
• Combine end results of dc and ac analysis to
  yield total voltages and currents in the network.

Chap13 - 8
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DC Equivalent for BJT Amplifier

• All capacitors in original amplifier circuits are
  replaced by open circuits, disconnecting vI, RI,
  and R3 from circuit.

Chap13 - 9
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AC Equivalent for BJT Amplifier
Chap13 - 10
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DC and AC Equivalents for MOSFET Amplifier
dc equivalent
Simplified ac equivalent
ac equivalent
Chap13 - 11
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Small-Signal Operation of Diode

• The slope of the diode characteristic at the
  Q-point is called the diode conductance and is
  given by
• gd is small but non-zero for ID 0 because slope
  of diode equation is nonzero at origin.
• Diode resistance is given by

For IDgtgtIS
Chap13 - 12
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Small-Signal Operation of Diode (contd.)
Subtracting ID from both sides of the equation,
For id to be a linear function of signal voltage
vd, This represents the requirement for
small-signal operation of the diode.
Chap13 - 13
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Current-Controlled Attenuator
Magnitude of ac voltage vo developed across diode
can be controlled by value of dc bias current
applied to diode.
From ac equivalent circuit,
From dc equivalent circuit ID I,
For RI 1 kW, IS 10-15 A, If I 0, vo vi,
magnitude of vi is limited to only 5 mV. If I
100 mA, input signal is attenuated by a factor of
5, vi can have a magnitude of 25 mV.
Chap13 - 14
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Small Signal Model of BJT
Using 2-port y-parameter network, The port
variables can represent either time-varying part
of total voltages and currents or small changes
in them away from Q-point values.
bo is small-signal common-emitter current gain of
the BJT.
Chap13 - 15
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Hybrid-Pi Model of BJT
Transconductance
Input resistance

• The hybrid-pi small-signal model is the intrinsic
  low-frequency representation of the BJT.
• Small-signal parameters are controlled by the
  Q-point and are independent of geometry of BJT

Output resistance
Chap13 - 16
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Small-Signal Current Gain and Amplification
Factor of BJT
Amplification factor is given by For VCE ltlt
VA, mF represents maximum voltage gain
individual BJT can provide and doesnt change
with operating point.
bo gt bF for iC lt IM, and bo lt bF for iC gt IM,
however, bo and bo are assumed to be equal.
Chap13 - 17
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Equivalent Forms of Small-Signal Model for BJT

• Voltage -controlled current source gmvbe can be
  transformed into current-controlled current
  source,
• Basic relationship icbib is useful in both dc
  and ac analysis when BJT is in forward-active
  region.

Chap13 - 18
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Small Signal Operation of BJT
For linearity, ic should be proportional to vbe
Change in ic that corresponds to small-signal
operation is
Chap13 - 19
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Small-Signal Model for pnp BJT

• For pnp transistor
• Signal current injected into base causes decrease
  in total collector current which is equivalent to
  increase in signal current entering collector.

Chap13 - 20
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Small-Signal Analysis of Complete C-E Amplifier
AC Equivalent

• Ac equivalent circuit is constructed by assuming
  that all capacitances have zero impedance at
  signal frequency and dc voltage source is ac
  ground.
• Assume that Q-point is already known.

Chap13 - 21
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Small-Signal Analysis of Complete C-E Amplifier
Small-Signal Equivalent
Overall voltage gain from source vi to output
voltage across R3 is
Terminal voltage gain between base and collector
is
Chap13 - 22
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C-E Amplifier Voltage Gain Example

• Problem Calculate voltage gain
• Given data bF 100, VA 75 V, Q-point is(1.45
  mA, 3.41 V), R1 10 kW, R2 30 kW, R3 100 kW,
  RC 4.3 kW, RI 1kW.
• Assumptions Transistor is in active region, bO
  bF. Signals are low enough to be considered small
  signals.
• Analysis

Chap13 - 23
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Small-Signal Model Simplification

• If we assume
• Generally R3 gtgt RC and load resistor ltlt ro. If we
  assume IC RC ?VCC with 0lt?lt1
• For this case, ?1/3 since common design
  allocates one-third power supply across RC. To
  further account for other approximations leading
  to this result, we use
• Also, if load resistor is forced to approach ro,
  RC and R3 are infinite, voltage gain is limited
  by amplification factor, mf of BJT itself.

This implies that total signal voltage at input
appears across rp.
Chap13 - 24
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C-E Amplifier Input Resistance

• Input resistance, the total resistance looking
  into the amplifier at coupling capacitor C1
  represents total resistance presented to source.

Chap13 - 25
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C-E Amplifier Output Resistance

• Output resistance is the total equivalent
  resistance looking into the output of the
  amplifier at coupling capacitor C3. Input source
  is set to 0 and test source is applied at output.

But vbe0.
As rogtgt RC.
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Sample Analysis of C-E Amplifier
Analysis To find the Q-point, dc equivalent
circuit is constructed.

• Problem Find voltage gain, input and output
  resistances.
• Given data bF 65, VA 50 V
• Assumptions Active-region operation, VBE0.7 V,
  small signal operating conditions.

Chap13 - 27
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Sample Analysis of C-E Amplifier (contd.)

• Next we construct the ac equivalent and
  simplify it.

Chap13 - 28
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Small Signal Model of MOSFET
Using 2-port y-parameter network, The port
variables can represent either time-varying part
of total voltages and currents or small changes
in them away from Q-point values.
Chap13 - 29
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Small Signal Parameters of MOSFET
Transconductance
Output resistance

• Since gate is insulated from channel by
  gate-oxide input resistance of transistor is
  infinite.
• Small-signal parameters are controlled by the
  Q-point.
• For same operating point, MOSFET has higher
  transconductance and lower output resistance that
  BJT.

Amplification factor for lVDSltlt1
Chap13 - 30
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Small Signal Operation of MOSFET
for
For linearity, id should be proportional to
vgs Since MOSFET can be biased with (VGS - VTN)
equal to several volts, it can handle much larger
values of vgs than corresponding values of vbe
for BJT.
Change in drain current that corresponds to
small-signal operation is
Chap13 - 31
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Body Effect in Four-terminal MOSFET
Drain current depends on threshold voltage which
in turn depends on vSB. Back-gate
transconductance is 0lt?lt1 is
called back-gate tranconductance
parameter. Bulk terminal is a reverse-biased
diode. Hence, no conductance from bulk terminal
to other terminals.
Chap13 - 32
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Small-Signal Model for PMOS Transistor

• For pnp transistor
• Positive signal voltage vgg reduces source-gate
  voltage of the PMOS transistor causing decrease
  in total current exiting drain, equivalent to
  increase in signal current entering drain.

Chap13 - 33
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Small-Signal Analysis of Complete C-S Amplifier
AC Equivalent

• Ac equivalent circuit is constructed by assuming
  that all capacitances have zero impedance at
  signal frequency and dc voltage sources represent
  ac grounds.
• Assume that Q-point is already known.

Chap13 - 34
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Small-Signal Analysis of Complete C-E Amplifier
Small-Signal Equivalent
Overall voltage gain from source vi to output
voltage across R3 is
Terminal voltage gain between gate and drain is
Chap13 - 35
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C-S Amplifier Voltage Gain Example

• Problem Calculate voltage gain
• Given data Kn 0.5 mA/V2, VTN 1V, l 0.0133
  V-1, Q-point is (1.45 mA, 3.86 V), R1 430 kW,
  R2 560 kW, R3 100 kW, RD 4.3 kW, RI 1 kW.
• Assumptions Transistor is in active region.
  Signals are low enough to be considered small
  signals.
• Analysis

Chap13 - 36
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Small-Signal Model Simplification

• If we assume
• Generally R3 gtgt RD and load resistor ltlt ro.
  Hence, total load resistance on drain is RD. For
  this case, common design allocates half the power
  supply for voltage drop across RD and (VGS - VTN
  ) 1V
• Also, if load resistor is forced to approach ro,
  RD and R3 are infinite, voltage gain is limited
  by amplification factor, mf of MOSFET itself.

This implies that total signal voltage at input
appears across gate-source terminals.
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C-S Amplifier Input Resistance

• Input resistance of C-S amplifier is much larger
  than that of corresponding C-E amplifier.

Chap13 - 38
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C-S Amplifier Output Resistance

• For comparable bias points, output resitances of
  C-S and C-E amplifiers are similar.

In this case, vgs0.
As rogtgt RD.
Chap13 - 39
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Sample Analysis of C-S Amplifier
Analysis Dc equivalent circuit is constructed.

• Problem Find voltage gain, input and output
  resistances.
• Given data Kn 500 mA/V2, VTN 1V, l 0.0167
  V-1

Chap13 - 40
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Sample Analysis of C-S Amplifier (contd.)

• Next we construct the ac equivalent and
  simplify it.

Chap13 - 41
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Small Signal Parameters of JFET
for
Chap13 - 42
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Small Signal Model of JFET
For small signal operation, condition on input
is Amplification factor is given by
Since JFET is normally operated with gate
junction reverse-biased,
Chap13 - 43
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Sample Analysis of JFET C-S Amplifier
Analysis Dc equivalent circuit is constructed.
IG 0, IS ID. Choose VGS less negative than
VP.

• Problem Find voltage gain, input and output
  resistances.
• Given data IDSS 1 mA, VP -1V, l 0.02 V-1
• Assumptions Pinch-off region of operation.

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Sample Analysis of JFET C-S Amplifier (contd.)

• Next we construct the ac equivalent and
  simplify it.

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Amplifier Power Dissipation

• Static power dissipation in amplifiers is
  determined from their dc equivalent circuits.

Total power dissipated in transistor
is Total power supplied is
Total power dissipated in C-B and E-B junctions
is where Total power supplied is
Chap13 - 46
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Amplifier Signal Range
But
Also
But
Similarly for MOSFETs and JFETs,
Chap13 - 47