Chapter 13Small-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 VBE and VCC = 10 V. The DC Q-point is set at, (VCE, IC) = (5 V, 1.5 mA) with IB

= 15 A.

Total base-emitter voltage is: bevBEVBEv +=

Collector-emitter voltage is: This produces a load line.

CR

Ci

CEv −=10

BJT Amplifier (continued)

An 8 mV peak change in vBE gives a 5 A change in iB and a 0.5 mA change in iC.

The 0.5 mA change in iC gives a 1.65 V change in vCE .

If changes in operating currents and voltages are small enough, then IC and VCE 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 1800 phase shift between input and output signals.

€

˜ A v =˜ v ce˜ v be

=1.65∠1800.008∠0

=206∠180=−206

A Simple MOSFET Amplifier

The MOSFET is biased in the saturation region by dc voltage sources VGS and VDS = 10 V. The DC Q-point is set at (VDS, IDS) = (4.8 V, 1.56 mA) with VGS = 3.5 V.

Total gate-source voltage is: gsvGS

VGS

v +=

A 1 V p-p change in vGS gives a 1.25 mA p-p change in iDS and a 4 V p-p changein vDS. 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, C1 and C3 are large coupling capacitors or dc blocking capacitors, their reactance (XC

= |ZC| = 1/C) at signal frequency is negligible. They are effective open circuits for the circuit when DC bias is considered.

C2 is a bypass capacitor. It provides a low impedance path for ac current from emitter to ground. It effectively removes RE (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 vI, RI, and R3 from the circuit and leaving RE 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)

€

RB

= R1

R2

=10kΩ 30kΩ

R= RC

R3

= 4.3kΩ100kΩ

• By combining parallel resistors into equivalent RB 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:

€

gm =IC

VT

≅ 40IC

Input resistance:

€

rπ =βoV

TIC

=βogm

Output resistance:

€

ro=V

A+V

CEIC

Small-signal Current Gain and Amplification Factor of the BJT

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

∂

∂−

==

int

11

poQCiF

FC

I

Frmgoβ

β

βπβ

o > F for iC < IM, and o < F for iC > IM, however, o and F are usually assumed to be about

equal.

The amplification factor is given by:

For VCE << VA,

F represents the maximum voltage gain an individual BJT can provide, independent of the operating point.

€

F≅

VA

VT

≅ 40VA

€

F ≡vce

vbe

,vce = −rogmvbe

€

F

= gmro=IC

VT

VA

+VCE

IC

=V

A+V

CEV

T

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.

€

o= gmrπ =β

F

1− IC

1β

F

∂βF

∂iC

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟Q − po int

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

€

o≅β

F

1− IC

1β

F

ΔβF

ΔIC

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟Q − po int

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

€

ΔF

ΔIC

≅ 200−100

10−2 −10−3=5.6x103 at about IC = 5 mA and 25 °C

€

o≅180

1−5x10−3 5.6x103

180

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

= 1801−0.15 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

=212 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 gmvbe can be transformed into a current-controlled current source,

• The basic relationship ic=ib is useful in both dc and ac analysis when the BJT is biased in the forward-active region.€

vbe

= ibrπ = i

bβogm

∴gmvbe

= gmibrπ =βoi

b

ic = gmvbe

+vcero

≅ gmvbe

=βoib

Small Signal Operation of BJT

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

++++=

=+=∴

...3

61

2

211

expexp

TVbe

v

TVbe

v

TVbe

vC

I

TVbe

v

TVBEV

SIciC

ICi

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

TVBE

v

SI

Ci exp

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+++=−=∴ ...3

61

2

21

TVbe

v

TVbe

v

TVbe

vC

IC

ICici

For linearity, ic should be directly proportional to vbe.

€

vbe

<<2VT

=50 mV

€

∴ic≅ IC

vbe

VT

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟=

IC

VT

vbe

= gmvbe

If we limit vbe to 5 mV, the relative change in ic compared to IC that

corresponds to small-signal operation is:

€

icIC

=gmv

beIC

=vbe

VT

≤ 0.0050.025

=0.200

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

€

RL

=ro RC

R3

Overall voltage gain from source vi to output voltage vo across R3 is:

€

Av =vovi

=vovbe

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

vbevi

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

∴Av =−gmRL

RB

rπ

RI

+ RB

rπ( )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

€

vo =−gmvbe

RL

and

vbe

=vi

RB

rπ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

RI

+ RB

rπ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

Capacitor Selection for the CE Amplifier

€

Zc = 1jωC

Capacitive Reactance Xc ≡ Zc = 1ωC

where ω=2πf

€

Xc1

<<RB

rπ ∴Make Xc1

≤ 0.01 RB

rπ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ for < 1% gain error.

€

Xc2⇒ 0 ∴Make X

c2≤1Ω for <1% gain error.

€

Xc3

<<R3 ∴Make X

c3≤ 0.01 R

3 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ for <1% gain error.

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 C1, represents the total resistance presented to the AC source.

ππ

πrRRrBRR

rBR

21xixv

in

)(xixv

===

=

C-E Amplifier Output Resistance

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

CRorC

RR

mgorC

R

≅==∴

++=

xixv

out

bevxvxvxi But vbe=0.

since ro is usually >> RC.

CE Amplifier Design Example

Using LabVIEW Virtual Instruments

Amplifier Power Dissipation

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

€

PD

=VCE

IC

+VBE

IB

Total power dissipated in C-B and E-B junctions is:

where

Total power supplied is:

€

PS

=VCC

IC

+ I2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ where I

2= I

1+ I

B

BEVCB

VCE

V +=

€

I1

=V

CCR1

+R2

and IB

=V

EQ−V

BE

REQ

+ βF

+1 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟RE

The difference is the power dissipated by the bias resistors.