Chap13-Small-Signal Modeling and Linear Amplification

7/27/2019 Chap13-Small-Signal Modeling and Linear Amplification

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Jaeger/Blalock 7/1/03

Microelectronic Circuit DesignMcGraw-Hill

Chapter 13Small-Signal Modeling and Linear

AmplificationMicroelectronic 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 outputresistances 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 V BE . Q-point is set at ( I C ,V CE )=(1.5 mA, 5 V) with I B = 15 mA.

Total base-emitter voltage is: bev BE V BE v

Collector-emitter voltage is: This gives the load lineC RC iCE v 10

Chap13 - 4


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BJT Amplifier (contd.)

8 mV peak change in v BE gives 5 mAchange in i B and 0.5 mA change in iC .

0.5 mA change in iC gives 1.65 Vchange in vCE .

If changes in operating currents andvoltages are small enough, then I C and V CE waveforms are undistortedreplicas of input signal.

Small voltage change at base causeslarge voltage change at collector.Voltage gain is given by:

Minus sign indicates 180 0 phaseshift between input and outputsignals.

2061802060008.0

18065.1

bev

cevv A

Chap13 - 5


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MOSFET Amplifier

MOSFET is biased in active region by dc voltage source V GS . Q-point is setat ( I D, V DS )=(1.56 mA, 4.8 V) with V GS =3.5 V.

Total gate-source voltage is: gsvGS V GS v

1 V p-p change in vGS gives 1.25 mA p-p change in i D and 4 V p-p changein v DS .

Chap13 - 6


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Coupling and Bypass Capacitors

AC coupling through capacitors isused to inject ac input signal and

extract output signal withoutdisturbing Q-point Capacitors provide negligible

impedance at frequencies of interestand provide open circuits at dc.

C 1 and C 3 are large coupling capacitorsor dc blocking capacitors, their reactance at signal frequency isnegligible.

C 2 is bypass capacitor, provides lowimpedance path for ac current fromemitter to ground, removing RE (required for good Q-point stability)from circuit when ac signals are

considered.

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 connectionsand 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 opencircuits, disconnecting v I , R I , and R3 from circuit.

Chap13 - 9


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AC Equivalent for BJT Amplifier

k 100k 3.43

k 30k 1021

RC

R R

R R B

R

Chap13 - 10


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DC and AC Equivalents for MOSFETAmplifier

dc equivalent

ac equivalentSimplified 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 isgiven by:

g d is small but non-zero for I D = 0 because

slope of diode equation is nonzero at origin. Diode resistance is given by:

D I D

I

T V D

I

d g

T V

S I D I

T V

DV

T V

S I

poQ Dv

Di

d

g

40V025.0

exp

int

For I D>> I S

d g d

r 1

Chap13 - 12


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Small-Signal Operation of Diode(contd.)

...3

61

2

21exp1exp

1exp

T V

d v

T V

d v

T V

d v

T V Dv

S I

T V

DV

S I

T V

d v

DV

S I d i D I

1exp

T V D

vS I Di

Subtracting I D from both sides of the equation,

...3

61

2

21)(

T V d

v

T V d

v

T V d

vS I D I d

i

For id to be a linear function of signal voltage vd ,This represents the requirement for small-signal operation of the diode.

V05.02 T V d v

d v

d g D I DiT V

d vS I D I d

i

)(

Chap13 - 13


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Current-Controlled Attenuator

Magnitude of ac voltage vo developedacross diode can be controlled by valueof dc bias current applied to diode.

From dc equivalent circuit I D = I ,

From ac equivalent circuit,

d r I

R I Rd r

d r

1

1ivivov

T V

I RS I I )(

1

1iv

For R I =1 k W, I S =10 -15 A,

If I = 0, vo = vi, magnitude of vi islimited to only 5 mV.If I = 100 mA, input signal isattenuated by a factor of 5, vi canhave 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 andcurrents or small changes in them awayfrom Q-point values.

cev22 bev21cicev12 bev11 bi

y y

y y

T V oC

I

BE v Bi

y pointQ0cev be

v bi

11

00

bevce

v bi

12 po intQCE

v Bi y

T V C I

BE v

C i

y

pointQ0cev bev

ci21

CE V AV C I

CE v

C i

y

pointQ0 bevcev

ci22

o is small-signal common-emittercurrent gain of the BJT.

Chap13 - 15


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Hybrid-Pi Model of BJT

The hybrid-pi small-signalmodel is the intrinsic low-frequency representation of theBJT.

Small-signal parameters arecontrolled by the Q-point andare independent of geometry of BJT

Transconductance:

C I

T V

C I

ym g 4021

Input resistance:

m g o

C I

T V o

yr

21

1

Output resistance:

C I

AV

C I

CE V AV yo

r

22

1

Chap13 - 16


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Small-Signal Current Gain andAmplification Factor of BJT

int

11

poQC i F

F

C I

F r m g o

o > F for iC < I M , and o < F for iC > I M , however, o and o areassumed to be equal .

T V

CE V

AV

C I

CE V

AV

T V

C I

or m g F m

Amplification factor is given by:

For V CE


18/47



Equivalent Forms of Small-Signal Modelfor BJT

Voltage -controlled current source g mvbe can be transformed intocurrent-controlled current source,

Basic relationship i c= i b is useful in both dc and ac analysis when BJTis in forward-active region.

bice

v bici

bi

bi

bev

bi

bev

oor

oo

r m

g m

g

r

Chap13 - 18


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Small Signal Operation of BJT

...3

61

2

211

expexp

T V bev

T V bev

T V bev

C I

T V bev

T V BE V

S I ciC I C i

T V BE

vS I C i exp

...3

612

21

T V bev

T V bev

T V bev

C I C I C ici

For linearity, ic should be proportional to vbe V005.02 T V bev

bevm g C I be

v

T V

C I C I

T V bev

C I ci

1

Change in ic that corresponds to small-signal operation is:

200.0025.0005.0

T V be

v

bev

C I

m g

C I

ci

Chap13 - 19


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Jaeger/Blalock 7/1/03 Microelectronic Circuit DesignMcGraw-Hill

Small-Signal Model for pnp BJT

For pnp transistor

Signal current injected into basecauses decrease in total collector current which is equivalent toincrease in signal current enteringcollector.

bi F B I F ci-C I C ibi- B I Bi

Chap13 - 20


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Small-Signal Analysis of Complete C-EAmplifier: AC Equivalent

Ac equivalent circuit isconstructed by assuming that allcapacitances have zeroimpedance at signal frequencyand dc voltage source is ac

ground. Assume that Q-point is already

known.

21 R R B R

Chap13 - 21


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Small-Signal Analysis of Complete C-EAmplifier: Small-Signal Equivalent

3 RC Ror L R

L Rm g bevov

bvcv

vt A

Terminal voltage gain between base and collector is:

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

r

B R

I R

r B

R L Rm g v A

ivbev

vt Aivbev

bevov

ivovv A

Chap13 - 22


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C-E Amplifier Voltage Gain: Example

Problem: Calculate voltage gain Given data: F =100, V A =75 V, Q-point is(1.45 mA, 3.41 V), R1 = 10

k W, R2 = 30 k W, R3 = 100 k W, RC = 4.3 k W, R I = 1k W. Assumptions: Transistor is in active region, O = F . Signals are low

enough to be considered small signals. Analysis:

dB3.42130

r B

R I

R

r B

R L Rm g v A

mS0.58)mA45.1(4040 C I m g

k 72.1mA45.1

)V025.0(100

C I

T V or

k 1.54mA45.1

V14.3V75

C I

CE V

AV

or

k 5.721 R R B R

k 83.33

RC Ror L R

mV57.8)(

V)005.0(

r

B R

r B

R I

Riv

Chap13 - 23


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Small-Signal Model Simplification

If we assume

Generally R3 >> RC and load resistor


25/47


C-E Amplifier Input Resistance

Input resistance, the total resistancelooking into the amplifier atcoupling capacitor C 1 representstotal resistance presented to source.

r R Rr B R R

r B R

21xixv

in

)(xixv

Chap13 - 25


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C-E Amplifier Output Resistance

Output resistance is the total equivalent

resistance looking into the output of theamplifier at coupling capacitor C 3. Inputsource is set to 0 and test source isapplied at output.

C Ror C R R

m g or C R

xixv

out

bevx

vxvxi But v be=0 .

As r o>> R C .

Chap13 - 26


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Sample Analysis of C-E Amplifier

Problem: Find voltage gain, inputand output resistances.

Given data: F = 65, V A =50 V Assumptions: Active-region

operation, V BE =0.7 V, small signaloperating conditions.

Analysis: To find the Q-point,dc equivalent circuit isconstructed.

A24566

A24165A71.3

B I E I B I C I

B I

5)4106.1()1(510 B I F BE V B I

V67.3

0)5()4106.1(4105

CE V E I CE V C I

Chap13 - 27


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Sample Analysis of C-E Amplifier (contd.)

Next we construct the acequivalent and simplify it.

0.84

in

in)3out( R I

R

R R Rm g

ivovv A

S31064.940 C I m g

k 64.6

C I T

V or

k 223

C I

CE V

AV

or

k 23.6in r B R R

k 57.9out o

r C

R R

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 andcurrents or small changes in them awayfrom Q-point values.

dsv22gsv21didsv12gsv11gi

y y

y y

0

0ds

vgsv

gi11

po intQGS v

Gi y

0

0gsvdsv

gi12

po intQ DS v

Gi

y

TN V GS V D I

GS v

Di

y2

0ds

vgsv

di

21 pointQ

DS V D

I

DS v Di

y

10gsv

dsv di

22 pointQ

Chap13 - 29


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Small Signal Parameters of MOSFET

Since gate is insulated fromchannel by gate-oxide inputresistance of transistor is infinite.

Small-signal parameters arecontrolled by the Q-point.

For same operating point,MOSFET has higher transconductance and lower outputresistance that BJT.

Transconductance:

D I n K

TN V

GS V

D I

ym g 2

2

21

Output resistance:

D I

D I

DS V

yor

1

1

22

1

Amplification factor for V DS


31/47


Small Signal Operation of MOSFET

2222 gsvTN V

GS V gsvTN

V GS

V n K d i D I Di

22

TN V GS vn K

Di

For linearity, id should be proportional to v gs

Since MOSFET can be biased with ( V GS - V TN ) equal to several volts, itcan handle much larger values of v gs than corresponding values of vbe for BJT.

TN V

GS V gsv 2.0

Change in drain current that corresponds to small-signal operation is:

4.0

2

)(2.0

TN V

GS V

TN V

GS V

gsv

D I

m g

D I

d i

222 gsvTN V

GS

V gsvn K

d i

for TN V GS v DS v

Chap13 - 31


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Body Effect in Four-terminal MOSFET

Drain current depends on threshold voltage whichin turn depends on vSB. Back-gatetransconductance is:

0


33/47


Small-Signal Model for PMOSTransistor

For pnp transistor

Positive signal voltage v gg reduces

source-gate voltage of the PMOStransistor causing decrease in totalcurrent exiting drain, equivalent toincrease in signal current enteringdrain.

d i- D I C i gg v-GGV SGv

Chap13 - 33


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Small-Signal Analysis of Complete C-SAmplifier: AC Equivalent

Ac equivalent circuit isconstructed by assuming that allcapacitances have zeroimpedance at signal frequencyand dc voltage sources represent

ac grounds. Assume that Q-point is already

known.

21 R RG R

Chap13 - 34


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Small-Signal Analysis of Complete C-EAmplifier: Small-Signal Equivalent

3 R D Ror L R

L Rm g gsvov

g vd v

vt A

Terminal voltage gain betweengate and drain is:

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

G R

I R

G R

L Rm g v A

iv gsv

vt Aiv gsv

gsvov

ivovv A

Chap13 - 35


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C-S Amplifier Voltage Gain: Example

Problem: Calculate voltage gain Given data: K n = 0.5 mA/V 2, V TN = 1V, = 0.0133 V -1, Q-point is

(1.45 mA, 3.86 V), R1 = 430 k W, R2 = 560 k W, R3 = 100 k W, R D = 4.3k W, R I = 1 k W.

Assumptions: Transistor is in active region. Signals are low enough to

be considered small signals. Analysis:

dB4.1369.4

G R

I R

G R

L Rm g v A

mS23.1)1(2 DS

V DS

I n K m g

k 5.54

1

D I

DS V

or

k 24321

R RG R

k 83.33

R D Ror L R

V48.02

2.02.0

n K D

I

TN V

GS V iv

Chap13 - 36


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Small-Signal Model Simplification

If we assume

Generally R3 >> R D and load resistor


38/47


C-S Amplifier Input Resistance

Input resistance of C-S amplifier ismuch larger than that of corresponding C-E amplifier.

G R RG R

in

xixv

Chap13 - 38


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C-S Amplifier Output Resistance

For comparable bias points, outputresitances of C-S and C-E amplifiers aresimilar.

D Ror D R Rxix

vout

In this case, v gs=0 .

As r o>> R D.

Chap13 - 39


40/47


Sample Analysis of C-S Amplifier


Given data: K n = 500 mA/V 2, V TN = 1V, = 0.0167 V -1

Analysis: Dc equivalent circuitis constructed.

61051DS V I

)1(410210 I D I DS V

V52)4.0(2

DS V TN V DS V n K D I

V2GS V A025 m D I

Chap13 - 40


41/47


Sample Analysis of C-S Amplifier (contd.)


93.7

in

in)3out( R I

R

R R Rm g

ivovv A

M 121in G

RG R R

k 2.183out G

R D

Ro

r R

S41020.5)1(2 DS

V DS

I n K m g

k 260

1

D I

DS V or

Chap13 - 41


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Small Signal Parameters of JFET

T V

SG I

G I

GS v

Gi

yr

pointQ

111

)(2

2

2 pointQ21

P V GS V P V

DSS I

P V GS V D I

GS v

Di

ym g

DS V D I

DS v

Di

yor

1221

pointQ

DS v

P V GS

v DSS I Di 1

2

1

for P V GS v DS v

1exp

T V GS

vSG I Gi

Chap13 - 42


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Small Signal Model of JFET

Since JFET is normally operatedwith gate junction reverse-

biased,

g r

SG I G I

For small signal operation,condition on input is:

Amplification factor is given by:

P V

GS V gsv 2.0

D I DSS I

P V P

V GS

V DS

V

or m g f

m 2

1

2

Chap13 - 43


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Sample Analysis of JFET C-S Amplifier


Given data: I DSS = 1 mA, V P = -1V,= 0.02 V -1

Assumptions: Pinch-off region of operation.

Analysis: Dc equivalent circuit isconstructed. I G = 0, I S = I D.

Choose V GS less negative than V P .

D I GS V 2000

mA250.0

V5.0

D I GS V

2

)1(1)3101)(3102(

GS

V

GS V

V75.4

2000000,2712

DS V S I DS V D I

Chap13 - 44


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


3.20

in

in)3out( R I

R

R R Rm g

ivovv A

M 1in G

R R

k 0.24out D

Ro

r R

mS05.1)1(2 DS

V DS

I DSS

I

P V

m g

k 219

1

DS I

DS V

or

Chap13 - 45


46/47



Amplifier Power Dissipation

Static power dissipation in amplifiers is determined from their dcequivalent circuits.

B I BE V C I CE V D P

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

where

Total power supplied is:

E I EE V C I CC V S P

BE V CBV CE V

Total power dissipated intransistor is:

Total power supplied is: D I DS V G I GS V D I DS V D P

D I DDV S P

Chap13 - 46


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Jaeger/Blalock Microelectronic Circuit Design

Amplifier Signal Range

BE V CE V M V

t M V CE V CE v sin V7.0CE vBut

0sin)( t M

V C

RC

I t C R

v

BE V

CE V

C R

C I M V ,min

Also

C RC I M V But

Similarly for MOSFETs and JFETs,))((,min

TN V

GS V

DS V

D R

D I M V

))((,min P

V GS

V DS

V D

R D

I M V

Chap13 - 47

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Chap13-Small-Signal Modeling and Linear Amplification