BJT Small-Signal Analysis

Contents

Common-Emitter fixed-bias configuration Voltage divider bias CE Emitter bias Emitter-follower configuration Common-base configuration Collector-feedback configuration Hybrid equivalent circuit and model

3

Typical amplifier operation.

RB

RC

Q1

VCCVB(ac)

IB(ac)

VCE(ac)

IC(ac)

4

A generic dc load line.

IC

VCE

(sat)CC

CC

VI

R

(off )CE CCV V

CC CEC

C

V VI

R

5

RB

RC2 k

Q1

+12 V

VCE2 4 6 8 10 12

2

4

6

8

IC

IC(sat)

VCE(off)

Plot the dc load line for the circuit shown in Fig.

6

Plot the dc load line for the circuit shown in Fig. Then, find the values of VCE for IC = 1, 2, 5

mA respectively.

RB

RC1 k

Q1

+10 V

VCE2 4 6 8 10

2

4

6

8

IC

10IC (mA) VCE (V)

1 9

2 8

5 5

CE CC C CV V I R

7

Optimum Q-point with amplifier operation

βC BI I

CE CC C CV V I R

VCEIB = 0 A

IB = 10 A

IB = 20 A

IB = 30 A

IB = 40 A

IB = 50 A

IC

Q-Point

VCCVCC/2

IC(sat)

IC(sat)/2

IB

8

Base bias (fixed bias).

CC BEB

B

V VI

R

βC BI I

CE CC C CV V I R

RC

RB

+0.7 V

IC

IB

IE

Input

Output

VBE

VCC

Q1

= dc current gain = hFE

9

Example

RC2 k

RB360 k

+0.7 V

IC

IB

IEVBE

+8 V

hFE = 100

0.7V 8V 0.7V

360kΩ

20.28μA

CCB

B

VI

R

100 20.28μA

2.028mAC FE BI h I

8V 2.028mA 2kΩ

3.94V

CE CC C CV V I R

The circuit is midpoint biased.

10

Example

Construct the dc load line for the circuit shown in Fig, and plot the Q-point from the values obtained in Example Determine whether the circuit is midpoint biased.

VCE (V)2 4 6 8 10

1

2

3

4

IC (mA)

Q

(sat )

8V4mA

2kΩCC

CC

VI

R

off 8VCCCEV V

11

Example (Q-point shift.)

The transistor in Fig. 7.12 has values of hFE = 100 when T =

25 °C and hFE = 150 when T = 100 °C. Determine the Q-point values of IC and VCE at both of these temperatures.

RC2 k

RB360 k

+0.7 V

IC

IB

IEVBE

+8 V

hFE = 100 (T = 25C)hFE = 150 (T = 100C)

Temp(°C) IB (A) IC (mA) VCE (V)

25 20.28 2.028 3.94

100 20.28 3.04 1.92

12

Base bias characteristics. (1)

RC

RB

+0.7 V

IC

IB

IE

Input

Output

VBE

VCC

Q1 Advantage: Circuit simplicity.

Disadvantage: Q-point shift with temp.

Applications: Switching circuits only.

Circuit recognition: A single resistor (RB) between the base terminal and VCC. No emitter resistor.

13

Base bias characteristics. (2)

RC

RB

+0.7 V

IC

IB

IE

Input

Output

VBE

VCC

Q1

(sat )

(off )

CCC

C

CE CC

VI

R

V V

Load line equations:

Q-point equations:

CC BEB

B

C FE B

CE CC C C

V VI

R

I h I

V V I R

14

Voltage divider bias. (1)

R1

R2 RE

RC

+VCC

Input

Output

I1

I2 IE

IB

IC

Assume that I2 > 10IB.

2

1 2B CC

RV V

R R

0.7VE BV V

EE

E

VI

R

Assume that ICQ IE (or hFE >> 1). Then

CEQ CC CQ C EV V I R R

15

Which value of hFE do I use?

Transistor specification sheet may list any combination of the following hFE: max. hFE,

min. hFE, or typ. hFE. Use typical value if there is one. Otherwise, use

(ave) (min) (max)FE FE FEh h h

16

Stability of Voltage DividerBias Circuit

The Q-point of voltage divider bias circuit is less dependent on hFE than that of the base bias (fixed

bias).

For example, if IE is exactly 10 mA, the range of hFE is 100 to 300. Then

10mAAt 100, 100μA and 9.90mA

1 101E

FE B CQ E BFE

Ih I I I I

h

10mAAt 300, 33μA and 9.97mA

1 301E

FE B CQ E BFE

Ih I I I I

h

ICQ hardly changes over the entire range of hFE.

17

Load line for voltage divider bias circuit.

2 4 6 8 10 12

5

10

15

20

25

IC (mA)

VCE (V)

(sat )

10V20mA

260Ω+240ΩCC

CC E

VI

R R

(off ) 10VCE CCV V

Circuit values are from Example 7.9.

18

Emitter bias.

RC

RE

RB

IC

IE

IBQ1

Input

Output

+VCC

-VEE

0.7V

1EE

BB FE E

VI

R h R

19

Load Line forEmitter-Bias Circuit

(sat )

( )CC EE CC EEC

C E C E

V V V VI

R R R R

( )CE off CC EE CC EEV V V V V

VCE

IC

IC(sat)

VCE(off)

20

Emitter-bias characteristics

RC

RE

RB

IC

IE

IBQ1

Input

Output

+VCC

-VEE

Circuit recognition: A split (dual-polairty) power supply and the base resistor is connected to ground.

Advantage: The circuit Q-point values are stable against changes in hFE.

Disadvantage: Requires the use of dual-polarity power supply.

Applications: Used primarily to bias linear amplifiers.

21

Collector-feedback bias.

RB

RC

+VCC

IC

IE

IB

CC C B C B B BEV I I R I R V

( 1)CC BE

BFE C B

V VI

h R R

CQ FE BI h I

1CEQ CC FE B C

CC CQ C

V V h I R

V I R

22

Circuit Stability ofCollector-Feedback Bias

RB

RC

+VCC

IC

IE

IB

hFE increases

IC increases (if IB is the same)

VCE decreases

IB decreases

IC does not increase that much.

Good Stability. Less dependent on hFE and temperature.

23

Collector-FeedbackCharacteristics (1)

RB

RC

+VCC

IC

IE

IB

Circuit recognition: The base resistor is connected between the base and the collector terminals of the transistor.

Advantage: A simple circuit with relatively stable Q-point.

Disadvantage: Relatively poor ac characteristics.

Applications: Used primarily to bias linear amplifiers.

24

Emitter-feedback bias.

RB RC

+VCC

RE

IB

IE

IC

1CC BE

BB FE E

V VI

R h R

CQ FE BI h I

CEQ CC C C E E

CC CQ C E

V V I R I R

V I R R

1E FE BI h I

25

Circuit Stability ofEmitter-Feedback Bias

hFE increases

IC increases (if IB is the same)

VE increases

IB decreases

IC does not increase that much.

IC is less dependent on hFE and temperature.

RB RC

+VCC

RE

IB

IE

IC

26

Emitter-FeedbackCharacteristics (1)

Circuit recognition: Similar to voltage divider bias with R2 missing (or base bias with RE added).

Advantage: A simple circuit with relatively stable Q-point.

Disadvantage: Requires more components than collector-feedback bias.

Applications: Used primarily to bias linear amplifiers.

RB RC

+VCC

RE

IB

IE

IC

27

Emitter-FeedbackCharacteristics (2)

RB RC

+VCC

RE

IB

IE

IC

Q-point relationships:

( 1)CC BE

BB FE E

V VI

R h R

CQ FE BI h I

CEQ CC CQ C EV V I R R

• re transistor model – employs a diode and controlled current source to duplicate the behavior of a transistor in the region of interest.

• The re and hybrid models will be used to analyze small-signal AC analysis of standard transistor network configurations.

Ex: Common-base, common-emitter and common-collector configurations.

• The network analyzed represent the majority of those appearing in practice today.

BJT Small Signal Analysis

AC equivalent of a network is obtained by:

1. Setting all DC sources to zero

2. Replacing all capacitors by s/c equiv.

3. Redraw the network in more convenient and logical form

Common-Emitter (CE) Fixed-Bias Configuration

The input (Vi) is applied to the base and the output (Vo) is from the collector.

The Common-Emitter is characterized as having high input impedance and low output impedance with a high voltage and current gain.

Removing DC effects of VCC and Capacitors

Common-Emitter (CE) Fixed-Bias Configuration

re Model

Determine , re, and ro: and ro: look in the specification sheet for the transistor or test the transistor using a curve tracer.re: calculate re using dc analysis:

Ee I

26mVr

Common-Emitter (CE) Fixed-Bias Configuration

36

The Norton Equivalent Circuit

• Get the Norton Equivalent Circuit from the Thevenin by Source Transformation.

Impedance Calculations

Input Impedance: Output Impedance:

eBi r||RZ

eB ei r10RrZ

Or||RZ Co

c o 10roZ RRc

Common-Emitter (CE) Fixed-Bias Configuration

Gain Calculations

Voltage Gain (Av):

Current Gain (Ai):

Current Gain from Voltage Gain:

e

oC

i

ov r

)r||(R

V

VA

Coe

Cv 10Rrr

RA

)r)(RR(r

rR

I

IA

eBCo

oB

i

oi

eBCoi r10R ,10RrA

C

ivi R

ZAA

Common-Emitter (CE) Fixed-Bias Configuration

Voltage Gain

e

CvCo

e

oC

eb

oCbv

eb i

oCbO

i

Ov

r

RA 10Ror r if

r

)r||(R

βrI

)r||(RβIA

βrIV

)r||(RβIV

V

VA

Common-Emitter (CE) Fixed-Bias Configuration

Current gain

C

ivi

Bo

Bo

i

oi

eBCo

eBCo

Bo

i

oi

eB

B

Co

o

i

b

b

o

i

oi

eB

B

i

b

eB

iBb

Co

o

b

o

Co

boo

R

ZAA

ooequation t thisusecan or we

βRr

βRr

I

IA

,βr10R and 10R r if

βrRRr

βRr

I

IA

βrR

R

Rr

βr

I

I

I

I

I

IA

βrR

R

I

I and

βrR

IRI

Rr

βr

I

I and

Rr

βIrI

circuitsoutput andinput the toruledivider -current

theapplyingby determined isgain current The

Common-Emitter (CE) Fixed-Bias Configuration

Phase Relationship

The phase relationship between input and output is 180 degrees. The negative sign used in the voltage gain formulas indicates the inversion.

Common-Emitter (CE) Fixed-Bias Configuration

CE – Voltage-Divider Bias Configuration

re Model

You still need to determine , re, and ro.

CE – Voltage-Divider Bias Configuration

Impedance Calculations

Input Impedance: Output Impedance:

21

2121

RR

RRR||RR

er||RZi

oC r||RZo

C C 10RroRZo

CE – Voltage-Divider Bias Configuration

Gain Calculations

Voltage Gain (Av):

Current Gain (Ai):

Current Gain from Voltage Gain:

e

oC

i

ov r

r||R

V

VA

Coe

C

i

ov 10Rrr

R

V

VA

)rR)(R(r

rR

I

IA

eCo

o

i

oi

Coei

oi 10RrrR

Rβ

I

IA

eCoi

oi r10R ,10RrI

IA

C

ivi R

ZAA

CE – Voltage-Divider Bias Configuration

Voltage Gain

e

C vCo

e

oC v

oCe

io

e

ib

oCbO

r

RA 10Ror r if

r

)r ||(RA

)r ||(Rβr

VβV

βr

VI

)r ||)(RI (βV

CE – Voltage-Divider Bias Configuration

Current gain

e

eo

o

i

oi

Co

eCo

o

i

oi

B21

βrR'

βR'

βrR'r

rβR'

I

IA

,R10rfor

βrR'Rr

rβR'

I

IA

RR||RR'

format. same thehave

gain willcurrent for theequation the,R' the

for except ion,configurat bias-fixedemitter -

common that similar to so isnetwork thesince

CE – Voltage-Divider Bias Configuration

C

iVi

i

oi

i

oi

e

R

ZAA

optionan as

I

IA

R'

βR'

I

IA

,r10R' if And

CE – Voltage-Divider Bias Configuration

Phase Relationship

A CE amplifier configuration will always have a phase relationship between input and output is 180 degrees. This is independent of the DC bias.

CE – Voltage-Divider Bias Configuration

CE Emitter-Bias Configuration

Unbypassed RE

re Model

Again you need to determine , re.

CE Emitter-Bias Configuration

Impedance Calculations

Input Impedance: Output Impedance:

Eeb 1)R(rZ

)R(rZ Eeb

eE Eb rRRZ

bBi Z||RZ Co RZ

CE Emitter-Bias Configuration

Phase RelationshipA CE amplifier configuration will always have a phase relationship between input and output is 180 degrees. This is independent of the DC bias.

CE Emitter-Bias Configuration

Bypassed RE

This is the same circuit as the CE fixed-bias configuration and therefore can be solved using the same re model.

CE Emitter-Bias Configuration

Emitter-Follower Configuration

You may recognize this as the Common-Collector configuration. Indeed they are the same circuit. Note the input is on the base and the output is from the emitter.

re Model

You still need to determine and re.

Emitter-Follower Configuration

Impedance Calculations

Input Impedance:

bBi Z||RZ

Eeb 1)R(rZ

)R(rZ Eeb

Eb RZ

Emitter-Follower Configuration

Calculation for the current Ie

Ee

ie

eee

Ee

i

Ee

ie

b

b

ibe

b

ib

Rr

VI

rβ

βr1)β(

βr and

β1)β(but R1)β(

βrV

1)Rβ(βr

1)Vβ(I

gives for Z gsubtitutin

Z

V1)β(1)Iβ(I

Z

VI

Emitter-Follower Configuration

Impedance Calculations (cont’d)Output Impedance:

eEo r||RZ eE

eo rRrZ

Ee

ie Rr

VI

ionconfiguratfollower emitter for the impedenceoutput theDefining

Emitter-Follower Configuration

Gain CalculationsVoltage Gain (Av):

Current Gain (Ai):

Current Gain from Voltage Gain:

eE

E

i

ov rR

R

V

VA

EeEeEi

ov RrR ,rR 1

V

VA

bB

Bi ZR

RA

E

ivi R

ZAA

Emitter-Follower Configuration

Voltage gain

1V

VA

RrR

,ran greater thmuch usually R

rR

R

V

VA

rR

VRV

i

ov

EeE

eE

eE

E

i

ov

eE

iEo

Emitter-Follower Configuration

Current Gain

E

ivi

bB

Bi

bB

B

i

b

b

o

i

oi

b

o

beo

bB

B

i

b

bB

iBb

R

ZAAor

ZR

RA

,)1( since

ZR

R)1(

I

I

I

I

I

IA

)1(I

I

I)1(II

ZR

R

I

I

ZR

IRI

Emitter-Follower Configuration

Phase RelationshipA CC amplifier or Emitter Follower configuration has no phase shift between input and output.

Vo

Emitter-Follower Configuration

Common-Base (CB) Configuration

The input (Vi) is applied to the emitter and the output (Vo) is from the collector.

The Common-Base is characterized as having low input impedance and high output impedance with a current gain less than 1 and a very high voltage gain.

re Model

You will need to determine and re.

Common-Base (CB) Configuration

Impedance Calculations

Input Impedance: Output Impedance:

eEi r||RZ Co RZ

Common-Base (CB) Configuration

Gain Calculations

Voltage Gain (Av):

Current Gain (Ai):

e

C

e

C

i

ov r

R

r

R

V

VA

1I

IA

i

oi

Common-Base (CB) Configuration

Voltage & Current gain

e

C

e

C

i

oV

Ce

io

e

ie

Ce

CcCoo

r

R

r

Rα

V

VA

Rr

VαV

r

VI

RαI

)RI(RIV

1I

IA

III

II

i

oi

ieo

ie

Common-Base (CB) Configuration

Phase Relationship

A CB amplifier configuration has no phase shift between input and output.

Vo

Common-Base (CB) Configuration

Approximate Hybrid Equivalent Circuit

The h-parameters can be derived from the re model:

hie = re hib = rehfe = hfb = -hoe = 1/ro

The h-parameters are also found in the specification sheet for the transistor.

Hybrid equivalent model re equivalent model

Approximate Common-Emitter Equivalent Circuit

Hybrid equivalent model re equivalent model

Approximate Common-Base Equivalent Circuit

Chapter 7: BJT Transistor ModelingChapter 7: BJT Transistor Modeling

84

Disadvantages

• Re model– Fails to account the output impedance level of device

and feedback effect from output to input

• Hybrid equivalent model– Limited to specified operating condition in order to

obtain accurate result

85

VS

VCC

C1

C2

C3

+

-

Vo

RS

Vi

+

-RE

RCR1

R2

VS

+

-

Vo

RS

Vi

+

-

RCR1

R2

•I/p coupling capacitor s/c• Large values• Block DC and pass AC signal • Bypass

capacitor s/c•Large values

DC supply “0” potential

Voltage-divider configuration under AC analysis

Redraw the voltage-divider

configuration after removing dc supply and

insert s/c for the capacitors

• O/p coupling capacitor s/c• Large values• Block DC and pass AC signal

86

VS

RSR2 R1

Rc

Transistor small-signal ac

equivalent cct

Vo

Zi

Ii

Zo

Io

Vi

+ +

- -

B

E

C

Redrawn for small-signal AC analysis

Modeling of BJT begin

HERE!

VS

+

-

Vo

RS

Vi

+

-

RCR1

R2

87

AC bias analysis :

1) Kill all DC sources

2) Coupling and Bypass capacitors are short cct. The effect of there capacitors is to set a lower cut-off frequency for the cct.

3) Inspect the cct (replace BJTs with its small signal model:re or hybrid).

4) Solve for voltage and current transfer function, i/o and o/p impedances.

88

IMPORTANT PARAMETERS

• Input impedance, Zi

• Output impedance, Zo

• Voltage gain, Av

• Current gain, Ai

Input Impedance, Zi(few ohms M)

The input impedance of an amplifier is the value as a load when connecting a single source to the I/p of terminal of the amplifier.

89

VS Two-portsystem

Vi

Rsense

IiZi

+

-

Determining Zi

+

-

sense

isi

R

VVI

i

ii

I

VZ

Two port system-determining input impedance Zi

• The input impedance of transistor can be approximately determined using dc biasing because it doesn’t simply change when the magnitude of applied ac signal is change.

90

Demonstrating the impact of Zi

VS=10mVTwo-portsystem

Vi

Rsource

Zi

+

-

+

-1.2 kΩ

600Ω

mV6.6600k2.1

)m10(k2.1

RZ

VZV

Ω600R impedance, sourceWith

system the toapplied 10mV Full

0ΩR source, Ideal

sourcei

sii

source

source

91

Example 6.1: For the system of Fig. Below, determine the level of input impedance

VS=2mV Two-portsystem

Vi=1.2mV

RsenseZi

+

-

+

-

1 k Ω

A8.0k1

m8.0

k1

m2.1m2

R

VVI

sense

isi

:Solution

k5.18.0

m2.1

I

VZ

i

ii

92

Output Impedance, Zo (few ohms 2M)

The output impedance of an amplifier is determined at the output terminals looking back into the system with the applied signal set to zero.

Two-portsystem

Rsource

Vs=0V

Rsense

V

+

-

+

-

Io

Zo

Vo

Determining Zo

sense

oo

R

VVI

o

oo

I

VZ

cctopen become ZRZ oLo RLZo=Ro

Iamplifier

IRo

IL

RoL

Lo

II

RRFor

93

Voltage Gain, AV

• DC biasing operate the transistor as an amplifier. Amplifier is a system that having the gain behavior. • The amplifier can amplify current, voltage and power.• It’s the ratio of circuit’s output to circuit’s input.• The small-signal AC voltage gain can be determined by:

i

ov

V

VA

94

VS AvNLVi

Rsource

Zi

+

-

+

-Vo

+

-

Determining the no load voltage gain

By referring the network below the analysis are:

cct)(open ΩRi

oLvNL

V

VA

load no

vNLARZ

Z

V

VA

:resistance sourcewith

si

i

s

ovs

95

Current Gain, Ai

• This characteristic can be determined by:

i

oi

I

IA

BJTamplifier

Vi

Zi

+

-

Vo

+

-

Ii

RL

Determining the loaded current gain

Io

L

ivi

R

ZAA

Li

io

ii

Lo

RV

ZV

Z/V

R/V

L

oo

RV

I

96

re TRANSISTOR MODEL

• employs a diode and controlled current source to duplicate the behavior of a transistor.• BJT amplifiers are referred to as current-controlled devices.

Common-Base Configuration

Common-base BJT transistorre modelre equivalent cct.

97

E

BB

C

Common-base BJT transistor - pnp

Ic Ie

e

b b

c

ec I αI

IcIe

re model for the pnp common-baseconfiguration

e

b b

c

ec I αI

IcIe

common-base re equivalent cct

re

current emitter

of level DC the isII

26mVr E

E(dc)

e

isolation part,Zi=re

e

b b

c

A0Ic

IcIe=0A

Determining Zo for common-base

reVs=0V

Zo

Therefore, the input impedance, Zi = re

that less than 50Ω.

For the output impedance, it will be as follows;

98

The common-base characteristics

99

e

b b

c ec I αI Ie

re

Defining Av=Vo/Vi for the common-base configuration

BJT common-basetransistor amplifier

Vi Vo

+

-

+

-

Zi

oZ RL

Io

LeLcLoo RIRIRIV

e

L

e

Lv

r

R

r

RA

gain, Voltage

eeiei rIZIV

ee

Lev

rI

RI

Vi

VoA

100

1A

gain,Current

i

e

e

e

c

i

oi

I

I

I

I

I

IA

e

b b

c ec I αI Ie

re

Defining Ai=Io/Ii for the common-base configuration

BJT common-basetransistor amplifier

Vi Vo

+

-

+

-

ZioZ RL

Io

105

Common-Emitter Configuration

Common-emitter BJT transistorre modelre equivalent cct.Still remain controlled-current source (conducted between collector and base terminal)Diode conducted between base and emitter terminal

Input Output

Base & Emitter terminal Collector & Emitter terminal

106

common-emitter BJT transistor

EE

B

C

Ib

Icbc I I

c

e e

b

Ic

Ib

re model npn common-emitter configuration

bc I I

c

e e

b

Ic

Ii=Ib

Determining Zi using re equivalent model

re

Ie+

-

Vbe

+

-

Vi

(1) Ii

ViZi

gives (1) into subtitute

and IbreIereVbeVi

b

eb

b

be

I

rI

I

VZi

erZi 7k~6 to hundred between ranges iZ

107

The output graph

108

bI

c

e

bIi=Ib

re model for the C-E transistor configuration

rero

e

0AbI

c

e

bIi=Ib

rero

e

Vs=0V

= 0A

oZ

impedance)high cct,(open ΩZ

the thusignored is r if

rZ

o

o

oo

Output impedance Zo

109

e

b b

cbco I II Ii=Ib

re

Determining voltage and current gain for the common-emitter amplifier

BJT common-emittertransistor amplifier

Vi Vo

+

-

+

-

oZ RL

Io

ei rZ

e

Lv

r

RA

Ib

Ib

Ib

Ic

Ii

IoA

gain,Current

i

LbLcLoo RIRIRIV

ebiii rIZIV

eb

Lb

i

ov

rI

RI

V

VA

gain, Voltage

iA

110

Hybrid Equivalent Model

• re model is sensitive to the dc level of operation that result input resistance vary with the dc operating point

• Hybrid model parameter are defined at an operating point that may or may not reflect the actual operating point of the amplifier

111

Hybrid Equivalent Model

The hybrid parameters: hie, hre, hfe, hoe are developed and used to model the transistor. These parameters can be found in a specification sheet for a transistor.

112

Determination of parameter

0VVo

i12

0VVi

i11

o12i11i

o

o

VV

h

IV

h

VhIhV

0AIo

o22

0VVo

i21

o

o22i21O

o

o

VI

h

II

h

, 0VV Solving

VhIhI

H22 is a conductance!

113

General h-Parameters for any Transistor Configuration

hi = input resistancehr = reverse transfer voltage ratio (Vi/Vo)hf = forward transfer current ratio (Io/Ii)ho = output conductance

114

Common emitter hybrid equivalent circuit

115

Common base hybrid equivalent circuit

116

Simplified General h-Parameter Model

The model can be simplified based on these approximations:

hr 0 therefore hrVo = 0 and ho (high resistance on the output)

Simplified

117

Common-Emitter re vs. h-Parameter Model

hie = rehfe = hoe = 1/ro

118

Common-Emitter h-Parameters

[Formula 7.28]

[Formula 7.29]

acfe

eie

h

rh

119

Common-Base re vs. h-Parameter Model

hib = rehfb = -

120

Common-Base h-Parameters

1fb

eib

h

rh