Unit 3: Transistor at Low Frequencies

BJT Transistor Modeling

• A model is an equivalent circuit that represents the AC characteristics of the transistor.

• A model uses circuit elements that approximate the behavior of the transistor.

• There are two models commonly used in small signal AC analysis of a transistor:

– re model

– Hybrid equivalent model

The re Transistor Model

• BJTs are basically current-controlled devices; therefore the re model

uses a diode and a current source to duplicate the behavior of the

transistor.

• One disadvantage to this model is its sensitivity to the DC level. This

model is designed for specific circuit conditions.

Common-Base Configuration

ee I

mV 26r =

ei rZ =

Ω∞≅oZ

e

L

e

LV r

RrRA ≅=

α

1−≅−= αiA

( ) bbe III ββ ≅+= 1

ee I

mV 26r =

∞==oriA β

eIα Ic =

input impedance:

Output impedance:

Voltage gain:

Current gain:

Common-Emitter Configuration

The diode re model can be replaced by the resistor re.

Common-Emitter Configuration

ei rZ β=

Ω∞≅= oo rZ

e

LV r

RA −=

∞==oriA β

ei rZ )1( += β

Eeo RrZ ||=

eE

EV rR

RA+

=

1Ai += β

Input impedance:

Output impedance:

Voltage gain:

Current gain:

Common-Collector Configuration

Input impedance:

Output impedance:

Voltage gain:

Current gain:

The Hybrid Equivalent Model

The following hybrid parameters are developed and used for modeling the transistor. These parameters can be found on the specification sheet for a transistor.

• hi = input resistance

• hr = reverse transfer voltage ratio (Vi/Vo) ≅ 0

• hf = forward transfer current ratio (Io/Ii)

• ho = output conductance

Simplified General h-Parameter Model

• hi = input resistance

• hf = forward transfer current ratio (Io/Ii)

acfe

eie

hrhββ

==

1−≅−==

αfb

eib

hrh

re vs. h-Parameter Model

Common-Emitter

Common-Base

The Hybrid p Model

The hybrid p model is most useful for analysis of high-frequency transistor applications.

At lower frequencies the hybrid p model closely approximate the re parameters, and can be replaced by them.

Common-Emitter Fixed-Bias Configuration

• The input is applied to the base

• The output is from the collector

• High input impedance

• Low output impedance

• High voltage and current gain

• Phase shift between input and output is 180°

eE r10Rei

eBi

rZ

r||RZ

ββ

β

≥≅

=

Co

O

10ro

Co

Z

r||RZ

RCR ≥≅

=Co 10Rr

e

Cv

e

oC

i

ov

r

RA

r)r||(R

VVA

≥−=

−==

eBCo r10R ,10Rri

eBCo

oB

i

oi

A

)r)(RR(rrR

IIA

ββ

ββ

≥≥≅

++==

Common-Emitter Fixed-Bias Configuration

Common-Emitter Fixed-Bias Calculations

Input impedance: Voltage gain:

Output impedance: Current gain:

C

ivi R

ZAA −=

ei

21

r||RZR||RRβ′=

=′

Co 10RrCo

oCo

RZ

r||RZ

≥≅

=

Co 10Rre

C

i

ov

e

oC

i

ov

rR

VVA

rr||R

VVA

≥−≅=

−==

Current gain from voltage gain:

Common-Emitter Voltage-Divider Bias

re model requires you to determine β, re, and ro.

Calaculations:

Input impedance:

Output impedance:

Voltage gain:

eCo

Co

r10R ,10Rri

oi

10Rrei

oi

eCo

o

i

oi

IIA

rRR

IIA

)rR)(R(rrR

IIA

ββ

ββ

ββ

≥′≥

≥

≅=

+′′

≅=

+′+′

==

C

ivi R

ZAA −=

Eb

Eeb

Eeb

bBi

RZ)R(rZ1)R(rZ

Z||RZ

ββ

ββ

≅+≅

++==

Co RZ =

Current gain:

Current gain from voltage gain:

Common-Emitter Emitter-Bias Configuration

Impedance Calculations

Input impedance:

Output impedance:

Eb

Eeb

RZE

C

i

ov

)R(rZEe

C

i

ov

b

C

i

ov

RR

VVA

RrR

VVA

ZR

VVA

β

β

β

≅

+=

−≅=

+−==

−==

bB

B

i

oi ZR

RIIA

+==

β

C

ivi R

ZAA −=

Gain Calculations

Voltage gain:

Current gain:

Current gain from voltage gain:

Emitter-Follower Configuration

• This is also known as the common-collector configuration.

• The input is applied to the base and the output is taken from the emitter.

There is no phase shift between input and output.

eE rReo

eEo

rZ

r||RZ

>>≅

=

Eb

Eeb

Eeb

bBi

RZ)R(rZ1)R(rZ

Z||RZ

β≅

+β≅

+β+β=

=

EeEeE RrR ,rRi

ov

eE

E

i

ov

1VVA

rRR

VVA

≅+>>≅=

+==

bB

Bi ZR

RA+

−≅β

Impedance Calculations

Input impedance:

Output impedance:

Gain Calculations

Voltage gain:

Current gain:

E

ivi R

ZAA −=

eEi r||RZ =

Current gain from voltage gain:

Common-Base Configuration

• The input is applied to the emitter.

• The output is taken from the collector.

• Low input impedance.

• High output impedance.

• Current gain less than unity.

• Very high voltage gain.

• No phase shift between input and output.

Calculations

Input impedance:

Co RZ =

e

C

e

C

i

ov r

RrR

VVA ≅==

α

1IIA

i

oi −≅−== α

F

C

ei

RR1

rZ+

=

β

Output impedance:

Voltage gain:

Current gain:

Common-Emitter Collector Feedback Configuration

• This is a variation of the common-emitter fixed-bias configuration

• Input is applied to the base

• Output is taken from the collector

• There is a 180° phase shift between input and output

Calculations

Input impedance:

FCo R||RZ ≅

e

C

i

ov r

RVVA −==

C

F

i

oi

CF

F

i

oi

RR

IIA

RRR

IIA

≅=

+==

ββ

Output impedance:

Voltage gain:

Current gain:

Collector DC Feedback Configuration

• This is a variation of the common-emitter, fixed-bias configuration

• The input is applied to the base

• The output is taken from the collector

• There is a 180° phase shift between input and output

F

C

ei

RR1

rZ+

=

β

FCo R||RZ ≅

e

C

i

ov r

RVVA −==

C

F

i

oi

CF

F

i

oi

RR

IIA

RRR

IIA

≅=

+==

ββ

Calculations

Input impedance:

Output impedance:

Voltage gain:

Current gain:

Two-Port Systems Approach

ooTh RZZ ==

ivNLTh VAE =

L

ivi R

ZAA −=

vNLoL

L

i

ov A

RRR

VV

A+

==

This approach:

• Reduces a circuit to a two-port system

• Provides a “Thévenin look” at the output terminals

• Makes it easier to determine the effects of a changing load

With Vi set to 0 V:

The voltage across the open terminals is:

where AvNL is the no-load voltage gain.

Effect of Load Impedance on Gain

This model can be applied to any current- or voltage-controlled amplifier.

Adding a load reduces the gain of the amplifier:

si

sii RR

VRV+

=

vNLsi

i

s

ovs A

RRR

VVA

+==

L

ivi

oL

vNLL

i

ov

RR

AA

RRAR

VV

A

−=

+==

Effect of Source Impedance on Gain

The fraction of applied signal that reaches the input of the amplifier is:

The internal resistance of the signal source reduces the overall gain:

Combined Effects of RS and RL on Voltage Gain

Effects of RL:

L

isvsis

vNLoL

L

si

i

s

ovs

RRR

AA

ARR

RRR

RVV

A

+−=

++==

Co RZ =

ei rRRZ β|||| 21=

Effects of RL and RS:

Cascaded Systems

The output of one amplifier is the input to the next amplifier

The overall voltage gain is determined by the product of gains of the individual stages

The DC bias circuits are isolated from each other by the coupling capacitors

The DC calculations are independent of the cascading

The AC calculations for gain and impedance are interdependent

R-C Coupled BJT Amplifiers

Input impedance, first stage:

Output impedance, second stage:

21

2

211

||||||

vvv

e

CV

e

eCv

AAArRA

rrRRRA

=

=

=β

Voltage gain: