SMALL-SIGNAL HYBRID-Π EQUIVALENT

CIRCUIT

Content

BJT – Small Signal AmplifierBJT complete Hybrid equivalent circuit

BJT approximate Hybrid model

ObjectivesDevelop the small-signal models of transistor that are used in analysis of linear amplifier.

Basic knowledge..

Ohm’s Law

Kirchoff’s Law

Thevenin and Norton’s Theorem

All electronic circuit analysis require these for mathematical manipulation.

Small signal hybrid- equivalent circuit of bipolar transistor

Need to develop a small-signal equivalent cct -- use hybrid- model because is closely related to the physic of transistor.

Treat transistor as two-port network.

2-port system

AC analysis require simplification of transistors as 2-port system.

Simplification leads to new parameters / definitions.

2-port system cont..

‘Single ended’ 2-port system has 1 input port shorted to 1 output port.

Alternative view =>system has a common input/output port.

Three terminal device device which only three connection leads, i.e transistor falls into this category.

Single-ended 2-port network

Differential 2-port network

The ‘differential 2-port’ network will be the basis for forthcoming analysis of all types of transistors (BJT and FET).

Port variables2-port network analysis is all about current and voltage by breaking down voltage direction (-ve to +ve or +ve to –ve) and current direction (to or from).

Each current and voltage has 2 possible

directions.

2-port variables

Below are the equations for BJT’s derived from 2-port network simplification.

ceoebfec

cerebiebe

VhIhI

VhIhV

Small signal hybrid-π equivalent circuit

Based on 2-port network, 1 input port and 1 output port shorted together to form a common port of both input and output.

Transistor has input and output ports shorted (emitter) resulting a small-signal 2-port hybrid- π network.

Cont..

Figure shows iB vs. vBE with small-time varying signal superimposed at Q-pt.

Since sinusoidal signals are small, the slope at Q-pt treated as a constant, has units of conductance.

The inverse of this conductance is small-signal resistance, rπ

Cont..We can relate small-signal input base current to small-signal input voltage by:

Finding rπ from Q-point slope lead to:

rπ also known as diffusion resistance and is a function of Q-point parameters.

riv bbe

CQ

T

BQ

T

b

be

I

V

I

Vr

i

v

Cont.. Now, we consider the output terminal characteristic of BJT.

Assume o/p collector current is independent of collector-emitter voltage collector-current is a function of base-emitter voltage, so the equation:

From eq 5.2 in Chapter 5 Neaman,

BE

ptQBE

CC v

v

ii

.

T

BESC V

vIi exp

Cont..After substitution and rearrange the above, we obtain:

The term ICQ / VT is a conductance. Since this term relates current in collector to a voltage in B-E circuit, it is called transconductance and is written:

Transconductance also a function of Q-pt parameters and directly proportional to dc bias current.

T

CQ

ptQT

BES

TptQBE

C

V

I

V

vI

Vv

i

exp.1

T

CQm V

Ig

Cont..

Using these new parameters develop a simplified small-signal hybrid-π equivalent cct for npn BJT.

Phasor components given in parentheses.

This circuit can be inserted into ac equivalent circuit shown previously.

Small-signal hybrid- equivalent circuit using transconductance

Transconductance parameter

gm=ICQ/VT

r=VT/ICQ

Cont..We can relate small-signal collector current to small-signal base current for o/p of equivalent cct.

Where

β is called ac common-emitter current gain.

Thus:

b

ptQB

Cc i

i

ii .

ptQB

C

i

i

bc ii

ib(Ib )

Current gain parameter

Small-signal hybrid- equivalent circuit using common-emitter current gain

Small-signal circuit parameters

Small-signal voltage gain

Combine BJT equivalent cct to ac equivalent cct.

Small-signal voltage gainVoltage gain, Av = ratio of o/p voltage to i/p voltage.

Small-signal B-E voltage is called control voltage, Vbe or V.

The dependent current source is gmV flows through RC produce –ve C-E voltage at the output.

Cbemceo RVgVV

Cont..

From the input portion of the circuit:

The small-signal voltage gain is:

sB

be VRr

rV

B

Cms

ov Rr

rRg

V

VA

Example 1

Given : = 100, VCC = 12V

VBE = 0.7V, RC = 6k, VT=0.026V, RB = 50k and

VBB = 1.2V

Calculate the small-signal

voltage gain.

Solutions

AR

VVI

B

onBEBBBQ 10

50

7.02.1)(

mAAII BQCQ 1)10(100

VRIVV CCQCCCEQ 6)6)(1(12

kI

Vr

CQ

T 6.21

)026.0)(100(

VmAV

Ig

T

CQm /5.38

026.0

1

4.11

B

Cms

ov Rr

rRg

V

VA

1.

2.

3.

4.

5.

6.

Example 2

Given VCC=5V, VBB=2V, RB=650kΩ, RC=15kΩ, β=100 and VBE(on)=0.7V. Determine a) Q-points, b) gm and r

c) voltage gain.

Early effect

Early Voltage (VA)

Early voltageFigure above show current-voltage characteristic for constant values of B-E voltage.

The curves are linear with respect to C-E voltage in forward-active mode.

The slope is due to base-width modulation effect Early Effect.

When the curves extrapolated at zero current, they meet a point on –ve voltage axis, vce = -VA. VA --- Early voltage with typical value in range of 50 < VA

< 300V.

Hybrid-π equivalent circuit with Early Effect

Early Effect => collector current, iC is dependent to collector-emitter voltage, vCE (refer Chapter 5-Neaman):

The output resistance, rO:

Substitute and rearrange both equation,

A

CE

T

BESC V

v

V

vIi 1.exp

ptQC

CEO i

vr

A

CQ

ptQAT

BES

O V

I

VV

vI

r

1.exp

1

Cont..

Hence, small-signal transistor output resistance, rO become:

rO is equivalent to Norton resistance rO is parallel with dependent current sources.

CQ

AO I

Vr

Modified bipolar equivalent circuits including rO

due to Early Effect.

Transconductance parameter

Current gain parameter

ro=VA/ICQ

Self study for pnp transistor

From Neaman textbook,Ac equivalent circuit – pg 386

Transconductance and current gain – pg 386 & 387

Small-signal hybrid-π equivalent circuit – pg 387

Do example 6.3

Expanded hybrid-π equivalent circuit

Include 2 additional resistance, rb and rμ.

rb series resistance of semiconductor material.Since rb << rμ., rb is neglected (short cct) at low freq.

rμ reverse-biased diffusion resistance of B-C junction. Typically in megaohms and neglected (open cct). Normally, in hybrid-π model, we neglect both rb and rμ.

Other small-signal parameters -h parameter

h-parameter -> relate small-signal terminal currents and voltages of 2-port network.The linear r/ship between terminal currents and voltages are:

Where:i for inputr for reversef for forwardo for outpute for common-emitter

ceoebfec

cerebiebe

VhIhI

VhIhV

h-parameter

These equations represent KVL at input and KCL at output applied to h-parameter model for common-emitter BJT.

h-parameter in relation to hybrid-π