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
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