Download small-signal hybrid-π equivalent circuit

SMALL-SIGNAL HYBRID-Π
EQUIVALENT CIRCUIT
Content
BJT – Small Signal Amplifier
BJT complete Hybrid equivalent circuit
BJT approximate Hybrid model
Objectives
Develop 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 variables
2-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.
Vbe  hie I b  hreVce
I c  h fe I b  hoeVce
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:
v be  i b r
Finding rπ from Q-point slope lead to:
v be
VT
 VT
 r 

ib
I BQ
I CQ
rπ also known as diffusion resistance and is a
function of Q-point parameters.
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:
i C 
i C
v BE
.v BE
Q  pt
From eq 5.2 in Chapter 5 Neaman,
iC
 v BE
 I S exp
 VT



Cont..
After substitution and rearrange the above, we
obtain:
iC
v BE
Q  pt
 v BE
1

. I S exp
VT
 VT
I CQ



 Q  pt VT
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:
gm 
I CQ
VT
Transconductance also a function of Q-pt
parameters and directly proportional to dc bias
current.
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
gm=ICQ/VT
r=VT/ICQ
Transconductance parameter
Cont..
We can relate small-signal collector current to
small-signal base current for o/p of equivalent cct.
i C
ic 
i B
Where
i C
i B
.i b
Q  pt

Q  pt
β is called ac common-emitter current gain.
Thus:
i c  i b
Small-signal hybrid- equivalent circuit
using common-emitter current gain
ib
(Ib )

Current gain parameter
Small-signal circuit parameters
Small-signal voltage gain
Combine BJT equivalent cct to ac
equivalent cct.
Small-signal voltage gain
Voltage 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.
Vo  Vce  g mVbe RC
Cont..
From the input portion of the circuit:
 r 
Vs
Vbe  
 r  RB 
The small-signal voltage gain is:
 r 
Vo

Av   g m RC 
Vs
 r  RB 
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
1.
2.
I BQ 
CEQ
4. r

6.
RB
1.2  0.7

 10 A
50
I CQ  I BQ  100(10A)  1 mA
3. V
5.
VBB  VBE ( on)
 VCC  I CQ RC  12  (1)(6)  6V
VT
(100)(0.026)


 2.6 k
I CQ
1
I CQ
1
gm 

 38.5 mA / V
VT
0.026
 r

Vo
  11.4
Av 
 g m RC 
Vs
 r  RB 
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 voltage
Figure 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 5Neaman):

 v BE  
v CE 




i C  I S exp
 . 1 

VA 
 VT   

The output resistance, rO:
v CE
rO 
i C Q  pt
Substitute and rearrange both equation,

 v BE
1
 I S exp
rO
 VT

 1
.
 V A

Q  pt
I CQ
VA
Cont..
Hence, small-signal transistor output resistance, rO
become:
VA
rO 
I CQ
rO is equivalent to Norton resistance  rO is
parallel with dependent current sources.
Modified bipolar equivalent circuits including rO due to
Early Effect.
Transconductance
parameter
ro=VA/ICQ
Current gain
parameter
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:
Vbe  hie I b  hreVce
I c  h fe I b  hoeVce
Where:
i for input
r for reverse
f for forward
o for output
e for common-emitter
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