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Circuit Models
for Amplifiers
The two most important amplifier circuit models explicitly use
the open-circuit voltage gain Avo :
Iin
Iout
Zout


Z in
Vin


Avo Vin

Vout

And the short-circuit current gain Ais :
Iin
Iout

Vin

Jim Stiles

Zout
Z in
Ais Iin
The Univ. of Kansas
Vout

Dept. of EECS
4/30/2017
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In addition, each equivalent circuit model uses the same two
impedance values—the input impedance Z in and output
impedance Zout .
Q: So what are these models good for?
A: Say we wish to analyze a circuit in which an amplifier is but
one component. Instead of needing to analyze the entire
amplifier circuit, we can analyze the circuit using the (far)
simpler equivalent circuit model.
For example, consider this audio amplifier design:

Vout

Vin


Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/30/2017
582721780
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Say we wish to connect a source (e.g., microphone) to its input,
and a load (e.g., speaker) to its output:
R1


L1
R2
Vg
L2

Vout

Let’s say on the EECS 412 final, I ask you to determine Vout in
the circuit above.
Q: Yikes! How could we possibly analyze this circuit on
an exam—it would take way too much time (not to
mention way too many pages of work)?
A: Perhaps, but let’s say that I also provide you with
the amplifier input impedance Z in , output impedance
Zout , and open-circuit voltage gain Avo .
You thus know everything there is to know about the amplifier!
Just replace the amplifier with its equivalent circuit:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/30/2017
R1


Vg
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L1
Zout

Vin

Z in


R2
L2
Avo Vin

Vout

From input circuit, we can conclude (with a little help from
voltage division):


Zin
Vin Vg 

R

j
ωL

Z
1
in 
 1
And the output circuit is likewise:
Vout


R2 j ωL2


 Avo Vin
 Zout  R2 j ωL2 


where:
R2 j ωL2 
j ω R2L2
R2  j ωL2
Q: Wait! I thought we could determine the output voltage from
the input voltage by simply multiplying by the voltage gain Avo . I
am certain that you told us:
oc
Vout
 Avo Vin
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/30/2017
582721780
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A: I did tell you that! And this expression is exactly correct.
oc
However, the voltage Vout
is the open-circuit output voltage of
the amplifier—in this circuit (like most amplifier circuits!), the
output is not open!
oc
Hence Vout  Vout
, and so :


R2 j ωL2

Vout  Avo Vin 
 Zout  R2 j ωL2 




R2 j ωL2
oc

Vout 
 Zout  R2 j ωL2 


oc
 Vout
Now, combining the two expressions, we have our answer:
Vout

R2 j ωL2





  Zout  R2 j ωL2 



Zin
j ω R2L2

 AvoVg 



R

j
ωL

Z
1
in   Z out R2  j ωL2   j ω R2L2 
 1

Zin
Vg Avo 
 R1  j ωL1  Zin
Now, be aware that we can (and often do!) define a voltage gain
Av , a value that is different from the open-circuit voltage gain
of the amplifier.
For instance, in the above circuit example we could define a
voltage gain as the ratio of the input voltage Vin and the output
voltage Vout :
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/30/2017
582721780
Av
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Vout
Vin


R2 j ωL2

 Avo 
 Zout  R2 j ωL2 




j ω R2L2


 Avo
 Zout R2  j ωL2   j ω R2L2 


Or, we could alternatively define voltage gain as the ratio of the
source voltage Vg and the output voltage Vout :
Av
Vout
Vg

Zin
 Avo 
 R1  j ωL1  Zin


j ω R2L2




Z
R

j
ωL

j
ω
R
L
  out  2
2
2 2 
Q: Yikes! Which result is correct; which voltage gain is “the”
voltage gain?
A: Both are!
We can define a voltage gain Av in any manner that is useful to
us. However, we must make this definition explicit—precisely
what two voltages are involved in the definition?
 No voltage gain Av is “the” voltage gain!
Note that the open-circuit voltage gain Avo is a parameter of the
amplifier—and of the amplifier only!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/30/2017
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Contrast Avo to the two voltage gains defined above (i.e., Vout Vin
and Vout Vg ).
In each case, the result—of course—depends on amplifier
parameters ( Avo , Zin , Zout ). However, the results likewise depend
on the devices (source and load) attached to the amplifier (e.g.,
L1 , R1 , L2 , R2 ).
 The only amplifier voltage gain is its open-circuit voltage
gain Avo !
Now, let’s switch gears and consider low-frequency (e.g., audio
and video) applications.
At these frequencies, parasitic elements are typically too small
to have any practical significance. Additionally, low-frequency
circuits frequently employ no reactive circuit elements (no
capacitor or inductors).
As a result, we find that the input and output impedances
exhibit almost no imaginary (i.e., reactive) components:
Zin ω   Rin  j 0
Zout ω   Rout  j 0
Likewise, the voltage and current gains of the amplifier are
(almost) purely real:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/30/2017
582721780
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Avo ω   Avo  j 0
Ais ω   Ais  j 0
Note that these real values can be positive or negative.
The amplifier circuit models can thus be simplified—to the
point that we can easily consider arbitrary time-domain signals
(e.g., vin t  or iout t  ):
iout t 
iin t 
Rout

vin t 
Rin



Avo vin t 


Jim Stiles
vout t 

iout t 
iin t 
vin t 

Rin
Rout
Ais iin t 
The Univ. of Kansas

vout t 

Dept. of EECS
4/30/2017
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For this case, we find that the (approximate) relationships
between the input and output are that of an ideal amplifier:
v
oc
out
t
t    A δ t  t  v t   A
vo
in
vo
vin t 

i
sc
out
t
t    A δ t  t  i t   A i t 
is
in
is
in

Specifically, we find that for these low-frequency models:
Rin 
Avo 
vin t 
iin t 
oc
vout
t 
vin t 
Rout 
Ais 
oc
vout
t 
sc
iout
t 
sc
iout
t 
iin t 
One important caveat here; this “low-frequency” model is
applicable only for input signals that are likewise lowfrequency—the input signal spectrum must not extend beyond
the amplifier bandwidth.
Now one last topic.
Frequently, both the input and output voltages are expressed
with respect to ground potential, a situation expressed in the
circuit model as:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/30/2017
582721780
10/10
iout t 
iin t 
Rout

vin t 
Rin



Avo vin t 

vout t 

Now, two nodes at ground potential are two nodes that are
connected together! Thus, an equivalent model to the one above
is:
iout t 
iin t 
Rout

vin t 
Rin



Avo vin t 

vout t 

Which is generally simplified to this model:
iout t 
iin t 
Rout

vin t 

Jim Stiles
Rin
The Univ. of Kansas


Avo vin t 

vout t 

Dept. of EECS