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Amplifiers
An ideal amplifier is a two-port circuit that takes an input signal vin t  and
reproduces it exactly at its output, only with a larger magnitude!
iin t 

 
vin t

Avo


vout t  Avo vin t
I


The real value Avo is the open-circuit voltage gain of this ideal amplifier, and
has a magnitude much larger than unity ( Avo
Jim Stiles
1 ).
The Univ. of Kansas
Dept. of EECS
2/9/2016
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We actually can find g(t) !
Now, let’s express this result using our knowledge of linear circuit theory!
Recall, the output vout (t ) of a linear device can be determined by convolving its
input vin (t ) with the device impulse response g (t ) :
vout (t ) 
t
 g (t t ) v
in
(t ) dt 

Q: Yikes! What is the impulse response of this ideal amp? How can we
determine it?
A: It’s actually quite simple!
Remember, the impulse response of linear circuit is just the output that results
when the input is an impulse function δ (t ) .
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/9/2016
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Every function an Eigen function
Since the output of an ideal amplifier is just the input multiplied by Avo , we
conclude if vin (t )  δ (t ) :



g t  vout t  Avo δ t
Thus:
vout (t ) 
t
 g (t t ) v
in
(t ) dt 


t
A
vo
δ (t t ) vin (t ) dt 

t
 Avo  δ (t t ) vin (t ) dt 

 Avo vin (t )
 Any and every function vin (t ) is an Eigen function of an ideal amplifier!!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/9/2016
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And now the Eigen value
Now, we can determine the Eigen value of this linear operator relating input to
output:
  

vout t  L vin t
Recall this Eigen value is found from the Fourier transform of the impulse
response:
G ω  

 jωt
h
t
e
dt






 jωt
A
δ
t
e
dt


 vo

 Avo  j 0
 Avo e j 0
This result, although simple, has an interesting interpretation…
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/9/2016
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DC to daylight
…it means that the amplifier exhibits gain of Avo for sinusoidal signals of any
and all frequencies!
G ω 
Avo
ω
BUT, there is one big problem with an ideal amplifier:
They are impossible to build!!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/9/2016
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Real amplifier have finite bandwidths
The ideal amplifier has a frequency response of G ω   Avo .
Note this means that the amplifier gain is Avo for all frequencies 0  ω  
(D.C. to daylight!).
The bandwidth of the ideal amplifier is therefore infinite!
*
Since every electronic device will exhibit some amount of inductance,
capacitance, and resistance, every device will have a finite bandwidth.
*
In other words, there will be frequencies  where the device does not
work!
*
From the standpoint of an amplifier, “not working” means G ω 
Avo (i.e.,
low gain).

Jim Stiles
Amplifiers therefore have finite bandwidths.
The Univ. of Kansas
Dept. of EECS
2/9/2016
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Amplifier bandwidth
There is a range of frequencies ω between ωL and ωH where the gain will
(approximately) be Avo.
For frequencies outside this range, the gain will typically be small (i.e.
G ω  Avo ):

G ω   

 Avo
Avo
ωL  ω  ωH
ω  ωL , ω  ωH
The width of this frequency range is called the amplifier bandwidth:
Bandwidth
ωH  ωL (radians/sec)
fL  fH
Jim Stiles
(cycles/sec)
The Univ. of Kansas
Dept. of EECS
2/9/2016
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Wideband is desirable
One result of a finite bandwidth is that the amplifier impulse response is not an
impulse function !
h (t ) 

 H (ω ) e
 jωt
dt  Avo δ (t )

therefore generally speaking:
vout t   Avo vin t  !!
However, if an input signal spectrum Vin ω  lies completely within the amplifier
bandwidth, then we find that will (approximately) behave like an ideal amplifier:
vout t   Avo vin t 
if Vin ω  is within the amplifier bandwidth
As a result, maximizing the bandwidth of an amplifier is a typically and
important design goal!
Jim Stiles
The Univ. of Kansas
Dept. of EECS