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The Gain of
Real Op-Amps
The open-circuit voltage gain Aop (a differential gain!) of a real (i.e., nonideal) operational amplifier is very large at D.C. (i.e., ω  0 ), but gets smaller as
the signal frequency ω increases!
In other words, the differential gain of an op-amp (i.e., the open-loop gain of a
feedback amplifier) is a function of frequency ω .
We will thus express this gain as a complex function in the frequency domain
(i.e., Aop (ω ) ).

vd ω 

Jim Stiles

vout ω   Aop ω  vd ω 
Aop ω 

The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
2/9
Gain is a complex function frequency
Typically, this op-amp behavior can be described mathematically
with the complex function:
Aop (ω ) 
A0
1 j
 
ω
ωb
or, using the frequency definition ω  2πf , we can write:
Aop (f ) 
A0
1  j  f 
 fb 
where ω is frequency expressed in units of radians/sec, and f is signal
frequency expressed in units of cycles/sec.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
3/9
DC is when the signal frequency is zero
Note the squared magnitude of the op-amp gain is therefore the real function:
A0
2
Aop (ω ) 

1 j
A0
  1 j  
ω
A02
1
ω
ωb
ωb
 
ω
2
ωb
Therefore at D.C. ( ω  0 ) the op-amp gain is:
Aop (ω  0) 
A0
1 j
 
0
 A0
ωb
and thus:
2
Aop (ω  0)  A02
Where:
A0  op-amp D.C. gain
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
4/9
The break frequency
Again, note that the D.C. gain A0 is:
1) an open-circuit voltage gain
2) a differential gain
3) also referred to as the open-loop D.C. gain
The open-loop gain of real op-amps is very large, but fathomable —typically
between 105 and 108.
Q: So just what does the value ωb indicate ?
A: The value ωb is the op-amp’s break frequency.
Typically, this value is very small (e.g. fb  10Hz ).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
5/9
The 3dB bandwidth
To see why this value is important, consider the op-amp gain at ω  ωb :
Aop ω  ωb  
A0
1 j
 
ω

ωb
A0
1 j

A0
2
j
A0
2

A0
2
e
jπ
4
The squared magnitude of this gain is therefore:
2
Aop (ω  ωb ) 
A0
A0
1 j 1 j

A02
1 j
2

A02
2
As a result, the break frequency ωb is also referred to as the “half-power”
frequency, or the “3 dB” frequency.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
6/9
This value is very important!
2
If we plot Aop (ω ) on a “log-log” scale, we get something like this:
Aop (ω )
2
(dB)
A0 (dB)
2
20 dB/decade
logω
0 dB
ωb
ωt
Q: Hey! You have defined a new frequency — ωt .
What is this frequency and why is it important?
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
7/9
The unity gain frequency
A: Note that ωt is the frequency where the magnitude of the gain is “unity”
(i.e., where the gain is 1). I.E.,
2
Aop (ω  ωt )  1
Note that when expressed in dB, unity gain is:
10 log10 Aop (ω  ωt )  10 log10 1   0 dB
2
Therefore, on a “log-log” plot, the gain curve crosses the horizontal axis at
frequency ωt .
We thus refer to the frequency ωt as the “unity-gain frequency” of the
operational amplifier.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
8/9
It’s the product of the
gain and the bandwidth!
Note that we can solve for this frequency in terms of break frequency ωb and
D.C. gain Ao:
A02
2
1  Aop (ω  ωt ) 
meaning that:

1
ωt 2  ωb 2 A02  1
But recall that A0
 
ωt
2
ωb

1 , therefore A02  1  A02 and:
ωt  ωb A0
Note since the frequency ωb defines the 3 dB bandwidth of the op-amp, the
unity gain frequency ωt is simply the product of the op-amp’s D.C. gain A0 and
its bandwidth ωb .
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687273424
9/9
It’s not rocket science!
As a result, ωt is alternatively referred to as the gain-bandwidth product!
ωt
Unity Gain Frequency
ωt
Gain - Bandwidth Product
and
This is so simple perhaps even I can
remember it:
The gain-bandwidth-product is the
product of the gain and the bandwidth!
Jim Stiles
The Univ. of Kansas
Dept. of EECS