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The Gain of Real Op-Amps
The open-circuit voltage gain Aop (a differential gain!) of a real (i.e., non-ideal) 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., Aopω( )).
Typically, this op-amp behavior can be described mathematically with the complex function:
A0 Aopω( ) = ω 1 + j ( ω ) b
or, using the frequency definition ω= 2 πf , we can write:
A A( f ) = 0 op 骣 1 + j 琪f 桫 fb where ω is frequency expressed in units of radians/sec, and f is signal frequency expressed in units of cycles/sec. 5/9/2018 02f01aa304d779d327354e8bfebad035.doc 2/6
Note the squared magnitude of the op-amp gain is therefore the real function:
2 A0 A 0 Aopω( ) = ω ω 1+j( ω) 1 - j ( ω ) b b A 2 = 0 2 ω 1 + ( ω ) b
Therefore at D.C. (ω = 0) the op-amp gain is:
A0 Aopω(= 0) = A = 0 0 1 + j ( ω ) b and thus: 2 2 Aopω(= 0) A = 0
Where:
A0 = op-amp D.C. gain
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 5/9/2018 02f01aa304d779d327354e8bfebad035.doc 3/6
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 10 Hz ).
To see why this value is important, consider the op-amp gain at
ω= ωb :
A0 Aopω( ω= b ) = ω 1 + j ( ω ) b A = 0 1 + j A A =0 - j 0 2 2
A - j π = 0 e 4 2
The squared magnitude of this gain is therefore:
2 A A Aω( ω= ) = 0 0 op b 1+j 1 - j A 2 = 0 1 - j 2 A 2 = 0 2 5/9/2018 02f01aa304d779d327354e8bfebad035.doc 4/6
As a result, the break frequency ωb is also referred to as the “half-power” frequency, or the “3 dB” frequency.
2 If we plot Aopω( ) on a “log-log” scale, we get something like this:
2 Aop (ω ) (dB)
2 A0 (dB)
20 dB/decade
logω 0 dB ω b ωt
QQ:: Hey!Hey! YouYou havehave defineddefined aa newnew frequencyfrequency —— .. WhatWhat isis thisthis frequencyfrequency andand whywhy isis itit important?important?
A: Note that ωt is the frequency where the magnitude of the gain is “unity” (i.e., where the gain is 1). I.E., 5/9/2018 02f01aa304d779d327354e8bfebad035.doc 5/6
2 Aopω( ω= t ) = 1
Note that when expressed in dB, unity gain is:
2 10 log10Aopω ( ω= t ) = 10 log 10 ( 1) = 0 dB
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.
Note that we can solve for this frequency in terms of break frequency ωb and D.C. gain Ao:
2 1=Aopω ( ω = t ) A 2 = 0 2 ωt 1 + ( ω ) b meaning that: 2= 2 2 - ωt ω b ( A0 1)
2 2 But recall that A0 ? 1, therefore A0- 1 A 0 and:
ωt= ω b A0 5/9/2018 02f01aa304d779d327354e8bfebad035.doc 6/6
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 bandwidthωb .
As a result, ωt is alternatively referred to as the gain- bandwidth product!
∐ ωt Unity Gain Frequency
and
∐ ωt Gain -Bandwidth Product
ThisThis is is so so simple simple perhaps perhaps even even I I can can rememberremember it: it:
TheThe gain-bandwidth-product gain-bandwidth-product is is the the productproduct of of the the gain gain and and the the bandwidth bandwidth! !