<p>5/9/2018 02f01aa304d779d327354e8bfebad035.doc 1/6</p><p>The Gain of Real Op-Amps</p><p>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!</p><p>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( )).</p><p>Typically, this op-amp behavior can be described mathematically with the complex function:</p><p>A0 Aopω( ) = ω 1 + j ( ω ) b</p><p> or, using the frequency definition ω= 2 πf , we can write:</p><p>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</p><p>Note the squared magnitude of the op-amp gain is therefore the real function: </p><p>2 A0 A 0 Aopω( ) = ω ω 1+j( ω) 1 - j ( ω ) b b A 2 = 0 2 ω 1 + ( ω ) b</p><p>Therefore at D.C. (ω = 0) the op-amp gain is:</p><p>A0 Aopω(= 0) = A = 0 0 1 + j ( ω ) b and thus: 2 2 Aopω(= 0) A = 0</p><p>Where:</p><p>A0 = op-amp D.C. gain </p><p>Again, note that the D.C. gain A0 is:</p><p>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</p><p>The open-loop gain of real op-amps is very large, but fathomable —typically between 105 and 108.</p><p>Q: So just what does the value ωb indicate ?</p><p>A: The value ωb is the op-amp’s break frequency. Typically, this value is very small (e.g. fb  10 Hz ). </p><p>To see why this value is important, consider the op-amp gain at</p><p>ω= ωb :</p><p>A0 Aopω( ω= b ) = ω 1 + j ( ω ) b A = 0 1 + j A A =0 - j 0 2 2</p><p>A - j π = 0 e 4 2</p><p>The squared magnitude of this gain is therefore:</p><p>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</p><p>As a result, the break frequency ωb is also referred to as the “half-power” frequency, or the “3 dB” frequency.</p><p>2 If we plot Aopω( ) on a “log-log” scale, we get something like this:</p><p>2 Aop (ω ) (dB)</p><p>2 A0 (dB)</p><p>20 dB/decade</p><p> logω 0 dB ω b ωt</p><p>QQ:: Hey!Hey! YouYou havehave defineddefined aa newnew frequencyfrequency —— .. WhatWhat isis thisthis frequencyfrequency andand whywhy isis itit important?important?</p><p>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</p><p>2 Aopω( ω= t ) = 1</p><p>Note that when expressed in dB, unity gain is:</p><p>2 10 log10Aopω ( ω= t ) = 10 log 10 ( 1) = 0 dB</p><p>Therefore, on a “log-log” plot, the gain curve crosses the horizontal axis at frequency ωt . </p><p>We thus refer to the frequency ωt as the “unity-gain frequency” of the operational amplifier.</p><p>Note that we can solve for this frequency in terms of break frequency ωb and D.C. gain Ao:</p><p>2 1=Aopω ( ω = t ) A 2 = 0 2 ωt 1 + ( ω ) b meaning that: 2= 2 2 - ωt ω b ( A0 1)</p><p>2 2 But recall that A0 ? 1, therefore A0- 1 A 0 and:</p><p>ωt= ω b A0 5/9/2018 02f01aa304d779d327354e8bfebad035.doc 6/6</p><p>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 .</p><p>As a result, ωt is alternatively referred to as the gain- bandwidth product!</p><p>∐ ωt Unity Gain Frequency</p><p> and</p><p>∐ ωt Gain -Bandwidth Product</p><p>ThisThis is is so so simple simple perhaps perhaps even even I I can can rememberremember it: it:</p><p>TheThe gain-bandwidth-product gain-bandwidth-product is is the the productproduct of of the the gain gain and and the the bandwidth bandwidth! !</p>