UNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences EE105 Lab Experiments Bode Plot Tutorial Contents 1 Introduction 1 2 Bode Plots Basics 2 1 Magnitude 2 2 Phase 1 1 2 3 Combining Poles and Zeroes 3 1 Introduction Although you should have learned about Bode plots in previous courses such as EE40 this tutorial will give you a brief review of the material in case your memory is rusty 2 Bode Plots Basics Making the Bode plots for a transfer function involve drawing both the magnitude and phase plots The magnitude is plotted in decibels dB and the phase is plotted in degrees For both plots the horizontal axis is either frequency f or angular frequency measured in Hz and rad s respectively The horizontal axis should be logarithmic i e increasing by powers of 10 Most of the transfer functions we ll deal with in this class can be separated into a general that resembles the following H j A j z1 1 j z2 1 j z3 j p1 1 j p2 1 j p3 1 A is an arbitrary constant and j is 1 As you can see the basic component of this transfer function appears to be 1 j c where c is some constant with the slight variation j c Let s analyze this basic component first before we analyze the entire transfer function 2 1 Magnitude Recall that the definition of magnitude measured in dB is as follows q 2 2 20 log H j 20 log H j H j Let s apply this definition to our basic transfer function component this is called a zero when it appears in the numerator of a transfer function q 20 log 1 j c 20 log 1 c 2 For small we have 20 log 1 j c 0 dB For large 20 log 1 j c When c the magnitude of the transfer function is approximately 3 dB Since there s so little change from 0 to c we approximate the magnitude in this region as a constant 0 dB 1 2 2 BODE PLOTS BASICS 2 For c the c dominates the magnitude expression allowing us to approximate the magnitude as 20 log c From this expression it s clear that if we increase by a factor of 10 we increase the magnitude by 20 dB Thus our Bode plot approximation for the zero is a constant 0 dB for c and a line constantly increase by 20 dB decade for c illustrated in Figure 1 Figure 1 also illustrates the Bode plot for a DC zero of the form j c This differs only slightly from the normal zero in that is lacks the additional 1 Thus instead of having the constant magnitude region for c is simply always increases at 20 dB decade We draw its intersection with the frequency axis where c since that s where the magnitude is 0 dB 40 Magnitude dB 20 20 log 1 j c 20 log j c 0 20 40 103 104 105 rad s 106 107 Figure 1 Bode plots magnitude for a normal zero and a DC zero for c 105 rad s the plots overlap for c The basic transfer function component 1 j c can also appear in the denominator in which case it is called a pole Although this may seem like an entirely different problem recall that we take the logarithm of our transfer function because our result is expressed in decibels Taking the logarithm of the inverse of a function simply gives the negative logarithm of the function meaning we simply have to negate the results of our zero analysis to get the appropriate expressions for poles The same argument applies with DC poles of the form j c so we can negate our DC zero analysis to get the DC pole results A normal pole will have a constant 0 dB value for c and will drop by 20 dB decade for c A DC pole will drop by 20 dB decade for any and will intersect the frequency axis 0 dB at c The results are shown in Figure 2 2 2 Phase Let s take a look at the phase of a zero DC zero pole and DC pole Recall the definition of phase Arg H j tan 1 H j H j 3 3 COMBINING POLES AND ZEROES 40 Magnitude dB 20 0 20 log 1 1 j c 20 log 1 j c 20 40 103 104 105 rad s 106 107 Figure 2 Bode plots magnitude for a normal pole and a DC pole for c 105 rad s the plots overlap for c Let s apply this to the normal zero first Arg 1 j c tan 1 c For 0 Arg 1 j c 0 For Arg 1 j c 90 For c Arg 1 j c 45 Thus our approximation for the phase of a zero is 0 for 0 1 c 45 for c and 90 for 10 c with a straight line connecting these points We can also look at the phase of a DC zero which is always 90 These results are shown in Figure 3 Similar to our analysis of the magnitude we can also consider poles and DC poles in our phase plots It can be shown that tan 1 tan 1 meaning our phase plots for poles and DC poles will simply be negated versions of the zero plots These are shown in Figure 4 3 Combining Poles and Zeroes Generally a transfer function may involve many poles and zeroes and their DC counterparts In order to make it easier to draw Bode plots your first step should be to factor the transfer function into the canonical form shown in Equation 1 This makes it easy to identify all of the poles and zeroes First you ll have to handle the constant A in front if present The magnitude of A will affect your magnitude plot and the sign of A will affect your phase plot Your magnitude plot must be shifted up by 20 log A For example if A 10 then your magnitude plot must be shifted up by 20 dB Similarly if A 1 10 then your magnitude plot must be shifted down by 20 dB If A 0 then your phase plot must be shifted up or down it s the same in this case by 180 Second you need to draw each pole and zero individually on the same set of axes whether you re making a magnitude or phase plot 3 4 COMBINING POLES AND ZEROES 100 Phase degrees 80 60 Arg 1 j c Arg j c 40 20 0 3 10 104 105 rad s 106 107 Figure 3 Bode plots phase for a normal zero and a DC zero for c 105 rad s the plots overlap for 10 c Third you simply add the curves that you ve drawn at each point to obtain the final Bode plot Remember to shift your plots accordingly based on the constant A as mentioned previously This …
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