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Lecture 45 Bode Plots of Transfer Functions A. Goals

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Lecture 45 Bode Plots of Transfer Functions A. Goals Lecture 45 Bode Plots of Transfer Functions A. Goals : Design Oriented Analysis 1. Design Oriented Analysis Overview 2. Inspection of T(s) in normalized form 3. Five points in design oriented analysis B. Review of Bode Plots 1. db values 2. Log - Log Slopes vs f 3. Normalized Form 1 + s/w a. Regular Pole b. Right Half Plane Zero flyback / Buck - Boost c. Inverted Forms 1 + w/s 1. Pole d. Complex T(s) Plots versus Frequency 4. Low-Pass Filter Resonant Circuit 5. Series Resonant R-L-C Circuit 1 Lecture 45 Bode Plots of Transfer Functions A. Goal : Design Oriented Analysis 1. Design Oriented Analysis Our aim in design is to always keep the whole system view at hand so that we can alter some parts of the system and immediately see the results of the change in other parts of the system. The keystone will be the Bode plots of each part of the system that comprise the open loop response. From the open loop response Bode plots various design changes may be explored. We seek simple intuitive understanding of a transfer function via Bode Plots vs f 2. Inspection of T(s) in normalized form to: a. Guess / estimate pole and zero location b. Determine asymptotic behaviors 2 3. The five points in design oriented analysis we will emphasize are given below: Before we can do this we need to review how to construct and make Bode plots as they are the key to our design oriented approach. Asymptotic approximations to the full Bode plots are key to rapid design and analysis. Depending on whether or not we know the high frequency or low frequency behavior of the transfer function we may choose either normal pole/zero from or inverted pole/zero forms as we will discuss below. Each has it’s advocates and we need to be familiar with both. What we don’t want to do is to mix the forms in one T(s) expression, if at all possible. For those students to whom Bode plots are hazy or even new we will give a brief review. 3 B. Bode Plot Review Usually in a transfer function Vo/Vin has a value at each applied frequency. We use db for the transfer function magnitudes, as it will allow for easy asymptotic approximations to the curves. 1. db values º 20 log10 G To employ a db scale we always need a BASE value. For example 50kW on a base of 10 kW, is considered as 14 db. 20 db W is 20 db greater than a base of 10kW, or 100kW 60 db mA is a current 60 db greater than a base of 1mA or one Ampere. Do you get the new way of thinking in db? Actual Magnitude Magnitude in dB ½ -6 dB 1 0 dB 2 6 dB 5 = 10/2 20 dB - 6 dB = 14 dB 10 20dB 1000 = 103 3 * 20dB = 60 dB What does -3db mean? What does +3 db mean? Can you tell the value of the T(s) ratio that gives 3db? Three db will be a useful rule of thumb to place the crude amplitude T(s) plots closer to the real ones as we will see later. 4 2. Log - log Plots and G vs f/fo slopes in db units fo corresponds to a characteristic pole or zero of the transfer function 3. Normalized Forms for Transfer Functions a. Single Isolated Pole G(s) = 1/(1+s/wp) wp º Pole break frequency in radians/sec. Bode Plots for Single Pole Crude Amplitude Plot Accurate Amplitude fo is the characteristic frequency When f is fo /2 or 2fo we go off the crude plot by only 1db to achieve the actual values. Likewise @fo we are off the crude plot by 3db as shown above. 5 Crude Phase Plot Accurate Phase Plot Real phase crosses Real phase crosses asymptote only once asymptote 3 times These break frequencies above and below f0 will be useful. 6 b. Isolated Right Half Plane Zero G(s) = 1 - s/wz Bode Plot Right Half Plane Zero vs. Left Half Plane Zero G(s) = 1+ s/wz G(s) = 1 - s/wz Very usual in T(s) Flyback / Buck-Boost for many converters T(s) have this unique right half zero feature On page 8 we compare and contrast the right and left plane zeros behavior versus applied frequency. While the amplitude plots cannot tell the difference between the two, the phase plots certainly do. What is the difference? 7 Gain up vs. f Gain up vs. f Phase up away from -180° Phase down toward - 180° These asymptotic plots of phase for left and right plane zeroes tell us the whole story. c. Inverted G(s) forms Have Unique Bode Plots When we focus on high f response of T(s) or G(s) we sometimes utilize w/s forms for the poles or zeros. 1. Inverted pole 1 G(s) - 1+ w p / s Bode plot of inverted pole has some unique properties: Low f amplitude vs w decreases to -¥ at low f, f < fo, unlike RHP zero where low f response saturates at some f, f < fo. 8 For an inverted pole we the gain up and the phase down vs. f for f < fo 2. Inverted Zero On page 10 we plot the frequency response of an inverted zero, with all the details of an asymptotic approach to both gain and phase plots. 9 d. Various complex T(s) Plots versus Frequency 1. G(s) = 32 db or 40 on a linear scale s s (1+ )(1+ ) 2p100Hz 2p 2kHz We plot this below by asymptotes. For 1+s/wo, wo is the break point. Note phase break frequencies are 10 times off from 10 amplitude break frequencies. 2. Well separated pole and zero means by a factor > 100 1+ s/ w z A(s) = Ao wp > 100 wz fz = 100f1(below) 1+ s/ w p When f ® 0 we obtain A(DC) = Ao. Why is A(f = ¥) higher in value? By exactly how much? Ao s/ w z w p limit s ® 0 A0 1/1 limit s ® ¥ = Ao = s/ w p w z fz Ao f1 Phase starts early at fz/10 with 45°/decade slope. Phase ends late at fp*10 with 45°/decade slope. 11 3. Given log - log plots Find T(s) Two possible answers occur depending on the form chosen for T(s): normal pole and zero form or inverted poles and zeros. Consider the two T(f) amplitude plots below Low f saturation identifies inverted pole Inverted w1 pole at w1 Midband Gain Gm Pole at w0 w1 -20 db/dec Gm w3 +20 db/dec -20 db/dec Inverted w2 Inverted zero at Pole at w1 -20 db/dec pole at w0 Midband Gain Consider the inverted pole/zero form first and then normal form. w G (1+ 2 ) m G Þ G(s) = s Þ G(s) = m w s w s (1+ 1)(1+ ) (1+ o )(1+ ) s w3 s w1 w G (s + w ) w G s OR: 3 m 2 OR: 1 m (s + w1)(s + w3 ) (s + wo)(s + w1) 4. Low-Pass Filter Resonant Circuit These Bode plots are unique and different from all prior ones. The circuit diagram is at the bottom of the page: v2 (s) 1 = (s) s s 2 v1 1 + + ( ) Qwo wo Q º Q factor of the resonant circuit. For Q £ 1/2 roots are real while for Q ³ 1/2 roots are complex. Q is in linear units. L + v1(s) C R v2(s) - 12 1 1 wo º Corner radian Frequency = or f = LC 2p LC C We will find below that Q(low pass filter) º R . L This may seem wrong as higher R values mean higher Q for the low-pass filter. Q for the series R-L-C circuit, as distinct from the low-pass filter, differs in it’s R dependence as shown below in section 5. In the low-pass filter case L ® 0, Q ® ¥ reallymeans we have only a RC filter. The 13 transfer function of the low-pass filter is easily found: V2 (s) 1 V2 | | = For w << wo: =1 2 V1(s) w (w/ w ) V1 [1- ( )2 ]2 + o 2 wo Q -2 V2 æ w ö For w >> wo: » ç ÷ V1 è wo ø We plot this response including the resonant bump below. The resonant bump near f=f0 is asymmetric in shape. Crude Plot Accurate Plot with “Q Bump” ||G(jw)|| db ||G|| 0db 0 db sharp -2 slow -20db (f/fo) 0 db |Q|db -40 db/dec fo -40db -60db -40 db/dec 0.1fo fo 10fo f V (s) 1 1 2 = = V (s) 1+ sR C+ s2 LC s s2 1 L 1+ + 2 Q wo wo 1 = (1+ s/ w1)(1+ s/ w2) For the resonant bump to occur w1, w0 are complex, not real! We do not have a simple Bode plot with only straight line 1 asymptotes, as this is a resonant circuit with wo = ! LC Rather we have to learn the proper way to treat resonant circuits which involves the linear asymptotes at frequencies far from f0 and a resonant bump near f=f0. 14 L 1 1 C = Rl C Q = = In series R-L-C R¯ Q• Qwo Rl Cwo R ¯ 1 LC 1 w V (s) Q Ð 2 = -1 | wo | = f(f) tan w V1(s) 1- ( )2 wo In summary for the low-pass filter: We now need to see some trends of T(s) phase plots for the low-pass filter near f=f0 as well as far from resonance.
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