Lecture 24 Examples of Bode Plots Process Control Prof. Kannan M. Moudgalya IIT Bombay Thursday, 26 September 2013 1/47 Process Control Examples of Bode Plots
Outline 1. First order transfer function - recall 2. Gain, integral and derivative 3. Adding Bode plots 3.1 Two first order systems in series 3.2 Lead transfer function 3.3 First order system with delay 2/47 Process Control Examples of Bode Plots
Recall: First order transfer function 1 1 ◮ G(s) = τ s + 1, G(j ω ) = j ωτ + 1 1 ◮ | G(j ω ) | = √ ω 2 τ 2 + 1 ◮ ω ≪ 1, | G(j ω ) | = 1, 3/47 Process Control Examples of Bode Plots
Recall: First order transfer function 1 1 ◮ G(s) = τ s + 1, G(j ω ) = j ωτ + 1 1 ◮ | G(j ω ) | = √ ω 2 τ 2 + 1 ◮ ω ≪ 1, | G(j ω ) | = 1, M = 20 log | G(jw) | = 0 ◮ Asymptote is M = 0 1 ◮ ω ≫ 1, | G(j ω ) | = ωτ , M = − 20 log ωτ ◮ Asymptote is M = − 20 log ωτ ◮ ω = ω 1 ⇒ M = − 20 log ω 1 τ ◮ ω = 10 ω 1 ⇒ M = − 20 log ω 1 τ − 20 ◮ Slope of − 20 dB per decade 3/47 Process Control Examples of Bode Plots
Corner Frequency 1 ◮ G(j ω ) = j ωτ + 1 1 ◮ | G(j ω ) | = √ ω 2 τ 2 + 1 ◮ For ω ≪ 1, the asymptote is | G(j ω ) | = 1 1 ◮ ω ≫ 1, the asymptote is | G(j ω ) | = ωτ ◮ Two asymptotes intersect at ω = 1 /τ ◮ w = 1 /τ is known as the corner frequency 4/47 Process Control Examples of Bode Plots
1 Bode plot of 10s+1 in semilog scale Semilog -0 -5 -10 Magnitude (dB) -15 -20 -25 -30 -35 -40 -45 -50 -3 -2 -1 0 1 2 10 10 10 10 10 10 w(rad/sec) -0 -10 -20 Phase(deg) -30 -40 -50 -60 -70 -80 -90 -3 -2 -1 0 1 2 5/47 Process Control Examples of Bode Plots 10 10 10 10 10 10
Value at the corner frequency 1 ◮ | G(j ω ) | = √ ω 2 τ 2 + 1 ◮ ω = 1 /τ is known as the corner frequency ◮ At ω = 1 /τ , what is M? √ ◮ M = − 20 log 2 = − 10 log 2 ≃ − 3 dB 6/47 Process Control Examples of Bode Plots
1 Bode plot of 10s+1 in semilog scale Semilog -0 -5 -10 Magnitude (dB) -15 -20 -25 -30 -35 -40 -45 -50 -3 -2 -1 0 1 2 10 10 10 10 10 10 w(rad/sec) -0 -10 -20 Phase(deg) -30 -40 -50 -60 -70 -80 -90 -3 -2 -1 0 1 2 7/47 Process Control Examples of Bode Plots 10 10 10 10 10 10
Phase relations for a simple pole 1 1 ◮ G(s) = τ s + 1, G(j ω ) = j ωτ + 1 ◮ ω ≪ 1, G(j ω ) = 1, φ = ∠ G(jw) = 0 1 j ωτ , φ = − 90 ◦ ◮ ω ≫ 1, G(j ω ) = 1 ◮ For ω = 1 /τ , G(j ω ) = j1 + 1 ◮ φ = − 45 ◦ 8/47 Process Control Examples of Bode Plots
1 Bode plot of 10s+1 in semilog scale Semilog -0 -5 -10 Magnitude (dB) -15 -20 -25 -30 -35 -40 -45 -50 -3 -2 -1 0 1 2 10 10 10 10 10 10 w(rad/sec) -0 -10 -20 Phase(deg) -30 -40 -50 -60 -70 -80 -90 -3 -2 -1 0 1 2 9/47 Process Control Examples of Bode Plots 10 10 10 10 10 10
MCQ: First order system Bode plot of a first order system has the following properties: A Slope = -20dB/decade for large frequency B w = 1 /τ at corner frequency C φ = − 45 ◦ at corner frequency D Phase reached at large frequencies = − 90 ◦ Choose the correct answer: 1. A and B only 2. A and C only 3. A, B and C only 4. All four are correct Answer: 4 10/47 Process Control Examples of Bode Plots
2. Gain, integral and derivative 11/47 Process Control Examples of Bode Plots
Effect of Gain on Magnitude Bode Plot △ ◮ G(s) = 100G 1 (s) ◮ M = 20 log | G(j ω ) | and M 1 = 20 log | G 1 (j ω ) | ◮ Both M and M 1 are plotted in the same graph, in dB (decibel units) M and M 1 are related in the following way: 1. M is higher than M 1 by 100 units 2. M is higher than M 1 by 40 units 3. M is lower than M 1 by 100 units 4. The slopes of M and M 1 are different by 100 units Answer: 2 12/47 Process Control Examples of Bode Plots
Effect of Gain on Phase Bode Plot △ ◮ G(s) = 100G 1 (s) ◮ φ = ∠ G(j ω ) and φ 1 = ∠ G 1 (j ω ) ◮ Both φ and φ 1 are plotted in the same graph φ and φ 1 are related in the following way: 1. φ is higher than φ 1 by 100 units 2. φ is higher than φ 1 by 40 units 3. Both φ and φ 1 plots are identical 4. There is no relation between φ and φ 1 Answer: 3 13/47 Process Control Examples of Bode Plots
Effect of gain △ ◮ G(s) = KG 1 (s), K > 0 ◮ M = 20 log | G(j ω ) | = 20 log | KG 1 (j ω ) | ◮ M = 20 log K+ 20 log | G 1 (j ω ) | , K > 0 ◮ Example: K = 100 ◮ M = 40 + 20 log | G 1 (j ω ) | ◮ At every frequency, add 40 dB! ◮ Phase plots of G 1 and G are identical 14/47 Process Control Examples of Bode Plots
Effect of integral mode or pole at zero ◮ G(s) = 1 s ◮ G(j ω ) = 1 j ω ◮ M = 20 log | G(j ω ) | = − 20 log ω ◮ Has a slope of − 20 dB per decade ◮ φ = ∠ G(j ω ) = − 90 ◦ 15/47 Process Control Examples of Bode Plots
Scilab code bode-5.sce exec ( ’ bodegen − 1. s c i ’ ) ; 1 2 s = %s; 3 4 num = 1 ; 5 den = s ; 6 7 w = 0 . 0 1 : 0 . 0 0 2 : %pi ˆ0; 8 LF = ” s e m i l o g ” 9 10 bodegen (num , den ,w, LF ) ; 16/47 Process Control Examples of Bode Plots
Bode plot of a pole at zero Semilog 40 35 Magnitude (dB) 30 25 20 15 10 5 0 -2 -1 0 10 10 10 w(rad/sec) -80 -82 -84 -86 Phase(deg) -88 -90 -92 -94 -96 -98 -100 -2 -1 0 17/47 Process Control Examples of Bode Plots 10 10 10
Bode plot of pure derivative action ◮ G(s) = s ◮ G(j ω ) = j ω ◮ M = 20 log | G(j ω ) | = 20 log ω ◮ Has a slope of +20 dB per decade ◮ φ = ∠ G(j ω ) = +90 ◦ 18/47 Process Control Examples of Bode Plots
Scilab code bode-5.sce Exchange the values of num and den and execute 19/47 Process Control Examples of Bode Plots
3. Adding Bode Plots 20/47 Process Control Examples of Bode Plots
3a. Two first order systems in series 21/47 Process Control Examples of Bode Plots
Product of two first order systems 1 1 G(s) = s + 1 0 . 01s + 1 ◮ Plot M for each transfer function separately ◮ What are the corner frequencies? For the first, ◮ it is 1 ◮ For the second, it is 1/0.01 = 100 ◮ Add the two ◮ Draw φ for each transfer function separately ◮ Add the two ◮ Scilab code and the plots are given next 22/47 Process Control Examples of Bode Plots
Magnitude Bode Plot 23/47 Process Control Examples of Bode Plots
Phase Bode Plot 24/47 Process Control Examples of Bode Plots
Scilab code bode-2.sce Scilab code: exec ( ’ bodesum − 2. s c i ’ ) ; 1 s = %s; 2 3 G1 = 1/( s +1) ; gai n = 1/(0.01 ∗ s +1) ; 4 d e l a y = 0 ; 5 6 w = 0 . 0 1 : 0 . 0 0 8 ∗ %pi :1000 ∗ %pi ; 7 bodesum 1 (G1 , delay , gain ,w) ; 25/47 Process Control Examples of Bode Plots
Scilab code bodesum-2.sci I Scilab code: / / B o d e p l o t a s a s u m o f c o m p o n e n t s 1 2 f u n c t i o n bodesum 1 (G1 , delay , gain ,w) 3 4 G1 freq = horner (G1 , %i ∗ w) ; 5 6 G1 mag = 20 ∗ log10 ( abs ( G1 freq ) ) ; g a i n f r e q = horner ( gain , %i ∗ w) ; 7 gain mag = 20 ∗ log10 ( abs ( g a i n f r e q ) ) ; 8 9 x s e t ( ’ window ’ ,0) ; c l f ( ) ; 10 s u b p l o t ( 3 , 1 , 1) 11 26/47 Process Control Examples of Bode Plots
Scilab code bodesum-2.sci II p l o t 2 d (w, G1 mag , l o g f l a g=’ l n ’ , s t y l e = 12 2) ; x g r i d ( ) ; 13 x t i t l e ( ’ Magnitude Bode p l o t as sum 14 of component p l o t s ’ , ’ ’ , ’G1 (dB) ’ ) ; s u b p l o t ( 3 , 1 , 2) 15 p l o t 2 d (w, gain mag , l o g f l a g=” l n ” , s t y l e 16 = 2) ; x g r i d ( ) ; 17 x t i t l e ( ’ ’ , ’ ’ , ’ g a i n d e l a y (dB) ’ ) ; 18 s u b p l o t ( 3 , 1 , 3) 19 27/47 Process Control Examples of Bode Plots
Scilab code bodesum-2.sci III p l o t 2 d (w, G1 mag+gain mag , l o g f l a g=” l n 20 ” , s t y l e = 2) ; x g r i d ( ) ; 21 x t i t l e ( ’ ’ , ’ Phase ( deg ) ’ , ’G1+ 22 g a i n d e l a y ’ ) ; 23 24 G1 ph = phasemag ( G1 freq ) ; g a i n p h = phasemag ( g a i n f r e q ) − d e l a y 25 ∗ w ∗ 180/%pi ; 26 x s e t ( ’ window ’ ,1) ; c l f ( ) ; 27 s u b p l o t ( 3 , 1 , 1) 28 28/47 Process Control Examples of Bode Plots
Scilab code bodesum-2.sci IV p l o t 2 d (w, G1 ph , l o g f l a g=’ l n ’ , s t y l e = 29 2) ; x g r i d ( ) ; 30 x t i t l e ( ’ Phase Bode p l o t as sum of 31 component p l o t s ’ , ’ ’ , ’G1 ( phase ) ’ ) ; s u b p l o t ( 3 , 1 , 2) 32 p l o t 2 d (w, gain ph , l o g f l a g=” l n ” , s t y l e 33 = 2) ; x g r i d ( ) ; 34 x t i t l e ( ’ ’ , ’ ’ , ’ g a i n d e l a y ( phase ) ’ ) ; 35 s u b p l o t ( 3 , 1 , 3) 36 29/47 Process Control Examples of Bode Plots
Scilab code bodesum-2.sci V p l o t 2 d (w, G1 ph+gain ph , l o g f l a g=” l n ” , 37 s t y l e = 2) ; x g r i d ( ) ; 38 x t i t l e ( ’ ’ , ’ Phase ( deg ) ’ , ’G1+ 39 g a i n d e l a y ’ ) ; e n d f u n c t i o n ; 40 30/47 Process Control Examples of Bode Plots
Magnitude Bode Plot 31/47 Process Control Examples of Bode Plots
Phase Bode Plot 32/47 Process Control Examples of Bode Plots
3b. Lead Transfer Function 33/47 Process Control Examples of Bode Plots
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