Lecture 23 Introduction to Bode Plots CL-417 Process Control Prof. Kannan M. Moudgalya IIT Bombay Wednesday, 25 September 2013 1/51 CL-417 Process Control Introduction to Bode Plot
Outline 1. Recall frequency response, including s = j ω substitution 2. Definition of Bode plot 3. Why do we use semilog plots? 4. An example 2/51 CL-417 Process Control Introduction to Bode Plot
1. Recall frequency response and s = j ω 3/51 CL-417 Process Control Introduction to Bode Plot
Recall: Frequency response sin ω 0 t y ( t ) G ( s ) ◮ Excited a linear system G(s) with sin ω 0 t ◮ y ss (t) = | G( ω 0 ) | sin ( ω 0 t + φ ( ω 0 )) ◮ Input is sinusoid ⇒ output is sinusoid ◮ Frequencies of input and output are same ◮ | G( ω 0 ) | multiplies output amplitude ◮ Output sinusoid shifts by ∠ G(j ω 0 ) = φ ( ω 0 ) with respect to input 4/51 CL-417 Process Control Introduction to Bode Plot
Relation between s and j ω sin ω 0 t y ( t ) G ( s ) ◮ y ss (t) = | G( ω 0 ) | sin ( ω 0 t + φ ( ω 0 )) ◮ How did this term | G( ω 0 ) | come about? ◮ Recall that △ | G( ω 0 ) | = | G(j ω 0 ) | = | G( − ω 0 ) | ◮ So, we substitute s = j ω in all frequency related discussions 5/51 CL-417 Process Control Introduction to Bode Plot
Example of Frequency Response: Did you try? ◮ What is the long time response of the transfer function 1 G(s) = (s + 1)(s + 2) when excited by sin 10t? ◮ Solve it using sin ω 0 t y ( t ) G ( s ) ◮ y ss (t) = | G(j ω 0 ) | sin ( ω 0 t + φ ( ω 0 )) 6/51 CL-417 Process Control Introduction to Bode Plot
Frequency response helps define filters We also discussed how this concept helps define filters. 7/51 CL-417 Process Control Introduction to Bode Plot
2. Definition of Bode plot 8/51 CL-417 Process Control Introduction to Bode Plot
Result of sine testing on SBHS ◮ Fan speed is constant at 100 ◮ Oscillated at 25 heater units (40units = 100%) at an amplitude of 5 units 9/51 CL-417 Process Control Introduction to Bode Plot
Result of sine testing on SBHS ◮ Same conditions as before, but for a lower frequency 10/51 CL-417 Process Control Introduction to Bode Plot
Magnitude and phase plots ◮ Make use of sin ω 0 t y ( t ) G ( s ) y ss (t) = | G(j ω 0 ) | sin ( ω 0 t + φ ( ω 0 )) ◮ Repeat this for a large number of ω ◮ Evaluate | G(j ω ) | for every ω ◮ Note down ∠ G(j ω ) = φ ( ω ) at every ω ◮ Plot these ◮ | G(j ω ) | vs. ω is known as the magnitude plot ◮ ∠ G(j ω ) = φ ( ω ) vs. ω is known as the phase angle plot ◮ Black plot in the next figure 11/51 CL-417 Process Control Introduction to Bode Plot
Bode plot obtained using SBHS 12/51 CL-417 Process Control Introduction to Bode Plot
Bode plot is also an analysis tool ◮ Bode plot is developed for plant experiments ◮ Can also be used as an analysis tool ◮ Recall our mathematical experiment: sin ω 0 t y ( t ) G ( s ) ◮ y ss (t) = | G(j ω 0 ) | sin ( ω 0 t + φ ( ω 0 )) ◮ Can excite G(s) for different frequencies ◮ Obtain Bode plots of G(s) ◮ In the previous page, compared ◮ Experimentally obtained Bode plot with ◮ that obtained using fitted transfer function 13/51 CL-417 Process Control Introduction to Bode Plot
Bode plots created through expts. & G(s) 14/51 CL-417 Process Control Introduction to Bode Plot
Example 1 ◮ Draw the Bode plot for G(s) = 10s + 1 ◮ Magnitude plot: draw | G(j ω ) | vs. ω ◮ Phase angle plot: draw ∠ G(j ω ) vs. ω 15/51 CL-417 Process Control Introduction to Bode Plot
1 Magnitude, phase of 10s+1 vs. freq. Normal scale 1.0 0.9 0.8 0.7 Magnitude 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25 30 35 w(rad/sec) -0 -10 -20 Phase(deg) -30 -40 -50 -60 -70 -80 -90 0 5 10 15 20 25 30 35 Cannot see details 16/51 CL-417 Process Control Introduction to Bode Plot
There is another difficulty ◮ G(s) = N 1 (s) N 2 (s) D 2 (s) · · · N m (s) D 1 (s) D m (s) ◮ G(j ω ) = N 1 (j ω ) D 2 (j ω ) · · · N m (j ω ) N 2 (j ω ) D 1 (j ω ) D m (j ω ) ◮ Plots of composite functions cannot be easily obtained from those of constituent functions 17/51 CL-417 Process Control Introduction to Bode Plot
The plot can be improved ◮ Cannot see the behaviour well at high frequencies ◮ Solution: use log scale ◮ Log scale on the y-axis provides the addition rule 18/51 CL-417 Process Control Introduction to Bode Plot
We use log scale for drawing Bode plots ◮ G(s) = N 1 (s) N 2 (s) D 2 (s) · · · N m (s) D 1 (s) D m (s) ◮ G(j ω ) = N 1 (j ω ) N 2 (j ω ) D 2 (j ω ) · · · N m (j ω ) D 1 (j ω ) D m (j ω ) ◮ log | G(j ω ) | = log | N 1 (j ω ) | + · · · + log | N m (j ω ) | − log | D 1 (j ω ) | − · · · − log | D m (j ω ) | ◮ ∠ G(j ω ) = ∠ N 1 (j ω ) + · · · + ∠ N m (j ω ) ◮ − ∠ D 1 (j ω ) − · · · − ∠ D m (j ω ) ◮ Large frequency range is covered ◮ Addition rule is available with log scale ◮ Can do constituent to composite plots 19/51 CL-417 Process Control Introduction to Bode Plot
1 Mag., phase plot of 10s+1 in loglog scale Loglog 0 10 -1 Magnitude 10 -2 10 -3 10 -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 10 10 10 10 10 10 20/51 CL-417 Process Control Introduction to Bode Plot
Scilab code bode-1.sce Scilab code: exec ( ’ bodegen − 1. s c i ’ ) ; 1 2 s = %s; 3 4 num = 1 ; 5 den = (10 ∗ s +1) ; 6 7 w = 0 . 0 0 1 : 0 . 0 0 2 : 1 0 ∗ %pi ; 8 LF = ” normal ” / / W a r n i n g : C h a n g e t h i s a s n e c e s s a r y 9 10 bodegen (num , den ,w, LF ) ; 21/51 CL-417 Process Control Introduction to Bode Plot
Scilab code bodegen-1.sci I Scilab code: / / B o d e p l o t 1 / / N u m e r a t o r a n d d e n o m i n a t o r a r e 2 p a s s e d a s i n p u t a r g u m e n t s / / B o t h a r e p o l y n o m i a l s i n p o w e r s 3 o f s ( s a y ) 4 f u n c t i o n bodegen (num , den ,w, l f ) 5 6 7 G = num/den ; 8 G1 = horner (G, %i ∗ w) ; 9 G1p = phasemag (G1) ; 22/51 CL-417 Process Control Introduction to Bode Plot
Scilab code bodegen-1.sci II 10 x g r i d ( ) ; 11 i f LF == ” normal ” then 12 x s e t ( ’ window ’ ,0) ; c l f ( ) ; 13 s u b p l o t ( 2 , 1 , 1) 14 p l o t 2 d (w, abs (G1) , l o g f l a g=”nn” 15 , s t y l e = 2) ; x t i t l e ( ’ Normal s c a l e ’ , ’ ’ , ’ 16 Magnitude ’ ) ; x g r i d ( ) ; s u b p l o t ( 2 , 1 , 2) 17 plot2d1 (w, G1p , l o g f l a g=”nn” , 18 s t y l e = 2) ; 23/51 CL-417 Process Control Introduction to Bode Plot
Scilab code bodegen-1.sci III x g r i d ( ) ; 19 x t i t l e ( ’w( rad / sec ) ’ , ’ ’ , ’ Phase 20 ( deg ) ’ ) ; e l s e i f LF == ” s e m i l o g ” then 21 x s e t ( ’ window ’ ,1) ; c l f ( ) ; 22 s u b p l o t ( 2 , 1 , 1) 23 p l o t 2 d (w,20 ∗ log10 ( abs (G1) ) , 24 l o g f l a g=” l n ” , s t y l e = 2) ; x g r i d ( ) ; 25 x t i t l e ( ’ Semilog ’ , ’ ’ , ’ 26 Magnitude (dB) ’ ) ; s u b p l o t ( 2 , 1 , 2) 27 24/51 CL-417 Process Control Introduction to Bode Plot
Scilab code bodegen-1.sci IV plot2d1 (w, G1p , l o g f l a g=” l n ” , 28 s t y l e = 2) ; x g r i d ( ) ; 29 x t i t l e ( ’w( rad / sec ) ’ , ’ ’ , ’ Phase 30 ( deg ) ’ ) ; e l s e i f LF == ” l o g l o g ” then 31 x s e t ( ’ window ’ ,2) ; c l f ( ) ; 32 s u b p l o t ( 2 , 1 , 1) 33 p l o t 2 d (w, abs (G1) , l o g f l a g=” l l ” 34 , s t y l e = 2) ; x g r i d ( ) ; 35 25/51 CL-417 Process Control Introduction to Bode Plot
Scilab code bodegen-1.sci V x t i t l e ( ’ Loglog ’ , ’ ’ , ’ Magnitude 36 ’ ) ; s u b p l o t ( 2 , 1 , 2) 37 plot2d1 (w, G1p , l o g f l a g=” l n ” , 38 s t y l e = 2) ; x g r i d ( ) ; 39 x t i t l e ( ’w( rad / sec ) ’ , ’ ’ , ’ Phase 40 ( deg ) ’ ) ; 41 end 42 e n d f u n c t i o n ; 43 26/51 CL-417 Process Control Introduction to Bode Plot
MCQ: Best Magnitude Plot The best way to draw magnitude (y-axis) vs frequency (x-axis) plot is 1. Both axis in normal scale 2. x-axis in normal and y-axis in logarithmic scale 3. x-axis in logarithmic scale and y-axis in normal scale 4. Both axis in logarithmic scale Answer: 4 27/51 CL-417 Process Control Introduction to Bode Plot
Bode Plot refers to Logarithmic Scale ◮ Bode plots mean the following: ◮ Magnitude and phase vs. frequency plots ◮ x axis (frequency): should be in log scale ◮ Magnitude plot should be in logarithmic scale ◮ The phrase Bode Plot implies logarithmic scale 28/51 CL-417 Process Control Introduction to Bode Plot
3. Why do we use semilog plots? 29/51 CL-417 Process Control Introduction to Bode Plot
1 Recall Bode plot of 10s+1 Loglog 0 10 -1 Magnitude 10 -2 10 -3 10 -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 10 10 10 10 10 10 30/51 CL-417 Process Control Introduction to Bode Plot
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