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![Bode Plot GM & PM
• Gain Margin: It is the amount of gain in db that can be
added to the system before the system become
unstable
– GM in dB = 20log10(1/|G(jw|) = -20log10|G(jw|
– Gain cross-over frequency: Frequency where magnitude plot
intersect the 0dB line (x-axis) denoted by wg
• Phase Margin: It is the amount of phase lag in degree
that can be added to the system before the system
become unstable
– PM in degree = 1800+angle[G(jw)]
– Phase cross-over frequency: Frequency where phase plot
intersect the 1800 dB line (x-axis) denoted by wp
– Less PM => More oscillating system](https://image.slidesharecdn.com/notestee602bodeplot-141201003423-conversion-gate01/85/bode_plot-By-DEV-12-320.jpg)
![Bode Plot GM & PM
• Gain Margin: It is the amount of gain in db that can be
added to the system before the system become
unstable
– GM in dB = 20log10(1/|G(jw|) = -20log10|G(jw|
– Gain cross-over frequency: Frequency where magnitude plot
intersect the 0dB line (x-axis) denoted by wg
• Phase Margin: It is the amount of phase lag in degree
that can be added to the system before the system
become unstable
– PM in degree = 1800+angle[G(jw)]
– Phase cross-over frequency: Frequency where phase plot
intersect the 1800 dB line (x-axis) denoted by wp
– Less PM => More oscillating system](https://image.slidesharecdn.com/notestee602bodeplot-141201003423-conversion-gate01/75/bode_plot-By-DEV-12-2048.jpg)
Uploaded byDevchandra Thakur
bode_plot By DEV
This document discusses Bode plots, which are used to analyze the stability of linear time-invariant control systems. Bode plots graphically represent a system's transfer function and consist of a magnitude plot and a phase plot versus frequency. The magnitude plot shows the gain in decibels and the phase plot shows the phase angle. Together these plots can determine the gain and phase margins of a system, which indicate its stability. Examples are provided to demonstrate how to construct Bode plots from transfer functions and analyze system stability.
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bode_plot By DEV
- 1. Bode Plot NNaaffeeeessAAhhmmeedd Asstt. Professor, EE Deptt DIT, DehraDun
- 2. Poles & Zerosand Transfer Functions Transfer Function: A transfer function is defined as the ratio of the Laplace transform of the output to the input with all initial conditions equal to zero. Transfer functions are defined only for linear time invariant systems. Considerations: Transfer functions can usually be expressed as the ratio of two polynomials in the complex variable, s. Factorization: A transfer function can be factored into the following form. = + + + ( ) ( )( )...( ) 1 2 m s p s p s p ( )( ) ... ( ) 1 2 n G s K s z s z s z + + + The roots of the numerator polynomial are called zeros. The roots of the denominator polynomial are called poles.
- 3. Poles, Zeros andS-Plane An Example: You are given the following transfer function. Show the poles and zeros in the s-plane. G s s s = + + s s s ( ) ( 8)( 14) + + ( 4)( 10) S - plane origin o x o x x -14 -10 -8 -4 0 s axis jw axis
- 4. Bode Plot •It is graphical representation of transfer function to find out the stability of control system. • It consists of two plots – Magnitude (in dB) Vs frequency plot – Phase angle Vs frequency plot
- 5. Bode Plot… •Consider following T.F = + + + 2 ( ) ( )( )...( ) s s p s p s p s • Put s=jw ö çè 1 1 2 ( )( ) ... ( ) 1 2 s = + + + 2 ( ) 1 ( 1 )( 2 )...( ) jw jw p jw p jw p jw ( ) ( )( ) ... ( ) 1 2 • Arrange in following form ïþ ïý ü ïî ïí ì ö çè ÷ø + + + + + æ 1 2 jw n n n N m w w G jw K jw z jw z jw z x ïþ ïý ü ïî ïí ì ÷ø + + + + +æ 1 2 n n n N m w w G s K s z s z s z x = + + + 2 ( ) 1 (1 1 )(1 2 )...(1 ) jw jwT jwT jwT jw G( jw) =|G( jw) |ÐG( jw) ïþ ïý ü ïî ïí ì ö çè ÷ø + + + + + æ ( ) (1 )(1 ) ... (1 ) 1 2 jw n n a b n N m w w G jw K jwT jwT jwT x
- 6. Bode Plot… •So G( jw) =|G( jw) |ÐG( jw) Magnitude Phase Angle 20log | ( ) | 10 Magnitude in dB = G jw • Hence Bode Plot consists of two plots 20log | ( ) | 10 jw G) – Magnitude ( dB) Vs frequency plot (w) – Phase angle ( ÐG( jw )Vs frequency plot (w)
- 7. Bode Plot… Magnitudein dB 20log10 |G( jw) |=20log10 | K | +20log10 |1+ jwT1 | +20log10 |1+ jwT2 | ...+20log10 |1+ jwTn | +20log10 |1+ jwTa | +20log10 |1+ jwTb | ...+20log10 |1+ jwTm | +etc Phase Angle ( ) ( ) ( ) ÐG( jw)=Ð(K) +Ð1+ jwT1 +Ð1+ jwT2 ...+Ð1+ jwTn -Ð(1+ jwTa ) -Ð(1+ jwTb )...-(1+ jwTm ) -etc ( ) ( ) ( ) jwT jwT jwTn 1 2 =900 +tan-1 - 1 - 1 +tan...+tan-tan-1( jwTa ) -tan-1( jwTb )...-tan-1( jwTm ) -etc
- 8. Bode Plot… Typeof System Initial Slope Intersection with 0 dB line 0 0 dB/dec Parallel to 0 axis 1 -20dB/dec =K 2 -40dB/dec =K1/2 3 -60dB/dec =K1/3 . . . . . . . . . N -20NdB/dec =K1/N
- 9. Bode Plot Procedure • Steps to draw Bode Plot 1. Convert the TF in following standard form & put s=jw = + + + 2 ( ) 1 (1 1 )(1 2 )...(1 ) jw jwT jwT jwT jw 2. Find out corner frequencies by using ïþ ïý ü ïî ïí ì ö çè ÷ø + + + + + æ ( ) (1 )(1 ) ... (1 ) 1 2 jw n n a b n N m w w G jw K jwT jwT jwT x 1 , 1 , 1 ... 1 , 1 , 1 / sec 1 2 3 Rad etc T T T Ta Tb Tc
- 10. Bode Plot Procedure… 3. Draw the magnitude plot. The slope will change at each corner frequency by +20dB/dec for zero and -20dB/dec for pole. For complex conjugate zero and pole the slope will change by ±40dB/ dec 4. Starting plot i. For type Zero (N=0) system, draw a line up to first (lowest) corner frequency having 0dB/dec slope of magnitude (height) 20log10K ii. For type One (N=1) system, draw a line having slope -20dB/dec from w=K and mark first (lowest) corner frequency. iii. For type One (N=2) system, draw a line having slope -40dB/dec from w=K1/2 and mark first (lowest) corner frequency.
- 11. Bode Plot Procedure… 5. Draw a line up to second corner frequency by adding the slope of next pole or zero to the previous slope and so on…. i. Slope due to a zero = +20dB/dec ii. Slope due to a pole = -20dB/dec 5. Calculate phase angle for different value of ‘w’ and draw phase angle Vs frequency curve
- 12. Bode Plot GM& PM • Gain Margin: It is the amount of gain in db that can be added to the system before the system become unstable – GM in dB = 20log10(1/|G(jw|) = -20log10|G(jw| – Gain cross-over frequency: Frequency where magnitude plot intersect the 0dB line (x-axis) denoted by wg • Phase Margin: It is the amount of phase lag in degree that can be added to the system before the system become unstable – PM in degree = 1800+angle[G(jw)] – Phase cross-over frequency: Frequency where phase plot intersect the 1800 dB line (x-axis) denoted by wp – Less PM => More oscillating system
- 13. Bode Plot GM& PM
- 14. Bode Plot &Stability Stability by Bode Plot 1. Stable If wg<wp => GM & PM are +ve 2. Unstable If wg>wp => GM & PM are –ve 3. Marginally stable If wg=wp => GM & PM are zero
- 15. Bode Plot Examples Example 1:Sketech the Bode plot for the TF Determine (i) GM (ii) PM (iii) Stability ( ) 1000 s s (1 0.1 )(1 0.001 ) G s + + =
- 16. Bode Plot Examples… Solution: 1. Convert the TF in following standard form & put s=jw ( ) 1000 jw jw + + (1 0.1 )(1 0.001 ) G jw = 2. Find out corner frequencies 1 = 10 1000 0.1 1 = 0.001 So corner frequencies are 10, 1000 rad/sec
- 17. Bode Plot Examples… • How to draw different slopes
- 18. Bode Plot Examples… • Magnitude Plot
- 19. Bode Plot Examples… • Phase Plot S.N o W Angle (G(jw) 1 1 ---- 2 100 -900 3 200 -980 4 1000 -134.420 5 2000 -153.150 6 3000 -161.360 7 5000 -168.570 8 8000 -172.790 9 Infi -1800
- 20. Bode Plot Examples… • Phase Plot
- 21. Bode Plot Examples… • Phase Plot …
- 22. Bode Plot Examples… • So Complete Bode Plot
- 23. References • AutomaticControl System By Hasan Saeed – Katson Publication
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