INC 342 Lecture 5: Bode plot Dr. Benjamas Panomruttanarug - PowerPoint PPT Presentation

INC 342 Lecture 5: Bode plot Dr. Benjamas Panomruttanarug Benjamas.pan@kmutt.ac.th BP INC342 1

3 expressions of sinusoidal signal    cos( ) sin( ) Starts from a sinusoidal signal, , which A t B t can be    2 2 1    cos tan ( / ) rewritten as A B t B A • Polar form (showing magnitude and phase shift):   M i i 2 2   M A B i  tan 1    ( / ) B A i BP INC342 2

2 expressions of sinusoidal signal (cont.) A  • Rectangular form (complex number): jB          cos( ) cos( ) cos( ) sin( ) sin( ) t t t          cos( ) cos( ) cos( ) sin( ) sin( ) M t M t M t i i   i    i   i    i A B  • Euler’s formula (exponential): j M i e i BP INC342 3

Frequency response of system (  • Magnitude response: ) M – ratio of output mag. To input mag.  (  ) • Phase response: – difference in output phase angle and input phase angle • Frequency response:     ( ) ( ) M BP INC342 4

Basic property of frequency Response ‘mechanical system’ input = force output = distance sinusoidal input gives sinusoidal output with same damped  (  ) frequency  (  ) shifted by , mag. expanded by (  ) M BP INC342 5

The HP 35670A Dynamic Signal Analyzer obtains frequency response data from a physical system. BP INC342 6

Finding frequency response from differential equation • Get transfer function ( s ) T s   • Set j • Write   ( ) ( ) M T s     ( ) ( ) T s • Then the output is composed of     ( ) ( ) ( ) M M M o i         ( ) ( ) ( ) o i              ( ) ( ) ( ) ( ) [ ( ) ( )] M M M o o i i BP INC342 7

Finding frequency response from transfer function  s j 1 Substitute with  ( ) G s  ( 2 ) s 1 1    ( ) G j     ( 2 ) ( 2 ) j j 0.5 ∟0 ω = 0, G = 0.5 ω = 2, G = 0.25 – j0.25 0.35 ∟‐ 45 ω = 5, G = 0.07 ‐ 0.17i 0.19 ∟‐ 68.2 ω = 10, G =0.019 ‐ j0.096 0.01 ∟‐ 78.7 ω = ∞ , G = 0 0 ∟‐ 90 BP INC342 8

What’s next? After getting magnitude and phase of the system, we need to plot them but how??? BP INC342 9

Types of frequency response plots • Polar plot (Nyquist plot): real and imaginary part of open ‐ loop system. • Bode plot : magnitude and phase of open ‐ loop system (begin with this one!!). • Nichols chart : magnitude and phase of open ‐ loop system in a different manner (not covered in the class). BP INC342 10

1  s ( ) Polar plot of G s  ( 2 ) so called ‘Nyquist plot’ BP INC342 11

Bode plot Magnitude Phase Note: log frequency and log magnitude BP INC342 12

1 What about ???  ( ) G s   ( 2 )( 3 ) s s • plot each term separately and sum them up • log magnitude (s+2) added with log magnitude (s+3) It’s convenient for calculation to plot magnitude in log scale!!! • phase (s+2) added with phase (s+3) BP INC342 13

Example  ( 3 ) s  ( ) sketch bode plot of G s   ( 1 )( 2 ) s s s break frequency at 1,2,3 BP INC342 14

Example  ( 3 ) s sketch bode plot of  ( ) G s 2    ( 2 )( 2 25 ) s s s s   j • Set then   ( 3 ) j   ( ) G j 2       (( ) 2 )(( ) 2 ( ) 25 ) j j j 3 • At DC, set s=0,  ( 0 ) G 50 • Break frequency at 2, 3, (or 5) 25 BP INC342 15

Conclusions Drawing Bode plot • Get transfer function ( s ) T s   • Set j • Evaluate the break frequency • Approximate mag. and phase at low and high frequencies, and also at the break frequency  for 1 st order, 20 dB / – Mag. plot: slope changes dec  for 2 nd order (at break frequency) 40 dB / dec – Phase plot: slope changes for 1 st order,  45  / dec  for 2 nd order 90  / dec BP INC342 16