Sketching Asymptotic Bode Plots Type II Systems with Zero: Mag

Frequency Response Outline
Examples
• Sketching Bode plots: asymptotes.
M. Sami Fadali • Sketching polar plots.
Professor of electrical Engineering • Examples.

University of Nevada 2

1 Type II Systems with Zero: Mag

Sketching Asymptotic Bode Plots G2 ( s)  10 s 2 s 2 1 1 Zero Break 2
10  Frequency 0,0,10
Type l, n poles, m zeros s
Pole Break
Magnitude plot Frequencies
1. Type l : Low frequency slope = 20 l dB/dec =
Magnitude plot
6 l dB/oct with intercept 20 log K at  =1 1. Type 2 : Low frequency slope = 40 dB/dec =
rad/s or 0 dB at (K)1/l.
2. Change slope at each break frequency (+20 12 dB/oct with intercept 20 dB at  =1 rad/s or
db/dec for a zero, 20 db/dec for a pole). 0 dB at (10)1/2.
Phase:
1. Approach 90 l º at low frequencies & 2. Change slope to 20 db/dec at 2 rad/s, then to
90(nm) º at high frequencies. 40 dB/dec at 10 rad/
2. Use calculator for a few additional points
(asymptotes slope roughly 45 º /dec) 4

3

Bode Plots: Zero and Bode Plots: Zero, , and Pole

50 Bode Diagram Bode Diagram
50
0 2+1
Magnitude (dB) 2+1 Magnitude (dB) 0 2+1
-50 1/
-50
90
45 -100
-90
0
Phase (deg) -45 Phase (deg) -135 2+1
-90 -180 10 + 1
-135
-180 100 101 102 5 10-1 100 101 102 103 6
Frequency (rad/s) Frequency (rad/s)
10-1

Type II Systems with Zero:  Type II Systems with Zero: Bode

s 2 1 Zero Frequency 0.2, 20 100 Bode Diagram
10  50
G2 ( s)  10 s 2 s 1 Magnitude (dB) 0
-50
Pole 1, 100 10 /
Frequencies -100 /
--115305
Phase:
-180
1. Approach 180º at low frequencies & 180º 10 /
at high frequencies. /
Phase (deg)
2. Use calculator for a few additional points
-225 100 101 102 103
(asymptotes give approximate shape: slope 10-1 Frequency (rad/sec)
+45º/dec at 0.2, 0º/dec at 1,  45º/dec at 20,
0º/dec at 100: skip). 8

7

Type II Systems with Zero: Polar Polar Plot: Vicinity of Origin

Nyquist Diagram
0

25 Nyquist Diagram

s 10 1 -0.01

20 G1 (s )  10 s 2 s 2 1
15
-0.02

10 -0.03

Imaginary Axis 5

-0.04

0 Imaginary Axis

-0.05

-5

-10  -0.06 G2 ( s)  10 s 2 1
s 10 
-15 s 2 1 -0.07 s 2 1
10 
-20 G2 (s)  10 2 s 1

s -0.08

-25 -350 -300 -250 -200 -150 -100 -50 0
-400

Real Axis -0.09

-0.-10.1 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0
Real Axis

9 10

Root Locus Polar Plot: Imaginary-axis Poles

Root Locus Root Locus 400 Nyquist Diagram 2+
14 10
12 300 0+ 50
10 8
6 200 0 /
8 4 Real Axis /
6 2
4 0
2 -2
0 -4
-2 -6
-4 -3 -2 -1 0 1 2 3 4 -8
-10
Real Axis -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
Imaginary Axis Imaginary Axis 100
Imaginary Axis Real Axis

0

-100

0 -200

G1 ( s)  10 s 10 1 G2 (s)  10 s 2 1 -300 2
10  -400
2 s 2 1 s -50
s s 2 1

11 Type 1, 4th order / 12

Nyquist Plot: Imaginary-axis Poles Transport Delay

400 2 Nyquist Diagram 2+ • Magnitude is unity.
300 • Phase is a linear function of frequency.
Imaginary Axis 200 2 0
100 14
-50 0+ 2+
0 System with Time Delay
-100 0 50
-200 Real Axis G(s)  10 e0.2s
-300 / (s 1)(s  5)
-400 /
>> g=zpk([],[-1,-5],10,'inputdelay',0.2)
Type 1, 4th order / Zero/pole/gain:

13 10
exp(-0.2*s) * -----------
Bode Plot: Transport Delay
(s+1) (s+5)
. >> bode(g)

1 Bode Diagram 2 Bode Diagram 16
0.5 Frequency (rad/s) 1.5
Magnitude (dB) Magnitude (abs)
0 1
-0.5 0.5

-10 00
-45
Phase (deg) -90 Phase (deg) -45
-135
-90
100
-135 2 468 10
101 Frequency (rad/s) 15

Bode Plot Bode Diagram RHP Poles: Bode Plot
Bode Diagram
0 20

G(s)  10 e0.2s Magnitude (dB) System: g Magnitude (dB) 10 = 10
(s 1)(s  5) 0 − 10 + 1
-20 Frequency (rad/s): 2.01
Magnitude (dB): -1.7

-40 -10

Phase (deg) -600 System: g Phase (deg) -20 = 10
-360 Frequency (rad/s): 2 90 10 + 1
-720 Phase (deg): -108 45
103
-1080 0 18
-45
-1440 10-1 100 101 102 -90 100 101 102
10-2 Frequency (rad/s) 17 Frequency (rad/s)
10-1

RHP Poles: Polar Plot

Nyquist Diagram
5

= 10
− 10 + 1

Imaginary Axis 0

= 10
10 + 1

-5-2 0 2 4 6 8 10

Real Axis 19