Scribd
Upload
80 views8 pages

LEC10-E1236 - Control 1

1) The document discusses specifications for open loop frequency response analysis as frequency changes from 0 to infinity. It defines corner frequency, gain crossover frequency, phase crossover frequency, gain margin, and phase margin. 2) It describes using a Nichols plot to represent open loop frequency response graphically by plotting magnitude in dB vs phase in degrees as frequency varies. 3) It provides an example problem to calculate gain margin and phase margin from a given open loop transfer function and verifies the results on a Nichols chart.

Uploaded by

Mandolin
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
80 views8 pages

LEC10-E1236 - Control 1

1) The document discusses specifications for open loop frequency response analysis as frequency changes from 0 to infinity. It defines corner frequency, gain crossover frequency, phase crossover frequency, gain margin, and phase margin. 2) It describes using a Nichols plot to represent open loop frequency response graphically by plotting magnitude in dB vs phase in degrees as frequency varies. 3) It provides an example problem to calculate gain margin and phase margin from a given open loop transfer function and verifies the results on a Nichols chart.

Uploaded by

Mandolin
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 8

LEC10-E1236- Open loop frequency Response analysis as ω changes from

zero to infinite - Nichols-plot


Specifications of open loop frequency Response analysis as the frequency ω changes
from zero to infinite there are:
1-Corner frequency ωc rad/sec: it is the frequency at which the magnitude of the the open
loop frequency response is changed sharply. It may be (0, 1, 1/T, ωn)
2-Gain crossover frequency ωg rad/sec.: it is the frequency at which the magnitude of the
open loop frequency response is equal to one. G(j 𝛚𝐠)H(j 𝛚𝐠) = 1 or G(j 𝛚𝐠)H(j 𝛚𝐠)
= 0𝑑𝑏
3-Phase crossover frequency ωp rad/sec.: it is the frequency at which the
phase of the open loop frequency response is equal to (-180) degrees or Imag. [
G(j ωp)H(j ωp)]=0
4-Gain margin Gm: it is reciprocal of the magnitude of the open loop frequency
response at the Phase crossover frequency ωp, Gm=1/[Real of G(j ωp)H(j ωp)]=
1/ G(j ωp)H(j ωp) =Kc/K, GM=20log Gm db
5-Phase margin γm: it is the angle of the open loop frequency response at the gain
crossover frequency plus 180 degrees. 𝛾𝑚 = ∠G j 𝜔𝑔 H j 𝜔𝑔 + 180 deg.
6- Stability analysis:
1- 𝛾𝑚 = +𝑣𝑒, 𝐺𝑚 = +𝑣𝑒 𝑑𝑏 system is stable
2- 𝛾𝑚 = +, 𝐺𝑚 = 0 𝑑𝑏 system is critical stable
3- 𝛾𝑚 𝑜𝑟 𝐺𝑚 = −𝑣𝑒 𝑑𝑏 system is unstable
7- Critical gain Kc can be obtained analytically by four methods:
1 -Routh arrary , 2- G(j 𝛚𝐠)H(j 𝛚𝐠) = 1,
3- Real of [G(j ωp)H(j ωp)]= -1, 4- Gm=20log[Kc/K] db=0db

Graphical representations of the open loop freq. response as frequency changes


from zero to infinite are:
1-polar plot(Nyquist)
2-Bode diagram(Margin)
3-Nichols plot
2-Nichols plot : is the Plot of the magnitude of M=
G(j ω)H(j ω) in db on the vertical axis against 𝚽
= ∠G j 𝜔 H j 𝜔 in degrees on the horizontal axis in the
X-Y plane as the frequency 𝜔 changes from zero to
infinity rad/sec.
>> n=[----];d=[----]; >>nichols(n,d) 20
Nichols Chart

-20

Open-Loop Gain (dB)


-40

-60

-80

-100

-120
-270 -225 -180 -135 -90 -45 0
Open-Loop Phase (deg)
The main steps are:
1-Find the open loop TF= G(S)H(S)
2-Find the freq.open loop TF= G(jω)H(jω)=MejΦ = 𝑀∟𝛷= 𝑅𝑒𝑎𝑙 + 𝑗 𝑖𝑚𝑎𝑔
3- Gain crossover frequency ωg rad/sec. as M= G(j ωg)H(j ωg) = 1
4- Phase crossover frequency ωp rad/sec. ∟ G(j ωp)H(j ωp)= = 180
or Imag.[ G(j ωp)H(j ωp)]=0
5- Calculate this table
ω 0.1 0 ωp ωg 10 100 ∞
Φ -180
M 1
20 log M(db) 0
Real
imag 0
The gain and the phase margins from the plot
Examp.
• Consider a unity feed- back control system has
K K
•G s = =
S+1+j S+2 S+1−j S3 +4S2 +6S+4
• a-Prove that as K=8 the gain margin=7.96 db at 2.45 rad/sec. and the phase
margin= 44.1 degrees at 1.56 rad/sec.?
• Find the freq.open loop TF=
8
• G(jω)H(jω)= 3 2 = MejΦ = 𝑀∟𝛷 =Re+j imag
S +4S +6S+4
𝟖
• 𝑴= ,
(𝟒−𝟒𝛚𝟐 )𝟐 +(𝟔𝛚−𝛚𝟑 )𝟐
• 𝚽 = − 𝐭𝐚𝐧−𝟏 ((𝟔𝛚 − 𝛚𝟑 )/(𝟒 − 𝟒𝛚𝟐 ))
𝟖
• 𝑴= =1
(𝟒−𝟒𝛚𝟐 )𝟐 +(𝟔𝛚−𝛚𝟑 )𝟐
8
• 𝑀= = 1, then ωg = 1.56rad/sec.
(4−4(1.56)2 )2 +(6(1.56)−(1.56)3 )2

• Φ = − tan−1 ((6ω − ω3 )/(4 − 4ω2 )) = − tan−1 ((6 ∗ 2.453 )/(4 − 4 ∗ 2.452 )) =


− 180 deg.
• then 𝜔𝑝 = 2.45𝑟𝑎𝑑Τ𝑠𝑒𝑐.
8 1
• 𝑀= = 0.4, then 𝐺𝑀 = 20𝑙𝑜𝑔 = 7.96𝑑𝑏
(4−4(2.45)2 )2 +(6(2.45)−(2.45)3 )2 0.4

• Φ 𝑎𝑡 𝜔𝑔 = 1.56𝑟𝑎𝑑/𝑠𝑒𝑐.
= − tan−1 ((6 ∗ 1.56 − 1.563 )/(4 − 4 ∗ 1.562 )) = −136deg .
• 𝛾𝑚 = ∠G j 𝜔𝑔 H j 𝜔𝑔 + 180 deg. = 180 − 136 = 44.1𝑑𝑒𝑔.
ω 0 1.56 2.45 ∞

Φ 0 -136 -180 -270

M 2 1 0.4

20logM 6.02 0 -7.96

Real G(jω)H(jω)

Imag G(jω)H(jω)
• Prog. >>n=[8]; d=[1 4 6 4]; >> nichols(n,d)

Nichols Chart
20

-20
Open-Loop Gain (dB)

-40

-60

-80

-100

-120
-270 -225 -180 -135 -90 -45 0
Open-Loop Phase (deg)

You might also like