Nyquist analysis

1
Nyquist analysis
A bit of theory
H(s) Complex analytic function of the complex
variable s
Transformation
ImH(s)
Ims
x
?
x
ReH(s)
Res
s-plane
H(s)-plane
2
Nyquist analysis
Example
Application
j16
j4
-11
2
s
H(s)
3
Nyquist
H(s) is a function of the complex variable s
A curve (c) in the s plane excluding poles and
zeros A curve (?) is created in the H(s)
plane s turns once clockwise along c H(s)
describes the curve ? H(s) turns N times
clockwise along ? P number of poles
(multiplicity1) of H(s) inside c Z number
of zeroes of H(s) (multiplicity1) inside c
4
A few definitions and conventions
Inside
0
0
0a turns positively
0A turns positively
Positive circulation Clockwise Inside of the
closed curve on the right
5
A few definitions and conventions
Inside
0
0
0a turns positively
0A turns negatively
Positive circulation Clockwise Inside of the
closed curve on the right
6
Nyquist
Exemple
Application H(s)as
1
a
Zero
7
Nyquist
Exemple
Application H(s)as2
1
a
Zero
8
Nyquist
Exemple
Application H(s)a/s as-1
a
1
Pole
9
Nyquist
Exemple
Application H(s)as
u
U
H(s)
-a
R
t
r
a
1
v
V
Zero-a
10
Nyquist
Exemple
Application H(s)as
H(s)
-a
1
a
Zero
11
Nyquist
Problem
Open loop gain KG(s)H(s) Stability the zeroes
of 1KG(s)H(s) must have negative real
part The poles of KG(s)H(s) are the same as
those of 1 KG(s)H(s)
12
Nyquist path
R
c
Left s plane
Right s plane
s
13
Nyquist critical point
We plot
-1
o
Critical point
14
Nyquist plot of open loop transfer function
-1
Rule Necessary and sufficient condition For ?
running the complete Nyquist contour, the
KG(j?)H (j?) curve must turn counter-clockwise
about the critical point (-1) a number of
times equal to the number of instable poles
ZNP0
15
Example of Nyquist plot
-1
Critical point (-1j0)
N2
ZNPgt0 ? unstable
16
Nyquist for first order
y
X
.5
-1
x
Marked in u or ?
Circle
17
What to do at origin ?
s
18
Example
There is a pole at origin (s0)
d
-1
a
e
j
i
f
s
19
Second example
-1
s
20
Third example
-1
s
21
Gain and phase margins
-1
?