All loci start for K = 0 at the Open Loop Poles and finish for K = ∞ at either Open Loop Zeros or s = ∞.
2
There will always be a locus on the real axis to the LEFT of an ODD number of Open Loop Poles and Zeros.
3
If there are n Open Loop Poles and m Open Loop Zeros, there will be n-m loci ending at infinity on asymptotes at angles to the real axis of
180°
±
4
n−m
±
,
540° n−m
900°
±
,
n−m
, K
The asymptotes meet on the real axis at n
σ
∑= p
m
i
-
i 1
=
∑= z
i
i 1
n - m
where pi is the position of the i’th open loop pole and zi is the position of the i’th open loop zero.
5
Where two real loci meet on the real axis they “breakaway” from the real axis at ±90° to form two complex loci, symmetrical about the real axis. Where two complex loci meet on the real axis they “break-in” to form two real loci, moving in opposite directions along the real axis.
dK
The “breakaway” and “break-in” points are given by the roots of
6
= 0
ds
The points of intersection of a locus with the imaginary axis can be determined by solving the equation
D( jω )
+
KN( jω )
=
0
Remember both the real part and the imaginary parts must be satisfied in this equation. Hence this will give two simultaneous equations one that will give values of ω while the other gives the K value at the crossing point.
7
The angle of departure of a locus from a complex open loop pole is given by;
φd
= 180°
−
n
∑ φi
m
+
i=1 i≠ d
∑ψ
i
i =1
where φi is the angle from the i’th open loop pole and ψi is the angle from the i’th zero. The angle of arrival of a locus at a complex zero is given by;
ψa
= 180°
+
n
∑φ i=1
6
i
−
m
∑ψ i =1 i≠ a
i
n
Magnitude Condition
∏P
i
K =
i =1 m
∏Z
i
i =1
180° =
Angle Condition
n
∑φ
m
i
-
i =1
ω=±
Lines of constant damping
1- ζ
ζ
∑ψ
i
i =1
2
σ
where s = σ + jω
NB Straight line through the origin of the s plane. Also lines makes an angle cos-1 ζ with the negative real axis
σ 2 + ω 2 = ω 2n
Lines of constant undamped natural frequency
6.
NB Circle with centre on the origin of the s plane and radius ωn
