j

1. Draw the Nyquist plot for open loop T.F G(S) H(S) = 1 / 2(S (S+1) (S+0.5))
SOLUTION
The given open loop T.F G(S) H(S) = 1 /( 2(S (S+1) (S+0.5))
STEP 1: Convert the given transfer function into standard form
G(S) H(S) = 1 /( 2*0.5 (S (S+1) ((S/0.5)+1))
= 1 / (S (1+S) (1+2S)
STEP 2: The given transfer function has a pole at origin, so select the
Nyquist contour such that it encloses entire right side of S-plane except at
origin. Draw the Nyquist contour. It consists of four sections.
CONTOUR SECTION C1: (Varies from 0+
to +∞)
= 1 / (S (1+S) (1+2S)
Substitute S = jω
= = 1 / (S (1+S) (1+2S)
Separating magnitude and phase
Magnitude = 1 / (ω(√ ((ω)2
+1) ( √((2ω)2
+1))
Phase = -90 - tan-1
(ω) - tan-1
(2ω)
ω
rad/sec
0.3 0.35 0.4 0.45 0.5 0.6 0.7 0.8 1

2. M 2.74 2.22 1.82 1.51 1.27 0.91 0.68 0.51 0.31
Phase -137.66 -144.2 -150.4 -156.2 -161.5 -171.1 -179.45 -186.6 -198.4
The graph obtained is
CONTOUR SECTION C2:
The mapping of CONTOUR SECTION 2 is from S- plane to G(S) H(S) plane
by
S= limit(R --> ∞) R ejθ
and varying θ from + ∏/2 to - ∏ /2
As S is proportional to R and R tends to ∞ we can approximate (1+SΤ) = ST
G(S) H(S) = 1 / ((S (S+1) (2S+1))
= 0.5/ S3
G(S) H(S) |S=limit(R -->∞)R exp(jθ) =0.5 / S3
| S=limit(R -->∞)R exp(jθ)=0 e-3jθ
Substitute θ = + ∏/2 G(S) H(S) = 0 e-(3j∏)/2
θ = - ∏/2 G(S) H(S) = 0 e(3j∏)/2
So, when the contour of S- plane θ varies from + ∏/2 to - ∏ /2 then
G(S) H(S) varies from -3∏/2 to + 3∏ /2
Now how to encircle in G(s) H(s) plane

3. Start from -3∏/2 in the plane we can reach + 3∏ /2 by moving towards -∏,-
(∏/ 2),0, ∏/ 2,∏, 3∏ /2
CONTOUR SECTION C3: (Varies from -∞ to 0-
)
The graph is same as CONTOUR SECTION 1 but takes mirror of CONTOUR
SECTION 1 and reverses the direction of graph
CONTOUR SECTION C4
The mapping of CONTOUR SECTION 4 is from S- plane to G(S) H(S) plane
by
S= limit(R --> 0) R ejθ
and varying θ from - ∏/2 to + ∏ /2
As S is proportional to R and R tends to 0 we can approximate (1+SΤ) = 1
G(S) H(S) = 1 / ((S (S+1) (2S+1))
= 1/ S
G(S) H(S) |S=limit(R -->0)R exp (jθ) = 1 / S |S=limit(R -->0)R exp (jθ)=∞ e-jθ

4. Substitute θ = - ∏/2 G(S) H(S) = ∞ e (j∏)/2
θ = + ∏/2 G(S) H(S) = ∞ e-(j∏)/2
So, when the contour of S- plane θ varies from - ∏/2 to + ∏ /2 then
G(S) H(S) varies from ∏/2 to -∏ /2
Now how to encircle in G(s) H(s) plane
Start from ∏/2 in the plane we can reach -∏ /2 by moving towards 0, -∏/ 2
with radius ∞
The Nyquist plot is combinations of all contours