This document provides the steps to draw the Nyquist plot for an open loop transfer function G(S)H(S) = 1 / 2(S (S+1) (S+0.5)). It first converts the transfer function to standard form, then selects the Nyquist contour to enclose the right half of the S-plane except at the origin. It maps four sections of the contour to the G(S)H(S) plane, providing the equations for magnitude and phase or path along each section. Combining all four mapped sections provides the full Nyquist plot for the given transfer function.

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