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

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

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

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