GATE Electrical Engineering (EE) Test: Polar Plot - 2 Free Online Test

The gain margin is found by taking the reciprocal of the magnitude of the open-loop transfer function at the phase crossover frequency. If the polar plot intersects the real axis at −2, this means that the magnitude of the open-loop transfer function is 2 at the phase crossover frequency.

For M = 1 the constant M circle is a straight line at x = -1/2.

Gain margin of a polar plot passing through the critical point is zero.

Gain margin is nothing but the inverse of the gain. Hence if gain of open loop system is doubled, then the gain margin will becomes half.

The system is all pass system and the gain of the system at frequency of 1 rad/sec.

Gain margin of a system is calculated at the phase cross over frequency and expressed in decibels.

Phase crossover frequency is calculated as by calculating the magnitude of the transfer function and equating it to 1 and the frequency calculated at this magnitude is phase cross over frequency.

Concept

The phase margin is given by:

PM = 180° + (Phase of G(s) at gain cross-over frequency)

The frequency at which the magnitude of G(s) is unity is known as the gain cross-over frequency.

The gain margin is given by:

GM = 1/ (Gain of G(s) at phase cross-over frequency)

The frequency at which the phase angle of G(s) is -180 is known as the phase cross-over frequency.
Calculation

Point P gives gain cross-over frequency whereas point R gives phase cross-over frequency.
The phase of G(s) at gain cross-over frequency is given by:

ϕ = 39.73° 
Point P lies in the 3 quadrants, hence net angle is:
 

ϕ = 180° + 39.73° = 219.73° 

PM = 180° + 219.73° = 399.73° 

PM = 399.73° = 360° + 39.73 

PM = 39.73°

Point R = −0.21 + 0.00��

GM =  4.76

Polar plot:

The polar plot is a plot, that is drawn between the magnitude and the phase angle of G(jω) by varying ω from zero 0 to ∞.

Concept:

The characteristic equation is:

1 + G(s)H(s) = s³ + 3s² + 2s + 4 

where G(s)H(s) is the open loop transfer function.

Calculation:

There is one pole at the origin, so the polar plot will start from -90° and end at -270° due to order = 3.

The frequency at which the phase angle of OLTF becomes -180° or the imaginary part of OLTF becomes zero is known as the phase cross-over frequency.

ω_pc = √2 rad/sec
Hence, option d is correct.