Polar Plot

The polar plot is a graphical representation of the open-loop transfer function G(jw) in the complex plane as the frequency w varies from zero to infinity. Unlike the Bode plot which separates magnitude and phase into two graphs, the polar plot traces a single continuous locus that simultaneously shows both magnitude and phase at every frequency. This makes it a compact and powerful tool for stability analysis.

Core Concept Explanation

To construct a polar plot, the frequency w is swept from 0 to infinity. At each frequency, G(jw) is a complex number with a specific magnitude and angle. This complex number is plotted as a point in the complex plane, and the collection of all such points as w varies traces a continuous curve called the G(jw) locus.

In polar form, G(jw) = |G(jw)| at angle phi(w). The magnitude is the distance from the origin to the plotted point and the angle is the argument measured from the positive real axis. As frequency increases, the magnitude generally decreases and the phase becomes more negative for typical physical systems, causing the locus to spiral inward toward the origin.

The key reference point in the polar plot is the critical point (-1 + j0). This point corresponds to a loop gain of exactly -1, which is the condition for marginal stability in a unity feedback system. The position of the G(jw) locus relative to this point determines stability and forms the basis of the Nyquist stability criterion.

Mathematical Expression

For a transfer function G(s), substituting s = jw gives G(jw) = G_real(w) + j * G_imag(w). Separating real and imaginary parts allows the locus to be plotted parametrically with w as the parameter.

For a simple example G(s) = K / (1 + sT), substituting s = jw: G(jw) = K / (1 + jwT) = K(1 - jwT) / (1 + w^2 * T^2). This gives a semicircle in the lower half of the complex plane as w goes from 0 to infinity.

Magnitude: |G(jw)| = K / sqrt(1 + w^2 * T^2), Phase: angle(G(jw)) = -arctan(wT). At w=0, magnitude = K and phase = 0. At w = infinity, both magnitude and phase approach 0 and -90 degrees respectively.

Practical Understanding

The polar plot of a type-0 system (no poles at origin) starts at a finite point on the positive real axis at w=0. A type-1 system has one pole at the origin, so at w=0 the magnitude is infinite, causing the locus to start from infinity along the negative imaginary axis. A type-2 system starts from infinity along the negative real axis. Recognizing the starting point and direction of the locus immediately tells you the system type.

The polar plot is particularly useful for applying the Nyquist stability criterion because the number of encirclements of (-1, j0) directly gives stability information through the Nyquist formula N = Z - P, where N is the number of encirclements, Z is the number of closed-loop right-half-plane poles and P is the number of open-loop right-half-plane poles.

Given:
G(s) = 10 / [(s+1)(s+2)], H(s) = 1
Draw key points of the polar plot

Why this formula applies:
Substitute s = jw and compute magnitude and phase at key frequencies

Formula:
G(jw) = 10 / [(jw+1)(jw+2)]
|G(jw)| = 10 / [sqrt(1+w^2) * sqrt(4+w^2)]
angle(G(jw)) = -arctan(w) - arctan(w/2)

Substitution:
At w=0: |G| = 10/(1*2) = 5, angle = 0 deg => point (5, 0)
At w=1: |G| = 10/(sqrt2 * sqrt5) = 10/3.16 = 3.16, angle = -45-26.6 = -71.6 deg
At w=2: |G| = 10/(sqrt5 * sqrt8) = 10/6.32 = 1.58, angle = -63.4-45 = -108.4 deg
At w=10: |G| = 10/(sqrt101 * sqrt104) = 10/102.5 = 0.097, angle = -84.3-78.7 = -163 deg

Calculation:
As w -> inf: |G| -> 0, angle -> -90 - 90 = -180 deg
Locus starts at (5,0) and spirals to origin approaching -180 deg

Final Answer:
Polar plot is a curve from (5,0) spiraling into origin. Phase crossover occurs near w = infinity for this system.

Exam Tip: For GATE, a type-N system has a polar plot that starts from infinity with an initial angle of -N*90 degrees. This lets you quickly identify the system type from the polar plot starting behavior without computing anything.

Polar Plot Characteristics Summary

• Type-0 system: polar plot starts at a finite positive real value at w=0 and ends at origin.
• Type-1 system: locus starts from infinity along the -90 degree direction (negative imaginary axis).
• Type-2 system: locus starts from infinity along the -180 degree direction (negative real axis).
• For stability using the Nyquist criterion: count the net clockwise encirclements of (-1, j0).
• The phase crossover on a polar plot is the point where the locus crosses the negative real axis.

Quick Revision

• Polar plot is the locus of G(jw) in the complex plane as w goes from 0 to infinity.
• Each point on the plot represents magnitude (distance from origin) and phase (angle) at a specific frequency.
• Critical point is (-1, j0); encirclements of this point determine stability via Nyquist criterion.
• Starting direction: Type-N system starts at angle -N*90 degrees from infinity.
• Phase crossover on polar plot corresponds to where locus crosses the negative real axis.
• GATE trap: The polar plot for a type-1 system does NOT start at a finite point; it comes from infinity.