Polar Plots - Control Systems - Electrical Engineering (EE)

Polar plot is a graphical representation of the frequency response of a linear time-invariant system obtained by plotting the complex value of the open-loop transfer function G(jω) (or G(jω)H(jω)) in polar coordinates as the angular frequency ω varies from zero to infinity. The polar form of the frequency response is

G(jω)H(jω) = |G(jω)H(jω)| ∠ G(jω)H(jω)

Definition and basic idea

Polar plot of a sinusoidal transfer function is the locus of points whose radial coordinate equals the magnitude |G(jω)| and whose angular coordinate equals the phase ∠G(jω), for ω varying from 0 to ∞. Each plotted point therefore gives both the magnitude and the phase of the frequency response at a particular frequency.

Polar sheet and interpretation

The polar plot is drawn on a polar sheet (polar graph paper) made of concentric circles and radial lines.

• The concentric circles represent magnitude values.
• The radial lines represent phase angles measured about the reference (0°) axis.
• Each point on the polar plot simultaneously indicates the magnitude and the phase of the complex frequency response at a given ω.

Polar Graph Sheet

Example of polar plotting conventions

• Positive phase angles are measured anti-clockwise from the 0° reference axis; negative phase angles are measured clockwise.

Complex representation and separation

The frequency response in rectangular form is

G(jω) = GR(jω) + j GI(jω)

• GR(jω) is the real part of G(jω).
• GI(jω) is the imaginary part of G(jω).

Note: The polar plot is drawn as ω varies from 0 to ∞. Do not confuse a polar plot with a Nyquist plot: the Nyquist plot is the extension of the polar plot obtained by plotting the frequency response for ω from -∞ to +∞ (or by appending the symmetric negative-frequency locus to the positive-frequency polar locus).

Advantages and limitations

• Advantage: A polar plot shows the entire frequency response (magnitude and phase) on a single curve for ω from 0 to ∞.
• Limitation: Because the response is plotted as a single locus, the separate contributions of individual factors (poles, zeros, gains) in the open-loop transfer function are not explicitly separated on the plot; they must be inferred by analysis.

Simple factors: integrator and differentiator

The integrator and differentiator have characteristic loci on the imaginary axis.

• Integrator: G(s) = 1/s. With s = jω, G(jω) = 1/(jω). This corresponds to a point on the negative imaginary axis (phase -90°) with magnitude ∝ 1/ω.
• Differentiator: G(s) = s. With s = jω, G(jω) = jω. This corresponds to a point on the positive imaginary axis (phase +90°) with magnitude ∝ ω.

Effect of adding a pole or a zero

• The addition of a single pole (first-order factor in the denominator) tends to rotate the end of the polar locus by about -90° as frequency increases from 0 to ∞.
• The addition of a single zero (first-order factor in the numerator) tends to rotate the end of the polar locus by about +90° as frequency increases from 0 to ∞.

Type and order of the system and their effect on start and end angles

The system type (number of integrators or poles at the origin) determines the behaviour of the polar plot as ω → 0 (the start of the locus). The order (total number of poles) determines the net rotation of the plot as ω → ∞ (the end of the locus). Illustrations of start and end quadrants for various pole configurations are commonly used to anticipate locus direction.

Order of the system determines the quadrant where the polar plot ends; the final angle is the sum of contributions from all poles and zeros as ω → ∞.

Procedure to sketch a polar plot

1. Determine the open-loop transfer function G(s).
2. Substitute s = jω to obtain G(jω).
3. Evaluate the magnitude |G(jω)| at ω = 0 and ω → ∞ to find the start and end radii of the locus.
   
   
4. Evaluate the phase ∠G(jω) at ω = 0 and ω → ∞ to find the start and end angles.
    
   
5. Algebraically separate the real and imaginary parts of G(jω) by rationalising, so that G(jω) = GR(ω) + j GI(ω).
6. Find frequencies where the imaginary part is zero (GI(ω) = 0). These ω give points where the locus intersects the real axis. Compute |G(jω)| at those ω to get the intersection points.
7. Find frequencies where the real part is zero (GR(ω) = 0). These ω give points where the locus intersects the imaginary axis. Compute |G(jω)| at those ω to get the intersection points.
8. Combine the start and end points, axis intersections and intermediate points to sketch the polar plot smoothly (respecting monotonic variation of phase and magnitude with ω).

Polar plots of some standard transfer functions

We classify examples by system type (number of integrators) and order (total pole count).

(a) Type 0 systems

Order 1

Let G(s) = 1/(1 + sT).

Substitute s = jω to obtain

G(jω) = 1/(1 + jωT)

Writing magnitude and phase explicitly,

|G(jω)| = 1 / sqrt(1 + ω² T²)

∠G(jω) = - tan⁻¹(ωT)

• At ω = 0: |G| = 1, ∠G = 0°.
• At ω → ∞: |G| → 0, ∠G → -90° (since tan⁻¹ ∞ = 90°).

Order 2

Let G(s) = 1/(1 + sT1)(1 + sT2).

With s = jω,

|G(jω)| = 1 / [sqrt(1 + ω² T1²) sqrt(1 + ω² T2²)]

∠G(jω) = - tan⁻¹(ωT1) - tan⁻¹(ωT2)

• At ω = 0: |G| = 1, ∠G = 0°.
• At ω → ∞: |G| → 0, ∠G → -180° (sum of two × -90° contributions).

(b) Type 1 systems

Order 1 (pure integrator)

Let G(s) = 1/s.

With s = jω,

G(jω) = 1/(jω) = (1/ω) ∠ -90°

• At ω = 0: |G| → ∞, ∠G = -90°.
• At ω → ∞: |G| → 0, ∠G = -90°.

Order 2 (integrator plus one pole)

Let G(s) = 1/[s(1 + sT)].

With s = jω,

G(jω) = 1 / [jω (1 + jωT)]

Writing magnitude and phase,

|G(jω)| = 1 / [ω sqrt(1 + ω² T²)]

∠G(jω) = -90° - tan⁻¹(ωT)

• At ω = 0: |G| → ∞, ∠G = -90°.
• At ω → ∞: |G| → 0, ∠G → -180° (-90° from integrator and -90° from high-frequency pole contribution).

Order 3 (integrator plus two poles)

Let G(s) = 1/[s(1 + sT1)(1 + sT2)].

With s = jω,

|G(jω)| = 1 / [ω sqrt(1 + ω² T1²) sqrt(1 + ω² T2²)]

∠G(jω) = -90° - tan⁻¹(ωT1) - tan⁻¹(ωT2)

• At ω = 0: |G| → ∞, ∠G = -90°.
• At ω → ∞: |G| → 0, ∠G → -270° (-90° per pole at high frequency including the integrator).

(c) Type 2 systems

Order 4 (two integrators plus two poles)

Consider a typical Type 2 structure with two poles at origin and two additional poles of first order.

With s = jω, a factor j² gives ∠ -180° contribution from the two integrators at low frequency.

G(jω) = 1 / [ (jω)² (1 + jωT1)(1 + jωT2) ]

|G(jω)| = 1 / [ω² sqrt(1 + ω² T1²) sqrt(1 + ω² T2²)]

∠G(jω) = -180° - tan⁻¹(ωT1) - tan⁻¹(ωT2)

• At ω = 0: |G| → ∞, ∠G = -180°.
• At ω → ∞: |G| → 0, ∠G → -360° (sum of -180° from the two integrators and -90° from each of the two additional poles at high frequency).

Gain margin and phase margin from polar plot

• Phase crossover frequency: the frequency at which the polar plot crosses the line at ∠ = -180°.
• Gain crossover frequency: the frequency at which the polar plot intersects the unit-magnitude circle (|G(jω)| = 1).
• Gain margin (GM): if the polar plot crosses the -180° line at point B whose magnitude is GB, then the gain margin in linear scale is Kg = 1 / GB and in decibels is GM(dB) = -20 log10(GB). The gain margin is positive if point B lies inside the unit circle (GB < 1), otherwise
• Phase margin (φm): at the gain crossover frequency the phase of G(jω) is φ (usually negative). The phase margin is φm = 180° + φ, where φ is the phase at the gain crossover frequency (φ is measured with sign conventions where anti-clockwise is positive).

The gain margin is sometimes denoted by Kg and given by

Kg = 1 / GB

• GB is the magnitude of G(jω) at the point where the locus crosses the -180° axis.
• Kg > 1 (positive in dB) indicates the locus crosses -180° inside the unit circle; Kg < 1 (negative in db) indicates a crossing outside the unit

The phase margin is written as

φm = 180° + θ

• Here θ is the phase (in degrees) of G(jω) at the gain crossover frequency (where magnitude equals unity). θ can be negative; evaluate φm accordingly.

Worked example

Example: The open loop transfer function of a unity-feedback system is

G(s) = 1 / [ s (s + 1) (2s + 1) ]

The transfer function can be recognised as a Type 1 system of total order 3 with factors that match the form

G(s) = 1 / [ s (1 + sT1) (1 + sT2) ]

Substitute s = jω to obtain

G(jω) = 1 / [ jω (jω + 1) (2 jω + 1) ]

Express magnitude and phase by separating each factor:

|G(jω)| = 1 / [ ω sqrt(1 + ω²) sqrt(1 + 4 ω²) ]

∠G(jω) = -90° - tan⁻¹(ω) - tan⁻¹(2ω)

Values at the extremes:

When ω = 0:

G(jω) → ∞ ∠ -90°

When ω → ∞:

G(jω) → 0 ∠ -270°

Compute |G(jω)| and ∠G(jω) at a range of ω values and plot the locus to obtain the polar plot.

After evaluating the locus for representative ω values (plotting magnitudes and phases), the polar plot appears as:

From the plotted polar locus identify the intersection with the -180° axis. In this example the magnitude at the -180° crossing is

GB = 0.7

Therefore the gain margin in linear scale is

Kg = 1 / GB

Kg = 1 / 0.7

Kg = 1.428

Next locate the point where the locus intersects the unit circle (|G(jω)| = 1). At that point measure the phase φ of the locus; in the plotted example the phase at the gain-crossover point is

φ = -168°

Thus the phase margin is

φm = 180° + φ

φm = 180° - 168°

φm = 12°

Hence for the given open-loop transfer function the gain margin is 1.428 (≈ 3.09 dB) and the phase margin is 12°.

Practical notes and tips for sketching polar plots

• Always determine start (ω → 0) and end (ω → ∞) points by considering the lowest- and highest-power contributions of s.
• Compute axis intersections by setting imaginary or real parts to zero after rationalising G(jω). These intersection points anchor the sketch.
• Plot a number of intermediate frequency points (especially near corner frequencies 1/T, 1/(2T), etc.) to produce an accurate locus.
• Use the monotonic nature of arctangent functions to determine how phase accumulates with increasing ω: each first-order pole contributes up to -90° and each first-order zero up to +90° as ω goes from 0 to ∞.
• When estimating stability margins from the polar plot, remember that the Nyquist criterion (with the full Nyquist contour) is required for drawing firm conclusions about closed-loop stability when encirclements of -1 are possible; for many practical minimum-phase cases, margins read from the polar locus are sufficient guides.

Summary: A polar plot directly shows how magnitude and phase of an open-loop transfer function change with frequency. By analysing start and end angles, axis intersections, and crossings of the unit circle and the -180° line, one can determine important stability margins such as gain margin and phase margin. System type and order determine the characteristic start/end behaviour and are therefore useful checks when sketching or interpreting polar plots.