Rules to draw Polar Plot - Control Systems - Electrical Engineering (EE)

Introduction

The polar plot of a sinusoidal transfer function is the locus of the complex number G(jω) when the angular frequency ω varies from 0 to ∞. The plot is drawn on a polar sheet formed by concentric circles (representing magnitude) and radial lines (representing phase angles). Each point on the polar plot represents the magnitude |G(jω)| as the radial distance from the origin and the phase ∠G(jω) as the angle measured from the 0° reference axis.

• The concentric circles on the polar sheet indicate magnitude values; the radial lines indicate phase angles.
• The positive phase angle is measured anti-clockwise from the 0° axis; the negative phase angle is measured clockwise.
• In rectangular form, the frequency response may be written as G(jω) = G_R(jω) + j G_I(jω), where G_R is the real part and G_I the imaginary part.
• The polar plot shows the frequency response for ω ∈ [0, ∞). It is related to, but not identical with, the Nyquist plot: the Nyquist plot typically shows G(jω) for ω from -∞ to +∞ (or maps a contour in the s-plane) while the polar plot shows ω ≥ 0 only.
• The polar plot is useful because it displays magnitude and phase over the whole frequency range on one plot; a limitation is that it does not directly show the separate contributions of individual factors of the open-loop transfer function.
• Standard factors and their polar behaviour:
  • The integrator G(s) = 1/s (put s = jω) gives G(jω) = 1/(jω); its locus lies on the negative imaginary axis (phase -90°), and magnitude decreases as 1/ω.
  • The differentiator G(s) = s (put s = jω) gives G(jω) = jω; its locus lies on the positive imaginary axis (phase +90°), and magnitude increases proportionally to ω.
• Effect of adding poles and zeros:
  • Each additional pole at the origin or in the right half of the s-plane contributes negative phase; a single simple pole changes the final phase contribution by -90° (in the idealised limit) relative to the factorless case.
  • Each additional zero contributes positive phase; a single simple zero gives +90° phase contribution in the corresponding limit.

Type and order of the system

The type and order of an open-loop transfer function determine where the polar plot begins (as ω → 0) and where it ends (as ω → ∞).

Polar plot of standard transfer functions

Type 0 systems (no pure integrator at origin)

Order 1 (single first-order factor)

Consider G(s) = 1/(1 + sT). Put s = jω.

G(jω) = 1/(1 + jωT)

Separate magnitude and phase:

|G(jω)| = 1 / √(1 + ω²T²)

∠G(jω) = -tan⁻¹(ωT)

Limiting values:

• At ω = 0: |G(jω)| = 1 and ∠G(jω) = 0°.
• As ω → ∞: |G(jω)| → 0 and ∠G(jω) → -90°.

Order 2 (product of two first-order factors)

Consider G(s) = 1/(1 + sT₁)(1 + sT₂). Put s = jω.

|G(jω)| = 1 / ( √(1 + ω²T₁²) · √(1 + ω²T₂²) )

∠G(jω) = -tan⁻¹(ωT₁) - tan⁻¹(ωT₂)

Limiting values:

• At ω = 0: |G(jω)| = 1 and ∠G(jω) = 0°.
• As ω → ∞: |G(jω)| → 0 and ∠G(jω) → -180°.

Type 1 systems (one integrator at origin)

Order 1 (pure integrator)

Consider G(s) = 1/s. Put s = jω.

G(jω) = 1/(jω) = (1/ω) ∠-90°

Limiting values:

• At ω = 0: magnitude → ∞ and phase = -90°.
• As ω → ∞: magnitude → 0 and phase = -90°.

Order 2 (integrator × first-order factor)

Consider G(s) = 1 / [s(1 + sT)]. Put s = jω.

|G(jω)| = 1 / [ ω · √(1 + ω²T²) ]

∠G(jω) = -90° - tan⁻¹(ωT)

Limiting values:

• At ω = 0: magnitude → ∞ and phase → -90°.
• As ω → ∞: magnitude → 0 and phase → -180°.

Order 3 (integrator × two first-order factors)

Consider G(s) = 1 / [ s(1 + sT₁)(1 + sT₂) ]. Put s = jω.

|G(jω)| = 1 / [ ω · √(1 + ω²T₁²) · √(1 + ω²T₂²) ]

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

Limiting values:

• At ω = 0: magnitude → ∞ and phase → -90°.
• As ω → ∞: magnitude → 0 and phase → -270° (i.e., -90° - 90° - 90°).

Type 2 systems (two integrators at origin)

Order 4 example

Consider G(s) = 1 / [ s² (1 + sT₁)(1 + sT₂) ]. Put s = jω.

|G(jω)| = 1 / [ ω² · √(1 + ω²T₁²) · √(1 + ω²T₂²) ]

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

Limiting values:

• At ω = 0: magnitude → ∞ and phase → -180°.
• As ω → ∞: magnitude → 0 and phase → -360°.

Gain margin and phase margin on the polar plot

• The gain margin (commonly denoted K_g) is determined from the point where the polar locus crosses the negative real axis (i.e., the 180° line). Let the magnitude at that crossing be G_B (the radial distance of that crossing point). Then the gain margin is K_g = 1 / G_B.
• The gain margin is considered positive if the crossing point lies inside the unit circle (G_B < 1), and negative if the crossing point lies outside the unit circle (g_b > 1). When the plot does not cross the 180° axis, formal gain-margin determination requires continuation or other methods (e.g., Nyquist).
• The phase margin (Φ_m) is given by Φ_m = 180° + θ, where θ is the phase of G(jω) at the gain-crossover frequency (the frequency at which |G(jω)| = 1). On the polar plot, find the point where the locus intersects the unit circle; draw a line from the origin to that point and read the phase angle θ (negative or positive). Then compute Φ_m = 180° + θ.
• Both margins can be read directly from a well-drawn polar plot; they indicate how much gain or phase variation the closed-loop system can tolerate before losing stability.

Example: The open loop transfer function of a unity feedback system is given by G(s) = 1/s(s + 1)(2s + 1). Sketch the polar plot and also determine the gain margin and the phase margin.

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

Identify type and order.

The system is Type 1 (one integrator) and of order 3.

Replace s by jω.

G(jω) = 1 / [ jω (jω + 1) (2jω + 1) ]

Separate magnitude and phase expressions.

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

∠G(jω) = -90° - tan⁻¹(ω) - tan⁻¹(2ω)

Limiting behaviour:

At ω = 0: |G(jω)| → ∞ and ∠G(jω) = -90°.

As ω → ∞: |G(jω)| → 0 and ∠G(jω) → -270°.

For sketching, evaluate |G(jω)| and ∠G(jω) over a set of ω values (or use a computational table) and plot the locus on polar paper. Key features to plot are the start at infinite radius along -90°, the gradual rotation with increasing ω by the contributions of the two first-order factors, and the end approaching -270° at the origin.

Reading margins from the plotted polar locus:

The polar locus crosses the 180° axis (i.e., negative real axis) at a radial value G_B = 0.7.

Gain margin K_g = 1 / G_B = 1 / 0.7 = 1.428

The intersection of the locus with the unit circle occurs at a point whose phase is θ = -168°.

Phase margin Φ_m = 180° + θ = 180° - 168° = 12°

Therefore, for this system the gain margin is 1.428 and the phase margin is 12°.

Practical notes and applications

• Polar plots are particularly useful in control design for assessing closed-loop stability and robustness, since gain and phase margins are read directly.
• When sketching by hand, mark principal angles (0°, ±90°, ±180°, ±270°) and concentric circles for magnitudes (e.g., 0.1, 0.2, 0.5, 1.0 etc.) to aid accuracy.
• For systems with non-minimum-phase zeros or right-half-plane poles, the polar plot will reflect the additional phase shifts; interpret margins with care and prefer Nyquist methods for rigorous stability conclusions when the plot does not satisfy simple crossing conditions.
• Computer tools (MATLAB, Python control libraries) produce accurate polar plots and numerical gain/phase margins; hand sketches remain valuable for intuition and for quick approximate design checks.