Angle and Magnitude condition of root locus (with Examples) - Control Systems - Electrical

The root-locus method is a graphical technique for studying how the closed-loop pole locations of a feedback control system change as a single parameter (usually the loop gain K) varies. Two fundamental conditions determine whether a point s in the complex s-plane lies on the root locus of the loop transfer function G(s)H(s): the angle condition and the magnitude condition. The angle condition tells us whether a point is on a root-locus branch. The magnitude condition gives the numerical value of the loop gain K for a point already known to be on the root locus by the angle condition.

Basic characteristic equation and polar form

• The closed-loop characteristic equation is 1 + G(s)H(s) = 0.
• Equivalently, G(s)H(s) = -1.
• Writing -1 in polar form gives -1 = 1∠(±(2q + 1)·180°), where q = 0, 1, 2, ...
• Therefore G(s)H(s) must have magnitude 1 and angle equal to an odd multiple of 180° for s to be on the root locus.

Angle condition

For a point s to lie on the root locus, the phase (angle) of the loop transfer function G(s)H(s) must equal an odd multiple of 180°:

∠G(s)H(s) = (2q + 1)·180°, where q = 0, 1, 2, ...

Equivalently, ∠G(s)H(s) = ±180°, ±540°, ±900°, ... . In practice we check whether the net angle contributed by zeros and poles (and any constant sign) yields an odd multiple of 180°.

How to evaluate the angle

• Write G(s)H(s) in factorised form so that the phase is a sum of contributions from zeros and poles: ∠G(s)H(s) = ∠K + Σ∠(s - z_i) - Σ∠(s - p_j).
• For a real positive K, ∠K = 0°; for a negative K, add 180° accordingly.
• Compute each phasor angle ∠(s - a) as the angle of the vector from point a in the s-plane to the test point s (use atan2 to preserve the correct quadrant).
• Sum the angles from zeros, subtract the angles from poles, and check if the result equals an odd multiple of 180° (within numerical tolerance).

Uses of the angle condition

• Test whether a given point s in the s-plane lies on the root locus (preliminary check).
• Locate portions of the real axis that belong to the root locus (sign rule based on number of poles and zeros to the right of a test point).
• Determine departure/arrival angles at complex poles/zeros (using phase sums to find directions of branches near poles/zeros).

Magnitude condition

Once the angle condition is satisfied for a point s, the magnitude condition determines the loop gain K that places a closed-loop pole at that s. The magnitude condition is

|G(s)H(s)| = 1.

If G(s)H(s) = K·N(s)/D(s), where N(s) and D(s) contain the factors due to zeros and poles (excluding the gain K), then

K = 1 / |N(s)/D(s)| = |D(s)/N(s)|.

Graphically, this becomes the product-of-distances rule:

• K = (product of distances from the test point to all open-loop poles) / (product of distances from the test point to all open-loop zeros).

If there are no open-loop zeros, the denominator is unity and K equals the product of distances to the poles.

Uses of the magnitude condition

• Compute the numerical gain K that places a closed-loop pole at a specific s already validated by the angle condition.
• Find gains corresponding to intersections of root-locus branches with lines (e.g., imaginary axis crossings) or specified damping ratios/natural frequencies.

Example: Test whether the point -2 + 5j is on the root locus and find K

Example: Test whether the point -2 + 5j is present on the root locus or not. Consider the system with G(s)H(s) = K / [s(s + 4)].

Solution.

Evaluate the angle condition for s = -2 + 5j.

G(s)H(s) = K / [s(s + 4)].

Angle of numerator ∠K = 0° (assume K > 0).

Angle of denominator = ∠s + ∠(s + 4).

Compute ∠s for s = -2 + 5j: ∠(-2 + 5j) = atan2(5, -2) = 111.801°.

Compute ∠(s + 4) for s + 4 = 2 + 5j: ∠(2 + 5j) = atan2(5, 2) = 68.199°.

Sum of denominator angles = 111.801° + 68.199° = 180.000°.

Net angle ∠G(s)H(s) = 0° - 180.000° = -180.000°, which is an odd multiple of 180°.

Therefore the angle condition is satisfied and the point s = -2 + 5j lies on the root locus.

Now apply the magnitude condition to find K.

Compute magnitudes:

|s| =

|s + 4| =

Product of distances to poles = |s| · |s + 4| = 5.3852 · 5.3852 = 29.

There are no finite open-loop zeros, so product of distances to zeros = 1.

Therefore |G(s)H(s)| = K / 29 = 1.

K = 29.

Hence s = -2 + 5j is a closed-loop pole when K = 29.

Graphical method to determine the value of K

When a point on the root locus is known (from angle condition or by inspection of the plot), the gain K can be determined from the ratio of phasor lengths:

• K = (product of distances from the point to open-loop poles) / (product of distances from the point to open-loop zeros).

Example 1. Find the value of system gain K for G(s)H(s) = K / [s(s + 4)] given that s = -2 + 5j is on the root locus

Example 1. Find the value of system gain K for the system G(s)H(s) = K/s(s + 4) Given that a point -2 + j5 is already present on the root locus plot.

Sol.

The open-loop poles are found from s(s + 4) = 0, hence poles at s = 0 and s = -4.

Distance from s = 0 to the point (-2 + 5j) is

Distance from s = -4 to the point (-2 + 5j) is

Product of distances to poles = √29 · √29 = 29.

No finite zeros, so denominator = 1.

K = 29.

Example 2. Root-locus nature for G(s)H(s) = K / [s(s + 2)]

Example 2. To find the nature of the root locus of the given system G(s)H(s) = K/s(s + 2)

Sol.

The characteristic equation is 1 + K / [s(s + 2)] = 0.

Multiply through: s(s + 2) + K = 0.

Thus s² + 2s + K = 0.

Solve for s: 

For different values of K the nature of roots changes:

• When K < 1, √(1 - k) is real and both roots are real and distinct (two real closed-loop)
• When K = 1, the roots coincide at s = -1 (repeated real root).
• When K > 1, sqrt(1 - K) is imaginary and the closed-loop poles form a complex conjugate pair symmetric about the real axis, moving off the real axis.

From pole locations and the polynomial order we conclude that there are two branches of the root locus (equal to the number of open-loop poles) originating at s = 0 and s = -2 and moving according to the rules of root-locus construction as K varies.

Practical notes and common procedures

• Always apply the angle condition first to check whether a test point belongs to the locus; only then use the magnitude condition to compute K.
• Use the real-axis rule to mark portions of the real axis that are on the root locus: a point on the real axis belongs to the root locus if the number of real poles and zeros to its right is odd.
• Find asymptotes (when number of poles n > number of zeros m) using angles and centroid formula: asymptotes number = n - m; centroid = (sum of poles - sum of zeros)/(n - m); asymptote angles = (2k + 1)·180°/(n - m), k = 0,1,...
• Compute breakaway and break-in points by differentiating the characteristic equation with respect to s and solving dK/ds = 0 for real s.
• For complex arithmetic use atan2 and magnitude formulas to avoid quadrant errors and sign mistakes.

Summary

The angle condition identifies whether a point s is on the root locus by requiring ∠G(s)H(s) to be an odd multiple of 180°. The magnitude condition determines the loop gain K at that point by requiring |G(s)H(s)| = 1. Together these two conditions allow analytical checks of root-locus points and computation of the specific gain values that cause closed-loop poles to occur at chosen locations in the s-plane.