Electronics Guide

Closed-Loop Poles and the Characteristic Equation

In the standard single loop, a gain K and a forward path G(s) drive the output, and a sensor H(s) feeds the output back for subtraction from the reference. The closed-loop transfer function is

T(s) = KG(s) / [1 + KG(s)H(s)]

and its poles are the roots of the characteristic equation

1 + KG(s)H(s) = 0

Write G(s)H(s) = N(s)/D(s), where N(s), of degree m, has the open-loop zeros z₁ through z_m as its roots, and D(s), of degree n, has the open-loop poles p₁ through p_n. Scale both polynomials to leading coefficients of 1, folding the constant into K. Clearing the fraction gives

D(s) + KN(s) = 0

whose n roots are the closed-loop poles. The root locus plots them as K runs from 0 toward infinity.

The closed-loop zeros, which are the zeros of G(s) and the poles of H(s), do not move with K. The locus therefore shows the poles, which fix stability and each mode's decay rate and ringing frequency, but not the zeros, which set how strongly each mode appears in a response.

A second-order loop shows the idea. With G(s)H(s) = 1/[s(s + 4)], the characteristic equation s² + 4s + K = 0 has the roots

s = −2 ± √(4 − K)

At K = 0 the poles sit at 0 and −4. As K rises to 4, they approach each other along the real axis and meet at −2. Above K = 4 they become −2 ± j√(K − 4) and climb vertically. The natural frequency, √K, grows and the damping ratio, 2/√K, falls, but the real part stays at −2, so no gain makes this loop unstable. Larger loops rarely have roots in closed form, and the rules below sketch their loci without them.

The method assumes a lumped, linear, time-invariant loop with real coefficients, negative feedback, a real gain K ≥ 0, and n ≥ m.

The Angle and Magnitude Conditions

A point s is a closed-loop pole for some gain exactly when KG(s)H(s) = −1. This complex equation splits into two real conditions. The angle condition decides whether a point lies on the locus, and the magnitude condition gives the gain that puts a pole there.

Draw a vector from each open-loop zero and pole to the test point s, with angles measured counterclockwise from the positive real direction. Let θ_z denote the angles of the vectors from the zeros and θ_p those from the poles. For K > 0 the angle condition is

∑θ_z − ∑θ_p = (2q + 1) × 180°, for some integer q

so the net angle must be an odd multiple of 180°. At a point that satisfies it, the magnitude condition gives the gain:

K = (product of the vector lengths from the poles) / (product of the vector lengths from the zeros)

with the denominator taken as 1 when the loop has no zeros.

Test two points against 1/[s(s + 4)]. At s = −2 + j3, the vectors from the poles at 0 and −4 have angles of 123.7° and 56.3°, a net angle of −180°, so the point lies on the locus. Both vectors have length √13, so K = 13, and s² + 4s + 13 = 0 indeed has the roots −2 ± j3. At s = −1 + j3 the angles sum to 153.4°, so no positive gain places a pole there.

Because angles add and lengths multiply, points can be tested with a protractor and a scale. Evans also invented the Spirule, an arm and a disk joined at a pivot. With the pivot on the test point, rotating the arm through each vector angle summed the angles, and a logarithmic spiral on the device turned each vector length into an angle, so adding those angles multiplied the lengths, as on a slide rule.

Sketching Rules: Branches, the Real Axis, and Asymptotes

The two conditions yield rules that fix the shape of a locus before any point on it is calculated.

1. Branches. The locus has n branches, one per closed-loop pole. They vary continuously with K, and because the coefficients are real, the locus is symmetric about the real axis.
2. Start and end points. At K = 0 the equation reduces to D(s) = 0, so the branches start at the open-loop poles. As K grows without bound, m branches end at the open-loop zeros, and the other n − m go to infinity.
3. Real-axis segments. A point on the real axis lies on the locus exactly when the number of real poles and zeros to its right is odd. Each one to the right contributes 180°, each one to the left 0°, and the angles from a complex-conjugate pair cancel.
4. Asymptotes. The n − m branches that go to infinity approach lines at angles of (2q + 1) × 180°/(n − m), for q = 0, 1, and so on up to n − m − 1. The lines radiate from a point on the real axis, the centroid σ_a = (sum of the pole locations − sum of the zero locations)/(n − m).
5. Constant sum. When n − m ≥ 2, the coefficient of sⁿ⁻¹ in D(s) + KN(s) does not depend on K, so the closed-loop poles always sum to the sum of the open-loop poles. Branches that bend right must be balanced by branches that move left.

For the motor loop of the worked example, K/[s(s + 10)(s + 100)], three branches start at 0, −10, and −100. The locus covers the real axis between 0 and −10 and to the left of −100. The asymptotes leave the centroid, (0 − 10 − 100)/3 = −36.7, at 60°, 180°, and −60°, so two branches must eventually cross into the right half-plane, and the three poles always sum to −110.

Sketching Rules: Breakaway Points, Axis Crossings, and Departure Angles

Three further rules each take a short calculation.

Reading Damping, Natural Frequency, and Gain from the Locus

A complex closed-loop pole at s = −σ + jω_d, together with its conjugate, has

• a natural frequency ω_n equal to its distance from the origin, √(σ² + ω_d²);
• a damping ratio ζ = cos θ, where θ is the angle between the vector to the pole and the negative real axis;
• a decay rate σ = ζω_n, its distance from the imaginary axis; and
• a damped natural frequency ω_d, its distance from the real axis.

Rays from the origin therefore mark constant damping, with ζ = 0.5 at an angle of 60° from the negative real axis and ζ = 0.707 at 45°. Circles about the origin mark constant natural frequency, and vertical lines mark constant decay rate. Time-Domain Response of Control Systems converts overshoot and settling-time limits into such boundaries, which enclose a wedge about the negative real axis cut off on the right by a vertical line. To choose a gain:

1. Draw the ray for the required damping ratio, or the boundary of the required region.
2. Find where the locus crosses it, from the angle condition or from computed points.
3. Read the gain at the crossing from the magnitude condition.
4. Find the other closed-loop poles, for example with the constant-sum rule, and confirm that they lie far to the left or close to zeros.
5. Check the error constant, and confirm the response by simulation, because zeros and the other poles shift overshoot and settling time.

For a loop that loses stability as gain rises, the locus also shows the gain margin, the ratio of the crossing gain to the design gain. For the motor loop under proportional control, designed for ζ = 0.5 at K = 8,340, the ratio is 110,000/8,340 ≈ 13.2, or 22.4 dB. A Bode plot gives the same value, because the loop's phase passes through −180° at the crossing frequency. The locus does not show phase margin or sensitivity to dynamics left out of the model; Stability Analysis and Compensation covers those measures.

How Added Poles and Zeros Reshape the Locus

Compensation adds poles and zeros, so it pays to see what one added element does. Start from K/[s(s + 10)], a motor under proportional position control with negligible armature inductance. Its branches meet at −5 and rise vertically, so every gain is stable, and K = 100 gives poles at −5 ± j8.66 with ζ = 0.5. The table adds one element at a time, each with unity gain at DC, so the velocity constant K_v = K/10 compares the cases directly.

Compensator Design on the Locus

If the locus misses the region the specifications require, lead or PD compensation reshapes it to pass through a chosen point, and the magnitude condition sets the gain there. If the locus passes through the region but the error constant there is too small, a lag or PI pair near the origin raises the low-frequency gain while leaving the locus almost unchanged.

Pole-Zero Cancellation: Uses and Hazards

Placing a compensator zero on a stable plant pole removes the pair from the loop transfer function and simplifies the locus that remains, and the worked example cancels the motor's mechanical pole this way. The shortcut has three hazards.

Worked Example: A Motor Position Loop

A permanent-magnet DC motor turns a heavy load through a gearbox. A PWM amplifier drives it as a voltage source limited to ±24 V, and an encoder measures the load angle. The motor's voltage-to-speed response has a mechanical pole at 10 rad/s, set mainly by the load inertia, and an electrical pole at 100 rad/s, set mainly by the armature inductance and resistance. With a speed gain of 2 (rad/s)/V at the load, the plant from drive voltage to load angle is

G(s) = 2,000 / [s(s + 10)(s + 100)] rad/V

A proportional controller of K_P volts per radian gives the loop K/[s(s + 10)(s + 100)] with K = 2,000K_P. The specification calls for overshoot below 20 percent on a small step, settling within 2 percent in less than 0.5 s, and a following error of at most 0.01 rad while the command moves at 1 rad/s, which requires K_v ≥ 100 s⁻¹.

The earlier sections have sketched this locus. The branches from 0 and −10 meet at −4.87, where K = 2,377 (K_P = 1.19), then rise as a complex pair toward the ±60° asymptotes and cross the imaginary axis at ±j31.6 rad/s when K = 110,000 (K_P = 55). Meanwhile the branch from −100 moves left, keeping the sum of the poles at −110. To find the gain for ζ = 0.5, write the dominant pair as s² + rs + r², where r = ω_n. Matching (s² + rs + r²)(s + c) to s³ + 110s² + 1,000s + K gives r + c = 110 and r(r + c) = 1,000, so r = 9.09 and c = 100.9. The poles are −4.55 ± j7.87 and −100.9, and K = r²c = 8,340, or K_P = 4.17 V/rad.

Simulated, the design overshoots by 16.2 percent, close to the second-order prototype's 16.3 percent, because the third pole lies more than 20 times farther left than the dominant pair. It settles within 2 percent in 0.90 s. The gain margin is 13.2 (22.4 dB), and the phase margin is 51.6°, near the prototype's 51.8° for ζ = 0.5. But K_v = K/1,000 = 8.3 s⁻¹, so the load trails a 1 rad/s command by 0.12 rad. No proportional gain meets the 0.5 s settling requirement. Beyond the breakaway point, more gain moves the dominant poles toward the imaginary axis, so their decay rate never exceeds the 4.87 s⁻¹ they have at that point. The fastest 2 percent settling comes just below K = 3,800 (K_P ≈ 1.9), where ζ ≈ 0.78 holds the overshoot just under 2 percent, and even that response needs about 0.6 s.

Worked Example, Continued: Lead and Lag Compensation

Faster settling needs the dominant poles farther left, which calls for lead compensation:

1. Choose the target. Keep ζ = 0.5 and double the decay rate to 10 s⁻¹: s_d = −10 + j17.32, with ω_n = 20 rad/s.
2. Find the deficiency. At s_d, the vectors from the plant poles at 0, −10, and −100 have angles of 120°, 90°, and 10.9°, a total of 220.9°, so the compensator must supply 40.9° of lead.
3. Place the zero and pole. A zero at −10 cancels the mechanical pole's 90°, so the compensator pole must contribute 90° − 40.9° = 49.1°. The vector from a pole at −p to s_d is (p − 10) + j17.32, and tan 49.1° = 17.32/(p − 10) gives p = 25.
4. Set the gain. The magnitude condition gives K = |s_d| × |s_d + 25| × |s_d + 100| = 20 × 22.9 × 91.7 = 42,000. Since K = 2,000K_c, the compensator is C(s) = 21(s + 10)/(s + 25).
5. Check the result. The characteristic polynomial of the loop 42,000/[s(s + 25)(s + 100)] factors exactly as (s² + 20s + 400)(s + 105), and K_v = 42,000/2,500 = 16.8 s⁻¹.

The lead design overshoots by 16.0 percent and settles within 2 percent in 0.41 s, with a gain margin of 7.4 (17.4 dB) and a phase margin of 51.8°. A step of r radians demands 21r volts at the first instant, so steps above about 1.1 rad saturate the amplifier, and larger moves need the motion profiling described in Automatic Control Circuits. The following error, 1/16.8 ≈ 0.06 rad at 1 rad/s, is still six times the limit.

A lag network can multiply K_v by 10. A first attempt puts its zero at −1, a tenth of the dominant poles' real part, and its pole at −0.1. K_v rises to 168 s⁻¹, and the dominant poles barely move, to −9.54 ± j16.96. But the slow closed-loop pole at −1.06 carries 6.1 percent of the step, close to the 5.4 percent that (z − p)/K_v0 = 0.9/16.8 predicts. Overshoot grows to 22.7 percent and the 2 percent settling time to 1.05 s, so both transient requirements now fail.

Moving the pair five times closer to the origin, to a zero at −0.2 and a pole at −0.02, keeps K_v at 168 s⁻¹ and shrinks the tail to 1.1 percent, with a time constant of about 5 s. Overshoot is 17.3 percent. The response stays within 2 percent after 0.30 s, sooner than with the lead alone, because the small positive tail lifts the first undershoot inside the band, though it remains more than 1 percent from its final value until 0.66 s. The final controller is

C(s) = 21(s + 10)(s + 0.2) / [(s + 25)(s + 0.02)]

A last check tests the cancellation. If the load inertia doubles, the plant becomes 1,000/[s(s + 5)(s + 100)], while the compensator's zero stays at −10. The final design then has dominant poles at −5.42 ± j9.68 and a real pole at −16.5; it overshoots by 27.5 percent and settles within 2 percent in 0.71 s. Proportional control facing the same change overshoots by 30 percent and needs 1.7 s. A load that varies this much calls for checking the design at both extremes or scheduling the gains on the load.

The Root Locus in the z-Plane

With a digital controller, the characteristic equation becomes 1 + KG(z) = 0, where G(z) is the pulse transfer function of the zero-order hold, plant, and sensor together. It must come from transforming that cascade as a whole, because the product of separately transformed blocks is not the transform of their combination. The algebra is unchanged, so every construction rule carries over; only the reading changes. Stability requires every closed-loop pole inside the unit circle, which replaces the imaginary axis, and the Jury test, or Routh's array after the substitution z = (1 + w)/(1 − w), replaces Routh's criterion. Under the mapping z = e^sT, lines of constant damping become logarithmic spirals that wind toward the origin, and z = 1 corresponds to s = 0. Z-Transform develops the mapping and the unit-circle stability condition.

Sampling costs stability. Sample the base loop K/[s(s + 10)] every T = 20 ms through a zero-order hold, and the discrete plant becomes

G(z) = 1.873 × 10⁻⁴ (z + 0.9355) / [(z − 1)(z − 0.8187)]

Its poles, at 1 and e^−10T = 0.8187, are images of the continuous poles, and its zero at −0.9355 comes from the hold. Two real poles and a real zero beyond them again give a circle about the zero, here of radius 1.84. The branches break away at 0.907, swing around the zero, and leave the unit circle at K = 1,034, at angles of ±35.7° that correspond to oscillation at 31.1 rad/s, before rejoining the real axis at −2.78. The continuous loop is stable at every gain; the sampled loop is not. At K = 100, where the continuous loop has ζ = 0.5, the sampled poles at 0.900 ± j0.162 correspond to ζ = 0.45, and sampling every 50 ms lowers that damping ratio to 0.37 and the stability limit to K = 436.

Software Tools and Numerical Checks

Software computes a locus by solving D(s) + KN(s) = 0 over a grid of gains and joining the roots into branches:

• MATLAB: The Control System Toolbox function rlocus plots the closed-loop pole trajectories against the feedback gain, assuming negative feedback, and returns the roots at specified gains. The Control System Designer app, opened with controlSystemDesigner, adds an interactive root locus editor for placing compensator poles, zeros, and gain.
• Python: The Python Control Systems Library, python-control, provides root_locus_map to compute the locus and root_locus_plot to plot the roots of 1 + kG(s) as k varies.
• Scilab: The evans function draws what its documentation calls the Evans root locus of a single-input, single-output system.

The hand rules remain the best check on a computed plot:

• Match the feedback sign. The tools assume negative feedback around the model as entered, so pass the negated model to study positive feedback.
• Refine the grid near special points. Near a breakaway point, a double root splits in proportion to the square root of the change in gain, so a coarse grid can make branches appear to jump or swap. Request the breakaway and crossing gains explicitly.
• Keep canceled modes visible. Reducing a model to minimal form drops canceled pole-zero pairs, along with the modes that disturbances still excite, so keep the plant and controller as separate blocks.
• Test the plot against the rules. Branch counts, real-axis segments, the centroid, and the constant pole sum catch most model-entry errors, such as a pole entered with the wrong sign.

Strengths and Limits of the Method

The root locus shows closed-loop poles directly, so it suits requirements stated as damping, natural frequency, or decay rate, and design choices that come down to one gain or one compensator zero. Its limits follow from the same directness. Its rules need a rational model, so it cannot work directly from measured frequency-response data, and a time delay calls for a rational approximation or a numerical solution of the angle condition. It varies one parameter at a time. It shows poles but not closed-loop zeros, so its predictions still need simulation. And it assumes linearity, while saturation, backlash, and friction shape real hardware. Frequency-response methods complement it by reading margins from measured loops and treating delays exactly, and multivariable loops call for state-space methods.

Summary

The root locus traces the roots of 1 + KG(s)H(s) = 0 as the gain rises from zero. The angle condition decides which points lie on it, and the magnitude condition labels those points with gains. The rules follow from the two conditions: n branches from the poles to the zeros or to infinity, real-axis segments to the left of an odd count of poles and zeros, asymptotes from the centroid, breakaway points from dK/ds = 0, axis crossings from Routh's array, and departure and arrival angles.

Read against rays of constant damping, the locus turns specifications into gains. Added poles push it right and added zeros pull it left. Lead and PD compensation move it toward faster poles, while lag and PI compensation raise low-frequency gain at the cost of a slow tail. Canceling a stable pole is a shortcut that disturbances and parameter changes can expose, and canceling an unstable pole or a right-half-plane zero never works. In the motor example, a lead network doubled the decay rate that proportional control allowed, and a lag pair near the origin then raised the velocity constant tenfold. In the z-plane the same rules apply, with the unit circle as the stability boundary.

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