Root Locus

Part 10 · Control Loops

Key point: closed-loop poles are the roots of \(1+T=0\). \(K\) does not invent new poles — it walks the ones you already placed, along a path the open-loop poles and zeros already drew.

Gray = locus. Magenta × = this \(K\). The dashed line is the \(j\omega\) axis. Each step animates.

1
Three poles, HF on the log-\(\sigma\) axis.
Plant + LF + HF. No zero. Gold × is the HF pole — now on-scale. The gray locus already heads for the right-half plane.
2
Walk \(K\) onto \(j\omega\) — a pure oscillator.
Raise gain until the magenta × sit on the imaginary axis. Real part is zero. The step is a sine that never dies and never grows.
3
Past \(j\omega\) — it runs away.
One more click of \(K\) and the poles are in the right-half plane. The step grows. That is unstable.
4
Zero too high.
Same \(K\) that wanted to ring. A lead zero way above crossover barely pulls the locus. The pair still sits near \(j\omega\).
5
Zero in the middle.
Walk \(f_z\) down to where it belongs. The green ○ yanks the locus left. Damping comes back. That is the compensator doing its job.
6
Zero too low — poles pulled in.
Walk \(f_z\) down further. The pair is dragged toward the zero. Slow. You bought damping and spent bandwidth.
7
HF pole closes the phase boost.
Lead zero started the boost; phase would stay up. Walk in the gold HF pole a decade above \(f_z\) — that’s the top of the window. Bode phase comes back down.
8
Now write \(G=L/(1+L)\).
Clear the \((1+s/\omega)\) factors and you get a cubic over a cubic. The quadratic formula just retired.
9
Characteristic polynomial.
\(1+L=0\) is that denominator. Three coefficients. You will not guess the roots.
10
The roots were already on the plot.
One real pole, one complex pair. The locus computed them while \(K\) walked. No pencil required.
11
Low-entropy \(G(s)\).
Same zero you placed. Poles from the locus — one real, one quadratic. That is why we draw the path instead of expanding.

Last page: zeros stay, poles are \(1+T=0\). This page is that path as \(K\) and \(f_z\) walk. Bode is \(L(s)\); the locus is the closed-loop poles.

s-plane as \(K\) walks

One real pole. \(K\) only slides it left — faster, no ring.

T(s) = 20.0 dB × 1 + s 2π · 19.6 kHz 1 + s 2π · 5.0 kHz · 1 + s 2π · 0.8 kHz · 1 + s 2π · 180 kHz

Gain K (dB) Plant pole kHz LF pole kHz Lead zero kHz HF pole kHz

open-loop \(f_c\) → —

CL · —

step · —

Closed-loop definition · poles of \(G\) are the roots of \(1+L=0\)

Open-loop \(L(s)\) — what the Bode is plotting

Closed-loop \(G(s)\) · factored from the locus

Root locus · gray = path vs \(K\) · magenta × = this \(K\) · dashed = \(j\omega\)

× plant × LF × HF ○ zero × CL (this K) — locus

Open-loop Bode of \(L(s)\) · same colors as the s-plane

|T| phase × plant × LF ○ zero × HF

Closed-loop step · dominant poles

Open-loop poles — —

Zeros — —

Closed-loop poles (roots of \(1+T=0\), this \(K\)) — —

Damping, overshoot, settle — the numbers behind the poles

Phase margin: —° \(\mathrm{PM}=180^\circ+\angle L(j\omega_{\mathrm{gc}})\)

Open-loop bandwidth \(\omega_{\mathrm{gc}}\): — kHz (\(|L|=1\))

Damping \(\zeta\): — \(\zeta=\sigma/\sqrt{\sigma^2+\omega^2}\)

\(Q\): — \(Q=1/(2\zeta)\)

\(\sigma=|\mathrm{Re}(s)|\): — kHz

\(\omega=|\mathrm{Im}(s)|\): — kHz

Natural frequency \(\omega_n\): — kHz \(\omega_n=\sqrt{\sigma^2+\omega^2}\)

Damped frequency \(\omega_d\): — kHz

Oscillation period: — µs

Overshoot: —% \(e^{-\pi\zeta/\sqrt{1-\zeta^2}}\times 100\%\)

Time constant \(\tau\): — µs \(\tau=1/|\sigma|\)

Settling \(t_s\approx 4\tau\): — µs

Closed-loop poles: —

Closed-loop zeros (the ones you placed): —

Root Locus vs Bode Plots — What Each Tool Reveals

Bode plots are one of the most practical tools in control systems design because they are relatively easy to sketch and interpret directly from the open-loop transfer function \(L(s)\). By inspecting the magnitude and phase contributions of each pole and zero, a designer can quickly estimate gain crossover frequency (bandwidth) and phase margin — two quantities that strongly correlate with closed-loop stability and transient response.

However, Bode analysis has limitations. It primarily gives information at a single frequency (the crossover) and relies on approximations (such as the relationship between phase margin and damping). It does not directly show how the system’s natural modes evolve as you change gain.

Root locus tells a deeper story, at the cost of more complexity. Instead of looking at frequency response, it shows exactly how the closed-loop poles move in the s-plane as the gain \(K\) is varied. Because the closed-loop poles determine the actual time-domain behavior (oscillation frequency, damping, settling time, overshoot), the root locus gives direct insight into stability and performance across a range of gains.

One of the key insights root locus provides is that closed-loop poles are generally much harder to calculate analytically than open-loop poles and zeros. With Bode, you are mostly manipulating the open-loop poles and zeros you already know. With root locus, you are watching the closed-loop poles migrate — something that requires solving the characteristic equation \(1+L(s)=0\) at many different gains.

In short:

Bode is fast for frequency-domain design (bandwidth, phase margin, stability margins).
Root locus reveals how those frequency-domain choices actually move the system’s natural dynamics in the time domain.

The two tools are complementary. Many engineers use Bode plots to quickly arrive at a candidate compensator, then use root locus to verify that the closed-loop poles end up where they want for the desired transient response.