Poles and zeros

The formula

Write a linear system as a ratio of two polynomials in the complex frequency variable s:

        N(s)     (s - z1)(s - z2)...(s - zm)
H(s) = ------ = K --------------------------
        D(s)     (s - p1)(s - p2)...(s - pn)

zi are the zeros — roots of the numerator N(s), the values of s that force H(s) = 0.
pi are the poles — roots of the denominator D(s), the values of s where H(s) runs to infinity.
K is the overall gain. Poles, zeros, and K completely define the system up to that constant.

A first-order RC low-pass filter:

H(s) = 1 / (1 + sRC)     pole at  s = -1/(RC)

Reading: one real pole on the negative-real axis. Put R = 1 kΩ, C = 1 µF, and the pole sits at s = -1000 rad/s, i.e. a -3 dB corner at 1000/(2π) = 159 Hz.

A second-order system in standard form:

H(s) = ωn² / (s² + 2·ζ·ωn·s + ωn²)
poles: s = -ζ·ωn ± ωn·√(ζ² - 1)

Reading: for 0 < ζ < 1 the poles are a complex-conjugate pair at radius ωn from the origin, sitting at angle arccos(ζ) off the negative-real axis. The real part sets the decay rate; the imaginary part sets the ringing frequency.

Where you meet it

Control loop review board. Someone brings a plant model and a proposed PID gain set. The first thing anybody competent does is pull up the closed-loop pole-zero map. Any pole in the right half-plane and the loop is unstable — no gain margin argument needed, the geometry says it outright.
Filter design at the bench. You spec a Sallen-Key or an active filter by placing pole pairs. A Butterworth response is just poles spread evenly on a circle in the left half-plane; a Chebyshev pushes them onto an ellipse to buy sharper rolloff at the cost of passband ripple.
Vibration and structural test. A modal test on a fixture or a spacecraft panel returns natural frequencies and damping ratios. Each lightly-damped mode is a complex pole pair near the imaginary axis. The closer to the axis, the longer the structure rings after a hammer strike.
RF and power electronics. A switching converter's control-to-output response has a right-half-plane zero in boost and flyback topologies. It bites you as non-minimum-phase behavior: command more output, get less first, then more. You cannot design past it by cranking gain — you have to respect where that zero lives.

How it works

The whole trick is that pole locations map directly onto time-domain behavior, so you read the answer off a plot instead of solving the differential equation.

A real pole at s = -a contributes a term that decays like e^(-a·t). Farther left means faster decay. A pole on the imaginary axis (a = 0) oscillates forever without decaying. A pole in the right half-plane (a < 0, so the real part is positive) grows without bound — that is instability, full stop.

A complex pair at -σ ± jωd gives a decaying sinusoid e^(-σ·t)·cos(ωd·t). The real part σ is the envelope decay; ωd is the ring frequency. For the standard second-order form, the poles lie on a circle of radius ωn, and the damping ratio is the cosine of the angle off the negative-real axis:

ζ = 0.5 → poles at 60° off the negative-real axis → about 16.3% overshoot.
ζ = 0.707 → poles at 45° → about 4.3% overshoot, the usual "good enough" target.
ζ = 0.1 → poles 84° out, hugging the imaginary axis → roughly 73% overshoot and a long ring.

Zeros do not cause instability, but they shape the response. A zero near a pole partially cancels it. A zero in the right half-plane produces the initial-wrong-way step response mentioned above and adds phase lag where you wanted lead. Ignoring zeros is the classic mistake: engineers stare at the pole map, declare victory on stability, and get blindsided by overshoot or undershoot the zeros were driving all along.

Now the math heritage, because "pole" and "zero" come straight from complex analysis and the classification matters. Expand a function in a Laurent series (a power series that allows negative powers) around a singular point z0. Three cases:

Removable singularity — no negative-power terms survive; the function only looks undefined at z0 and you can patch a finite value in. sin(z)/z at z = 0 is the textbook case; define it as 1 and it is analytic.
Pole of order m — finitely many negative-power terms, the most negative being 1/(z - z0)^m. Multiply by (z - z0)^m and the singularity vanishes. This is the singularity every transfer function is built from.
Essential singularity — infinitely many negative-power terms. e^(1/z) at z = 0 is the standard example. Near it the function goes berserk: the Casorati–Weierstrass theorem says it comes arbitrarily close to every complex value, and Picard's Big Theorem says it actually hits every value (at most one exception) infinitely often. e^(1/z) never equals 0, and 0 is exactly that one allowed exception.

Limit of validity: this rational-function picture assumes a linear time-invariant system. Nonlinear systems (saturation, backlash, a valve that slams shut) have no fixed poles — the "pole locations" shift with operating point, and a linearized model is only good in a neighborhood. It also assumes a lumped, finite-dimensional system: a pure transport delay e^(-sT) — propellant riding down a feedline, a combustion lag, the run of tubing to a pressure sensor — is perfectly linear and time-invariant but not rational, so no finite set of poles and zeros describes it; you hand it a Padé approximation instead, and the poles and zeros you get back are artifacts of the approximation order, not physics. A lightly-damped pole pair sitting right on the imaginary axis in your model is a warning that small unmodeled effects decide stability; don't ship a design that lives on that line.

One more gotcha: pole-zero cancellation on paper is not cancellation in hardware. If you cancel an unstable plant pole with a controller zero, the mode is still there — it's just unobservable in the output while it quietly diverges internally. That is how a "stable" loop cooks a component.

History

The vocabulary is older than control theory and comes from pure complex analysis. Augustin-Louis Cauchy built the machinery of residues in his Exercices de Mathématiques starting in 1826, working out how a function behaves at the isolated points where it becomes infinite — the objects we now call poles ^[1]^[2]. The classification of singularities got sharper mid-century: Casorati and, independently, Sokhotski described the wild behavior near an essential singularity in 1868, and Weierstrass published the same result shortly after, which is why it carries the Casorati–Weierstrass name ^[3]^[4]. In 1879 Émile Picard proved the much stronger statement — that near an essential singularity a function takes essentially every value infinitely often — using elliptic modular functions ^[3]^[5].

The jump from pure math to something an engineer would recognize came through Oliver Heaviside. During the 1880s Heaviside, a self-taught British telegrapher, built his "operational calculus" to solve the differential equations of long transmission lines by treating the derivative operator p (later s) as an algebraic quantity you could multiply and divide, then published it systematically through the 1890s ^[6]^[7]. He coined impedance, inductance, and reluctance along the way, and he was the one who put the step function and the impulse to work on switching problems ^[6]. His methods were heuristic and drew fire from mathematicians, but Bromwich later gave them a rigorous footing with contour integration, tying the whole thing back to Cauchy's residues and to what we now call the Laplace transform ^[7]. Once you write a circuit or a plant as a ratio of polynomials in s, the roots of the denominator are Cauchy's poles and the roots of the numerator are the zeros — and the pole-zero map became the control engineer's fingerprint of a system.

Related tools

/tools/rc-filter
/tools/lc-resonance
/tools/series-rlc-impedance
/tools/q-factor-bandwidth
/tools/vibration-natural-freq
/tools/rc-snubber

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →