Polar Plots and Nyquist Diagrams

Chapter 14 treated \(G(j\omega)\) as two real functions of frequency and plotted each separately. That is one legitimate choice; there is another. At each frequency \(G(j\omega)\) is a single complex number, so it can be drawn as a single point in the complex plane, with \(\operatorname{Re}G\) horizontally and \(\operatorname{Im}G\) vertically. As \(\omega\) runs from \(0\) to \(\infty\) that point traces a curve. This curve is the polar plot, and it is the same data the Bode pair carried, merely reorganised.

The reorganisation costs something. Frequency is no longer an axis: it survives only as a parameter, marked as ticks along the curve, and reading \(|G|\) at a given \(\omega\) now requires measuring a length rather than glancing at a gridline. Individual factors no longer add, so the pencil-and-straight-edge construction of Chapter 14 has no analogue here.

What it buys is decisive. The critical condition \(G(j\omega)H(j\omega)=-1\) is a statement about a single complex number, and in the complex plane it names a single point: \(-1+j0\). Whether the loop is stable turns out to depend on how the curve is arranged around that point — not on where it passes any particular gridline. The Bode pair can only test that condition by cross-referencing two plots at two different frequencies, as Section 14-6 did with its two crossovers. The polar plot tests it by looking.

Nothing about a polar plot requires cleverness in principle: choose frequencies, compute \(|G|\) and \(\angle G\) at each, and plot the points. In practice a handful of structural facts settle the shape before any arithmetic is done, and a sketch built from them needs only three or four computed points to be trustworthy. The four questions to answer, in order, are these.

The two crossing questions are answered by the same technique: rationalise \(G(j\omega)\) so that its real and imaginary parts are explicit, then set one of them to zero. Rationalising means multiplying numerator and denominator by the conjugate of the denominator, which leaves a real denominator and a numerator whose real and imaginary parts can be read off:

Since \(|Q|^2\) is never zero for a physical system at real \(\omega\), each part vanishes exactly when the corresponding part of the numerator \(P\bar Q\) does. Real-axis crossings therefore come from a polynomial equation in \(\omega\), and there is usually only one root worth having.

One more structural fact is worth stating before any examples. For most loop transfer functions the phase decreases monotonically with frequency while the magnitude decreases too, so the locus spirals inward in the clockwise sense. That is the default picture. Departures from it — a locus that turns back outward, or one that loops — signal either a lead network's phase advance or the peculiar geometry of a conditionally stable system.

Write the loop transfer function in time-constant form, exactly as in Section 14-4:

The low-frequency end. As \(\omega\to0\) every \((1+j\omega T)\) tends to \(1\), so \(G(j\omega)\to K/(j\omega)^N\). For a type-0 system the limit is the finite real number \(K\): the locus starts on the positive real axis at a distance \(K_p\) from the origin, the position error constant of Chapter 9. For \(N\ge1\) the magnitude diverges and the starting phase is \(-90N^\circ\), so the locus comes in from infinity along the negative imaginary axis direction for type 1, and along the negative real axis direction for type 2.

"Along the direction of" is not the same as "asymptotic to the axis", and for type-1 systems the difference matters. Expand one step further, using \((1+j\omega T_1)^{-1}(1+j\omega T_2)^{-1}\approx1-j\omega(T_1+T_2)\) for small \(\omega\):

The high-frequency end. As \(\omega\to\infty\) each factor is dominated by its \(j\omega T\) term, so numerator and denominator behave as \((j\omega)^m\) and \((j\omega)^n\) where \(m\) and \(n\) count the finite zeros and all poles. Hence \(|G|\to0\) whenever \(n>m\), which is true of every physically realisable system, and the phase tends to \(-90^\circ(n-m)\). The locus therefore ends at the origin, approaching it from a fixed direction set by the pole–zero excess alone.

Because polar plots do not add, there is less to memorise here than in Chapter 14 — but the shapes of the simplest factors are worth knowing exactly, because larger loci are recognisably distortions of them.

The first-order lag. Take \(G(j\omega)=1/(1+j\omega T)\) and write \(u=\omega T\). Rationalising,

The locus is not merely semicircle-like; it is a semicircle exactly. Form \(X-\tfrac12=(1-u^2)/[2(1+u^2)]\) and add the square of \(Y\):

So every point lies on the circle of radius \(\tfrac12\) centred at \(\tfrac12+j0\). Since \(Y\le0\) for all \(\omega\ge0\), only the lower half is traced: from \(1\angle0^\circ\) at \(\omega=0\), through \(0.707\angle-45^\circ\) at the corner frequency \(\omega=1/T\), to the origin at \(\omega\to\infty\), arriving along the \(-90^\circ\) direction.

The integrator. \(1/(j\omega)=-j/\omega\) is purely imaginary and negative for all \(\omega>0\): its locus is the whole negative imaginary axis, traversed from \(-j\infty\) down to the origin. Two integrators give \(-1/\omega^2\), the negative real axis.

The quadratic lag. Its locus starts at \(1\angle0^\circ\) and ends at the origin along \(-180^\circ\), crossing the negative imaginary axis exactly at \(\omega=\omega_n\), where the value is \(1/(2\zeta)\) — the same number that produced the resonant bulge in Chapter 14. For small \(\zeta\) that crossing point is far out, and the locus bulges enormously; for \(\zeta\) near \(1\) it is modest.

Transport lag. The factor \(e^{-j\omega T_d}\) has magnitude \(1\) at every frequency and phase \(-\omega T_d\) radians, which grows without bound. Multiplying any locus by it leaves every radius unchanged while rotating each point clockwise by an amount proportional to \(\omega\). A locus that used to end tidily at the origin from a fixed direction now spirals in, wrapping around the origin infinitely many times. Every such wrap is a fresh opportunity to enclose the point \(-1\), which is why dead time is so corrosive to stability — and why a system that has infinite gain margin without it can have almost none with it.

Of the four structural questions in Section 15-2, one carries far more weight than the others: where the locus crosses the negative real axis. That crossing occurs at the phase crossover frequency of Section 14-6 — the phase is exactly \(-180^\circ\) there — and the distance from the origin to the crossing point is \(|G(j\omega_{pc})|\), the reciprocal of the gain margin. A single number decides how much gain the loop can absorb.

Work it out in general for the commonest examination case, the type-1 system with two lags:

A quotient \(K/D\) is real precisely when \(D\) is real. The imaginary part of the denominator vanishes when \(\omega(1-\omega^2T_1T_2)=0\), and discarding the trivial root \(\omega=0\) leaves one useful frequency. Substituting it back gives the crossing value directly.

Notice what this says about the shape of the whole family. Changing \(K\) does not change \(\omega_{pc}\) at all — the phase is independent of gain — and it scales every point of the locus radially by the same factor. The polar plot of a loop at gain \(2K\) is a photographic enlargement of the plot at gain \(K\), pinned at the origin. The critical point \(-1\), however, does not move. Stability analysis in this plane is therefore a question of when a fixed point is swallowed by an expanding curve.

The imaginary-axis crossings, obtained by setting \(\operatorname{Re}G=0\), carry no stability meaning but are useful for a faithful sketch. For a type-0 system with two lags, for instance, the locus crosses the negative imaginary axis when the two arctangents sum to \(90^\circ\), by the same tangent identity used in Example 3 of Chapter 14.

So far the curve has been drawn for \(\omega\) from \(0\) to \(\infty\) — a piece of the imaginary axis. Chapter 16 will need something stronger: the image of a closed path in the \(s\)-plane that encloses the entire right half-plane, because that is the region in which a closed-loop pole would be fatal. The path is called the Nyquist contour and it is assembled from three parts.

First, the imaginary axis itself, traversed from \(s=-j\infty\) up to \(s=+j\infty\). Second, a semicircle of radius \(R\to\infty\) closing back through the right half-plane, traversed clockwise from \(+j\infty\) round to \(-j\infty\). Together these enclose every point with \(\operatorname{Re}s>0\), and the clockwise sense is the convention that makes the encirclement count in Chapter 16 come out positive for unstable poles.

Third, and easy to forget, an indentation. If \(G(s)H(s)\) has a pole on the imaginary axis — and every type-1 or type-2 system has at least one, at the origin — the contour cannot pass through it, because \(GH\) is infinite there and has no image. The remedy is to detour around such a pole on a small semicircle of radius \(\varepsilon\to0\), taken to the right so that the pole is excluded from the enclosed region. Excluding it matters: the criterion counts open-loop poles strictly inside the contour, and a pole sitting on the boundary would make that count ambiguous.

What does the indentation map to? On it, \(s=\varepsilon e^{j\theta}\) with \(\theta\) running from \(-90^\circ\) through \(0^\circ\) to \(+90^\circ\). Near the origin a type-\(N\) loop behaves as \(GH\approx K/s^N\), so

The image is an arc of infinite radius. As \(\theta\) increases by \(180^\circ\) the image angle decreases by \(180N^\circ\), so for a type-1 system the image sweeps \(180^\circ\) clockwise, from \(+90^\circ\) through \(0^\circ\) to \(-90^\circ\), passing through the positive real direction at infinity. For a type-2 system it sweeps \(360^\circ\). This great arc is the piece that closes the diagram, and it is where beginners lose encirclements.

The large semicircle is easier. For every realisable system \(n>m\), so \(|GH|\to0\) uniformly as \(|s|\to\infty\): the entire infinite semicircle maps to the single point at the origin. It contributes no arc, no encirclement, and nothing to worry about.

Assembling the pieces requires one final observation. The coefficients of \(G(s)H(s)\) are real, so \(GH(-j\omega)=\overline{GH(j\omega)}\): the value at a negative frequency is the complex conjugate of the value at the corresponding positive frequency. The branch traced for \(\omega<0\) is therefore the mirror image, about the real axis, of the branch already drawn for \(\omega>0\). It never has to be computed — only reflected.

Two habits prevent most errors. Mark the direction of increasing \(\omega\) with arrowheads as the curve is drawn, because the encirclement count of Chapter 16 is signed and the sign is the direction. And check that the two branches meet: the \(\omega\to0^+\) end of the polar plot and the \(\omega\to0^-\) end of the mirror must be joined by the infinite arc, and the \(\omega\to\pm\infty\) ends must both arrive at the origin. A diagram with loose ends has a missing piece.

The margins of Section 14-6 have a purely geometric meaning here, and it is worth seeing them as distances and angles rather than as formulae.

The gain margin is read where the locus cuts the negative real axis. If it cuts at \(-a\), then \(|GH|=a\) at that frequency and \(\text{GM}=1/a\). Equivalently, the crossing point is at \(-1/\text{GM}\): the closer the crossing creeps toward \(-1\), the smaller the reserve. A crossing at \(-0.5\) means the gain may be doubled; a crossing at \(-2\) means the point \(-1\) has already been passed and, for a minimum-phase loop, the closed loop is unstable.

The phase margin is read where the locus cuts the unit circle, since that is where \(|GH|=1\). Draw the radius to that intersection; the angle between it and the negative real axis, measured anticlockwise, is the phase margin. It is the rotation that would be needed to carry the crossing point onto \(-1\) without changing its distance from the origin — which is exactly what an added phase lag does.

Sketch first, compute second. Fix the start point from the type, the end point from \(n-m\), and only then hunt for crossings. Assume \(K>0\) and unity feedback throughout. Difficulty rises down the list.