Closed-Loop Frequency Response and the Nichols Chart

Chapters 14 to 17 built an entire theory on one function: the open-loop transfer function \(G(j\omega)H(j\omega)\). There were good reasons for that. It factors into first- and second-order terms, so its Bode magnitude is a chain of straight-line asymptotes; its polar plot is drawn once and reused; and Nyquist's criterion turns closed-loop stability into a statement about how that open-loop curve encircles \(-1\). Gain margin and phase margin, from Chapter 17, are distances measured on the same open-loop curve.

But nobody buys an open loop. The quantity a servo user cares about is \(C(j\omega)/R(j\omega)\) — how faithfully the output follows a commanded sinusoid, at what frequency it starts to resonate, and beyond what frequency it simply stops responding. Gain and phase margins are indirect measures of those things: they say how far the loop is from breaking, not how the closed loop actually behaves. Chapter 17 ended by admitting exactly this. Here we close the gap.

The obvious approach — compute \(T(j\omega)=G/(1+G)\) numerically at a hundred frequencies — works but teaches nothing and destroys the structure that made the open-loop plot easy in the first place: \(1+G\) does not factor. So we do something cleverer. We leave the open-loop plot exactly as it is and lay over it a grid: a family of curves along which the closed-loop magnitude is constant, and a second family along which the closed-loop phase is constant. Once that grid is drawn, the closed-loop response is read off the open-loop curve by inspection, the way a contour map converts a walk across a hillside into a list of altitudes.

Take the unity-feedback loop of Chapter 9, whose closed-loop transfer function is \(T(s)=G(s)/(1+G(s))\). At a fixed frequency \(\omega\), the complex number \(G(j\omega)\) is one point in the plane. Write it in rectangular form,

Define the two things we want as functions of \(\omega\): the closed-loop magnitude \(M(\omega)=|T(j\omega)|\) and the closed-loop phase \(\alpha(\omega)=\angle T(j\omega)\). Taking magnitudes and arguments of the fraction gives them directly.

The magnitude expression deserves a second look, because it is not really algebra — it is geometry. The numerator \(\sqrt{X^2+Y^2}\) is the distance from the origin to the point \(G(j\omega)\). The denominator \(\sqrt{(1+X)^2+Y^2}\) is the distance from the point \(-1\) to that same point, since \(1+G = G-(-1)\). So the closed-loop magnitude at any frequency is simply a ratio of two lengths measured on the polar plot we already have.

Similarly, \(\alpha\) is the difference of two angles: the angle the point subtends at the origin minus the angle it subtends at \(-1\). Both quantities are therefore determined by position alone. Two different systems whose polar plots happen to pass through the same point at some frequency have the same closed-loop magnitude and phase at that frequency, whatever else they do. That observation is what makes a universal overlay possible: the value of \(M\) is a property of the point, not of the system.

Fix a value of \(M\) and ask which points \(X+jY\) produce it. Squaring the magnitude relation and clearing the denominator turns the question into a quadratic in \(X\) and \(Y\).

Two cases separate at once. If \(M=1\) the quadratic terms vanish and what remains is \(-2X-1=0\), a straight line. Otherwise divide by \((1-M^2)\) and complete the square in \(X\):

That is the equation of a circle. Writing the centre with \(M^2-1\) in the denominator makes the sign convention easier to remember: for \(M>1\) the centre sits on the negative real axis.

The two limits are worth checking against intuition. As \(M\to\infty\) the centre tends to \(-1\) and the radius to zero: infinite closed-loop magnitude occurs only at the point \(G=-1\), which is exactly the condition \(1+G=0\) for a closed-loop pole on the imaginary axis. As \(M\to 0\) the circle collapses onto the origin, where \(G=0\) and the loop passes nothing. The family therefore interpolates smoothly between "no response" and "resonance without limit", and the \(M=1\) line is the watershed between them.

The same treatment applied to the phase gives a second family. Let \(N=\tan\alpha\) and expand the tangent of the difference of the two arctangents that define \(\alpha\).

That both fixed points belong to every member of the family is not an accident. At \(G=0\) we have \(T=0\), and at \(G=-1\) the closed loop is singular; in both cases the phase is undefined, so every constant-phase curve is entitled to pass through them. The two special cases are equally quick: \(\alpha=90^{\circ}\) gives \(N=\infty\), a circle of radius \(\tfrac12\) centred at \(-\tfrac12\); \(\alpha\to 0\) sends the centre to infinity and the circle degenerates into the real axis itself.

One caution accompanies these curves. Since \(\tan\alpha=\tan(\alpha\pm 180^{\circ})\), a single N circle serves two phase angles differing by \(180^{\circ}\), and only an arc of it corresponds to each. In practice the correct branch is fixed by the quadrant in which \(G(j\omega)\) lies, and the charts printed in textbooks label the usable arcs directly.

With both families printed as a background grid, the polar plot of \(G(j\omega)\) becomes a route across a contour map. Where the plot crosses an M circle, the closed-loop magnitude at that frequency equals the label on the circle; where it crosses an N circle, the closed-loop phase equals that label. Sweeping \(\omega\) along the plot and recording the labels reconstructs the entire closed-loop frequency response without a single complex division.

Three readings matter more than the rest. The largest M circle the plot reaches — the one it touches tangentially rather than cuts — gives the resonant peak \(M_r\), and the frequency at the point of tangency is the resonant frequency \(\omega_r\). The crossing of the \(M=0.707\) circle, on the far side of the peak, marks the bandwidth \(\omega_b\), because \(0.707\) is \(-3\) dB.

The construction also answers a design question: what loop gain \(K\) makes \(M_r\) equal a specified value? The geometry is pleasing. The M circle for a given \(M>1\) has its centre at distance \(d = M^2/(M^2-1)\) from the origin and radius \(r = M/(M^2-1)\). A straight line drawn through the origin tangent to that circle makes an angle \(\psi\) with the negative real axis given by \(\sin\psi = r/d\), and the algebra collapses beautifully.

The classical procedure follows. Draw the polar plot of \(G(j\omega)/K\), the shape with unit gain. Draw the line through the origin at angle \(\psi=\sin^{-1}(1/M_r)\) below the negative real axis. Construct the circle that is tangent both to that line and to the unit-gain plot, with its centre on the negative real axis; if that circle meets the negative real axis at \(-c\) on the side nearer the origin, then, because the true M circle meets the axis at \(-M/(M+1)\), the required gain is

Nathaniel Nichols' contribution was a change of coordinates. Instead of plotting \(G(j\omega)\) in the complex plane, plot its log-magnitude in decibels as the ordinate against its phase in degrees as the abscissa, with frequency as the parameter running along the resulting curve. The point \(-1\) — magnitude 1, phase \(-180^{\circ}\) — becomes the single point \((-180^{\circ},\,0\text{ dB})\), and the M and N circles map into closed contours wrapped around it.

The payoff is immediate. Multiplying the loop gain by \(K\) multiplies every magnitude by \(K\), which adds \(20\log_{10}K\) decibels to every ordinate and leaves every phase untouched. On the Nichols chart a gain change is a rigid vertical translation of the whole curve — the shape is preserved, so one plot serves every gain.

Reading the chart is a matter of four habits. The gain margin is the vertical distance, in decibels, from the curve's crossing of the \(-180^{\circ}\) vertical down to the \(0\) dB horizontal. The phase margin is the horizontal distance from the \(-180^{\circ}\) vertical to the point where the curve crosses \(0\) dB. The resonant peak is the highest M contour the curve touches, and the frequency marked there is \(\omega_r\). The bandwidth is the frequency at which the curve crosses the \(-3\) dB contour on its way down. All four come from one drawing.

Three numbers summarise a closed-loop frequency response, and each has a plain engineering meaning.

The resonant peak \(M_r\) is the maximum of \(|T(j\omega)|\). It measures relative stability: a system with \(M_r=1.05\) is heavily damped and dull, one with \(M_r=3\) rings badly, and a design target of \(1.1\) to \(1.5\) — roughly \(0.8\) to \(3.5\) dB — is usual. The resonant frequency \(\omega_r\) is where that peak occurs and is a measure of speed. The bandwidth \(\omega_b\) is the frequency at which \(|T|\) has fallen to \(1/\sqrt2\) of its low-frequency value, that is by \(3\) dB; beyond it the system effectively stops responding. Bandwidth is the most useful single number in the set, because it governs both the speed of the transient and the amount of measurement noise the loop lets through.

For the standard second-order system of Chapter 8 all three follow in closed form. With \(u=\omega/\omega_n\),

The magnitude peaks where the quantity under the root is smallest. Differentiating that quantity with respect to \(u\) and setting the result to zero locates the peak.

The root \(u=0\) is the trivial one. The useful root exists only while \(1-2\zeta^2>0\), so a peak appears only for \(\zeta<1/\sqrt2\approx0.707\); above that damping the magnitude falls monotonically from unity and there is no resonance at all. Substituting \(u_r^2=1-2\zeta^2\) back into the denominator gives \((2\zeta^2)^2+4\zeta^2(1-2\zeta^2)=4\zeta^2(1-\zeta^2)\), so the peak height is

Bandwidth follows from setting \(|T|=1/\sqrt2\), which requires the denominator squared to equal 2:

Only the positive root of the quadratic in \(u^2\) is physical. At \(\zeta=0.707\) the bracket evaluates to \(0+1=1\), so \(\omega_b=\omega_n\) exactly; at \(\zeta=0.5\) it gives \(\omega_b=1.272\,\omega_n\). Across the whole useful damping range \(\omega_b\) sits between about one and one-and-a-half times \(\omega_n\), which is why bandwidth is so often quoted as a proxy for natural frequency.

A fourth quantity, the cut-off rate, is the slope of the magnitude curve beyond \(\omega_b\). A steep cut-off is good for rejecting high-frequency noise, but steepness and peaking travel together: a sharp corner at the band edge almost always implies a large \(M_r\). The design compromise is to keep the cut-off as steep as the peak specification allows, and no steeper.

The frequency-domain numbers are only worth having if they predict the step response, and for a dominant second-order pair they do so exactly. Two more relations complete the dictionary. The gain-crossover frequency of the open loop \(G(s)=\omega_n^2/[s(s+2\zeta\omega_n)]\) satisfies \(\omega_g=\omega_n\sqrt{\sqrt{1+4\zeta^4}-2\zeta^2}\), and the phase margin measured there is

Read down the table and the story is one story. As damping rises the phase margin widens, the resonant peak falls towards unity, the resonance migrates to lower frequency and finally disappears, the bandwidth shrinks, and the overshoot subsides. Every column is a different instrument reading the same physical fact.

Two rules of thumb come out of the numbers and are worth carrying. For phase margins up to about \(60^{\circ}\), \(\zeta \approx \gamma/100\) with \(\gamma\) in degrees — at \(\gamma=51.8^{\circ}\) the true value is \(0.5\) against an estimate of \(0.518\). And a rough peak estimate \(M_r \approx 1/\sin\gamma\) is often quoted; it errs on the pessimistic side, giving \(1.27\) where the exact value is \(1.155\) at \(\zeta=0.5\), which makes it a safe design bound rather than a prediction.

The practical attraction of these measures is that they can be obtained on hardware without a model. Drive the closed loop with a sinusoidal reference, sweep the frequency, and record the amplitude ratio. The peak of that record is \(M_r\), its location is \(\omega_r\), and the \(-3\) dB crossing is \(\omega_b\). No transfer function need ever be identified — which is precisely why frequency-domain specification survived into an era of computers, and why the Nichols chart is still the surface on which the lead and lag compensators of Part 5 are designed.

Assume unity feedback and a stable closed loop throughout. For the second-order problems, identify \(\zeta\) and \(\omega_n\) before reaching for any formula; for the rest, sketch the geometry before computing.