The phase the magnitude already knows

Everybody who has drawn a Bode plot knows the rule: twenty decibels per decade of slope goes with ninety degrees of phase. Forty with a hundred and eighty. It is used to read a phase margin off a magnitude plot, and it works often enough that the question of why rarely comes up.

The answer is that for a large and useful class of networks the phase is not an independent thing at all. Given the magnitude at every frequency, the phase at every frequency follows — completely, with nothing left over. The rule of thumb is one line of that statement.

And the interesting part is the class. It is not “all networks”, and the ones outside it are not exotic.

The network, and why it is a lead rather than a low-pass #

The network is a resistor with a capacitor across it, feeding a second resistor: nine kilohms and 17.68 nanofarads in parallel, over one kilohm. It has a zero at 1.00 kHz and a pole at 10.0 kHz, so its magnitude runs flat at −20 dB, rises at twenty decibels per decade for one decade, and goes flat again at 0 dB.

A low-pass would have done, and this is better, because its phase goes up and comes back:

The peak is 54.903°, at the geometric mean of the pole and the zero, and it is $\arcsin(9/11)$ to three decimals — the closed form for a lead of ten to one. A reconstruction that reproduces a hump with a maximum in the middle of it is saying considerably more than one that reproduces a monotone fall.

The integral, and what its weighting function looks like #

Bode’s relation is

$$\varphi(\omega_0) = \frac{1}{\pi}\int_{-\infty}^{\infty} \frac{dM}{du}\,\ln\coth\frac{|u|}{2}\,du$$

where $M = \ln|H|$ and $u = \ln(\omega/\omega_0)$. In words: the phase at a frequency is a weighted average of the slope of the log-magnitude, taken over all frequencies, with the weight $\ln\coth(|u|/2)$ centred on the frequency being asked about.

Two things about that weight matter, and both are measurable.

It has a logarithmic singularity at the centre. The slope near the frequency of interest counts for much more than the slope far away, which is why the rule of thumb works locally.

Its tails are long. The weight falls off, but slowly, and it never stops. Half a decade either side captures 74.07 per cent of it; one decade, 91.89; two decades, 99.19; three, 99.92.

The total weight on one side is exactly $\pi^2/4$, and that number is where the rule of thumb comes from. If the slope is a constant $-n$ over all frequencies — $n$ times twenty decibels per decade — then the integral is $\tfrac{1}{\pi}\cdot(-n)\cdot\tfrac{\pi^2}{2} = -n\pi/2$, which is $-n$ times ninety degrees. The famous rule is the constant-slope case of the integral, not a separate fact, and everything that makes it inaccurate is the difference between a real magnitude curve and a straight line.

What the reconstruction actually does here #

The figure samples the solved network’s magnitude on a grid of a hundred points per decade over sixteen decades, throws the phase away, and evaluates the integral by convolution: the weight of each cell depends only on its offset from the frequency being asked about, so it is computed once and reused.

The weight is integrated in closed form rather than sampled, and that was not optional. Expanding $\ln\coth(u/2)$ as $\ln(1+e^{-u}) - \ln(1-e^{-u})$ and integrating term by term gives

$$F(a) = \int_0^a \ln\coth\frac{u}{2}\,du = \frac{\pi^2}{4} - 2\sum_{k\ \mathrm{odd}} \frac{e^{-ka}}{k^2}$$

so each cell’s weight is a difference of two of those. A midpoint rule that samples the weight instead loses the mass under the singularity: at four thousand cells it returned −44.96° where the answer is −45.00°, and the error fell as $1/n$ rather than $1/n^2$ — which is the signature of a singularity being sampled rather than integrated, and is the kind of thing that reads as a small physical discrepancy if nobody checks the convergence rate.

With the weights exact, the recovered phase at the corner is 39.289° and the solved phase is 39.289°. The worst disagreement anywhere in the three decades drawn is 0.019°.

The edge, and it is not a frequency #

Every figure in this collection carries the frequency, amplitude or size at which its model stops being true. This one carries something else: the span of the measurement.

The integral runs over all frequencies. A real magnitude sweep does not. So the recovery is wrong by whatever the truncated tails were carrying, and because the weight is known that error is computable before any data is collected:

A degree of phase margin is worth having, so the practical reading of that table is: about two decades either side of the frequency in question, and three if the number matters. Which is a statement about how much of a network has to be measured before its phase can be inferred, and it is the sort of requirement that is invisible until the weighting function is written down.

The networks the magnitude cannot see #

The second curve in the figure is the same lead network followed by a first-order all-pass — a network whose magnitude is exactly one at every frequency and whose phase runs from zero to −180°.

The all-pass here is built as $2v_C - v_{\mathrm{in}}$, from an RC low-pass and two dependent sources, which is $(1-sRC)/(1+sRC)$ exactly and therefore of unit magnitude by construction rather than by adjustment. There is a buffer in front of it, and it matters: without one, the all-pass’s input resistor loads the lead network and the two magnitudes are no longer the same response at all. The first version of this figure measured them 35% apart and the difference was entirely that.

With the buffer, the two networks’ magnitudes agree to the last bits of a double — the worst relative difference over sixteen decades and 1,601 sampled frequencies is $10^{-15}$ — while their phases differ by as much as 180°.

So no procedure that reads only the magnitude can tell them apart. The reconstruction lands on the minimum-phase network and is wrong about the other by exactly the all-pass’s own phase, which is not an error: that difference is the definition of excess phase. A network’s phase is its minimum-phase part, which the magnitude determines, plus an excess part, which the magnitude cannot see at all.

Why there is a relation at all #

It is worth saying where the integral comes from, because it is not a fact about circuits.

A network’s transfer function $H(s)$ is analytic in the right half of the complex plane for the plainest possible reason: a response that started before its cause would need a pole there. Analytic functions are not free to have arbitrary real and imaginary parts — Cauchy’s theorem ties the two together — and applying that to $\ln H = \ln|H| + j\varphi$ relates the log-magnitude and the phase. Bode’s integral is that relation written for a logarithmic frequency axis.

Two consequences follow immediately and neither is obvious from the circuit.

The relation is about causality, not about components. The same integral holds for a refractive index and its absorption, where it is called the Kramers–Kronig relation and is used for exactly the same purpose: measure one, infer the other. Nothing in the derivation knows what a resistor is.

The zeros are the loophole. $\ln H$ is analytic in the right half-plane only if $H$ has no zeros there — a zero of $H$ is a pole of $\ln H$. So a right-half-plane zero is precisely the thing that breaks the tie, which is why “minimum-phase” is defined by where the zeros are and not by anything one can see on a plot.

That is also why the loophole cannot be closed by measuring more carefully. The magnitude of a network with a right-half-plane zero is identical to the magnitude of the network with that zero mirrored into the left half-plane, and the two are different circuits with different phases. There is no additional magnitude information anywhere to distinguish them, at any frequency, to any precision.

Which real networks are minimum-phase, and which are not #

The name is not a description of behaviour; it is a statement about where the zeros are. A network is minimum-phase when all of its transfer function’s zeros are in the left half-plane, and it is called that because among all networks with a given magnitude it is the one with the least phase lag.

Ladders of passive elements are minimum-phase. Every filter in this collection’s filters field qualifies, which is why a filter’s phase can be read off its magnitude and why the group-delay figures there are not carrying independent information about the components.

A bridged or lattice structure need not be. The all-pass above is the extreme case, and a real lattice equaliser is the same idea applied on purpose.

Anything with a forward path that can cancel is not. A common-emitter stage has a right-half-plane zero at exactly the transconductance over the collector-base capacitance, produced by the signal arriving at the collector through the capacitance and through the transistor with opposite signs. Its phase is therefore more lagging than its magnitude implies — which is a stability problem and is invisible on a magnitude plot.

Anything with a delay in it is not. A propagation delay is $e^{-s\tau}$: unit magnitude, unbounded phase. A metre of cable in a feedback loop contributes nothing whatever to the magnitude and all of the phase that closes the loop the wrong way.

What this buys, and where it stops #

The practical consequence is worth stating plainly, because it is the reason the relation is used rather than admired.

A magnitude measurement of a minimum-phase network is a complete measurement. If a network is known to be minimum-phase, its phase margin can be computed from a magnitude sweep, and a swept-magnitude instrument is very much easier to build than a phase-accurate one.

The rule of thumb is the same statement, done badly. Reading a slope off a plot and multiplying by ninety degrees is the constant-slope case, evaluated by eye. It is exactly right at a frequency where the slope is constant over the several decades the weight reaches, and near a corner — which is where a phase margin is read — it is not.

And it is wrong, silently, for anything that is not minimum-phase. The failure mode is the worst kind: a magnitude plot that looks fine, a phase margin computed from it that looks comfortable, and a loop that oscillates. The right-half-plane zero and the transport delay are the two common causes and neither appears on the magnitude at all.

Both causes are ordinary rather than exotic, and this collection meets each of them in a circuit nobody would flag. The frequency a device sets for itself finds a right-half-plane zero in a common-emitter stage — the plainest amplifier there is — arriving with the two picofarads between collector and base, and notes that the Miller approximation has no room for it at all while predicting 643 kHz against a solved 504. A magnitude sweep of that stage is a complete measurement of its magnitude and an incomplete measurement of the stage.

The staircase in time is the other cause in its purest form: a metre of cable is 4.83 nanoseconds during which the source has no information about what is on the far end. A delay has unit magnitude at every frequency and phase proportional to frequency without limit, so it is invisible to this integral by construction — and it is the reason a loop closed through any length of wire has a phase margin that no magnitude measurement of the loop can report.

The practical rule that follows is narrower than “check whether the network is minimum-phase”, which is not a question a bench can ask. It is this: the class is closed under series and parallel connection of passive elements and under feedback around them, and it is left by exactly three things — a delay, a controlled source with the wrong sign somewhere in its path, and an all-pass section somebody fitted on purpose. A network built only of resistors, capacitors, inductors and couplings is minimum-phase and the integral applies; anything containing a transistor, a length of transmission line or an equaliser needs the phase measured.

What would falsify it #

The check the figure makes is unusually strong for this collection, because it is a comparison between two quantities that share no arithmetic at all. One route solves a matrix at 1,601 frequencies and takes the argument of a complex number. The other takes the magnitudes of those same solves, discards the arguments, and evaluates an integral whose weights come from a series expansion of a hyperbolic cotangent.

If the network were not minimum-phase, they would disagree by the excess phase — which is why the figure draws a non-minimum-phase network beside it and reports 180° of disagreement as a result rather than as a failure. If the sweep were too narrow, they would disagree by the truncated tails, which the slider walks from 7.90° down to nothing. And if the quadrature were wrong, they would disagree by a fixed fraction of a degree that does not move when the network changes — which is what the first version did, at 0.04°, until the weights were integrated rather than sampled.

Three ways to be wrong, three signatures, and all three are on the same axes.

What the relation is worth to the rest of the collection #

A phase that is determined by a magnitude is not a curiosity about one network; it decides what several measurements elsewhere on this site are entitled to claim, and in two cases it explains a result that was otherwise a coincidence.

The straight lines, and where they are not the curve is the closest relative and the sharpest illustration. A two-slope sketch is a statement about a magnitude, and every reader who draws one also draws the phase sketch beside it — two decades wide, 45° at the corner — as though it were a second approximation. It is not a second approximation. Given the magnitude asymptotes, the phase is whatever the integral on this page returns, so a sketch that is 3.0103 decibels wrong at the corner has a phase error that follows from that and cannot be chosen independently. The two sketches are one sketch, and its two errors are one error.

The ideal amplifier, and where it stops being one is where the same identity does real work. An amplifier’s gain–bandwidth roll-off is minimum phase, so the lag it contributes at any frequency is not a second specification to be looked up — it is the magnitude specification restated. That is why the frequency that is not the formula can compute an oscillator’s frequency error from a gain–bandwidth ratio alone, and why the Q the amplifier decides finds a filter section’s quality factor high by exactly the fraction its pole frequency is low: two numbers moving together because there is only one number.

And the exception matters as much as the rule. Flat delay, bought with more delay is built entirely out of the class this integral cannot see — an all-pass section whose magnitude is one to $9\times10^{-16}$ over six decades and whose phase does whatever the design asks. That is the 180° of disagreement in this essay’s own figure, put to use: the only way to change a filter’s delay without touching its magnitude is to leave the minimum-phase class, and everything that can be done inside the class is already decided by the magnitude somebody has already specified.

Which is the useful way to hold the relation. It is not a computational shortcut — a solver has both quantities and needs neither integral — but a statement about how many independent design decisions a network contains. Choosing a magnitude response chooses a phase response, so the two cannot be traded against each other, and a specification that asks for both is asking for one thing twice or for something impossible. The only way to get a third decision is to leave the class, which costs an all-pass section and its delay, and that is a purchase rather than an adjustment.