Four readings of one distance

One solve, read four ways makes one substitution — allow the quantity multiplying each element to be complex, then set $s = j2\pi f$ — and shows that reactance, phase, the corner frequency and the roll-off are four readings of the number that comes out. Among the things it lists as following from the substitution is one it does not then use:

A natural frequency — a mode the circuit rings at with nothing driving it — is a value of s at which…

and then the essay returns to the axis, where the rest of it lives. The plane that sentence opened is where the four readings become one.

The response is a ratio of two distances #

A single-pole low-pass has one pole, at $p = -1/RC$ on the negative real axis, and its response is

$$H(s) = \frac{p}{s - p}$$

which is unity at $s=0$ and has the pole where the denominator vanishes. Take the magnitude of that on the imaginary axis and the two terms become distances in the plane:

$$|H(j\omega)| = \frac{|p|}{|j\omega - p|} = \frac{\text{the pole's distance from the origin}}{\text{its distance from the point on the axis}}$$

Both are lengths. And the phase is minus the angle the separation subtends at the real axis, which is the other thing a vector in a plane has.

Checked against the nodal solve at sixty frequencies over five decades, the two agree to a part in $10^{12}$ in magnitude and in phase. That is not a good agreement; it is the same arithmetic reached twice, and the point of checking it is to confirm that the geometric statement is the algebra rather than a picture of it.

The second column is the whole of the response and the rest of the table is arithmetic on it.

Each of the four readings, as a property of that distance #

The flat passband is the distance not yet changing. The point on the axis is close to the origin, the pole is $|p|$ away along the real axis, and moving the point a little up the imaginary axis changes a right-angled triangle’s hypotenuse by the square of the small leg over the large one. At a tenth of the corner the distance is 1.0050 times its smallest, which is 0.043 decibels — and the passband is flat because a hypotenuse is insensitive to a short perpendicular leg. It is the same reason a measurement at the bottom of a parabola is insensitive to where exactly it was taken.

The corner is where the distance is $\sqrt2$ times its smallest. Which happens when the two legs are equal — when $\omega = |p|$ — and that is the whole derivation of the corner frequency: it is not a definition and not a convention, it is the place where a right-angled triangle is isosceles. The three decibels follow: $20\log_{10}\sqrt2 = 3.0103$.

The roll-off is the distance becoming $\omega$. Once the point is far up the axis the pole’s own offset from the origin is negligible against it, so the distance is $\omega$ itself — 10.050 at ten corners, against $\omega/|p| = 10$ — and $|H| = |p|/\omega$, which is twenty decibels a decade. The asymptote is not an approximation to the response; it is the response with one leg of the triangle dropped.

And the phase is the angle. Forty-five degrees at the corner because the triangle is isosceles there, approaching ninety because the pole’s offset becomes negligible, and 5.71 degrees a decade below because $\arctan(0.1)$ is 5.71 degrees.

So that essay’s four readings are four questions about one right-angled triangle whose legs are $|p|$ and $\omega$. Reactance is the vertical leg, the corner is where the two are equal, the roll-off is the vertical leg winning, and the phase is the angle between them.

The same picture for the other three elements #

The single pole on the negative real axis is one circuit, and that essay’s substitution applies to every element in the collection — so it is worth reading three more off the same plane before leaving it, because each is a sentence a reader already knows arriving as a position.

A capacitor alone has its pole at the origin: $1/sC$ is infinite at $s = 0$. So the distance from a point on the axis to it is $\omega$ itself at every frequency, and the impedance falls at exactly twenty decibels a decade for ever with no corner — which is the ideal capacitor’s whole behaviour, read as a pole that has nowhere to move away from.

A resistor has no pole and no zero at all. Its impedance is a constant, and on the plane that is the absence of any point for a distance to be measured to. Which is the precise sense in which the essay before it can say that a resistance contributes $1/R$ with no $s$ in it: it is the one element with nothing on the plane.

And an inductor has a zero at the origin rather than a pole, so the distance appears in the numerator and the impedance rises at twenty decibels a decade. The symmetry between it and the capacitor, which the essay below states as a fact about admittances and impedances, is on the plane the same point read as a zero rather than as a pole.

The useful consequence is the series pair. A resistor and a capacitor in series have an impedance with a zero at $-1/RC$ and a pole at the origin, so their impedance is a ratio of two distances rather than one — which is why the capacitor that is an inductor can find an impedance that follows $1/2\pi fC$ for four decades and then turns round and climbs: the real capacitor has a third point on the plane, a pair off the real axis at its self-resonance, and above them the nearest point to the axis is a different one.

So the whole of a component’s behaviour is where its points are, and the whole of a response is which point the axis is nearest to. That is not a new fact and it is a compact one, and that essay’s four readings are the special case with exactly one point.

Why the contours are circles #

The faint circles in the figure are contours of constant magnitude, and their shape says something the axis cannot.

A constant $|H|$ means a constant $|s - p|$, and the locus of points at a constant distance from a fixed point is a circle centred on it. So the magnitude of a single-pole response, over the whole plane, is a set of concentric circles round the pole — and the frequency response is the imaginary axis cutting through them.

That gives that essay’s straight-line sketch a geometric reading. The sketch says flat, then twenty decibels a decade, with a corner between. On the plane, the axis starts far from the pole in the horizontal direction only, so the first few circles it crosses are nearly concentric with the origin and it crosses them slowly — the flat part. Then it passes the pole’s own latitude, after which every further step up the axis is a step directly away from the pole and the circles are crossed at a constant rate in log-log — the slope. The corner is where the axis stops moving around the pole and starts moving away from it.

Which also says why the sketch is exactly 3.0103 decibels wrong at the corner and nowhere worse. The straight lines, and where they are not the curve measures that: 3.0103 at a single real pole, at the corner and nowhere worse, with the error the same a factor above as the same factor below. On the plane that symmetry is the statement that the hypotenuse $\sqrt{|p|^2 + \omega^2}$ is symmetric in its two legs — swap them and the distance is unchanged — so the error a decade below the corner and a decade above it are the same number, and the sketch’s two straight lines are the two legs taken one at a time.

What a second pole does to the picture #

The single pole is the case where the geometry is simplest, and it is worth saying what survives when there are more, because that is where the plane earns its keep.

With several poles the magnitude is the product of the distances from the origin divided by the product of the distances from the point on the axis, and the phase is minus the sum of the angles. So everything above generalises by multiplication: the response is a product over poles, and each pole contributes its own distance and its own angle.

Which makes two things visible that the axis makes hard.

A pole pair near the axis produces a peak, because as the point on the axis passes the pair’s imaginary part, one of the two distances becomes small — smaller than $|p|$ — and the ratio exceeds one. How small it gets is how close the pair is to the axis, which is the damping. Where the behaviour is written down is where that is measured as a response, and on the plane it is a distance passing through a minimum.

And a zero is the same construction with the distance in the numerator. A transmission zero at a point on the axis means the distance from the axis to it goes to zero there, and the response with it — which is why the notch essays’ nulls are bottomless in principle and why what fills them is anything that moves the zero off the axis.

So the plane’s real value is not the single-pole case, where the arithmetic is simple anyway. It is that every response in this collection is a ratio of products of distances, and a great many questions about one — where it peaks, how sharply, what a tolerance does to it — are questions about how far some point is from some other point.

What the picture is not #

It is not a new solve. Everything on the plane comes from the same matrix the essay before it inverts, with $s$ allowed off the axis. The site’s check is that the two agree to a part in $10^{12}$, and the reason to do the check is that a geometric statement is easy to get slightly wrong in a way that still looks right.

It is not a claim that the pole is a physical thing. The pole is where the network’s matrix is singular — a value of $s$ at which a non-zero solution exists with nothing driving it, which is the natural frequency the essay before it names. That it is at $-1/RC$ for this circuit is a fact about this circuit, and drawing it as a point does not make it a location.

And the response off the axis is not a response to anything. $H(\sigma + j\omega)$ for $\sigma \ne 0$ is the response to a growing or decaying exponential, which is a mathematical object rather than a signal a instrument produces. The circles are contours of a function that is only measured along one line.

That last one is worth insisting on, because it is where the picture misleads. A reader who sees the magnitude as a surface over the plane may conclude that the parts of it away from the axis are physically meaningful in the way the axis is. They are meaningful as the thing that determines the axis: the response everywhere is fixed by the pole, and the pole is one point, so the axis carries everything. What the plane adds is not information — it is the shape of the information the axis already had.

What the plane is for, which is not this circuit #

A single pole needs no picture. The algebra is one line and the response is a shape every reader already carries, so a reader entitled to ask what the plane has bought deserves an answer, and it is not about this circuit at all.

It is that the plane is where the questions these essays keep asking have short answers, and three of them are worth naming because each takes a page elsewhere and a sentence here.

Where does a response peak, and how sharply. On the axis that is a search: sweep, find a maximum, bisect. On the plane it is the point on the axis nearest to a pole pair, and the peak’s height is the ratio of the pair’s distance from the origin to its distance from the axis. Which is $Q$, so resonance, and the bandwidth it sets exactly’s $f_0/Q$ half-power width is the statement that a circle of a given radius round the pair cuts the axis over that width.

What does a tolerance do. On the axis that is a derivative of a transfer function with respect to a component value, which is the arithmetic the tolerance that is not on any part uses. On the plane it is a displacement of a point, and the response’s sensitivity is how much the distance changes — which is largest when the displacement is along the separation and zero when it is perpendicular to it. That is the geometric reading of a stationary point, and it is why some errors cost first order and some second.

And is it stable. On the axis, nothing: a Nyquist plot is needed, and these essays have several essays about reading one. On the plane it is whether any pole is to the right of the imaginary axis — one inequality, with no sweep, no contour and no encirclement count. The whole of stability is a sign.

None of those is available from a magnitude plot, and each of them is one sentence once the points are drawn. That is what the plane is for, and the single-pole case in this essay is the one where it can be checked against arithmetic a reader can do by hand — which is the reason to start with it and not the reason to draw it.

Still open: the second pole drawn, and the sensitivity as a displacement #

The pole pair on the same axes. Everything above about peaks and damping is stated and not drawn. One figure with a complex pair, its two distances and their product would make the resonance essays’ quality factor a geometric quantity — the ratio of a pair’s distance from the axis to its distance from the origin — and that is a definition of $Q$ that no algebra makes as obvious.

A tolerance as a displacement. A component’s tolerance moves a pole, and what that does to the response is the change in a distance. So the sensitivity results is computed here by differentiating a transfer function are, on the plane, a question about how far a point moves and in which direction — and the tolerance that is not on any part would read as a displacement rather than as an exponent. Whether the geometric reading makes the second-order behaviour of a doubly terminated ladder of inductors and capacitors obvious is worth finding out.

And the settling time, which is not a distance. A pole’s real part is a decay rate and its imaginary part a ringing frequency, so a step response is read off the plane by projecting rather than by measuring distances. That is a different geometry on the same picture and this figure does not attempt it — the transients field’s own pole-plane work is where it lives, and the two readings of one diagram have never been drawn together.

What is checked #

The magnitude is required to be the ratio of the two distances, against the nodal solve, at sixty frequencies over five decades and to a part in $10^{12}$. That tolerance is the arithmetic’s floor rather than a bound, because the claim is an identity.

The phase is required to be minus the angle, to the same precision, because the geometric statement has two halves and requiring only the magnitude would leave the picture half-checked.

The corner is stated as a statement about the distance — that it is $\sqrt2$ times the smallest — and then the 3.0103 decibels is stated as following from it. The order matters: the decibels are the consequence and the distance is the claim.

And both asymptotes are stated as limits of the distance: that a thousandth of the corner gives the pole’s own distance, and that a thousand times it gives $\omega$ itself. The straight-line sketch the essay below measures against is those two limits, so requiring them here is requiring that the sketch is the geometry with one leg dropped.