The reading that does not care which way round it is

Take a network with two accessible pairs of terminals. Push a current into the first pair and measure the voltage that appears across the second. Now do it the other way round: push the same current into the second pair and measure the voltage across the first.

The claim is that the two measurements give the same number. Not similar, not close enough for practical purposes — the same number.

That is reciprocity, and it is stated in most textbooks as a theorem about networks of resistors, capacitors and inductors. It is worth stating instead as a fact about a matrix, because doing so says exactly what it depends on, exactly which components have it, and exactly what has to be added to take it away.

Why it is a fact about the matrix #

Modified nodal analysis assembles a network into a matrix equation. Each passive element contributes a small pattern of entries: a conductance $G$ between nodes $a$ and $b$ adds $+G$ at the two diagonal positions and $-G$ at the two off-diagonal ones, and a capacitance or an inductance does the same thing with a frequency-dependent value in place of $G$.

Every one of those patterns is symmetric. The entry the element puts at row $a$, column $b$ is the entry it puts at row $b$, column $a$. That is not a convention: it is the statement that the element’s current depends on the voltage across itself and on nothing else, which is what it means for a component to be a two-terminal component at all.

A sum of symmetric matrices is symmetric, and the inverse of a symmetric matrix is symmetric. The transfer impedance from port one to port two is one entry of that inverse; the transfer impedance the other way is the entry reflected across the diagonal. They are therefore the same entry, and the equality is exact in the same sense that a matrix equals its own transpose.

So the reason reciprocity holds is not that the components are passive, or linear, or lossless. It is that each of them minds its own business.

The measurement, and why it is a current source #

The figure injects a current rather than applying a voltage, and the choice is the whole measurement rather than a detail of it.

A first version drove one port with an ideal voltage source and shorted the other, then moved the source and the short. That is two different netlists: moving the short moves an element from one node to another, so what is being compared is two circuits rather than one circuit read two ways. The two answers came out $1.5\times10^{-6}$ apart, which is a true statement about two networks and no statement at all about the law being tested.

An ideal current source contributes nothing to the matrix. It appears only in the right-hand side vector. Injecting at the other port is therefore the same matrix with a different source vector, and the comparison is between two entries of one inverse — which is precisely what the theorem is about.

That distinction is the reason this essay has a figure rather than a paragraph. It is not visible in any statement of the theorem, and getting it wrong produces a number that looks like a small violation of a law of physics.

What the two readings actually are #

At ten kilohertz the ladder drawn above gives $5.965026\times10^{2}$ volts per ampere in one direction and $5.965026\times10^{2}$ volts per ampere in the other, differing in the sixteenth significant figure. Across four decades of frequency the worst difference is $4.9\times10^{-14}$ of the reading.

That is not a measurement of the network. It is a measurement of double-precision arithmetic: a matrix inversion of this size accumulates rounding at about that level, and the two entries being compared are computed by different eliminations of the same matrix. The network’s contribution to the difference is zero.

This is the first quantity on this site whose interesting property is that it has no magnitude. Every other figure in the collection carries a number at which a model stops being true; here the model is exact and the number to carry is the size of the arithmetic underneath it.

The element that looks as though it should have a direction #

A coupled pair of windings has a primary and a secondary. It is drawn with a dot on one side. It is described in terms of what the first winding does to the second. Everything about the way it is presented suggests an arrow.

It is reciprocal, and it is reciprocal to the last bit. Injecting into a one-millihenry winding coupled at $k = 0.7$ to a four-millihenry one, and then injecting into the four and reading the one, returns the same double — not a small difference, but bit-for-bit the identical floating-point number, at every frequency tried.

The reason is in the stamp again. A coupling adds an off-diagonal term $sM$ relating the two inductors’ currents, and it adds the same term in both places, because the flux linkage is mutual in the literal sense: the fraction of winding one’s flux that reaches winding two is the fraction of winding two’s flux that reaches winding one. That is Neumann’s formula, and it is symmetric in the two circuits by inspection.

Which is why a transformer works both ways round. Step-up and step-down are the same component used in two directions, and a designer’s decision about which side is the input is not a decision about the part.

What takes it away, and how little of it is needed #

Add one controlled source — a transconductance $g_m$ whose current depends on a voltage somewhere else — and the matrix acquires an entry at one off-diagonal position and nothing at its mirror. The matrix is no longer symmetric, and the two readings part.

The interesting part is the shape of the departure. It is exactly proportional to the transconductance, with a fitted exponent of 1.000 over four decades:

Read down that column and there is no threshold anywhere on it. There is no small amount of gain that a network can carry and remain reciprocal to some tolerance; there is a constant, 62.8 per siemens for this network at this frequency, and the departure is that constant times the transconductance all the way down.

Which gives the number this essay’s boundary is: the arithmetic’s floor is $4.9\times10^{-14}$, so the two readings differ by more than the arithmetic can resolve at a transconductance of 0.78 femtosiemens. A transistor biased at a microamp has a transconductance of forty microsiemens, which is eleven orders of magnitude above that.

So reciprocity is not approximately true of an amplifier. It is absent from it, by a factor with eleven zeros in it, and the same fact is the reason an amplifier has an input and an output while a filter has two ends.

The two theorems that are not this one #

Superposition and reciprocity are both consequences of linearity and they are not the same statement, which is worth separating because they are often introduced on the same page.

Superposition says that the response to two sources is the sum of the responses to each. It needs linearity and nothing else — a network full of controlled sources obeys it exactly. The superposition essay measures it at $10^{-14}$ on a network with two sources at arbitrary relative phase, and finds it holds while the power superposition fails by a factor that has a closed form.

Reciprocity says that swapping the source and the detector changes nothing. It needs symmetry, which is a stronger and quite different condition. Every network in this collection obeys superposition; the ones with transistors in them do not obey reciprocity.

The distinction has a practical form. A linear circuit may be analysed one source at a time. A reciprocal one may additionally have either end called the input — and a circuit with gain in it may not.

Where it is used, which is more places than it is stated #

Reciprocity is one of those results that is invoked far more often than it is named.

A four-terminal measurement can be read either way. A Wheatstone bridge with a source across one diagonal and a detector across the other gives the same reading with the two exchanged — so which diagonal is which is a matter of convenience, and the same bridge can be used with whichever of the two available instruments is the better source.

An antenna’s receiving pattern is its transmitting pattern. That is reciprocity applied to a pair of ports separated by a field rather than by a ladder, and it is the reason a single measurement characterises both.

A passive filter has no preferred direction. Turning a filter round does not change its transfer function, which is why a symmetrical layout is a matter of manufacture rather than of response.

A crosstalk measurement is symmetric. The voltage that track A puts on track B, per unit of drive, is the voltage B puts on A. Both couplings between two passive tracks are mutual terms, so measuring one direction measures both.

The circuits in this collection that do not have it #

It is worth naming them, because the list is short and every entry on it is a place where an engineer’s intuition about “which end is the input” is doing real work.

Every transistor stage. The transconductance is the whole of the gain, and it is the whole of the non-reciprocity. A common-emitter stage’s collector-base capacitance, by contrast, is reciprocal — it is a two-terminal component — which is exactly why it provides a reverse path and why the Miller effect exists.

Every operational amplifier circuit. The nullor this site uses for an ideal amplifier is the extreme case: it constrains its input pair to zero volts and zero current while sourcing whatever the output needs, which is as far from a symmetric stamp as an element can be.

Every current mirror. The reference side sets the output side and not the reverse, and the mirror’s two errors — the base currents and the Early effect — are both measurements of how incompletely it manages even that.

What this figure would look like if it were wrong #

The habit this collection runs on is that an assertion which has never rejected anything proves nothing, so it is worth saying what would make this one fail.

If the current source were not ideal — if it had a finite output conductance stamped between its own terminals — that conductance would be a symmetric element and reciprocity would survive it. If the detector loaded the network, the same. Neither of the two things that would look like experimental error in a laboratory can produce an asymmetry here.

What does produce one is any element whose current depends on a voltage that is not across itself. There are exactly four such elements in this site’s library — the voltage-controlled voltage source, the voltage-controlled current source, the nullor, and any of them in disguise — and the figure’s slider walks the second of them from zero to ten millisiemens so that the failure can be watched arriving rather than described.

What is left #

The measurement here is the cleanest kind this site can make: two numbers that agree to the last bits a double carries, with no tolerance chosen anywhere and nothing to argue about.

What it buys is a licence. Inside a passive network, the words “input” and “output” are labels applied by whoever drew the schematic, and a great many analyses may be done in whichever direction is more convenient. Outside one — and any circuit with a transistor in it is outside one — that licence is withdrawn completely, at a transconductance eleven orders of magnitude below anything a device actually has.

The boundary is not a frequency, an amplitude or a size, which are the three this collection’s rule names. It is a transconductance, and its value is 0.78 femtosiemens.

What the licence is used for #

Reciprocity is the kind of property that is easy to admire and hard to point at, so it is worth naming the three places in this collection where a result depends on it.

One number from two measurements is the sharpest, because it is a check that exists only because reciprocity does. Neither of this site’s two standing verifications can see a mutual inductance — a coupling adds no current anywhere, so the current law is unmoved by it, and a coupled pair dissipates nothing, so the energy balance is unmoved too — and a coupling stamped into the wrong row would produce a well-formed solution to a different circuit with both checks passing. The third route is the bench method: connect the windings in series one way, then the other, and the difference is four times the mutual inductance. That works because the two connections are the same network read in two directions.

Two ports from two one-ports is where the property is measured rather than used: the four impedance parameters of a coupled pair are recovered by four solves, each port driven with the other open, and two of the four come out equal to $2.8\times10^{-16}$ — which is reciprocity, and is the first property a coupling stamped into one row instead of two would break.

And every derivative, and the one that is zero is the one that could not exist without it. Transposing the matrix and solving once more returns the derivative of a response with respect to every element at once, exactly — and the transpose is the adjoint network, which is the same network with its sources and readings exchanged. For a reciprocal network the adjoint is the network itself, which is why one extra solve answers a question that would otherwise take one solve per component.

That is the licence spent rather than described: a measurement, a check and a derivative, each of them available only because the words “input” and “output” are labels.

Where it is withdrawn, and what replaces it #

Half the circuits in this collection contain a controlled source, so it is worth saying what is lost and what is gained where the property does not hold.

The clearest case is what gets through from the rail, where two circuits with identical loop gain, identical crossover and phase margins agreeing to a millionth of a degree differ by sixty decibels in one direction and not in the other — because the disturbance travels through a forward path that has no reverse. That asymmetry is not a defect; it is the entire value of an active device, and the sixty decibels is the pass transistor’s own intrinsic gain.

The device that never sees the swing is the case where directionality is bought deliberately: a cascode’s whole function is to stop a signal at the output from reaching the input, and what it buys is a bandwidth fourteen times better at a cost of two volts of a five-volt supply. A reciprocal network cannot do that, which is why every amplifier is built from things that are not reciprocal.

And the frequency a device sets for itself is what happens when a small reciprocal element is left inside a non-reciprocal one: two picofarads between collector and base is a perfectly reciprocal capacitor, and because the stage around it is not, that capacitance is drawn into the input multiplied by one plus the gain — with the Miller approximation predicting 643 kHz against a solved 504 and having no room for the right-half-plane zero the network also has.

The trade is the same one every time. Reciprocity buys checks, derivatives and the freedom to analyse in whichever direction is convenient; giving it up buys gain and isolation. This collection’s solver supports both, and the femtosiemens measured here is the line between them.