Circuit Response
every model has an edge, and the figure carries it
Networks, and how a solve is checked

The readings that come back the same

Reciprocity is usually stated as 'swap the source and the meter and the reading is unchanged', and that is true of some readings and false of others. On one passive ladder network the transimpedance, the transadmittance, and the reverse voltage gain against the forward current gain each come back the same to 10⁻¹³, while the voltage gain forwards and backwards differ by 13.1 times at 82.5 kHz — exactly z₂₂/z₁₁. A 600-to-50-ohm pad has voltage gains of 0.0426 and 0.511 and one power gain both ways. A controlled source breaks all three reciprocal pairs by the same factor.
15 min read 5 figures Two routes to a numberMeasured, not tabulatedThe schematic is a label

Assumes: The reading that does not care which way round it is · What a network answers, and how the answer is checked

The reading that does not care which way round it is injected a current at one end of a passive network and read the voltage at the other, then swapped them, and got the same number to five parts in 10¹⁴. The reading that does care which way round put a controlled source into the same ladder network and found the two readings parting by gmg_m times the impedance of the one branch between the source’s control node and its output.

Both measured one particular reading: a current in, a voltage out. That choice was not arbitrary — it is the reading the textbook statement of reciprocity is usually written for — but it hides something that matters as soon as the statement is used. A network can be read in four natural ways, depending on whether the drive is a voltage or a current and whether the far port is left open or shorted. Reciprocity does not say that all four come back the same. It says exactly which ones do, and the answer includes a pair that looks, at first, like a category mistake: a voltage ratio one way equal to a current ratio the other.

This essay measures all of them on the same ladder network, takes each reading by building the netlist it needs rather than by converting one parameter set into another, and then shows what the distinction means for a real component.

Five readings of one passive ladder network, each taken forwards and backwards. computed by solving, not by drawing; every reading built as its own netlist, never converted from another. Transimpedance (current in, open-circuit voltage out), transadmittance (voltage in, short-circuit current out) and the reverse voltage gain against minus the forward current gain all give a ratio of one, to 1.0e-13. The voltage gain forwards against backwards does not: 1.000 at 100 Hz, 2.129 at 1 MHz. Nor does the current gain.
Fig. 1 Five readings of one passive ladder network, each taken forwards and backwards, as a ratio. Transimpedance — current in, open-circuit voltage out — transadmittance — voltage in, short-circuit current out — and the reverse voltage gain against minus the forward current gain all give a ratio of one, to 10⁻¹³. The voltage gain forwards against backwards does not: 1.000 at 100 Hz and 2.129 at 1 MHz. Nor does the current gain.

The ladder network, and how each reading is taken

The network is the one both earlier measurements used: 470 ohms, a millihenry and 22 nanofarads to ground, then a kilohm and 4.7 nanofarads to ground — a two-section low-pass with port 1 at its input and port 2 at its output. It is solved by nodal analysis at each frequency, and every number here is a node voltage or a source current the solver returned.

The discipline that matters is that no reading is derived from another. A transimpedance is measured by connecting a one-ampere source to one port, leaving the other open, and reading the open port’s voltage. A transadmittance is measured by connecting a one-volt source to one port, replacing the other port with a zero-volt source — a short circuit that reports its own current — and reading that current. A voltage gain is a volt in and an open-circuit voltage out; a current gain is an ampere in and a short-circuit current out. Each direction of each reading is its own netlist, eight of them at every frequency.

That matters because the textbook relations between parameter sets — z to y by inversion, z to h by a few lines of algebra — would make any identity among them a consequence of arithmetic rather than a measurement. Here, if two readings agree, it is because two different circuits produced the same number. What a network answers makes the same distinction for a single solve: an answer is evidence only when it was not computed from the thing it is being compared with.

Three pairs that are reciprocal

The transimpedance comes back the same, as the earlier measurement found: an ampere into port 1 produces the same open-circuit voltage at port 2 as an ampere into port 2 produces at port 1. The transadmittance does as well: a volt at port 1 drives the same short-circuit current at port 2 as a volt at port 2 drives at port 1. Both are the same statement — the nodal matrix is symmetric — read with a different kind of source.

The third pair is the surprising one, and it gets its own figure.

A voltage ratio one way and a current ratio the other: the reverse voltage gain against the forward current gain. computed by solving, not by drawing. Dots: drive port 2 with a volt and read the open-circuit voltage at port 1. Line: inject an ampere at port 1, short port 2 and read the current in the short, with its sign reversed. They are one complex number at every frequency, to 5.6e-14 — a voltage gain equal to a current gain, 0.9999 at 100 Hz and 3.922e-5 at 1 MHz.
Fig. 2 A voltage ratio one way and a current ratio the other. Dots: drive port 2 with a volt and read the open-circuit voltage at port 1. Line: inject an ampere at port 1, short port 2 and read the current in the short, with its sign reversed. They are one complex number at every frequency, to 5.6 × 10⁻¹⁴ — 0.9999 at 100 Hz and 3.922 × 10⁻⁵ at 1 MHz.

Drive port 2 with a voltage and read the voltage that appears at an open port 1: that is the reverse voltage gain. Now inject a current into port 1 and read the current that flows in a short across port 2: that is the forward current gain. These are different kinds of quantity measured on different circuits, and they are equal in magnitude and opposite in sign at every frequency, to 5.6 parts in 10¹⁴.

In the language of two-port parameters this is h12=−h21h_{12} = -h_{21}, the hybrid form of reciprocity, and it follows from the symmetric matrix by the same argument as the other two. But it is worth seeing as a measurement because it is the form that is most useful and least remembered. It says that a passive network which attenuates a voltage travelling backwards by some factor must attenuate a current travelling forwards by exactly the same factor. A passive network cannot be good at passing current one way and bad at passing voltage the other; the two are the same property.

Two that are not

The voltage gain is not reciprocal, and neither is the current gain. At 100 hertz the two voltage gains agree, because the ladder network’s capacitors carry almost no current and an open port sees the whole drive in either direction. As the frequency rises they part.

The voltage gain of one ladder network forwards and backwards, and the ratio of its two port impedances. computed by solving, not by drawing. Forwards (port 1 driven, port 2 open) and backwards agree at low frequency, where no current flows, and part as the frequency rises: at 82.5 kHz they are 2.644e+0 and 2.020e-1, a ratio of 13.089. Their ratio (dots) is z22/z11, the open-circuit impedance at port 2 over that at port 1, read from two more netlists — to ten figures at every frequency. A voltage gain carries the impedance it is measured into, and the two ports have different ones.
Fig. 3 The voltage gain forwards (port 1 driven, port 2 open) and backwards, and the ratio of the ladder network’s two port impedances. At 82.5 kHz the gains are 2.644 and 0.2020, a ratio of 13.089. That ratio (dots) is z22/z11z_{22}/z_{11} — the open-circuit impedance at port 2 over that at port 1, read from two more netlists — to ten figures at every frequency.

At 82.5 kilohertz the forward voltage gain is 2.644 — the first section resonates and lifts the voltage — while the reverse gain is 0.202: a ratio of 13.1. The ratio is not arbitrary. Measured against the two ports’ own open-circuit impedances, read from two further netlists, it is z22/z11z_{22}/z_{11} to ten figures at every frequency. The voltage gain forward is z21/z11z_{21}/z_{11} and the reverse is z12/z22z_{12}/z_{22}; the transimpedances are equal, so what is left in the ratio is the ratio of the impedances the drive sees at each end.

That is the general shape of a non-reciprocal reading in a reciprocal network. A voltage gain is a transfer quantity divided by an impedance, and the two ends of a network present different impedances. The transfer part is the same both ways; the normalisation is not. So “swap the source and the meter” holds only when the source and the meter are of the right kinds — a current source with a voltmeter, a voltage source with an ammeter, or the crossed pair — and fails, by a ratio of port impedances, when both are voltages or both are currents.

A pad, read the way it is used

A two-port read in isolation, with open and shorted ports, is a laboratory exercise. In use, each port sees a source or a load of some definite impedance, and the reading of interest is usually a voltage gain or a power gain. A resistive pad between two different impedances shows what reciprocity says about those.

A minimum-loss pad from 600 Ω to 50 Ω, read in both directions. computed by solving, not by drawing, at DC: 574.5 Ω in series on the 600 Ω side and 52.2 Ω in shunt on the 50 Ω side, each direction driven from and loaded by the impedance the pad is designed for. The voltage gain is 0.0426 one way and 0.5109 the other. The power gain is 0.02175 both ways, a loss of 16.63 dB. Read with the 50 Ω side as port 1, the reverse open-circuit voltage gain and the forward short-circuit current gain are both 0.08333.
Fig. 4 A minimum-loss pad from 600 Ω to 50 Ω — 574.5 Ω in series on the 600 Ω side and 52.2 Ω in shunt on the 50 Ω side — each direction driven from and loaded by the impedance the pad is designed for. The voltage gain is 0.0426 one way and 0.5109 the other. The power gain is 0.02175 both ways, a loss of 16.63 dB. With the 50 Ω side as port 1, the reverse open-circuit voltage gain and the forward short-circuit current gain are both 0.08333.

The pad is the minimum-loss match between 600 ohms and 50, the kind used to join two pieces of equipment with different characteristic impedances. Driven from 600 ohms into 50, it delivers 4.26 per cent of the voltage; driven from 50 ohms into 600, it delivers 51.1 per cent. A voltmeter would report the pad as twelve times more transparent in one direction than the other.

The power gain is 0.02175 both ways — a loss of 16.63 decibels, which is the known minimum loss for that impedance ratio — and it is the same in both directions to nine figures. The power gain of a terminated passive network between matched terminations is a reciprocal quantity, and the voltage gain is not, for exactly the reason the ladder network showed: the two ports carry different impedances and a voltage ratio carries them with it. Power has them divided out. This is why an attenuator is specified in decibels of loss and not as a voltage ratio, and why that number does not have a direction.

The hybrid pair is the third reading, and it lands on a number with a meaning: 0.08333 is exactly 50/600. Driving the 600-ohm side and reading the open 50-ohm side, the pad divides the voltage by twelve, and injecting into the 50-ohm side with the 600-ohm side shorted, a twelfth of the current reaches the short. The noise a pad adds is its temperature measured what a pad like this does to noise; what it does to signal is fixed in both directions by the one symmetric matrix underneath.

What a controlled source does to all three

The earlier measurement of a controlled source’s effect read one pair. If reciprocity is one property showing up in three pairs, a controlled source should break all three by the same amount.

With a controlled source in the ladder network: the departure from reciprocity, read three ways, at 10 kHz. computed by solving, not by drawing. The forward-over-reverse ratio of the transimpedance (line), of the transadmittance (dots) and of the reverse-voltage-gain-against-forward-current-gain pair (rings) departs from one by the same amount at every transconductance, to six figures — gm times 2πfL = 62.832 Ω, the one inductor between the source's control node and its output. Reciprocity fails as one number, whichever pair of readings is used to see it.
Fig. 5 With a transconductance from the ladder network’s first node to its second, at 10 kHz: the forward-over-reverse ratio of the transimpedance (line), of the transadmittance (dots) and of the crossed voltage-and-current pair (rings) departs from one by the same amount at every transconductance, to six figures — gm times 2πfL = 62.832 Ω, the one inductor between the source’s control node and its output.

It does, to six figures at every transconductance from a nanosiemens to ten millisiemens. The three departures are equal to each other and to gm⋅2πfLg_m \cdot 2\pi fL — the 62.832 ohms of the inductor between the source’s control node and its output that the reading that does care which way round identified. That is no coincidence of this network. Each of the three pairs is a ratio of the matrix’s two off-diagonal entries for the two ports, divided by the same determinant or the same port impedance in both directions, so whatever unbalances those two entries unbalances all three ratios identically.

So reciprocity is not three facts that happen to coincide. It is one fact about one pair of matrix entries, visible through three different instruments, and it fails as one fact. A measurement that finds the transadmittances unequal has already found the transimpedances unequal by the same factor, and does not need to make the second measurement.

Why the matrix is symmetric, one element at a time

All three reciprocal pairs come from one fact, so it is worth seeing where the fact comes from. Nodal analysis builds the matrix by adding each element’s stamp, and an element’s stamp is symmetric when the current it carries between two nodes depends only on the voltage between those same two nodes. A resistor, a capacitor and an inductor each relate their own current to their own voltage, so each adds the same entry above and below the diagonal, and a sum of symmetric stamps is symmetric.

A pair of coupled windings looks as though it should break this, since it has a primary and a secondary and couples one branch to another. It does not, because the mutual inductance appears as the same entry in both directions — the voltage induced in the secondary by the primary’s current has the same coefficient as the voltage induced in the primary by the secondary’s. That symmetry is what lets one number from two measurements extract a single coupling coefficient from a series-aiding and a series-opposing reading, and it is why a transformer has the same turns ratio read from either side once the band a turns ratio holds over is respected.

A controlled source is the element whose current between two nodes depends on the voltage between two other nodes, so it stamps an entry in one off-diagonal position and nothing in the transposed one. That single asymmetric entry is the whole of the departure measured above, which is why the departure is proportional to the transconductance and to the impedance that connects the control node to the output.

The symmetry is also what makes a matched load a two-way statement. The load that takes the most found the load that draws the most power from a source, and a reciprocal network between them transfers that power with the same efficiency in either direction, which is why a matching network designed to present 50 ohms to an antenna on transmit presents the antenna’s impedance back to the receiver on receive. The match with no knob is the same symmetry used in the other order.

Where the distinction is spent

The practical content of all this is a rule about what may be swapped. A designer measuring a filter with a signal generator and an oscilloscope — a voltage source and a voltmeter — cannot assume the response is the same in reverse, because that is a voltage gain and the two ports have different impedances. The same designer measuring with a network analyser between matched terminations can assume it, because that is a power gain into equal references. The scattering parameters a network analyser reports, S21S_{21} and S12S_{12}, are equal for a passive reciprocal network for the same reason the pad’s power gains were: they are normalised to the same reference impedance at both ports.

It also says when reciprocity can be used as a shortcut in a measurement. If a transfer is hard to measure in one direction — the drive point is inaccessible, or the sensitive instrument only fits at one end — the reverse measurement gives the same number, provided the source and meter are swapped with their kinds. Inject current where the voltmeter was and read voltage where the current source was. Swapping a voltage source for a voltmeter is the mistake, and the ladder network above shows by how much it is a mistake: z22/z11z_{22}/z_{11}, thirteen times at one frequency on a two-section filter.

There is a quieter consequence for anyone checking a solver. A test that compares a network’s forward and reverse voltage gains will report a “failure” on a perfectly reciprocal network. The earlier measurement recorded exactly this trap in its own instrument: a voltage source at one port and a short at the other is two different networks, not one network read twice, and they disagreed by a part in a million for a reason that had nothing to do with reciprocity. The identities that hold to the last bit are the ones where the two directions share one matrix and differ only in where the source vector is non-zero.

What the measurement leaves out

Every network here is linear and solved in the sinusoidal steady state. Reciprocity in this form is a statement about linear networks; a diode or a saturating core has no symmetric matrix to speak of, and its small-signal version is reciprocal only if every controlled source in the linearised model happens to cancel, which in a device it does not.

The ports are ideal. A real port has a reference node that is not quite the ground of the other port, and a coupling between the two returns that the netlist does not hold — the common impedance the node that is not in the circuit is about. A two-port measured with a shared return that carries signal current is not the two-port the matrix describes, and can appear non-reciprocal for that reason alone.

Magnetic materials with a bias field — a ferrite in a static magnetic field, which is how circulators and isolators are made — are linear and passive and genuinely non-reciprocal, because their permeability is not a symmetric tensor. No lumped element in this netlist language has that property except a gyrator, and the reading that does care which way round measured the gyrator separately.

The number to keep

A passive network’s reciprocity is a statement about three pairs of readings: transimpedance, transadmittance, and the crossed pair in which a reverse voltage gain equals minus a forward current gain — measured equal here to 10⁻¹³ as separate netlists. A voltage gain and a current gain are not reciprocal; the forward and reverse voltage gains differ by exactly z22/z11z_{22}/z_{11}, 13.1 times at 82.5 kilohertz on this ladder network. Between matched terminations a pad’s power gain is the same both ways, 16.63 decibels of loss, while its voltage gains differ twelvefold. A controlled source breaks all three reciprocal pairs by the same factor, gmg_m times one branch impedance.

Still open: noise read backwards, the scattering parameters, and the bilateral amplifier

Noise, by the reverse transfer. If the transfer from a current injected across any element to the output voltage equals the transfer from a current injected at the output to the voltage across that element, then one solve from the output gives every resistor’s contribution to the output noise at once. Whether the sum is then 4kT times the real part of the output impedance — and what a controlled source does to that — is the natural next measurement.

Scattering parameters. S21=S12S_{21} = S_{12} is the form reciprocity takes on a network analyser. Measuring it directly, with unequal reference impedances at the two ports, would show how the normalisation decides which form of the identity holds.

An amplifier with feedback. An amplifier is not reciprocal, but feedback around it makes its port impedances something the designer chooses. A resistance made that way presents the same impedance as a resistor and need not make the same noise, which is a property only a non-reciprocal network can have.

Part 3 on reciprocity

One argument about Reciprocity, and one of 4 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

The ratio of the two readings, drawn where it lives. computed by solving, not by drawing. The same five-element RLC ladder at 10 kHz, read from each end, with the quotient of the two readings plotted in the complex plane. A reciprocal network sits at the point 1. One transconductance moves it along a straight ray, 1 − gm·Z, whose direction is the phase of the single branch between the source's control node and its output node and whose length is gm|Z| — 62.83 Ω for the 1 mH inductor, 120 Ω for a resistor, 72.34 Ω for 220 nF, all at 10 kHz. A mirrored pair of transconductances stays at 1 to 2.3e-15; reversing one of them runs the ratio around the unit circle to 8.9e-16, where the two readings are the same size and differ only in phase. The straight-ray law needs the two nodes joined by exactly one branch, and a second path between them takes it away by a factor rather than by a percentage. The reading that does care which way round Part 2 — An earlier essay measured a network's departure from reciprocity and left its size as a constant — 62.8 per siemens, for that network at that frequency. It is not a constant and it is not the network's: it is 2πfL for the single inductor between the controlled source's control node and its output node, 62.832 ohms at ten kilohertz, and the ratio of the two readings is 1 − gm·Z to 8.8 parts in 10¹⁴ over ninety-nine readings. So 4.5455 millisiemens across a 220 ohm branch makes a ladder that transmits a hard zero forwards at every frequency at once, and 206.13 ohms back at ten kilohertz. A terminated ladder network's output noise three ways: one solve per resistor, one solve from the output, and 4kT·Re{Z}. computed by solving, not by drawing, at 290 K. The line is 4kT·Re{Zout}. The dots sum each resistor's 4kT/R through its own transfer to the output, one solve per resistor. The rings get the same transfers from a single solve with an ampere injected at the output — reciprocity — and agree to 4.4e-16; the sum agrees with Nyquist's expression to 4.4e-16. 4.60 nV/√Hz at 100 Hz, 3.82 at 1 MHz. The resistance colder than the room Part 4 — In a passive network the noise at a port is 4kT times the real part of its impedance, and reciprocity is why: the transfer from each resistor to the port equals the transfer from the port to that resistor, so one solve from the output gives every resistor's share and the shares add to Nyquist's expression, here to 4.4 × 10⁻¹⁶. With gain in the network both fail. A 50-ohm termination made by feedback around an amplifier of gain A presents 50.000 Ω and has the noise of a 50-ohm resistor at 290/(1 + A) kelvin — 2.90 K at a gain of 99 — but only where the amplifier still has its gain.

Where this goes next

A passive network, read from each end — and the two readings are one number. computed by solving, not by drawing. A five-element RLC ladder with a current injected at one port and the voltage read at the other, then the two exchanged. With no controlled source the two readings agree to 4.9e-14 of themselves over four decades, which is the arithmetic's noise rather than a physical difference — the network cannot tell which way round it is being used. A mutual inductance keeps that: a 1 mH and a 4 mH winding at k = 0.7 give 0.0e+0, because the coupling puts the same entry in both halves of the matrix. A transconductance does not, and the departure is proportional to it with a fitted exponent of 1.000 over four decades — so there is no small amount of gain that is harmless. It passes the arithmetic's own floor at 0.781 femtosiemens, and the smallest transistor in this collection is nine orders above that. The reading that does not care which way round it is Inject a current at one end of a network and read the voltage at the other; then swap the two. A passive network gives back the same number — not a similar one, but the same to five parts in ten to the fourteenth over four decades of frequency, and for a coupled pair of windings the same double. Put a transconductance of eight tenths of a femtosiemens anywhere in it and the two readings part company. The ratio of the two readings, drawn where it lives. computed by solving, not by drawing. The same five-element RLC ladder at 10 kHz, read from each end, with the quotient of the two readings plotted in the complex plane. A reciprocal network sits at the point 1. One transconductance moves it along a straight ray, 1 − gm·Z, whose direction is the phase of the single branch between the source's control node and its output node and whose length is gm|Z| — 62.83 Ω for the 1 mH inductor, 120 Ω for a resistor, 72.34 Ω for 220 nF, all at 10 kHz. A mirrored pair of transconductances stays at 1 to 2.3e-15; reversing one of them runs the ratio around the unit circle to 8.9e-16, where the two readings are the same size and differ only in phase. The straight-ray law needs the two nodes joined by exactly one branch, and a second path between them takes it away by a factor rather than by a percentage. The reading that does care which way round An earlier essay measured a network's departure from reciprocity and left its size as a constant — 62.8 per siemens, for that network at that frequency. It is not a constant and it is not the network's: it is 2πfL for the single inductor between the controlled source's control node and its output node, 62.832 ohms at ten kilohertz, and the ratio of the two readings is 1 − gm·Z to 8.8 parts in 10¹⁴ over ninety-nine readings. So 4.5455 millisiemens across a 220 ohm branch makes a ladder that transmits a hard zero forwards at every frequency at once, and 206.13 ohms back at ten kilohertz. A network solved, and checked: a bridge, which no series-parallel reduction reaches. Node potentials from modified nodal analysis. The branch currents are then recomputed from each element's own law and summed at every node; the residual is 2.7e-16 of the largest current in the circuit, which is floating-point rounding and nothing else. What a network answers, and how the answer is checked A circuit has exactly one answer and a matrix finds it. The part that matters is not that the answer exists but that it can be checked twice, by routes that share no arithmetic — and that a circuit with no answer is refused by name rather than returned as a large plausible number.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Controlled sourceImpedance matchingInput impedanceInsertion lossModified nodal analysisReciprocitySymmetric matrixTransfer impedanceTwo-portVerification