Circuit Response
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Series

Reciprocity — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · networks
  2. 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.

    part 2 · networks
  3. 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.

    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.

    part 3 · networks
  4. 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

    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.

    part 4 · networks

All series