Circuit Response
every model has an edge, and the figure carries it
Series

Virtual earth — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What the summing junction of an inverting amplifier actually is, at 1.00 MHz of gain–bandwidth. computed by solving, not by drawing by driving a current into the node and reading the voltage. It is 100 mΩ at direct current, rises 1.000 decades per decade of frequency, and settles at 909.5 Ω — which is the 1 kΩ and 10 kΩ in parallel, with the amplifier contributing nothing. It passes one per cent of the input resistor at 995 Hz, a factor of 1,005 below the gain–bandwidth. The second route — the open-loop impedance over one plus the return ratio from the cut loop — agrees to 0.045%.

    The node that is at ground for a while

    An inverting amplifier's summing junction is held at ground by the loop, so it is at ground exactly as well as the loop is strong. Driven with a current and measured, it is a tenth of an ohm at direct current, ten ohms at a kilohertz, and 909 ohms above a megahertz — which is the two feedback resistors in parallel, with the amplifier contributing nothing.

    part 1 · feedback
  2. A summing junction with 8 inputs on it, and the ceiling that fell. computed by solving, not by drawing by driving a current into the node and reading the voltage, with the input source zeroed, at each of 6 channel counts. With one input the node's ceiling is 909.5 Ω — the input and feedback resistors in parallel — and with 32 it is 31.15 Ω, a factor of 29.19 stiffer for 32 times the inputs — less than the count, because the feedback resistor is in the parallel combination too. Only the ceiling moves: the direct-current value and the whole rise are the same to 5.4 per cent at every count, because the open-loop impedance and the loop gain both scale as 1/N and cancel. So what the channel count does is bring the ceiling down to meet the rise earlier, and the per-pair leakage saturates rather than reaching unity — -0.8 dB at one input against -30.1 dB at 32. The total into every other channel together does not saturate: it is (N − 1)·Z/Rin, and with Z ≈ Rin/N that tends to unity — 0.966 at 32 inputs, 0.476 at two. A summing junction above its loop hands over essentially all of one input's signal to the others collectively, and the count decides only how it is divided.

    The node that does not care how many

    Put thirty-two inputs on a summing junction instead of one and its ceiling falls from 909 ohms to 31.2 — the input resistors in parallel with the feedback resistor, twenty-nine times stiffer. Every other part of the curve is unchanged to five per cent, because the open-loop impedance and the loop gain both scale as one over the count and cancel. So the stiffer node buys nothing: the per-channel leakage saturates instead of reaching unity, and the leakage into every other channel together tends to exactly one, whatever the count.

    part 2 · feedback
  3. A summing junction with a second pole above it: 3.1 dB of peak, and inductive. computed by solving, not by drawing on a two-pole amplifier — a gain block, two sections and a buffer, so the feedback network loads the output exactly as it does with one pole. The faint curve is the one-pole case, which rises to 909.1 Ω and stops. With a second pole at 100 kHz the junction reaches 1304 Ω at 112 kHz — 3.13 dB above the two resistors in parallel — because the division is by |1 + T| and near crossover that is smaller than one. A third of a decade below the peak the node's impedance leads its current by 84.4°, which is an inductance: the node is no longer a resistance that a capacitance loads, it is a reactance that a capacitance resonates with. The slider puts capacitance on the node and moves the peak.

    The node that is an inductance

    A summing junction with a one-pole amplifier round it rises to the two feedback resistors in parallel and stops there, and that ceiling is the whole of what a virtual earth can be. Give the amplifier a second pole and the node overshoots it — 1074 ohms against 909 with the second pole at 300 kilohertz, 1868 with it at thirty — and below the peak the impedance leads its current by more than twenty degrees, which is an inductance. So a stray capacitance on the node is no longer a load on a resistance; it is a resonance, and the peak grows with it rather than being damped by it.

    part 3 · feedback
  4. An integrator's summing junction is flat at 15.7 Ω, and worst at direct current. computed by solving, not by drawing. A current is driven into the node and the voltage read. The dashed curve is the same amplifier with a 10 kΩ resistor as its feedback element: a tenth of an ohm at direct current, rising a decade per decade, 909.5 Ω at the top. With a 10 nF capacitor instead, the loop gain has no frequency in it — the amplifier's gain falls as 1/f while the feedback factor rises as f — so it is flat at 62.83 and the node is flat with it, at Rin/(1 + 2π·GBW·C·Rin) = 15.666 Ω, measured 15.668 Ω and holding to 0.50 per cent from 100 Hz to 5.31 kHz — between the amplifier's own pole and the capacitor's corner, which is where a loop gain with no frequency in it lives. Both ends are the other way round from the resistive case: at direct current the capacitor is an open circuit, there is no loop at all, and the node is 843.0 Ω — a factor of 8388 worse than the resistor's 100 mΩ. Above the amplifier's crossover the capacitor is a short and the node is 47.6 Ω against 909.5. The two cross at 1.56 kHz, which is the frequency above which an integrator is the better virtual earth of the two.

    The shelf a capacitor makes

    An inverting amplifier's summing node rises a decade per decade to the two resistors in parallel. Replace the feedback resistor with a capacitor and it does neither: an integrator's loop gain has no frequency in it, so the node is flat at Rin/(1 + 2π·GBW·C·Rin) — 15.67 ohms over four decades, to half a per cent. Both ends invert. At direct current the capacitor is an open circuit, there is no loop at all, and the node is 843 ohms against the resistor's tenth of one; above the amplifier's crossover the capacitor is a short and the node is nineteen times better.

    part 4 · feedback

All series