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

Nodal analysis — the series

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

    part 1 · networks
  2. The circuit does not care which node is called zero, and the matrix does. computed by solving, not by drawing. The condition number of the nodal matrix for a 12-section chain, against which of its nodes was taken as the reference. The network, its elements and its physics are identical in every case — only a label has moved — and every branch voltage and branch current comes back the same to 1.1e-13. The condition number runs from 5.25e+4 at "n6" to 1.72e+5 at "n12", a factor of 3.28, which is 0.52 decimal digits of the arithmetic's own margin.

    The node that is not in the circuit

    Nodal analysis needs a node to call zero and no circuit contains one. Moving it changes every node voltage by the same amount and no branch voltage or branch current at all — to a part in ten to the fourteenth on a well-behaved chain. What it does change is the matrix: the condition number of a twelve-section chain moves by a factor of 3.3 with the reference, and on a star whose resistances span nine decades by 8.0. Measured against the same matrix solved in twice the precision, the worst reference costs four decimal digits of the answer, and it is the reference the condition number named before the error was looked at.

    part 2 · networks
  3. The best reference is the middle of the chain below 2.37 kHz and ground above it — the node the capacitors return to. computed by solving, not by drawing. The condition number of the nodal matrix of an 8-section RC ladder — 1 kΩ in series, 10 nF and 1 MΩ to ground at each node — solved at s = jω with each of its 10 nodes in turn called zero, against frequency. Drawn: ground, the input, the middle node and the far end; the others lie between. At direct current n4 is best, near the middle as the earlier resistor chain found; from 2.37 kHz ground is — well below a single section's 15.9 kHz corner, where each capacitor's admittance is already 0.15 of a series resistor's conductance, and all 8 together about 1.2 of one.

    The reference a sweep should keep

    At direct current the best node to call zero in a resistor chain is its middle. Put a capacitor to ground at each node and solve at s = jω, and the best reference moves: the middle of an eight-section RC ladder below 2.37 kHz, ground above it, where every capacitor returns. So a solver that chooses its reference once for a whole sweep is choosing wrongly somewhere, as the earlier essay suspected — but the two wrong choices are not alike. Kept for the whole sweep, ground is never worse than 1.27 times the best condition number, a tenth of a digit. The middle node, best at direct current and kept, is 143 times worse at a megahertz, 2.2 digits. A fifth-order LC filter says the same: ground within 2.18 times throughout.

    part 3 · networks

All series