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

Nodal analysis — the series

2 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

All series