A series-parallel network is any network which can be constructed using just, well, series and parallel connections. For the formal definitions lovers out there,

A network is series-parallel iff:
  • It is simple (i.e. a single impedance: one R, one L, or one C); - OR -
  • It is two series-parallel networks connected in series; - OR -
  • It is two series-parallel networks connected in parallel.

It is very easy to calculate the effective resistance (or effective impedance, if you're into such twisted matters) of series-parallel networks. This makes people underestimate Thevenin's theorem, because they immediately say...

Huh? Aren't they all series-parallel?

NO!

The classic counterexample is Wheatstone's bridge:

      +--'\/\/\/\/`-+-`\/\/\/\/`--+
      |      1      |      2      |
      |             \             |
------+             / 3           +----
      |             \             |
      |      4      /      5      |
      +--'\/\/\/\/`-+-`\/\/\/\/`--+

OK, so how do I calculate the effective resistance of a bridge?

Linear algebra. Write a whole bunch of linear equations for voltage and current, and solve them.

Ummm, but how do I know these equations have a unique solution?

Didn't we just mention Thevenin's theorem?

So how do I really prove Thevenin's theorem, not the weak-kneed version for series-parallel networks?

Didn't we just mention linear algebra?

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