Gene Fuller wrote:
I am quite familiar with Kirchoff's equation in regard to voltages
around a loop and his equation about currents at a node. Did I miss a
third equation regarding currents around a loop?

Nope, you missed an implication of Kirchhoff's current law.
Unequal currents into and out of a passive black box implies
the existence of a node inside the box.
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73, Cecil http://www.qsl.net/w5dxp