Both the KCL and KVL equations are required to completely model a resistor network. However, the two main methods to analyze these circuits Nodal analysis and Mesh analysis only use KCL and KVL respectfully. In other words,these methods don't use the "complete model of the circuit" (i.e. Nodal analysis ignores the KVL part of the circuit model). Yet somehow they're able to completely analyze the entire circuit – how? How is something that doesn't take the entire model into account able to completely analyze the circuit?
Why do circuit analysis methods work
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That is not correct.
Nodal analysis appears to be using KCL only, where it actually uses KVL implicitly, absorbed in Ohm's law in the currents equations.
Mesh analysis appears to be using KVL only, where it actually uses KCL implicitly, absorbed in Ohm's law in the voltages equations.
By the way, KVL and KCL are just a consequence of the lumped-element abstraction of linear circuits at low frequencies (when the wavelength of the signal is much more larger than the physical lengths of the wires and components). You can derive it easily from Maxwell equations (which are more accurate models of the circuit behavior.)