A Theorem on Power Superposition in DC Networks

The superposition theorem, a particular case of the superposition
principle, states that in a linear circuit with several voltage and
current sources, the current and voltage for any element of the circuit
is the algebraic sum of the currents and voltages produced by each
source acting independently. The superposition theorem is not applicable
to power, because it is a non-linear quantity. Therefore, the total
power dissipated in a resistor must be calculated using the total
current through (or the total voltage across) it. The theorem proposed
and proved in this paper states that in a linear DC network consisting
of resistors and independent voltage and current sources, the total
power dissipated in the resistors of the network is the sum of the power
supplied simultaneously by the voltage sources with the current sources
replaced by open circuit, and the power supplied simultaneously by the
current sources when the voltage sources are replaced by short-circuit.
This means that the power is superimposed. The theorem can be used to
simplify the power analysis of DC networks. The analysis results are
validated via numerical examples.