Tellegen's theorem

Tellegen's theorem states that the algebraic sum of branch powers in a lumped network is zero when its currents and voltages satisfy Kirchhoff's laws. It holds regardless of the network elements’ characteristics.

Kirchhoff's current law: At each circuit node, the algebraic sum of currents is zero. This conservation constraint makes admissible branch currents orthogonal to node potentials.
Lumped-element model: A circuit model that treats components as localized elements connected by ideal wires. Tellegen's theorem applies to networks represented with finitely many branches and nodes.
Circuit analysis: The calculation of voltages, currents, and power in electrical networks. Tellegen's identity checks whether a proposed network solution respects its conservation constraints.
Conservation of energy: The principle that energy cannot be created or destroyed in an isolated system. The power identity is consistent with energy conservation, though it follows from network laws alone.
Ohm's law: A relation stating that voltage across a resistor is proportional to current through it. Tellegen's theorem needs no resistor law or other voltage-current relation.
Kirchhoff's voltage law: Around every closed circuit loop, the algebraic sum of voltage differences is zero. It constrains branch voltages to differences of node potentials, enabling the power-sum identity.
Branch (circuit theory): A connection between two nodes in a circuit graph, assigned a current and voltage. Branch quantities are the terms whose instantaneous powers the theorem sums.
Modified nodal analysis: A matrix-based method for solving circuit equations using node voltages and selected branch currents. Its equations inherit the network structure behind Tellegen's power balance.
Tellegen's lemma: A bilinear identity relating two sets of branch quantities satisfying compatible network constraints. This generalized form extends the theorem beyond pairing a network's voltages with its own currents.
Kirchhoff's circuit laws: The current and voltage conservation rules used to model lumped electrical networks. These are the theorem's premises; Tellegen's theorem is a consequence applying to all branch powers.

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GyratorRelated: Tellegen’s network theory provides a broader setting for analyzing ideal elements such as gyrators.