Power Conservation in Circuits

Module by: Don Johnson

Summary: Proof that circuits conserve power regardless of the elements used to construct them.

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Now that we have a formal method---the node method---for solving circuits, we can use it to prove a powerful result: KVL and KCL are all that are required to show that all circuits conserve power, regardless of what elements are used to build the circuit.

Figure 1

Part of a general circuit to prove Conservation of Power

First of all, define node voltages for all nodes in a given circuit. Any node chosen as the reference will do. For example, in the portion of a large circuit depicted here, we define node voltages for nodes a, b and c. With these node voltages, we can express the voltage across any element in terms of them. For example, the voltage across element 1 is given by v1=eb−ea v1 eb ea . The instantaneous power for element 1 becomes v1i1=eb−eai1=ebi1−eai1 v1 i1 eb ea i1 eb i1 ea i1 Writing the power for the other elements, we have v2i2 = eci2−eai2 v3i3 = eci3−ebi3 v2 i2 = ec i2 ea i2 v3 i3 = ec i3 eb i3 When we add together the element power terms, we discover that once we collect terms involving a particular node voltage, it is multiplied by the sum of currents leaving the node minus the sum of currents entering. For example, for node b, we have ebi3−i1 eb i3 i1 . We see that the currents will obey KCL that multiply each node voltage. Consequently, we conclude that the sum of element powers must equal zero in any circuit regardless of the elements used to construct the circuit. ∑kvkik=0 k k vk ik 0

The simplicity and generality with which we proved this results generalizes to other situations as well. In particular, note that the complex amplitudes of voltages and currents obey KVL and KCL, respectively. Consequently, we have that ∑kVkIk=0 k k Vk Ik 0 . Furthermore, the complex-conjugate of currents also satisfies KCL, which means we also have ∑kVkIk¯=0 k k Vk Ik 0 . And finally, we know that evaluating the real-part of an expression is linear. Finding the real-part of this power conservation gives the result that average power is also conserved in any circuit. ∑k12ℜVkIk¯=0 k k 1 2 Vk Ik 0

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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.

Last edited by Connexions on Jun 11, 2009 3:20 pm GMT-5.