2.8: A Conservation Law

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Page ID: 55527

James Kirtley
Massachusetts Institute of Technology via MIT OpenCourseWare

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It is possible to show that complex power is conserved in the same way as we expect time average power to be conserved. Consider a network with a collection of terminals and with a collection of internal branches. Instantaneous power flow into the network is:

\(\ p_{i n}=\sum_{\text {terminals }} v i\)

Note that this expression holds for voltage and current expressed over any complete set of terminals. That is, if it is possible to delineate the terminals of the network into a set of pairs, the voltages might correspond to voltages across the pair, while currents would flow between the terminals of each pair. Alternatively, the voltages might correspond to single node-to-datum voltage, while currents would then be single input node currents. Since power can go only into network elements, it follows that the sum of internal branch powers must be equal to the sum of terminal powers:

\[\ \sum_{\text {terminals }} v i=\sum_{\text {branches }} v i\label{67} \]

If this is true for instantaneous power, it is also true for complex power:

\[\ \sum_{\text {terminals }} \underline{V I}=\sum_{\text {branches }} \underline{V I}\label{68} \]

Now, if the network is made up of resistances, capacitances and inductances,

\[\ \sum_{\text {terminals }} \underline{V I}=\sum_{\text {resistances }} \underline{V I}+\sum_{\text {inductances }} \underline{V I}+\sum_{\text {capacitances }} \underline{V I}\label{69} \]

For these individual elements:

Resistances: \(\ \underline{V I}^{*}=R|\underline{I}|^{2}\)
Inductances: \(\ \underline{V I}^{*}=j \omega L|\underline{I}|^{2}\)
Capacitances: \(\ \underline{V I}^{*}=-j \omega C|\underline{V}|^{2}\)

Then equation 69 becomes:

\[\ \sum_{\text {terminals }} \underline{V I}=\sum_{\text {resistances }} R|\underline{I}|^{2}+j \sum_{\text {inductances }} \omega L|\underline{I}|^{2}-j \sum_{\text {capacitances }} \omega C|\underline{V}|^{2}\label{70} \]

Then, identifying individual terms:

\(\ \sum_{\text {terminals }} \underline{V I}=2(P+j Q) \quad \text { Total Complex Power into Network }\)

\(\ \sum_{\text {resistances }} R|\underline{I}|^{2}=2 \sum<p_{r}>\quad \text { Power Dissipated in Resistors }\)

\(\ j \sum_{\text {inductances }} \omega L|\underline{I}|^{2}=4 \omega \sum<w_{L}>\text { Energy Stored in Inductances }\)

\(\ j \sum_{\text {capacitances }} \omega C|\underline{V}|^{2}=4 \omega \sum<w_{C}>\text { Energy Stored in Capacitances }\)

So, for any RLC network:

\[\ P+j Q=\sum_{\text {resistors }}<p_{r}>+2 j \omega\left[\sum_{\text {inductors }}<w_{L}>-\sum_{\text {capacitors }}<w_{C}>\right]\label{71} \]