Basic
`Circuit
`Theory
`
`Charles A. Desoer
`and
`Ernest S. Kuh
`
`Department of Electrical Engineering
`and Computer Sciences
`University of California, Berkeley
`
`McGraw-Hill Book Company
`New York St. Louis San Francisco
`London Sydney Toronto
`Mexico Panama
`
`RPX-Farmwald Ex. 1047, p 1
`
`

`

`To the University of California
`on Its Centennial
`
`Basic Circuit Theory
`
`Copyright© 1969 by McGraw-Hill, Inc. All rights reserved.
`Printed in the United States of America. No part of this
`publication may be reproduced, stored in Ir-retrieval system,
`or transmitted, in any form or by any means, electronic,
`mechanical, photocopying, recording, or otherwise, without
`the prior written permission of the publisher.
`Library of Congress Catalog Card Number 68-9551
`
`ISBN 07-016575-0
`56789-MAMM-7 6543 2
`
`RPX-Farmwald Ex. 1047, p 2
`
`

`

`Chap. 9 Network Graphs and Tellegen's Theorem 396
`
`Thus, we have shown that given any set of branch voltages subject to KVL
`only and any set of branch currents subject to KCL only, the sum of the
`products v,Jk is zero. This concludes the proof.
`
`Exercise 1 Suppose that starting from the reference node and following a certain path
`to node @, we obtain (by adding appropriate branch voltages) for its
`node potential the value ea. Show that if following another path we were
`to obtain a potential e~ =:/=ea, then the branch voltages of these two paths
`would violate KVL.
`
`Exercise 2 Consider an arbitrary network driven by any number of sources of any
`kind. Let v1(t), v2(t), ... , v,,(t) and }i(t),)2(t), ... ,}b(t) be its branch voltages
`and currents at time t. If ta and tb are arbitrarily selected instants of time,
`what can you say about
`b L Vk(ta)}k(tb)
`
`k = l
`
`- Applications
`
`5.1
`
`Conservation of Energy
`
`for all t
`
`Considering an arbitrary network, we have, with the notations of
`Tellegen's theorem,
`b 2: vk(t)jk(t) = o
`k = l
`Since vk(t)}k(t) is the power delivered at time t by the network to branch k,
`the theorem may be interpreted as follows: at any time t the sum of the
`power delivered to each branch of the network is zero. Suppose the net(cid:173)
`work has several independent sources; separating in the sum the sources
`from the other branches, we conclude that the sum of the power delivered by
`the independent sources to the network is equal to the sum of the power ab(cid:173)
`sorbed by all the other branches of the network. From a philosophical point
`of view, this means that as far as lumped circuits are concerned, KVL and
`KCL imply conservation of energy.
`Let us briefly look into the interpretation of this conservation of energy
`as far as linear time-invariant RLC networks are concerned. The power
`delivered by the sources is the rate at which energy is absorbed by the
`network. The energy is either dissipated in the resistors at the rate
`R!fik2(t) for the kth resistor, or it is stored as magnetic energy in inductors
`[~L,Jk2(t)] or as electric energy in capacitors [~Ckvk2(t)). When ele(cid:173)
`ments are time-varying (as in electric motors and generators or in para(cid:173)
`mAtrir <>mnlifi"""' thP rli"""""inn 1c: mnr.h mnrP. r.nmn1icated and is dis-
`
`RPX-Farmwald Ex. 1047, p 3
`
`

`

`Sec. 5 Applications 397
`
`Remark Tellegen's theorem has some rather astonishing consequences. For ex(cid:173)
`ample, consider two arbitrary lumped networks whose only constraint is
`to have the same graph.
`In each one of these networks, let us choose the
`same reference directions and number the branches in a similar fashion.
`(The networks may be nonlinear and time-varying and include inde(cid:173)
`pendent sources as well as dependent sources.) Let uk,}k be the branch
`voltages and currents of the first network and ~.]k be corresponding
`branch voltages and currents of the second. Since the uk's and 'V"k's satisfy
`the same set of KVL constraints and since the ]k's and the }k's satisfy the
`same set of KCL constraints, Tellegen's theorem guarantees that
`b
`b
`2.: ulfik = 2.: 'V"Jk = o
`k = l
`and
`b
`2.: 'V"lfik = 2.: uJk = o
`k=•
`k = l
`Note that whereas the first two are expressions of the conservation of
`energy, the last two expressions do not have an energy interpretation be(cid:173)
`cause they involve voltages of one network and currents of another.
`
`k = l
`
`b
`
`5.2
`
`Conservation of Complex Power
`
`(5.1)
`
`Consider a linear time-invariant network. For simplicity let it have only
`one sinusoidal source in branch 1, as shown in Fig. 5.1. Suppose that the
`network is in the sinusoidal steady state. For each branch (still using
`associated reference directions), we represent the branch voltage uk by
`the phasor Vi and the branch current }k by the phasor Jk. Clearly,
`Vi, V2, . .. , v;, and Ji.J2, ... , Jb satisfy all the constraints imposed by KVL
`and KCL. However, the conjugates] 1, 12, . .. , Jb also satisfy all the KCL
`constraints; therefore, by Tellegen's theorem
`b L ~Vk]k=O
`k = l
`Since Vi is the source voltage and J 1 is the associated current measured
`with respect to the associated reference direction, ~Vi ] 1 is the complex
`power delivered to branch I by the rest of the network, and hence - ~Vi] 1
`is the complex power delivered by the source to the rest of the network.
`We rewrite Eq. (5.1) as follows:
`b
`-~ViJ1 = 2.: ~Vi]k
`k = 2
`Clearly, the above can be generalized to networks with more than one
`source. Thus. we state the theorem of conseruation of complex power as
`
`RPX-Farmwald Ex. 1047, p 4
`
`