`CIRCUIT THEORY
`
`Ernst A. Guillemin
`PROFESSOR OF ELECTRICAL COMMUNICATION
`DEPARTMENT OF ELECTRICAL ENGINEERING
`MASSACHUsms INSTITUTE OF TECHNOLOGY
`
`NEW YORK· JOHN WILEY & SONS, INC.
`LONDON· CHAPMAN & HALL, LIMITED
`
`RPX-Farmwald Ex. 1046, p 1
`
`
`
`Copyright, 1953
`By
`John Wiley & Sons, Inc.
`
`All Rights Reserved
`
`This book or any part thereof must not
`be reproduced In any form without the
`written permission of the publisher.
`
`Ubrary of Congress Catalog Card Number. 53-11754
`
`Printed In the United States of America
`
`RPX-Farmwald Ex. 1046, p 2
`
`
`
`520
`
`GENERALIZATION OF CIRCUIT EQUATIONS
`
`Of particular int.erest is the result 163 if the network is excited by a
`single source. Letting this one be E1, we have in this special case
`
`p =Pav+ Re [E;1 eJ'2wt]
`
`(164)
`
`Taking E 1 as phase reference and denoting the input admittance angle
`by"'' we have
`I E1I1 I
`COB (2<.it + 'I')
`p =Pav+
`2
`
`(165)
`
`However, noting Eq. 157,
`I E1I1 I
`2
`
`so that Eq. 165 can be written
`
`(166)
`
`P =Pav+ VPav2 + Qav2
`COS (2<.it +'I')
`(167)
`a result which shows that the amplitude of the double-frequency sinus(cid:173)
`oid equals the magnitude of the vector power.
`
`7 Equivalence of Kirchhoff and Lagrange Equations
`In this article we wish to show that Lagrange's equations, which
`express the equilibrium of a system in terms of its associated energy
`functions, are identical with the Kirchhoff-law equations so far as the
`end results are concerned. We need first some preliminary relations
`which can readily be seen from Eqs. 111, 112, 113 for the functions
`F, T, Vin terms of the loop currents. If we differentiate partially with
`respect to a particular loop current, we find
`aF
`z
`- . = E R;kik
`aii
`k=t
`aT
`z
`-. = E Likik
`av
`- = E S;kqk
`aq;
`
`ai;
`
`k-1
`
`1
`
`k-1
`
`(168)
`
`(169)
`
`(170)
`
`These results may most easily be obtained if one considers the perti(cid:173)
`nent function written out completely as T is in Eq. 121. It is then
`obvious that a particular loop current, say i 2 , is contained in all terms
`of the second row and second column, and only in these terms. Hence,
`
`RPX-Farmwald Ex. 1046, p 3
`
`
`
`EQUIVALENCE OF KIRCHHOFF AND LAGRANGE EQUATIONS 521
`
`if we differentiate partially with respect to i 2 , no other terms are in(cid:173)
`volved, and we find
`
`where we note that the term with L22 yields a factor 2 because the
`2 is involved. However, since L;k = Lk;, we can rewrite
`derivative of i 2
`this result as
`
`(171)
`
`from which Eq. 169 follows. Equations 168 and 170 are obtained in
`the same manner.
`In all three, the summation involved is a simple
`summation on the index k.
`If we differentiate Eq. 169 totally with respect to time, we have
`
`'
`d (aT\
`dik
`dt ai) = A:~ Li,, dt
`
`(173)
`
`(174)
`
`(175)
`
`and Eq. 170 can be rewritten as
`
`so that with Eq. 168 we obtain
`
`!:. (a~ + a~ + av = E (Lik !:. + Rik + sikfdt) iA:
`
`dt a~) aii
`
`aq,
`
`A:-1
`
`dt
`
`Reference to the Kirchhoff voltage-law Eqs. 108 now shows that these
`may alternatively be written
`d (aT\
`aF av
`dt ~) + ai, + aqi = eli,
`
`i = 1, 2, .. ·, l
`
`(176)
`
`This form, in which the voltage equilibrium equations are expressed
`in terms of the energy functions, is known as the Lagrangian equations.
`From the way in which they are here obtained, it is clear that they are
`equivalent to the Kirchhoff-law equations although their outward
`appearance does not place this fact in evidence.
`
`RPX-Farmwald Ex. 1046, p 4