Ieeeinitial Conditions

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY A ND APPLICATIONS, VOL. 40, NO. 12, DECEMBER 1993 885

Circuits and Systems ExpositionsAnalysis of L inear Networks with

Inconsistent Initial ConditionsJavier Tolsa and Miquel SalichsAbstract-This paper presents a new method to analyze time-invariant linear networks allowing the existence of inconsistentinitial conditions. This method is based on the use of distributionsand state equations. Any time-invariant linear network can beanalyzed. The network can involve any kind of pure or controlledsources. Also, the transferences of energy that occur at t=O aredetermined, and the concept of connection energy is introduced.The algorithms are easily implemented in a computer program.

I . INTRODUCTIONNITIAL conditions may be inconsistent when there occursI change in the network topology, as when two capacitorswith different initial voltagesareconnected in parallel to forma new network.The instant when a network forms a new topology will bet =0. Initial conditions at0- will be called initial conditionssimply, while the values immediately after switchingare theinitial conditions at O+. This paper will show an efficient andsimple method to analyze an electric network knowing theinitial conditions at 0-.Voltage and current values at O+ and 0- are related bycharge conservation in capacitive cutsets and flux conservationin inductive loops. Nevertheless, the application of these lawsdoes not always suffice for obtaining the initial values at O+from initial values at0-: in the network of the second exampleof Section VIII, there is only one inductor, and its flux at 0-is different from its flux at O f.Consider the two capacitors again. If initial voltages aredifferent, the total energy stored in capacitors atO+ is smallerthan the energy at 0- because at t =0, the capacitors havetransformed part of their energy stored in their electric fieldinto electric energy, which is not zero if initial conditions areinconsistent. This energy, which we call theconnection energy,is analyzed in detail in this paper.The problem of determining initial values at O+ has beenstudied before. However, the analysis of the electric energy

Manuscript received February 20, 1992; revised manuscript receivedMarch 20, 1993. The work of J . Tolsa was supported by the Departmentd'Ensenyament de la Generalitat de Catalunya under Grant DOGC n. 1514,6.11.1991. This paper was recommended by A ssociate Editor S . Karni.The authors are with the Department d'Enginyeria Elhctrica, UniversitatPol ithcnica de Catalunya, Barcelona 08028, Spain.IEEE L og Number 9210020.

absorbed in the network connection is a little studied subjectthat this paper treats in detail.Dervisoglu [3] developed an algorithm to calculate initialvalues at Of using a state variable approach. His methoddoes not involve distributions directly, but introduces the DiracimpulseS and its derivatives in the analysis. Murakam [4]pro-posed another way of determining the response of a networkwith inconsistent initial conditions. However, his method doesnot allow the existence of dependent sources and pure currentsources. His analysis is based on the state equation and theuse of distributions, although the application of distributionsis not as interesting as in our new formulation. Recently, Opaland Vlach [ 5 ] proposed a new method to calculate initialvalues at O+ without introducing state equations. They use anumerical Laplace transform inversion, exact for impulses andits derivatives. In their algorithm, it is necessary to integratea system of equations in a time interval At tending to zero.This fact introduces numerical errors difficult to measure.The problem of conservation of energy when initial con-ditions are inconsistent has been studied by Goknar [12]. Heconsiders networks consisting of capacitors or inductors only,without sources. He demonstrates that in those networks, thedifference of the energy stored in capacitors (or inductors)from 0- to O+ is always positive, and that this differenceequals the energy consumed in the interval [0,+cc[ by someresistors properly included into the network. However, he doesnot explain why the principle of conservation of energy seemsto be violated.Our approach is based on currents and voltages defined asdistributions. The method is simple: first, equations of thenetwork (K irchhoff's laws and Ohm's law) are stated forcurrents and voltages defined as distributions. Then, we obtaina singular system of differential equations of distributions. Thissystem is similar to classic singular systems of differentialequations for functions, which can be written as

ddt

where the matrix T may be singular. Next, this equation istransformed into the following pair of equations:

SY(t)+T-Y(t) =E( t )

1057-7122/93$03.00 0 1993 IEEE

Authorized licensed use limited to: Universitat de Barcelona. Downloaded on February 13, 2009 at 07:29 from IEEE Xplore. Restrictions apply.


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886 IEEE TRANSACTI ONS ON CIRC UITS AND SYSTEMS-I: FUNDAMENTA L THEORY AND APPLICATIONS, VOL. 40, NO. 12, DECEMBER 1993

Switches are not considered as elements of the network.Instead, we assume that the network topology for t 2 0 isdifferent from the topology for t


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TOLSA AND SALICHS: ANALYSIS OF LINEAR NETWORKS 887

Let us consider a network in the time interval - , +CO[.At t=0, all branches are connected to form a new networktopology. Currents and voltages are determined by Ohm's lawand Kirchhoffs laws.Ohm's law is assumed to be valid in thewhole interval ] - , +CO[. The equation is expressed as

d-u-t ={&t


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888 IEEE TRANSACT IONS ON CIRCUIT S AN D SYSTEMS-I: FUNDAMENT AL THEORY AND APPLICATI ONS, VOL. 40, NO. 12, DECEMBER 1993

Then, (12) is equivalent to[W]$with the following change of variables:

and the matrix F(d/dt)has been written asF - = -(i)

So the system (12) is equivalent to

z+=c - (E+fSK (O-)).(3Equation(15) is the network state equation for distributions.The components ofX+are the state variables. The derivativeswhich appear in (15)and (16) are in the sense of distributions.It is also interesting to observe thatE+can be any distribution,not only a continuous function.For instance,E+can includeany impulses ~(1, n 2 0.

v. S O L U T I O N OF THE STATE EQUATIONThe state equationisequivalent to the following system ofconvolution equations in the algebraD:

(Sld - SA)*X+=B - (E +SK(0-))(3(the symbol * stands for the convolution of distributions). Thisresult is due to the fact that for any distribution f ,

Also, we have(Sld - =h(t)etA

where h(t) s the Heaviside function (h(t)=0 if t


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TOLSA AND SALIC HS: ANALYSIS OF LINEAR NETWORKS 889

If the matrix C(d/dt) s written asCEO i(di/dti), e havem

2+(t)=CCi{E+(t)}i= O

VII. ENERGY NALYSISA. Multiplication of Distributions

Electric power is equal to the product of current and voltage.If currents and voltages are defined by distributions, theirproduct is not possible in general (only the product of adistribution and a C function is well defined). Due to thisfact, power and energy cannot be analyzed in the space ofdistributions.To multiply distributions, t is necessary to introduce anotherspace where multiplication is possible. Such a space is thespaceG of generalized functions defined by Colombeau [8].In this formulation, a distribution is a particular case ofgeneralized function. So the space 2) of distributions is asubspace of 6.The product of two generalized functions always exists inG, in particular if these generalized functions are distribu-tions. For example, the square of the Dirac impulse 6 is thegeneralized function i 2 , which is not a distribution.Electric energy is defined as the definite integral of power,which is a generalized function that depends on time. Thedefinite integral of a generalized function in an interval [a,b]is introducedin [8] too. It is always defined and is equal to ageneralized number (the set of generalized real numbers is anextension of R). Obviously, f a generalized function is definedby a continuous function, its definite integral as a generalizedfunction coincides with its usual definite integral as a function.Weassume that the space2) of distributions is included inthe spaceG of generalized functions, where power and energycan be analyzed correctly.

Thus, the total electric energy consumed by the network inthe time interval [a,b] is6W=2 W i = Pdt.

i=lIn the interval [a , b], 6


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890 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY A ND APPLICATIONS, VOL. 40, NO. 12, DECEMBER 1993

is also zero if we assume that there exists an initial networktopology in the interval ] - , 0[ (in general, this topology isdifferent from the final topology). However, this assumptionis not necessary in our analysis since we only require theexistence of a network topology for t 2 0.The energyW" s not zero in general. We defineW"as thenetwork connection energy. This electric energy is absorbed bythe network in t=0 as a consequence of the topology change.Consider the energyW: again. We can observe that the totalenergy absorbed by the branch ''i" in t =0 is not only W:,but the addition of W: and the part of W corresponding tot = 0, which we will denote as W,".However, the part of the energy absorbed by the wholenetwork in t =0 corresponding to W+,which we denote asWO, s zero [this is derived immediately from(21)]. Therefore,the total energy absorbed by the network in t =0 is equal toW".Let us assume that there exists an interval 10, E'[ wherethe restrictions of distributions I;' and U, are continuousfunctions with limt at O+. Then, operating as in Section 11,we obtain

U, =U," +u,oI;' =I,"+ ;o

whereU," and IT0arethe restriction of distributionsU, andI, to the open interval 30, +CO[ extended by zero and U,", I,"aredistributions with support in (0) (this is always possible ifthevector E of independent sources is defined by a piecewiseC" function). From [8], we derive

C. Determination of the Connection E nergyTo calculate the integrals of (23),we must take into accountthat U', I o, U - , and I - are generalized functions. Thefollowing results are derived from the formulation given byColombeau [8].If a


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TOLSA AND SALICHS: ANALYSIS OF LINEAR NETWORKS 891

E3'i ::I 1:Fig. 2. Circuit of first example.

a) Time-Domain Analysis with the State Equation: From (7)and (8) and operating as in Section 111 we obtain

After some elementary row and column transformations, weobtain the following state equation:

where

From (26), we deriveI,+=h(t) xp ( - - t )1+ R2

L 1+ L2

Therefore,

Also, from (26),

Fig. 3. Circuit of second example.

b)Analysis of the Network Connection Energy: The con-nection energy in branch 1 isWf =Jd I;U:dtb

L1L2 (11(0-)+12(o-))

In branch 2,1 L lL22 L 1+ L2

bWC=Jd I T U i d t =.(I 1(0-)+12(0-))12(0-).

In branch 3, current and voltage are finite in t=0. Therefore,w. 0.The network connection energy is3(26) wc =cw;= L 1 L 2 ( I l ( o - )+ 2(0-))'. (29)2 L 1+ L2i=l

From (27) and (28), we obtainU: =SLi(Il(O+)- l (O-))=6A4lU$ =SL 2(Iz(O+)- 2 ( 0 - ) ) =6A@2=U:.

That is to say, flux is conserved in the loop formed by bothinductors:A41 =A & =A $. Then,1 1

2"=Wf +WC=ZA4I l(O-) +-A$12(0-)12-A4( I i (O- ) + i(O+))1+p 4 ( ' 2 ( 0 + )+ z(0-1)

=-L J 1(0+)2 - L111(0-)2+-L 212(0+)2 - L zI2(0-)2.1 12 21 1

2 2Thus, W" is equal to the increment of the energy stored inthe magnetic field created by both inductors. From (26), wederive that this increment is always5 0.

Example 2: Given the network of Fig.3, let us suppose thatwe interconnect its branches in t =0. If capacitorsC, andC,have different initial voltages, a current impulse is producedin the loop formed by both capacitors. This current impulseis converted into a voltage impulse by the controlled voltage

(27)

~--6 L lL2 ( I ~( ( ) - ) 12(o-)). (28) sourceE l . Due to this voltage impulse, the flux of inductorL Z s not conserved and 12(0+)# 12(O-) .1+ L2



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892 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 12, DECE MBER 1993

a) Time-Domain Analysis with the State Equation: FromOhm's law and Kirchhoff s laws, we derive

0-c3U3(0- )- 4U4(0-

After some row and column elementary transformations, weobtain the following state equation:

L 0 Jwhere the new variables are

c3U3(O-)+C4 U4( 0 - )c3+ c 4

I z(0-1 +(&-&) AQ3$] =h(t)etd[The matrix A is obtained immediately from (30). From thisresult, it is easily derived that AQ3 is equal to the chargeincrement in the capacitor C3 between the instants 0- andO+.(22),we can derive that the network connection energy isb)Analysis of the Network Connection Energy: Applying

The total connection energy of both capacitors is1WC+WC= -AQ3(U3(0-) - u4(0-))212- u4( 0+) - u4(0-) )]2

= -AQ3[(U3(O-) +u3(0+))1-1c3u3(o+)25C3u3(0-)'

1 12 2- c4u4(0+) 2 - C4u4(0-) '

which is equal to the increment of the energy stored in theelectric field created by both capacitors.

El(72(b)same cir cuit with an additional resistor.Fig. 4. Networks of third example. (a) Circuit with a CE loop. (b) The

Example 3: Let us consider the network of Fig. 4(a). Theswitch S is open for t


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TOLSA AND SALICHS: ANALYSIS OF LI NEAR NETWORK S 893

C circuits without sources. However, if &(t) s not a constantfunction, the energy consumed by R3 in [0, +CO[ is generallydifferent from -W (for example, if El(t) is sinusoidal, theenergy consumed by R3 is+m).On the other hand, it can bechecked that our first interpretation of -W as the limit of theenergy consumed by R3 when R3 +0 in the finite interval[0,t]canbeextended to not constant functionsE1(t)f El (t)is derivable enough.Therefore, we think that our interpretation of -W is betterthan Goknars. It should be investigated if our hypothesis istrue for anyRLCM circuit. However, this is a difficult problemsince it involves singular perturbations in singular systems.

APPENDIXGiven a pencil of square matricesS+AT,we are going toexplain an algorithm to calculate a unimodular matrix F (A )and an invertible matrix D such that (13) holds. The matricesF (A ) and D will be calculated using elementary row andcolumn transformations. Our algorithm is purely algebraic,such as the algorithm given by Fettweis [9].The concept of elementary row and column transformationsof polynomial matrices can be found in [I 11. If these transfor-mations do not depend on A, they are said to be strict. If onematrix is obtained from another by elementary transformations,these matrices are equivalent. If all the transformations arestrict, then they are strictly equivalent.The concept of the row echelon of a matrix is introduced byCampbell [13]. He gives this definition: a rectangularmx nmatrix A which has rank T is said to be in row echelon formif A is of the form

crxn[o(m-r)xn]where the elementsc; j of C (=C,,,) satisfy the followingconditions:1) cij = 0 if i >j.2) The first nonzero entry in each row of C is 1.3) If cij =1 is the first nonzero entry of the ith column,then the jth column of C is the unit vector e; whoseonly nonzero entry is in the ith position. This column issaid to be a distinguished column.For example, the following matrix is in row echelon form:

1 3 0 - 2 0 4 0[!!i :%E].It is easy to program an algorithm to obtain the row echelonmatrix of any matrix by elementary row transformations. Wehave the following properties:

Any rectangular matrix B can always be row reduced torow echelon form by elementary row operations. That isto say, there always exists an invertible matrix G suchthat GB=A, where A is in row echelon form.The rank of the matrix B equals the rank of its row ech-elon formA and is equal to the number of distinguishedcolumns of A .

The algorithm to obtain (13) is shown in the followingexample. Let us consider the nonsingular polynomal matrixS +AT(1):Through strict elementary row transformations in thepolynomal matrix S+AT, the matrix T is transformedin its row echelon form. The resulting matrix is

If the rank of T equals the dimensions of S+AT(1),henwe have finished, since the submatrixT,(;)must be equalto the identity matrix and the submatricesS ~~ y S ~~)cannot exist.By means of strict elementary row transformations inS(l)+AT(1)we derive the row echelon form of thesubmatrix

Therefore, the pencil S(l)+XT(l)is transformed into

The rank of the submatrix

equals its number of rows, since det (S+AT) # 0.Thus, exchanging some columns in S2+AT2we obtain

1 Id22 J(the columns of the submatrix Ida2arethe distinguished).If the submatrices of S(3)+

AT(3)S;)+AT,(:), S;)+AT;;)

do not exist (due to the fact that S+ATis a unimodularmatrix), then we have finished too.Though (not strict) elementary row transformations, thepencil S(3)+AT(3)is transformed into

: Id22 JDue to the fact thatdet (S; +AT(4)ll)#0.



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894 IEEE TRA NSACTI ONS ON CIRCUITS AN D SYSTEM S-I: FUNDAM ENTAL THEORY AND APPLIC ATIONS, VOL. 40, NO. 12, DECEMBER 1993

Now we return to point 1of this algorithm, operatingin the submatrix Si;) +AT,(:), nstead of the matrixS +AT.However, all the elementary transformations ofpoints 1, 2, 3, and4 must bedone in the whole matrix~ ( 4 )+ ~ ( 4 1 ,ot only ins!;)+AT,(:).These operationswill finish in point 1 or 3 after a finite number ofloops, since in each loop the dimensions of submatrixSi;) +AT,(:) is strictly smaller than the dimension ofS +AT. In fact, the number of loops must be I rank(T )+l. At the end of this process we will obtain thematrix

-A+AId, :....................[ M :I]

If all of the same elementary row transformations ofpoints 1-5 are made in the identity matrix, the resultingmatrix is a unimodular matrix F(A). The only columntransformations n points 1-5 are the column exchangesof point 3.Exchanging the same columns in the identitymatrix, we derive an invertible matrixD. If we defineI d, :D = D I . . . . . . _ _ _ _ _ _ _ _ _ . . .

[ M i : b ]

REFERENCES[l ] N. Balabanian, T. A. B ickart, and S. Seshu,Electrical Nehoork Theory.New York: Wiley, 1969.[2] L . 0.Chua and P. Lin, Computer Aided Analysis of Electronic Cir-cuits: Algorithms and Computational Techniques. Englewood Cliffs,NJ : Prentice-Hall , 1975.[3] A. Dervisoglu, State equations and initial values in active RLC net-works, IEEE Trans. Circuit Theory, vol. CT -18, pp. 544-547, Sept.1971.[4] Y . M urakami, A method for the formulation and solution of circui tscomposed of switches and linear RLC elements, IEEE Trans. CircuitsSyst., vol. CA S-34, pp. 496-509, May 1987.[5] A. Opal and J. Vlach, Consistent initial conditions of li near switchednetworks, IE EE Trans. Circuits Syst., vol. 37, pp. 364-372, Mar. 1990.[6] L. Schwartz, TM orie des Distributions. Pari s: Hermann, 1966.[7] -, Mitodos Matemticospara l as Ciencias F isicas. Madrid: Selec-ciones Cientificas, 1969.[SI J . F. Colombeau, Elementary Introduction to New Generalized F unc-tions. Amsterdam: North-Holland, 1985.[9] A. Fettwels, On the algebraic derivation of state equations, IEEETrans. Circuit Theory, vol. CT-16, pp. 171-175, May 1969.[IO] G. C. Verghese, B. C. U vy , and T. Kailath, A generali zed state-spacefor singular systems, IEEE Trans. Automat. Contr., vol. AC-26, pp.

[I l l F. R. Gantmacher, The Theory of Matrices ( 2 vol.). New York:Chelsea, 1977 (vol. 1. 1989 (vol. 2).[I21 I. C. Goknar, Conservation of energy at initial time for passiveRLCMnetworks. IE EE Trans. Circuit Theory,pp. 365-367, July 1972.[13] S. L. Campbell and C. D . M eyer, Generalized Inverses o LinearTransformations. New Y ork: Dover, 1991.

811-831, A ug. 1981.

Javier Tolsa, photograph and biography not available at the time of publi-cation.(13) holds. Observe that D has been obtained throughexchanges of columns in the identity matrix. Then, it isx+must be a subsetof the original variables Y +n (12).Seenthat thestate Miquel Salichs, photograph and biography not available at the time ofpublication.

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