PAN INITIAL CONDITIONS IN LINEAR SWITCHED NETWORKS XVIII - SPETO pod patronatem. Summary

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1 PAE SEINAIU Z PODSTAW ELEKTOTEHNIKI I TEOII OBWODÓW 8 - TH SEINA ON FUNDAENTALS OF ELETOTEHNIS AND IUIT THEOY DALIBO BIOLEK ILITAY AADEY, BNO, ZEH EPUBLI XVIII - SPETO pod patronatem PAN INITIAL ONDITIONS IN LINEA SWITHED NETWOKS Summary The simple algorithm of initial condition computation immediately before and after switching is proposed in this paper. The algorithm aoids both limiting step t [] and the time expensie numerical Laplace inersion []. It is based on the solution of the set of linear equations. These equations can be easily compiled for arbitrary linear switched network. Proposed algorithm is included to the program S for the time-domain simulation of switched networks with external switching.

2 INTODUTION odeling of switched networks (switched-capacitor or switched-current networks, conerters D-D, switched modulators etc.) often leads to the discontinuity of networks ariables at the time instances of switches state change. This phenomenon is called as inconsistent initial conditions []. The computer algorithm how to search some transfer matrices for recounting initial conditions after switching from ones before switching is described in []. This algorithm is based on so-called twostep Laplace inersion []. Howeer, the computation can be time expensie due to searching of optimal step to minimize the error caused by incidental Dirac impulse contained in the calculated response. The necessity to use complex arithmetic can be further disadantage. The simple approach will be described in this contribution which consists in the solution of the set of linear algebraic equations. The algorithm is explained on two examples:. Ideal switched capacitor (S) network,. Arbitrary switched network consisting of both reactances and resistances. After generalization, the common algorithms for calculation of inconsistent initial conditions will be stated out. The time instances immediately after/before switching are signed as / -. Algorithm for idealized S networks Let us consider the simple S network in Fig.a. In the phase (or ), the network is described by the set of independent nodal oltages :, or :,, respectiely. Knowing ector ( ) immediately before switching, it is necessary to determine ector ( ) immediately after switching. The corresponding operator diagram in the phase is in Fig.b, considering initial charges,, and. Fig.b leads to the two-graph modified nodal approach (-graph NA) equations [] D W V s s( ) V Y s V. () s w W s s s a) b) Fig.. a) example of idealized two-phase S network, b) operator diagram for the phase. The original well-known admittance matrix Y is modified to the matrix Y due to influence of oltage source in the node.

3 For our network, the charge conseration law can be described as follows: ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( )] ( ) ( ) There are two ways how to determine ector in (): [ ]. ( ) ( ) ( ) ( ) ( ) ( ) () () atrices and arose by simple modification of original matrices and after resetting the row corresponding the input oltage source. Algorithm how to compile these matrices is described in []. In the case of networks, the principle is indicated in Tab.. phase element Y s : a* : c* e** b* d* f ** s -- s -- s s a* b * - - c* d * Tab.. The principle of circuit matrices compilation in case of circuit elements. The symbols * and ** denote the oltage and current nodal coefficients in conformity with the -graph NA []. Due to aforementioned modification, the matrix equation () consists of two scalar equations. : The first one is triial:. This equation can be replaced by the equality w ( ) ( ) w ( ) ( ) ( ) D e** f **. () The matrix is regular as a consequence of is the ector of independent nodal oltages. Soling () yields required transform equation ( ) ( ) P w ( ). (5) This equation stated that the network state at beginning phase is determined by the state at the end of phase and at the input signal alue at beginning phase. The transform matrix and ector P can be formally calculated as follows: For our network is ( ), P ( ) D. (6)

4 , P. After generalization of results mentioned aboe, we can state the algorithm of transform matrices creation in case of ideal S networks:. We compile matrices and according to Tab. or [5].. If the input oltage/current source is actie in phase, i.e. if the oltage/current coefficient a*/a** of input node is nonzero, we write "" to the corresponding row a** of ector D. In case of oltage source, we fill "" the row a** of matrices and. Then we write "" to the position: row a**, column a* of. After these operations, the creation of matrices and is finished. If the input sources are not actie in phase, then and.. We compute transform matrix and ector P using (6). Algorithm for arbitrary switched networks Let us consider switched network in Fig.a. In the phase, the network is described by the set of three independent oltages :,,. As a result of disconnecting input source in the phase, the set of ariables is then reduced to two independent oltages :,. The operator diagram for phase in Fig. b) yields D W V s( ) V Y s V (7) s w s a) b) Fig.. a) example of network with external switching, b) its operator diagram for phase. The charge conseration law leads to the equations

5 [ ], or (8) This is the set of two equations. The first one is again triial. Howeer, the reason consists now in the absence of accumulation elements connected to node. In this way, the first equation in the set (7) is algebraical (in the operator form without the operator s). That means this equation is also true for instantaneous alues and for the limits on the right:. We add this equation to the set (8): (9) omparing (9) and () yields following conclusions: D ( the network is not exited in the phase ). is not the square matrix (the numbers of independent nodal ariables in phases and are different). contains the parameters of both capacitie and resistie elements. The desired solution can be obtained using inersion of matrix : After generalization of aforementioned conclusions we can state the algorithm for the compilation of transform matrices in case of arbitrary switched networks:. We compile matrices, and according to Tab. or [5].. We scan all rows of matrix exception to the row corresponding to the input node. In case of empty rows they will be replaced by corresponding rows of matrix. This step is unnecessary in case of idealized S networks.. If the input oltage/current source is actie in phase, i.e. if the oltage/current coefficient a*/a** of input node is nonzero, we write "" to the corresponding row a** of ector D. In case of oltage source, we fill "" the row a** of matrices and. Then we write "" to the position: row a**, column a* of. After these operations and prospectie modification in accordance with the item, the creation of matrices and is finished.

6 . We compute transform matrix and ector P using (6). ONLUSION The presented algorithm of inconsistent initial condition computation in the switched networks is based on the two-graph modified nodal approach. Using this method, the network is described by the set of independent circuit ariables in each switching phase. As a result, the regular minimum set of equations is obtained. Noel algorithm utilizes this regularity and compiles transform matrices between the ectors of nodal ariables immediately before and after switching. The algorithm is implemented to the general program for the fast analysis of real switched networks [], [5]. This work is supported by the rant Agency of the zech epublic under grant No. /9/8. EFEENES [] ZUHAO,Z.: ZZ model ethod for Initial ondition Analysis of Dynamics Networks. IEEE Trans. on AS, ol.8, No.8, August 99, pp [] OPAL,A.-VLAH,J.: onsistent Initial onditions of Linear Switched Networks. IEEE Trans. on AS, ol.7, No., arch 99, pp.6-7. [] VLAH,J.-SINHAL,K.: omputer methods for circuit analysis and design. Van Nostrand einhold, N.Y., 98. [] BIOLEK,D.-ZAPLATILEK,K.: Frequency domain analysis of switched networks by generalized transfer functions. ETD'9 Daos, pp Switzerland. [5] BIOLEK,D.: omputer oriented Analysis of Switched Networks using TF approach. "ASE'9", Hyderabad, India. ILITAY AADEY BNO Department of Electrical Engineering K Assoc.Prof. Ing. Dalibor Biolek, Sc. Kounicoa 65, PS Horácké nám. 9/ 6 BNO, ZEH EPUBLI 6 BNO, ZEH EPUBLI