Değiştirilmiş düğüm analizi ile durum eşitliklerinin oluşturulması

Abstract

Devrelerin durum eşitliklerinin oluşturularak incelenmesi en çok kullanılan ve oldukça çok üstünlükleri olan bir yaklaşımdır. Değiştirilmiş Düğüm Analizi (DDA) yöntemi ortaya konmadan önceki yöntemler graf teorisine dayanmaktaydı. Bu yöntemlerde durum eşitliklerini oluşturmak için devrenin gra-Fının önceden belirlenmesi gerekiyordu. Halbuki değiştirilmiş düğüm analizi ile bu eşitlikler doğrudan elde edilebilir. Yöntemin uygulanması oldukça basittir, basit devrelerde eşitlikler elle bile oluşturulabilir. Daha karmaşık devrelerde kullanılmak için hazırlanan bilgisayar programı da bir çok üstünlükler içerir. Yöntem tüm akti-F-pasif lineer ve lineer olmayan devre elemanlarını içerir. Bu çalışmada, yöntemin açıklanmasında ve bilgisayar programının oluşturulmasında lineer devre elemanları kullanılmıştır. Yapılan ampirik incelemelerde yöntemin daha hızlı sonuç verdiği görülmüştür.

State space analysis has common use in network analysis programs -for its generality and advantages. In state space analysis network equations had been set by two basic methods until Modi-Fied Nodal Approach (MNA) was found : Graph Theoretical, Hybrid Matrix, Multiport Method. All these graph theoretic -Formulations of state equations require many matrix additions, multiplications and trans-Fers in memory to obtain initial -Forms o-F equations. These equations can be obtained directly by using modified nodal analysis. In graph theoretic methods state variables should be capacitor voltages and inductor currents. This limitation brings about many mathematical manipulations to form equations. Indeed state variables can be any set of variables. If MNA is used state vector mainly consists of node voltages. Network equations are obtained with MNA in the following forms G0 XQ + C0 dX* = B0 Ws(t) (1) dt Y0 = DQ Xfl(t) (2) Except when there are impulse sources at t=0 using continuity equation initial conditions at t=0* can be formulated as : C0 XQ(0+) = Wo,, (3) In this form of state equations G« and Ca ars nxn ( n is number of variables in X^ ) square matrices, Xq is a vector containing some network variables which form state vector, Ba is the order nxr (r is the number of inputs) and Wq. is the vector containing initial conditions. In output equation Yq is a vector containing required output variables, Dq is of order pxn o - r» "' A - - Ci 6<, B = Cj Bi j Bi - Cı Bn 9 D = Dj and E = Ej. initial conditions are given by Xt(0+) = W0 = Cj"1 W02 (15) viiI-F C, is not in triangular -form there will be some zero rows at the bottom of Cj. This indicates that some variables can be eliminated -From the Xj. The cycle can be repeated until a non-singular C matrix is -found. After repeated cycles circuit equations become dX =AX + BWs + B1dWs + B2 d2 W. +.. + B, dB W. (16) dt dt dt2 dt* and Y0 = D X + E01 Wg + E02 Wg + Ej dW, + E2 d2^ dt dt2 + B(B.U d^İJL (17) dt(H) The final form of equations must be got in one cycle. If the process is repeated derivatives of the source terms arise in the output equation. The presence of any derivatives of the source terms in the output equation indicates that it is an improper network. In these improper circuits natural frequencies are uncontrollable. Because the initial conditions depend not only on Xg(0+) and W (0+) but also derivatives of W at fc=0. These networks are unobservable and unstable too. Such situations only arise in active networks. The usefulness of such networks are very limited. A computer program is adapted for the entire algorithm in C computer language. Although method can handles all active, passive linear and nonlinear devices only linear devices are used in this program. Resistor, capacitor, inductor, all types of independent and controlled sources and operational amplifier are valid devices for the program. Integrator model of operational amplifier is chosen. Networks are let to have up to 30 devices. Only one independent source and five controlled sources can exist in networks. There are some rules to obey when circuit topology introduced to the program. If a mistake is done there is no way to correct it. Program checks the following points in circuits introduced : a) Circuits should have only one independent source. b) Circuits should be connected. For this control the program checks all nodes and every node should has been viiiintroduced at least two times. c) Circuits should have a reference node labelled '0'. If one o-F these controls fails the program stops running and the network should be introduced again. One more important point is that before controlled sources ar& entered, their controlling device should have been entered. Float type is used for matrix entities. The program takes network topology at the first step and sets the initial form of equations given with (1) and (2). If network is a proper network the program gives the final form of equations given with (13) and (14). If network is an improper network program gives the message " This is an improper network " and along with this message the intermediate form of equations is given too.