Değiştirilmiş düğüm yöntemi kullanılarak durum denklemlerinin elde edilmesi
Obtoining of state equations using modified nodal analysis
- Tez No: 66526
- Danışmanlar: PROF. DR. FUAT ANDAY
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Devreler ve Sistemler Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 81
Özet
ÖZET Bu tezde devre denklemlerinin durum gösterilimini elde etmek üzere yeni bir yöntem açıklanmıştır. Durum ve çıkış denklemlerini elde etmek için önceden bilinen metotlar devre grafi yaklaşımı üzerine kurulmuştur. Burada açıklanan yöntem ise değiştirilmiş düğüm analizi (MNA) kullanılarak geliştirilmiştir. Açıklanan yöntem için bir algoritma verilmiş ve bu algoritma programlanarak DURUM adında bir program geliştirilmiştir. Giriş bölümünde yöntem kısaca anlatılmış ve neden farklı bir gösterilim üzerinde durulduğu kısaca açıklanmıştır. İkinci bölümde yöntem ayrıntılı olarak incelenmiş ve birkaç tipik örnek üzerinde açıklanmıştır. Burada açıklanan yöntemin önceden bilinen yöntemlere karşı en önemli üstünlüğü, başlangıç denklemlerinin çok daha az işlem yapılarak bulunabilmesidir. Bunun sebebi, durum değişkenlerinin kondansatör gerilimleri ve self akımları olma zorunluluğunun olmamasıdır. Ayrıca bu bölümde eski durum denklemlerine nasıl geçileceği gösterilmiştir. Üçüncü bölümde ise geliştirilen program hakkında bazı bilgiler verilmiştir. Programın nasıl kullanılacağı ve program giriş dosyasının nasıl hazırlanacağı örneklerle açıklanmıştır. iv
Özet (Çeviri)
SUMMARY OBTAINING OF STATE EQUATIONS USING MODIFIED NODAL ANALYSIS In this thesis a new method for obtaining state representation of circuit equations are explained. Previously known methods for obtaining state and output equations were based on the graph theoretic approach. However the method explained in this thesis was developed using modified nodal analysis ( MNA). The advantages of using state representation of circuit equations are well known [1-4]. There are two basic approaches used to obtain the state representation. The so called initial state and output equations are obtained either using the graph theoretical approach [1-9], using the the hybrid matrix approach [2] or using the multiport method [3]. The multiport method presented in Reference 3 is similar to the hybrid matrix approach of Reference 2. All these graph theoretic formulations of state equations were developed before the development of modified nodal analysis (MNA) [6]. The modified nodal method has been widely used for formulating circuit equations in computer-aided network analysis and design programs. The formulation of circuit equations using modified nodal analysis is straightforward and easy to implement in computer formulation. Even if paper and pencil methods are used, for simple circuits, the equations can be obtained by inspection. All methods clearly involve many matrix additions, multiplications, transfers in memory locations etc., [4] even to obtain the initial state and output equations. This is one of the major limitations in the formulation of state equations despite other advantages. This limitation arises because the state variables are capacitor voltages and inductor currents. The state variables can actually be any set of variables and are not necessarily inductor currents and capacitor voltages. Therefore, if MNA is used, most variables are node voltages. These equations are in the form (Go + sCo) Xo(s) = BoWs(s) + W0 Y0=DoXo (1) where G0 and Co are two square matrices of order nxn, B0 is the order nxr ( r is the number of inputs) and Wo is the vector containing the initial conditions. Do is of order pxn where p is the number of outputs. Xo contains the node voltages, inductor currents, currents of voltage sources, currents of dependent current sources and output currents. These equations are already in the form of initial state and output equations and can be obtained by inspection. MNA can handle all types of active and passive elements including nonlineer elements so there is no difficulty in obtainingthese equations. All the initial mathematical manipulations required in previous methods can thus be avoided. Only linear networks are considered in this thesis. Co is usually very sparse and singular in the MNA based equation 1. Xo contains several variables that are not state variables and the excess variables must be eliminated from these initial state equations. The main purpose of this work is to provide a simple and systematic method of eliminating these excess variables. A similar elimination approach can be found in Reference 3 ( p.255) where an implicit assumption is made that the inverse of a square matrix exists. A similar problem exists in the proposed elimination process. A simple technique to solve this problem in the computer formulation of state equations is suggested. The equations for finding the initial conditions for the remaining variables are also given. The method is illustrated using some simple example circuits. The time domain equivalent of equations 1 are G^o + Co-^BoWsOO Y0 = D0Xo(t) (2) such that CoXo((T) = Woo (3) Last equation gives the initial conditions at t = 0“. Except when there are impulse sources at t = 0, even though XoCO*) may be different from Xo((T), using the continuity equation CoXoCOV Woo (4) The impulse sources do not arise in practical circuits except when the impulse response of circuits is sought. To find the impulse response, the initial conditions are set to zero. Therefore assume that equation 4 is valid. This assumption means that Xo(0~) equals to Xo^4). The method depends on the conversion of the Co matrix to the row echelon form [9]. The conversion of a matrix to the row echelon form involves the well known elemantary row operations and an algorithm is available for this purpose [9]. The same elemantary operations are also carried out on the Go, Bo and Woo matrices to maintain the equivalence of the circuit equations. If the rank of the Co matrix is equal to k, then the first k-rows of the equivalent matrix of C0 will contain at least one non-zero element in each row and the remaining (n-k) rows will contain only zero elements. Thus the equivalent circuit equations will be in the following form where the matrices are partitioned into different submatrices for the purpose of clarity. VIk rows (n-k) rows such that k rows (n-k) rows 'G”g12 Yx, VG21 ^22 A 2 Y0 = [Doi D02] /S y^X2 'c“ c12Yx,(o+)] ' t 0 0 Xx2(o+)j.0; f-v \ fn \ X vX2y B01 v”o2y [Wj (5) (6) (7) The bottom rows of the initial condition vector in equation 7 must be filled with zeros and if not, there is some inconsistency in the initial conditions. The bottom rows in equation 5 indicate that some variables are algebraically related and therefore, (n-k) variables can be eliminated from Xo without using any iterative process. Assume that it is desirable to eliminate X2(t) from the set. The circuit equations can then be described using the variable set of Xi(t) alone. The corresponding equation is dX, dWs G,Xi + Ci-r1- = B,WS + 62-*- dt dt (8) where Gi = G11 - G12G22" G21 Ci = Cn - C12G22 G21 Bi = B01 - G12G22 B02 B2 = - C12G22 B02 (9) The output equation becomes where Yo = DiXi + EiWs Di = D01 - Do2G22~ G21 Ei = D02G22 B02 (10) (ID The equation for the initial conditions becomes CiXi(0*) = W01 + 82X^3(0^ = W02 (12) Clearly equations 9 and 1 1 indicate the method of obtaining the matrices for the reduced system of equations. Equation 12 also indicates the method of calculating the Vllnew initial conditions for Xi. The above procedure can only be successful if G22 is non-singular, i.e., a set of (n-k) variables, X2 such that G22 is non-singular must be found. Otherwise this solution is not possible. Fortunately such a set of variables can always be found. Since the solution of a physical network always exists, the rank of the submatrix [G21 G22] must be (n-k). To find a set of variables X2 it is rearranged some of the variables such that G22 is non-singular. This identification can also be systematically performed as described below. If the circuit equations of equation 2 are obtained using MNA, then the first set of variables are node voltages and are followed by current variables. Therefore C12 and D02 are usually very sparse. The number of calculations can thus be minimised if as many current variables are eliminated as possible. It is usually necessary to find node voltages as the output variables. To identify the variables of X2, it must be found the row-echelon form of the submatrix [G21 G22] starting from the bottom row and going to the top ( k+1) row. This again involves elementary row operations. At this point, the same elemetary operations must be operated on the B02 matrix to keep the equivalence of the circuit equations. There is no need to operate on the Co matrix at this point since the correspending rows of this matrix are filled with zeros only. It is identified and rearranged the columns of the Co, Go and Do matrices such that G22 is non-singular from the echelon form of [G21 G22]. When the columns are rearranged, the submatrix corresponding to G22 will be triangular. As an example, when n = 6 and k = 3, the submatrix [G21 G22] will be y y y x 0 0 y y y y X 0.y y y y y x. G21 G22 where ys may be zeros or non-zeros but xs are non-zeros. At this point G22 is already in the triangular form. To find G22_1G2i and G22_1Bo2, it only requires some back substitutions. The new matrices can be stored in the same locations of G21 and B02. Then it is a simple matter to carry out the matrix manipulations as indicated in equations 9, 11 and 12. These manipulations are necessary in any reduction process [2,3]. The values of WsCO*) are required to find W02. The entire reduction process described above is summarised in the following steps. (a) Convert the Co matrix to a row-echelon form. Use the same elemantary operations on the Go, Bo and Woo matrices to maintain the equivalence of the circuit equations. (b) Convert the [G21 G22] matrix into a row-echelon form starting from the last row. Carry out the same row operations in the B0 matrix to maintain the equivalence of the circuit equations. At this point, Go, Co, Bo and Woo have been modified. vui(c) Interchange the columns in the G0, C0 and D0 matrices to identify X2(t) such that G22 is non-singular. (d) Use back substitutions to find G22_1G2i and G22-1B02. Then use equations 9 and 1 1 to find Gi, Ci, Bi, B2, Di and Ei. (e) Evaluate Ws(0+) and use equation 12 to find W02. (f) Repeat step a again to find whether Ci is non-singular. After step f, Ci will be triangular if it is non-singular. Thus only back substitutions are required to find the state and output equations in the form of dX, dWs -^-AXj+BWs + B!-^ (13) Yo = DXi+EWs (14) where A = -Cf'Gi, B = Cf 'Bi, Bi = CfIB2,D = D, and E = Ei. The initial conditions are given by X1(0+) = Wo = Cf1Wo2 (15) If Ci is singular after step f, the process can be repeated to eliminate some more variables from Xi. The cycle can be repeated as many times as necessary until a C matrix which is non-singular is found. The state and output equations are then found in the form dX dWs d2Ws dmWs - = AX + BWS + B,-^ + B2^ + + B.-^ (16) dWs d2Ws d^W. Y0 = DX + EWS + E,-^ + E2-^- + + Em.,-^f- (17) The presence of any derivatives of the source terms in the output equation indicates that it is an improper circuit (system) [4]. The initial conditions depend not only on Xo(0+) and WstO*) but also on the derivatives of Ws(t) at t = 0+. Such circuits are unobservable and uncontrollable [4]. In such improper circuits, the transfer and driving point functions relating the input and output variables are such that the degrees of the numerator polynomials are higher than the degrees of the denominator polynomials by more than one. The impulse response of such circuits will contain doublets and higher order singularity functions. These facts also imply that such circuits are unstable [10]. Such situations only arise in the case of active networks. The usefulness of such unstable circuits is very limited. For all useful and proper circuits, Ci should be non-singular and the iteration should stop in one cycle and the IXA, B, Bı, D, E and Wo matrices should be found in one cycle. The Ci matrix must be converted to a row-echelon form to determine whether the network is proper or improper. If Ci is non-singular, the state and output equations in the form of equations 13 and 14 and the initial conditions, Xi(0*), can be found. If Cj is singular, the network is identified as an improper network and the equivalent of matrices in equations 9, 1 1 and 12 are given. The entire algorithm is programmed and developed a computer programme called DURUM. In Chapter 3 knowledge is given about the programme in details.
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