Çok-uçlu alt-devrelerin bazı özel biçimlerde bağlanmalarıyle oluşturulan çok-uçlu cebirsel devrelerin uç-denklemlerinin varlığı ve bu denklemlerin belirlenmesi için geliştirilmiş yeni bir yöntem

Abstract

Ill ÖZET Bu tezde, cebirsel en genel elemanlardan oluşmuş ve daha genel olarak, sinusoidal sürekli halde aktif RLC çok-uçlu elemanlardan o- luşmuş çok-uçlu devrelere ilişkin uç-denklemlerinin elde edilmesi problemleri ele alınmıştır. Ayrıca, çok-uçlu ideal transformatörler den oluşan çok-uçlu devrelerin özelliklerinin belirlenmesi problem leri de değişik ve yeni bir yaklaşımla ele alınmıştır. Gözönline alınan en genel çok-uçlu devre elemanlarının, seri, paralel, hibrid, kaskat bağlanmalarıyle oluşturulan ve ayrıca bu e- lemanların bazı kapılarının kısa devre edilmeleriyle ya da bazı ka pılarının açık devre bırakılmalarıyle oluşmuş çok-uçlu devrelerin özelliklerini belirleyen uç-denklemlerinin, nasıl kurulabileceğine ilişkin ayrı ayrı yöntemler geliştirilmiştir. Bu yöntemler gelişti rilirken, bu elemanların en genel halde geçerli olan uç-denklemle rinin kullanılması öngörülmüş ve bu uç-denklemlerinden hareketle tanımlanan değiştirilmiş uç-denkiemieri 'nden yararlanılmıştır. Bun dan başka, p-ters (pseudo-inverse) kavramından ve simetrik ve yarı kesin pozitif (SYKP) ya da hermit ve yarı kesin pozitif (HYKP) mat rislerin tezde verilen biçimde çarpanlara ayrılmasından yararlanıl ması söz konusu olmuştur. Geliştirilen yöntemlerle uç-dervklemleri kurulurken, çok-uçlu elemanların uçtan-çözüleb/ilir (terminal sol vable) olmaları ya da Z, Y, H, K parametrelerinden birine sahip olmaları gibi varsayımlar yapılmamıştır. Bundan sonra, daha önceki çalışmalarda ele alınan pasif n-kapılı direnç devrelerine ilişkin uç-denklemlerinin tezdeki genel yöntemlerden kolayca nasıl buluna bileceği gösterilmiş ve elde edilen sonuçların daha önce bulunan sonuçlara uyumluluğu gözlenmiştir. Son olarak, çok-uçlu ideal transformatörlerden oluşmuş çok- uçlu devrelerin özelliklerini belirleyen, daha önce bilinenlerdenIV farklı yeni bir yöntem geliştirilmiştir. Bu yöntem, tezde tanımla nan değişti ri İmiş! uç-denkl emi erinden yararlanılmasına ve yine bu tezde ideal transformatörler için verilen yeni bir tanımın kulla nılmasına dayanmaktadır. Bu tür çok-uçlu devrelere ilişkin uç-denk- lemlerı kurulduğunda, geliştirilen yöntemle, bu uç-denklemlerinin de çok-uçlu ideal transformatörlere ilişkin olduğu gösterilmiştir.

SUMMARY When series, parallel, hybrid, cascade interconnection of two n-port networks containing only passive resistors are consi dered with the additional assumption that each network is augmen ted by including n ideal transformers each of which having a turns ratio 1:1 connected to each port, and instead of Y, H, C terminal matrices, terminal impedance matrices Z ( not necessarily nonsin- gular ) are utilized, then the matrices corresponding to series sum s parallel sum, hybrid sum and cascade sum defined by Anderson, Duff in and Trapp give explicit expressions for the terminal impe dance matrices of the resulting multi-port networks. These results on series, parallel, hybrid, cascade interconnection which are also valid for the case where a new multi -terminal network obtain ed by shorting or open circuiting some of the ports of a given n-port network, can not be directly applied to other kind of al gebraic n-port networks. On the other hand, a network to be inter connected or some of its ports are to be shorted or open circuited may not be terminal solvable or well-defined. In this thesis, it is shown that, even in the case of most general algebraic networks and in the case of sinusoidal steady-state active RLC n-port net works, the results of Anderson, Duffin and Trapp can be utilized, with some modifications, to obtain the terminal equations of the resulting n-port networks. However, the procedure considered in this thesis is kept more general so that it does not require the existence of the terminal Z, Y, H, C matrices for the description of n-port components. On the other hand, if the terminal graphs of n-port components are connected, then the augmentation of n- port components by ideal transformers at their ports becomes unnecessary.VI First, a n-port network having the terminal equations of the form,. (n) (n) V (m) [a İ B ] I - 0 (1) is defined as an algebraic n-port if the matrices A and B are real. Where 0 <m ^2n, rank ( [ A! B ]) = m, V and I rep resent the port vectors consisting, respectively, of terminal vol tages and terminal currents of n-port. If an active RLC n-port network is considered in the sinusoidal steady-state, it will also have the terminal equations as in (1), however this time A and B turn out to be complex matrices with V and I representing the port vectors consisting, respectively, of the terminal volta ges phasors and the terminal currents phasors. The multi -terminal components defined by the terminal equa tions given in (1) may not be terminal solvable. However this ge neral form of the terminal equations of a multi -terminal network obtained by interconnecting two n-ports in series, parallel, hyb rid and cascade forms produce some difficulties in defining the series sum, the parallel sum, the hybrid sum and the cascade sum of proper terminal matrices. Similar difficulties also arise in the case where the multi -terminal network is obtained from another such network some of whose ports are either short circu ited and/or open circuited. In this thesis it is found expedient to separate various cases and consider each of them separately. Indeed, it will beVII assumed that either 0 <m ^n or n <m ^2n. In there two cases, it is shown in the thesis that the coefficient matrix appearing in the terminal equations (1), if necessary, can be bordered by row of zeros to have the forms: i) 0 <m.£n (n) (n) (n) [ L Aj j L B, = 0 (2) ii) n <m C2n (n) (n) (n) (m-n) (n)(m^n)(n) (n)(n) 0 } U ^11 L' B II X Y = 0 (3) where. A>, Ayr, By, Bjj are some sub-matrices of. the matri ces A and B appearing in (1) while L and L' are nonsin- gular matrices of order n. In (2) and (3) the variables in the vectors X and Y are generally the mixtures of terminal currents and terminal voltages of the n-port network.VIII The use of these equivalent forms of terminal equations pro vides one to choose L and L' in such a way that the coefficient matrices: and (n) (n) (n) L A, L B, 0 <m ^n (n) (n) (n)(m-n)(n) (n)(n) (4) (n) (m-n) L' j 0 0 ! U *II L' B II (5) n <m ^2n have some desired properties. If it is let, L Bj = S or H, and L'Bn = S' or H' then it is required that S, (S1) be symmetric and positive semidefinite matrix (SSD) or H, (H1) be hermitian positive semidefinite matrix (HSD). In this thesis, the existence andIX the computation of the real (complex) matrix L (or L') are gi ven so that S, (S'), { H,( H' )} are indeed SSD, {HSD} mat rices. From this point on, it is shown that, even in the case of most general algebraic n-port networks or in the case of sinuso idal steady-state active RLC n-port networks, the results of Anderson, Duff in and Trapp can be utilized, with some modifica tions, to obtain the terminal equations of the resulting h-port networks. The foil owings are the part of contributions estab lished in this thesis: I- When two N-, and N` n-port algebraic components not necessarily terminal solvable, with the terminal equa tions given as in the form of (2) or (3) are consi dered ; ` it is shown that, if N, N, N. and N repre sent the algebraic multi-port networks obtained by interconnecting N, and N2 in series, parallel, hybrid and cascade forms respectively, then there exists a unique relation between the terminal vectors V and I, that is the terminal equations exist and they are in the form of (1) `. II- When a n-port algebraic component N which is not neces sarily terminal solvable is considered with the termi nal equations given as in the form of (2) or (3), it is shown that, if Ns is the multi-port network obtained from N by shorting or open circuiting some of the ports of N, then there exists a unique relation between the terminal vectors of V and I» that isthe terminal equations exist and they are in the form of (!). r III- The two. properties. stated above are proved to be also valid in the case of sinusoidal steady-state active RLC components. The results obtained by Anderson, Duff in and Trapp in their works, turn out to be the special cases of the properties estab lished in this thesis. These general results, in this thesis, are also applied to the interconnection problems of multi-terminal ideal transformers and all the known previous results [TO-2 ], [TO-4]] are reestablished. However here a new definition for the multi-terminal ideal transformer is introduced which provides simplifications in proving the theorems. In this thesis, in order to simplify the treatments for the interconnection of two multi-port networks, first the case of 0 <m <£n, and then the case of n <m ^2n are considered sepa rately for each multi-port network. However, if for one of the multi-ports the case 0 <m ^n is true while for the other multi- port the case n <m £2n is true, then the results obtained in this thesis are also shown to be valid for this general case. Generalization of series, parallel, hybrid and cascade intercon nections of two algebraic n-ports and in the case of sinusoidal steady-state n-port active RLC networks, brings automatically the consideration of pathological multi -terminal networks. The results established in this thesis may be extended to s-domain active RLC networks.