Basamaklı türden pasif devrelerin simulasyon yöntemleri

Abstract

Bu tezde basamaklı türden pasif RLC-devrelerinin si mülasyon yöntemleri incelenmiştir. Basamaklı türden pasif RLC devrelerinin düğüm denklemlerinde, bu devreler deki gerilim/akım ilişkilerinden hareketle bir takım dü zenlemeler yapılmış; aktif RC-devreleri ve ayrık zaman lı sistemlerle gerçeklenmeye uygun işaret akış diyag ramları elde etmeye yönelik matematiksel modeller ku rulmuştur. Bu matematiksel modellerden hareketle çizilen işaret akış diyagramlarından, uygun al t -devre al t -diyagram iki lilerinden yararlanarak, basamaklı türden pasif RLC dev relerinin aktif RC-devreleri, sayısal devreler ve SC- devreleriyle simülasyonu yapılmıştır. Tanıtılan simülasyon yöntemleri ile bir filtre fonk siyonunu gerçekleyen basamaklı türden bir pasif RLC-dey- resinin aktif RC-devreleri, sayısal devreler ve SC-dev- releriyle simülasyon, oldukça kolay olmaktadır. Simü lasyon sonucunda elde edilen devrenin eleman değerleri, simüle edilen devrenin eleman değerleri cinsinden kolay ca belirlenebilmektedir.

In this thesis» simulation methods of RLC ladders are presented. These methods are based on nodal equations of passive-RLC networks and available to the design of active-RC networks and discrete systems. Passive filter networks have no active elements which are composed of resistors» capacitors and inductors. Using inductors in passive filter networks causes many problems especially in low frequences and also increases dimension of network. Nowadays» filter networks can easily be realized as integrated circuits and using integrated circuits is very widespread. Realisation of active filter networks as integrated circuits is possible. Furthermore SC and digital filter networks is used commonly Simulation of a passive network by using active » SC or digital networks can easily be performed by following simulation methods which are presented in this thesis. The way followed in simulation methods can be summarized as rearranging the nodal equations of passive networks and building a mathematical model to obtain a suitable signal -flow graph» which can be realized by aetive-RC networks or discrete systems. For that purpose, relations between voltage and current in passive ladders are used. A passive-RLC network can be described by the nodal equation» CsC + s_1r + GDV - J C1D where C» r and G are capacitance» inverce inductance and conductance matrices and represent the contributions of capacitors» inductors and conductors respectively. V is defined by V = CV. -V,, Va, -V 1 C23 ± z a 4to make all elements of C, r and G matrices possitive. Inverse inductance matrix in equation C15 can be decomposed into where» T = A D AT C33 L L L D » diagCL"1,!/1»]-"1, 5. C40 l. i 2 a A new set of variables can be. defined to draw a signal -flow graph which is suitable for active-RC realisation: X = s-1D ATV C5D If equation C13> is arranged by using this set of variables, V can be expressed as V = -CsC + 0"Vx + CsC + G}"1! C63 Li If C has diagonal and off -diagonal elements; it will be more complicated to take the inverse of matrix CsC + G5 in equation C60. In this case matrix C can be written as the sum of two matrices containing diagonal and off -diagonal elements of matrix C. C = Cd + Cd where C contains diagonal elements of matrix C and C contains off -diagonal elements of matrix C. If equation C63 is arranged by usign equation C7D, can be rewritten as V = '-CsC -KSD'Sc V - CsC +G>_1CA X +J5 C83 d dd d L Equations C55 and C8) are neeeded equtaions to draw a suitable signal -flow graph for active-RC realisation. A new active-RC simulation method can be obtained by LU decomposition method. This simulation method can also be altered to the simulation with discrete systems. A square matrix can be decomposed into LU form. For this reason matrix C can be decomposed in LU form: C = LU C9D Matrix L is chosen as CviDL = UTD C1C» where D is diagonal matrix. Matrix U has diagonal elements and off -diagonal elements. Diagonal elements are equal to 1. For this reason matrix U can be expressed as the sum of two matrices : U = I + u CUD dd where U contains all off -diagonal elements of matrix U By using equation C£D; CI 5 can be arranged into sluv + s_1rv + gv = j cia:> form. Let define a new set of variables as X ¦ sDUV C13D From equations CUD, C185 and C133 X = -U* X -Cs"4r +GDV + J C14D dd can be obtained. On the other hand, from equation C13D V = s~1D"4X - U V C1SD dd can be written. Equations CI 4} and CI S3 are needed to draw a signal flow graph according to LU decomposition method. The simulation of passive networks using discrete systems is also possible. To do this» it is necessary to perform bilinear transform on equation C1D. If A is chosen as» A = C + T + G C16D equation CI D can be rewritten as CA + QW4r + Q2GDV = QCl+zDJ Cİ7D where, Q = z"Vci-z-1:> C185 W = İ/Cl-z'1? C195 As matrix A is always symmetric» it can be decomposed into the form, A = UTDU C20D CviiDwhere» U is equal to I + U if a new set of variables is defined as X = DUWQ caiD then. X «= -U* X - CW4r + 3GDV + Cl+zDJ C82D dd V = QD_iX - UJtV C83D dd can be obtained. These equations are needed to draw a signal -flow graph which is suitable for the simulation with discrete systems CSC or digital networks}. The component values of active, SC and digital filter networks which are obtained by using these simulation methods» can be determined in terms of passive networks component values. The simulation method based on LU decomposition method has more advantageous than the general simulation method. Because high frequency noise levels of active-RC networks which are obtained by using this simulation method are lower than networks which are obtained by using the general simulation method. Furthermore, the number of addition and multiplication elements of digital filter networks which is obtained by using LU decomposition method, is lower than canonical digital filter networks and their operation speeds are higher than the others. It is suggested that the matrix operations in the simulation methods presented of this thesis, can be performed by using computer. Doing this speeds up the simulation. The contents of chapters of this thesis can be summarized as follows: In the second chapter, the active-RC simulation of passive ladders has been explained. A general active-RC simulation method of passive ladders and the simulation method based on LU decomposition method have been introduced. Simulation examples for two different passive filter networks have been given. In the third chapter, the simulation method of passive ladders by using discrete time systems has been Cviii3introduced. As a simulation example a passive filter network has been taken and simulation of this passive filter network has been performed by using digital and SC networks. Conclusions and suggestions have been given in the last chapter