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PAPER PREPARATION GUIDELINES UDC 621.3 COMPARISON OF METHODS FOR ELECTRIC CIRCUITS SIMULATION J. Dziak, I. Tomčíková Technical university in Košice Park Komenského 3, 04200, Košice, Slovakia. E-mail: [email protected], [email protected] This paper deals with few methods of electrical circuit analysis. It points the advantages and disadvantages of each method and evaluates their use for electrical circuit simulation. In paper are outlined procedure for network analysis in general and also the procedure for each method. It establishes criteria for comparison of methods. Paper compares methods according to specified criteria. Кey words: circuit simulation, computer simulation, network analysis. ПОРІВНЯННЯ МЕТОДІВ МОДЕЛЮВАННЯ ЕЛЕКТРИЧНИХ КІЛ Д. Дзіак, І. Томчікова Технічний університет Кошице вул. Парк Коменскего, 3, м. Кошице, 04200, Словаччина. Е-mail: [email protected], [email protected] У статті розглянуто декілька методів аналізу електричних кіл. Було визначено переваги і недоліки кожного з розглянутих методів та виконано оцінювання можливості їх застосування для моделювання електричних кіл. В статті виокремлено загальну процедуру мережевого аналізу та процедуру аналізу для кожного з методів. Також представлено критерії порівняння методів. У статті методи моделювання електричних кіл порівняно згідно запропонованого критерію. Ключові слова: моделювання електричних кіл, комп’ютерне моделювання, мережевий аналіз. PROBLEM STATEMENT. Circuit simulation is a technique for checking and verifying the design of electrical and electronic circuits and systems prior to manufacturing and deployment. It is used across a wide spectrum of applications, ranging from integrated circuits and microelectronics to electrical power distribution networks and power electronics. Circuit simulation combines mathematical modeling of the circuit elements, or devices, formulation of the circuit equations and techniques for solution of these equations. We will focus mainly on the formulation and solution of the network equations in this paper. We will focus mainly on the formulation and solution of the network equations in this paper [1]. In a computer program the equations have to be formulated automatically in a simple, comprehensive manner. Once formulated, the system of equations has to be solved. There are two main aspects to be considered when choosing algorithms for this purpose: accuracy and speed. In principle, accuracy depends on mathematical modeling of the circuit elements and speed depends on method. Consider a network having n nodes and m branches. A network may be characterized of a set of network equations. There are two types of equations: branch equations and connection equations. Branch equations are also called element equations. These equations describe a circuit element by means of a current - voltage relationship. Element equations can by completely expressed as (1). Set of element equations consist of m equations. i (1) Z Y s . u Z – Matrix of branch (element) impedance Y – Matrix of branch (element) admittance i – Vector of branch currents u – Vector of branch voltages s – Vector of voltage and current sources Table 1 – Examples of branch relations of basic elements in linear electric circuits with harmonic sources in the complex form Element Branch relation Matrix values Y 1 resistor U – R.I = 0 Z –R s 0 Y 1 inductor U – jωL.I = 0 Z – jωL s 0 Y jωC capacitor jωC.U – I = 0 Z -1 s 0 Y 1 voltage U = US Z 0 source s Us Y 0 current I = IS Z 1 source s IS Connection equations that consist of Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL) from the structure of the network, they don’t due to the properties of any network element. Formulation of the connection equations is done by applying KCL and KVL to the network. It is possible to capture KCL and KVL in two forms. The first form is in terms of branch currents and branch voltages. Set of equations (2) consist of m independent equations in 2m unknowns. A 0 i 0 0 B u 0 . A –Incidence matrix of nodes B –Incidence matrix of meshes (2) PAPER PREPARATION GUIDELINES Matrix A expresses relationships between nonreference nodes and branches (elements). These relationships can take the forms as it shown in Figure 1 options a), b) and c). Matrix B expresses relationships between meshes and branches. They can take the forms as it shown in Figure 1 options d), e), and f). Matrix values for all combinations of network relationships are in Table 2 – Matrix values of network relationships. Figure 1 – Relations between node and branch and between mesh and branch. Table 2 – Matrix values of network relationships Option a) b) c) 1 -1 0 A Value A A Option Value d) 1 B e) -1 B f) 0 B The second form is more compact with more equations and variables which are branch currents, branch voltages and node voltages. Set of equations (3) consists of (m+n-1) independent equations with (2m+n-1) unknowns. i 0 0 A 0 0 E AT u 0 . (3) v E – Unit matrix AT – Transpose reduced incidence matrix of nodes v – Vector of node voltages The connection equations written in the both forms (2) and (3) are due to the network topology only, and independent of any element’s property. The form (2) is not preferred in practice for handling large networks, because this set of equations is not sparse. The form (3) is highly sparse and this is major advantage for solving large circuits. EXPERIMENTAL PART AND RESULTS OBTAINED. There are several methods for electrical circuit analysis. Six methods were selected for compared: superposition theorem, kirchhoff’s law analysis, nodal analysis, mesh analysis, modified nodal analysis, sparse tableau analysis. SUPERPOSITION THEOREM. Superposition theorem (ST) is used to simplify networks containing two or more sources. In a network containing more than one source, the current at any one point is equal to the alge- braic sum of the currents produced by each source independently (4) [2]. k i i . i (4) i 1 ii – Matrix of currents produced by each source k – Number of sources in network ST can be used only in linear circuits. If we use ST, then is not possible to formulate circuit equations automatically. Automatic formulation of equations is a very important for computer-based circuit simulation. Therefore we will not use this method for simulation. KIRCHHOFF’S LAW ANALYSIS. In Kirchhoff's law analysis (KLA) are used directly Kirchhoff's laws. Circuit equations consist of two sets of equations. Connection equations formed by using Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) (2) and element equations described the voltage - current relationship of elements (1). Aim of KLA is do find all unknown’s branch currents (5) [2] [3]. A.s I A B.Z i B.s . U sI – Vector of current sources sU – Vector of voltage sources (5) Circuit description consists of relations between the branches (elements) and nodes corresponding matrix A, and relations between branches and meshes corresponding matrix B. It complicates and prolongs the computation and thus adversely affects the speed of the algorithm. NODAL ANALYSIS. The nodal analysis (NA) is based on the following idea. Instead of finding for circuit variables, current and voltage of each element, we seek node voltages in this case, which automatically satisfy KVL. We do not need to write KVL equations we need only to apply KCL equations to nonreference nodes. The aim of nodal analysis is to determine the voltage at each node relative to the reference node (6) [2] [3]. Y v A s Y I . sU (6) NA is a fairly comprehensive method of solving circuits. But it has complication for computer circuit simulation. In NA we have a problem with the voltage sources. If all the voltage sources in network do not have a common node, an algorithm for automatic computation is difficult. It extends time of computation and negatively affects the speed of algorithm. Therefore, modified nodal analysis has been developed in order to remove problems of NA. MESH ANALYSIS. The mesh analysis (MA) is analogy of the NA. We solve for a new set of variables, mesh currents that automatically satisfy KCL. As such, mesh analysis reduces circuit solution to writing KVL PAPER PREPARATION GUIDELINES equations. The aim of mesh analysis is to determine the currents at each independent mesh (7) [2] [3]. B.Z.B i T M B s B.Z U . sI (7) iM – Vector of mesh currents BT – Transpose incidence matrix of meshes Lack of MA is that we can MA use just for planar circuits. Element or connecting wire can't intersect other element or connecting wire in plane. MA also has a problem with circuit description. Circuit is described by using relations between branches and meshes. Instead of simply matrix A we need more complicated matrix B. It adversely affects the speed of the algorithm. MODIFIED NODAL ANALYSIS. NA is used for formulating circuit equations. However, several limitations exist in this method including the inability to process voltage sources and current dependent circuit elements in a simple and efficient manner. A modified nodal analysis (MNA) retains the simplicity and other advantages of nodal analysis while removing its limitations [4]. To handle voltage sources, the key idea is to not insist on eliminating their currents, but to retain those currents as additional variables. For of these new variables, we add a new equation, namely the branch equation for that (voltage source) element. The size of the matrix equation will grow, by as many equations as we have voltage sources, compared to nodal analysis. We can retain more currents (other currents) than just the voltage source currents [1]: • All voltage source currents, be they independent or controlled. • Any current that is a control variable for current control voltage source (CCVS) or current control current source (CCCS) • Any current that is a user-specified simulation output. We have a system of equations for the currents according NA, a system of equations for the other currents. Merging both obtained MNA system (8). A.Y . A 1 T AO AO u A.sU . Z O iO s I (8) iO – Vector of other branch currents AO – Node incidence matrix of other currents AOT – Transpose node incidence matrix of other currents ZO – Matrix of branch impedance of other currents Level of MNA is between STA, where no currents were eliminated, and NA, where all were eliminated. This creates a disadvantage of MNA. MNA doesn’t contain information about all currents and voltages in electric circuit [5]. SPARSE TABLEAU ANALYSIS. In this approach the only matrix operation required is that of repeatedly solving linear algebraic equations of fixed sparse structure. For non-linear circuits the partial derivatives and numerical integration are done at the branch level leading to complete generality and maximum sparsity of the characteristic coefficient matrix [6]. Sparse tableau analysis (STA) described in [6], involves the following steps: write KCL, write KVL in form (2) and write the element equations (1). The combination of these three sets of algebraic equations leads to the sparse tableau system (9) [1]. 0 i 0 A 0 0 E AT u 0 . Z Y 0 v s (9) This formulation has some key features consisting in fact that it can be applied to any circuit in a systematic fashion. The equations can be assembled directly from the input (circuit specification). The coefficients matrix is very sparse with mostly zero elements, although it is larger in dimension than the MNA matrix [1]. STA makes possible a simple yet truly general purpose computer program. By combining the concept of variability type with sparse matrix techniques, this program can achieve practically optimum efficiency by choosing the order of elimination so as to minimize the total operations count required for simulation and/or automated optimization [6]. COMPARISON OF METHODS. In this paper, have been studied options of circuit analysis for computer simulation. All investigated methods of analyses could be divided into three groups. First is group of methods that are unsuitable for computer simulation, the second one is group of methods fewer suitable and the last one is group of methods that are very suitable for computer simulation. Unsuitable for computer simulation is superposition theorem. It is not possible to formulate circuit equations using these methods automatically. This method (called "ad hoc") is inappropriate to analysis of all circuits. Instead, we need a systematic and automatic approach for formulating and solving the circuit equations. Into group of methods fewer suitable were included Kirchhoff's laws analysis, nodal analysis and mesh analysis. They can be used for computer simulation, but their algorithm will be more complicated and / or must be used for circuits with restrictions. MA has both disadvantages. Circuit must be planar and relationships must be defined between branches and nodes and moreover between branches and meshes. KVL has the same complications as MA with circuit description. NA isn't a complex method because it has a problem with the voltage sources and the algorithm for solving all types of networks is tedious and difficult. Unlike previous methods, the methods in the group of methods very suitable can be used for a general solution to electrical circuits. It is a modified nodal analysis and sparse tableau analysis. MNA is more compact than STA, preparation and creation of analysis is faster than STA. MNA doesn’t contain information about all cur- PAPER PREPARATION GUIDELINES rents and voltages in electric circuit. Handicap of STA is that preparation of STA analysis is slower than MNA, but STA contains information about all voltages and currents. ACKNOWLEDGMENT. The paper has been prepared under support of Slovak grant projects KEGA No. 005TUKE-4/2012. REFERENCES 1. Najm F. N. – Circuit Simulation, John Wiley & Sons, Inc., Hoboken, United States of America, 2010, ISBN 978-0-470-53871-5. 2. Mayer D. – Úvod do teorie elektrických obvodů, SNTL / ALFA, Prague, Czech Republic, 1978, ISBN 04–536–78. 3. Šimko V., Kováč D.: Učebné texty z teoretickej elektrotechniky I, elfa, Košice, Slovakia, 1998, ISBN 80-88786-79-7. 4. Ho Ch., Ruehli A. E., Brennan P.A. – The modified Nodal Approach to Network Analysis. IEEE Transactions on circuit and systems, 1975 5. Vansáč M., Vince T. – Sparse tableau analysis of electrical circuits, XIV International PhD Workshop OWD, Poland, 2012. 6. G.D, Brayton R.K., Gustavson F.G.: The Sparse Tableau Approach to Network Analysis and Design. IEEE Transactions on circuit theory, 1971. СРАВНЕНИЕ МЕТОДОВ МОДЕЛИРОВАНИЯ ЭЛЕКТРИЧЕСКИХ ЦЕПЕЙ Д. Дзиак, И. Томчикова Технический университет Кошице ул. Парк Коменскего, 3, м. Кошице, 04200, Словакия. Е-mail: [email protected], [email protected] В статье рассмотрено несколько методов анализа электрических цепей. Определены достоинства и недостатки каждого из рассмотренных методов и проведена оценка возможности их применения для задач моделирования электрических цепей. В статье выделена процедура сетевого анализа и процедура анализа для каждого метода. Также представлены критерии сравнения методов. Методы моделирования электрических цепей, представленные в данной статье, были сравнены согласно предложенного критерия. Ключевые слова: моделирование электрических цепей, компьютерное моделирование, сетевой анализ.