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Zack Christensen Elementary Applications of Linear Algebra in Circuit Analysis For the beginning student of electrical engineering, or those interested in learning the basic working of a circuit, the task of analyzing even the simplest of circuits can seem daunting. However, once it is known that you can organize the information into a system of equations that can them be solved by linear algebra the task becomes far less intimidating. The resistor network. Beginning with the simplest collection of components we will construct a working model through which the process can be explained. Consider this circuit where V is some constant voltage source; R1, R2 and R3 are resistors of some known value; and the labeled points a nodes with “ref.” being the reference node. The object is to find the voltage at every node except the reference node. To do this first we apply Ohm’s law and Kirchoff’s Current Law [1] to each node. This gives us the equations: v1 V v 2 v1 v 2 v3 0 R1 R2 v3 v 2 v3 0 R2 R3 This can then be arranged in matrix for: 0 0 v1 V 1 R 2 R1 R 2 R1 v2 0 . 0 R3 R 2 R3 v3 0 And then these equations are solved giving the values v1 V , R 2( R 2 R3) * V R 2 * R3 * V , and v3 . This can be v2 ( R1 R 2)( R 2 R3) R1 * R3 ( R1 R 2)( R 2 R3) R1 * R3 applied to any resistive network so long as the equations are written correctly. Time dependent networks. With the introduction of circuit elements like inductor coils and capacitors voltage in a circuit no longer remains constant with a direct current source, it will vary over time. Let us look at our example circuit from before but this time we will replace R2 with an inductor with value L and R3 with a capacitor of value C. Zack Christensen Elementary Applications of Linear Algebra in Circuit Analysis Now we have a simple RLC circuit; with R standing for the resistor, L for the inductor, and C for the capacitor; where we will apply Ohm’s Law; the current equations for an dv 1 inductor and a capacitor, il vl dt and ic C c ; and Kirchoff’s Current Law [1]. L dt Applying these we get the equations: v1 V v 2 v1 1 v 2 v3 dt 0 R L dv 1 v3 v 2 dt C 3 0 L dt These equations can be written with indefinite intervals because in the next step they are going to be differentiated and written in matrix form so that a known time interval is not needed. Now rewrite the equations so that they involve only derivatives and put them in matrix form. 1 v 0 0 1 V d R d R v2 0 . L dt L dt 1 1 d 2 v3 0 0 LC LC dt 2 As it stands this set of equations cannot be solved because it will produce differential equations and there have been no initial conditions set forth. The method, though, is the same as it was for a resistor network only now incorporating differential equations to produce a more powerful tool for the analysis of simple electronic circuits. References 1. David Cunningham and John Stuller, Circuit Analysis, Houghton Mifflin, 1995.