3-Lesson_Notes_Lecture10_ee205 - Lecture 10 State Equations...

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Lecture 10 State Equations and Computer aided circuit Analysis This lecture will cover the following topics. 9 Introduction to the zero-input circuit. 9 Definition of state variables. 9 The matrix state equation. After finishing this lecture, you should be able to 9 Understand the concept of the zero-input circuit. 9 Understand the definition of state variables. 9 Derive matrix state equation of a zero-input circuit Zero-Input Circuit : R L v C I L + _ Figure 11-1 Series RLC zero-input circuit.
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Let us consider the series RLC circuit of Figure 11-1. ¾ At t = 0 , Let i L (0) 0 and V c (0) 0 ¾ We can say that the “Condition” or “state” of the circuit at time t = 0 is specified by the inductor current and capacitor voltage. For this reason we call the pair of numbers (0), (0) [] c L i V the initial state of the circuit. ¾ Extending this concept, we can refer to the pair () , () c L i tVt ⎡⎤ ⎣⎦ as the state of the circuit at time t . ¾ The variables i L and v c as the are called state variables of the circuit. Applying KVL to this circuit, we obtain 0 L c L di LV R i dt ++ = (11.1) With c L dv iC dt = (11-2) Substituting equation (11-2) into equation (11-1) after division by LC, we get 2 2 0 cc c dv v d v CR LC LC dt dt = (11-3) 2 2 1 ( ) 0 c d v R v L LC dt dt ++= (11-4) Since 2 R L α = , 1 n LC ω = 2 2 2 20 nc d v v dt dt αω = (11-5) Let us rewrite equations (11-1) and (11-2), such that the coefficients of the derivatives are unity and the derivatives alone are on one side of the equation.
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1 L c L di R iv LL dt =−
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This note was uploaded on 01/16/2010 for the course EE ee205 taught by Professor Profsyed during the Spring '09 term at King Fahd University of Petroleum & Minerals.

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3-Lesson_Notes_Lecture10_ee205 - Lecture 10 State Equations...

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