L-11(GDR)(ET) ((EE)NPTEL) - Module 3 DC Transient Version 2...

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Module 3 DC Transient Version 2 EE IIT, Kharagpur
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Lesson 11 Study of DC transients in R-L-C Circuits Version 2 EE IIT, Kharagpur
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Objectives Be able to write differential equation for a dc circuits containing two storage elements in presence of a resistance. To develop a thorough understanding how to find the complete solution of second order differential equation that arises from a simple RLC circuit. To understand the meaning of the terms (i) overdamped (ii) criticallydamped, and (iii) underdamped in context with a second order dynamic system. Be able to understand some terminologies that are highly linked with the performance of a second order system. L.11.1 Introduction In the preceding lesson, our discussion focused extensively on dc circuits having resistances with either inductor ( ) or capacitor ( ) (i.e., single storage element) but not both. Dynamic response of such first order system has been studied and discussed in detail. The presence of resistance, inductance, and capacitance in the dc circuit introduces at least a second order differential equation or by two simultaneous coupled linear first order differential equations. We shall see in next section that the complexity of analysis of second order circuits increases significantly when compared with that encountered with first order circuits. Initial conditions for the circuit variables and their derivatives play an important role and this is very crucial to analyze a second order dynamic system. L C L.11.2 Response of a series R-L-C circuit due to a dc voltage source Consider a series RL circuit as shown in fig.11.1, and it is excited with a dc voltage source C −− s V . Applying around the closed path for , KVL 0 t > () c di t LR i t v t dt ++= s V (11.1) The current through the capacitor can be written as Version 2 EE IIT, Kharagpur
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() c dv t it C dt = Substituting the current ‘ ’expression in eq.(11.1) and rearranging the terms, 2 2 cc c dv t d v t s L CR C v t dt dt ++ V = (11.2) The above equation is a 2 nd -order linear differential equation and the parameters associated with the differential equation are constant with time. The complete solution of the above differential equation has two components; the transient response and the steady state response . Mathematically, one can write the complete solution as cn vt cf ( 12 tt nc f vt v t v t Ae A αα =+ = + + ) (11.3) Since the system is linear, the nature of steady state response is same as that of forcing function (input voltage) and it is given by a constant value . Now, the first part of the total response is completely dies out with time while and it is defined as a transient or natural response of the system. The natural or transient response (see Appendix in Lesson-10) of second order differential equation can be obtained from the homogeneous equation (i.e., from force free system) that is expressed by A 0 R > 2 2 () 0 c d LC RC v t dt dt = 2 2 1 c d R dt L dt LC ⇒+ + = 2 2 c d ab c v t dt dt ++= (where 1 1, R a b and c LL == = C ) (11.4) The characteristic equation of the above homogeneous differential equation (using the operator 2 2 2 , dd
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This note was uploaded on 08/03/2010 for the course ELECTRICAL EE212 taught by Professor Shetty during the Spring '10 term at International Institute of Information Technology.

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L-11(GDR)(ET) ((EE)NPTEL) - Module 3 DC Transient Version 2...

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