19-SOC's I - Second Order Circuits Second Order Circuits I...

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econd Order Circuits I Second Order Circuits I © Fred Terry all, 2008 Fall, 2008 1
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Second Order Circuits • Exactly Two Energy Storage Elements (2 L’s, 2 C’s, or 1 L & 1 C) •R e s istors, Independent and Dependent Sources • Result with be a Second Order Ordinary ifferential Equation in t Differential Equation in t • Useful for Understanding/Modeling Real Circuits & Other Physical Systems (e.g. Transmission yy ( g Lines) • Useful for Filter Circuits 2
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General Approach • Use KVL, KCL, Circuit Element Laws to find differential equation for circuit problem • Find the Initial Conditions (same as boundary conditions) – Must Have 1 Initial Condition Per Energy Storage Element i d th t lSlt i l ti ) Find the Natural Solution (homogenous solution) – This is the same as the unforced circuit response – Will Have 2 Independent Solutions • Find the Forced Solution (particular solution) • Add the Forced+Natural Solutions to get the Complete Solution • Eliminate the Unknowns (2 unknowns in a SOC) using the Initial Conditions 3
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Simplest Possible SOC: d d L C Undamped LC Assume Initial Conditions L I L (t) C I C (t) + V + V L V C (0)=V 0 )=0 - C - I L (0) 0 Find V C , I C for t ¥ 0 I C (t) t=0 L I L (t) C + V C + V L 4 - -
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Simplest Possible SOC I L (t) I C (t) Find Differential Equation L C + - V C + - V L 5
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Simplest Possible SOC I L (t) I C (t) No Forcing Source (Function) so Natural oln=Complete Soln L C + - V C + - V L Soln Complete Soln Find it 6
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Simplest Possible SOC I L (t) I C (t) Match Boundary Conditions L C + - V C + - V L 7
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Vc(t)/V0 IL(t)/(CV0w0) I L (t) I C (t) + + 1 L C - V C - V L 0.5 L 0 lize V C & I - 5 Norma 0.5 9 -1 02468 1 0 ω 0 t
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Power & Energy Vs. Time L I L (t) C I C (t) + V C + V L - - 10
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I L (t) I C (t) + + EC/(0.5*CV0)^2 EL/(0.5*CV0)^2 L C - V C - V L 0.6 0.8 1 ored Energy 0.2 0.4 ormalized St o 0 02468 1 0 No ω 0 t Vc(t)/V0 IL(t)/(CV0w0) 0.5 1 ze V C & I L -1 -0.5 0 0 Normali z 11 0 2 4 6 8 10 ω 0 t
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Series RLC: • Analysis of Important Case troduce the concept of damping Introduce the concept of damping • Illustration of General Methods d • Systematic look at the only solutions for 2 nd order systems • We will start with a natural decay problem • Then a step response problem 12
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Natural Response L i(t) +- V L R Note i L =i C =i (Passive Sign Convention) ind Differential Equation for i(t) C + - V C Find Differential Equation for i(t) 13
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This note was uploaded on 01/31/2011 for the course EECS 215 taught by Professor Phillips during the Spring '08 term at University of Michigan.

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19-SOC's I - Second Order Circuits Second Order Circuits I...

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