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RLC Circuits Natural Response ECE 201 Circuit Theory I

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Parallel RLC Circuit iC iL iR + V0 - + v - I0 ECE 201 Circuit Theory I

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Parallel RLC Circuit iC iL iR + V0 - + v - I0 ECE 201 Circuit Theory I

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ECE 201 Circuit Theory I

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**Characteristic Equation**

Look at the roots ECE 201 Circuit Theory I

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Solved by ECE 201 Circuit Theory I

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**The general solution is given by**

The circuit behavior is determined by the values of s1 and s2. Rewrite them as Neper Frequency Resonant Radian Frequency ECE 201 Circuit Theory I

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**s1 and s2 are complex frequencies **

There are three possible outcomes for the roots – Real, distinct roots when ω02 < α2 “overdamped” Complex conjugate roots when ω02 > α2 “underdamped” Real and equal roots when ω02 = α2 “critically damped” ECE 201 Circuit Theory I

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**Overdamped Response Real, distinct roots Solution has the form**

Where s1 and s2 are the roots of the characteristic equation A1 and A2 are determined by initial conditions ECE 201 Circuit Theory I

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**The Solution If s1 and s2 are known determine A1 and A2 from**

Initial Voltage on the Capacitor Rate of change of the initial Capacitor voltage ECE 201 Circuit Theory I

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Initial Value of dv/dt ECE 201 Circuit Theory I

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**Initial Value of Capacitor current**

+ V0 - I0 iC(0+) ECE 201 Circuit Theory I

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Example 8.2 page 293 For the circuit shown, v(0+) = 12 Volts, and iL(0+) = 30 mA. + v - iR iC iL 50 mH 0.2 μF 200 Ω ECE 201 Circuit Theory I

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**Find the initial current in each branch**

For the inductor, iL(0-) = iL(0+) = 30 mA For the resistor, iR(0+) = 12V/200Ω = 60 mA For the capacitor, iC(0+) = -iL(0+) – iR(0+), or iC(0+) = -30 mA -60 mA = -90 mA ECE 201 Circuit Theory I

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**Find the initial value of dv/dt**

ECE 201 Circuit Theory I

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**Find the expression for v(t)**

Roots are real and distinct, therefore overdamped ECE 201 Circuit Theory I

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ECE 201 Circuit Theory I

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ECE 201 Circuit Theory I

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**Sketch v(t) for 0<= t <= 250μs**

ECE 201 Circuit Theory I

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