RLC Circuits Natural Response ECE 201 Circuit Theory

RLC Circuits Natural Response ECE 201 Circuit Theory I 1

Parallel RLC Circuit i. C + V 0 i. L i. R I 0 - + v - ECE 201 Circuit Theory I 2

Parallel RLC Circuit i. C + V 0 i. L i. R I 0 - + v - ECE 201 Circuit Theory I 3

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Characteristic Equation Look at the roots ECE 201 Circuit Theory I 5

Solved by ECE 201 Circuit Theory I 6

The general solution is given by The circuit behavior is determined by the values of s 1 and s 2. Rewrite them as Neper Frequency Resonant Radian Frequency ECE 201 Circuit Theory I 7

s 1 and s 2 are complex frequencies There are three possible outcomes for the roots – Real, distinct roots when ω02 < α 2 “overdamped” Complex conjugate roots when ω02 > α 2 “underdamped” Real and equal roots when ω02 = α 2 “critically damped” ECE 201 Circuit Theory I 8

Overdamped Response • Real, distinct roots • Solution has the form • Where s 1 and s 2 are the roots of the characteristic equation • A 1 and A 2 are determined by initial conditions ECE 201 Circuit Theory I 9

The Solution If s 1 and s 2 are known determine A 1 and A 2 from Initial Voltage on the Capacitor Rate of change of the initial Capacitor voltage ECE 201 Circuit Theory I 10

Initial Value of dv/dt ECE 201 Circuit Theory I 11

Initial Value of Capacitor current + i. C(0+) V 0 I 0 - ECE 201 Circuit Theory I 12

Example 8. 2 page 272 • For the circuit shown, v(0+) = 12 Volts, and i. L(0+) = 30 m. A. i. C 0. 2 μF + i. R i. L 50 m. H 200 Ω v - ECE 201 Circuit Theory I 13

Find the initial current in each branch • For the inductor, i. L(0 -) = i. L(0+) = 30 m. A • For the resistor, i. R(0+) = 12 V/200Ω = 60 m. A • For the capacitor, i. C(0+) = -i. L(0+) – i. R(0+), or i. C(0+) = -30 m. A -60 m. A = -90 m. A ECE 201 Circuit Theory I 14

Find the initial value of dv/dt ECE 201 Circuit Theory I 15

Find the expression for v(t) Roots are real and distinct, therefore overdamped ECE 201 Circuit Theory I 16

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Sketch v(t) for 0<= t <= 250μs ECE 201 Circuit Theory I 19