RLC Circuits Natural Response ECE 201 Circuit Theory I

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

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

ECE 201 Circuit Theory I

Characteristic Equation Look at the roots ECE 201 Circuit Theory I

Solved by ECE 201 Circuit Theory I

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

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

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

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

Initial Value of dv/dt ECE 201 Circuit Theory I

Initial Value of Capacitor current + V0 - I0 iC(0+) ECE 201 Circuit Theory I

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

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

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

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

ECE 201 Circuit Theory I

ECE 201 Circuit Theory I

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