1. RLC Circuits
Natural Response
ECE 201 Circuit Theory I

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

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

4. ECE 201 Circuit Theory I

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

6. Solved by
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7. 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

8. 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

9. 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

10. 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

11. Initial Value of dv/dt
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12. Initial Value of Capacitor current+ V0 - I0
iC(0+)
ECE 201 Circuit Theory I

13. 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

14. 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

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

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

17. ECE 201 Circuit Theory I

18. ECE 201 Circuit Theory I

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