1. RLC Circuits Dr. Iyad Jafar
Chapter 9
RLC Circuits
Dr. Iyad Jafar

2. RLC Circuits Analyze circuits containing R, L and C together
Series RLC
Parallel RLC
Similar to RL and RC circuits, RLC circuits has two parts
Source free or natural response
Forced / steady state response
Forced Response  a step input causes a step output.
Natural Response  Different and more difficult than RL, RC.

3. Source-free RLC Circuits
We study the natural response by studying source-free RLC circuits.
Parallel Source-free RLC Circuit
Second-order
Differential equation

4. This second-order differential equation can be solved by assuming solutions
The solution should be in form of
If the solution is good, then substitute it into the equation will be true.
which means
s=??

5. Use quadratic formula, we got
Both
and
are solution to the equation
Therefore, the complete solution is

6. From
Define resonant frequency
Damping factor
Therefore,
in which we divide into 3 cases according to the term inside the bracket

7. Solution to Second-order Differential Equations
α > ω0 (inside square root is a positive value) Overdamped case
α = ω0 (inside square root is zero) Critical damped case
α < ω0 (inside square root is a negative value) Underdamped case

8. 1. Overdamped case , α > ω0
Example: find v(t) if the initial conditions are vc(0) = 0, iL(0) = -10A
α > ω0 ,therefore, this is an overdamped case
 s1 = -1, s2 = -6

9. Then, we will use initial conditions to find A1, A2
From vc(0) = 0 we substitute t=0
… (1)
From KCL  At t =0+
… (2)

10. Solve the equations (1) and (2) A1 = 84 A2 = -84 and the solution is
v(t) t

11. 2. Critical damped case , α = ω0
Example: find v(t) if the initial conditions are vc(0) = 0, iL(0) = -10A
α = ω0 , this is an critical damped case
 s1 = s2 = -2.45
The complete solution of this case is in form of

12. Then, we will use initial conditions to find A1, A2
From vc(0) = 0 we substitute t=0
… (1)
Therefore A2 =0 and the solution is reduced to
Find A1 from KCL at t=0
… (2)

13. Solve the equation and we got A1 = 420 and the solution is
v(t) t

14. 3. Underdamped case , α < ω0
from
The term inside the bracket will be negative and s will be a complex number
define
Then
and

15. Use Euler’s Identity

16. 3. Underdamped case , α < ω0
Example: find v(t) if the initial conditions are vc(0) = 0, iL(0) = -10A
α < ω0 ,therefore, this is an underdamped case and v(t) is in form
where

17. Then, we will use initial conditions to find B1, B2
From vc(0+) = vc(0-) = 0 we substitute t=0
Therefore B1 =0 and the solution is reduced to
Find B2
from KCL
at t=0

18. we got and then the solution is
v(t) t

19. Series RLC
Overdamped
Critical damped
Underdamped
Solution

20. Forced RLC Circuits X(t) = Xn(t) + Xf(t)
Similar to RL and RC circuits, the total response is the sum of the transient/natural response and the forced/steady state response
X(t) = Xn(t) + Xf(t)
Where Xn is one of the three cases
Overdamped
Critical damped
Underdamped
And Xf(t) is the value as t ∞ and the constants are found from the initial conditions

21. Example
Find vc(t)
This is overdamped case, so the solution is in form

22. Consider the circuit we found that the initial conditions will be
vC(0) = 150 V
iL(0+) = iL(0-) = 5 A
And the steady state values
vC(∞) = 150 V
iL(∞) = 9 A
Using vC(0) = 150 V , we got
Using vC(∞) = 150 V , we got
Therefore,
1. Vf = 150
2. A1+A2 = (1)

23. Next, use KCL on the circle below
Use the initial condition iL(0) = 5 A, we have to change vc(t) to iL(t)
From
Next, use KCL on the circle below or

24. From
Substitute iL(0) = 5A
(2)
From (1), (2) A1 = 13.5, A2 = -13.5
Therefore,