BASICS OF ELECTRICAL CIRCUITS
EHB 211 E
RLC Circuits
Asst. Prof. Onur Ferhanoğlu
1
Natural Response of RLC circuits
Assume, the solution is in the exponential form
Rational assumption, since higher derivatives
yield to a similar form with the function itself
Natural response: KCL
Take derivative
Don’t use A = 0 as a solution -> voltage is 0
Characteristic equation
2nd order
Differential Equation
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
2
Natural Response of RLC circuits
Satisfies the equation,
regardless of the A value
Their sum is also a solution!
Characteristic equation
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
Natural response
Initial conditions
determine A1, A2
3
Natural Response of RLC circuits
Characteristic equation
Real roots -> overdamped response
Complex roots -> underdamped response
Critically damped response
Damping: the way voltage response
reaches its final value
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
4
Overdamped voltage response
Finding v(0+) and dv(0+)/dt:
KCL
dv(0+)/dt:
Finding overdamped response:
1. Find roots (s1,s2) using RLC values
2. Find v(0+) and dv(0+)/dt:
3. Find A1, A2 by solving:
(2 eq. 2 unknowns)
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
5
Example 1
For the circuit, v(0+) = 12 V, iL(0+) = 30mA
a) Find initial current at each branch
b) Find initial value of dv/dt
c) Find v(t)
d) Sketch v(t)
a)
b)
v(0+) = 12 V (capacitor voltage)
iR= 12/200 = 60mA
KCL:
iC = -90 mA (iL + iR + iC = 0)
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
Before time 0, dv/dt = 0
(stationary voltage)
6
Example 1 continued..
For the circuit, v(0+) = 12 V, iL(0+) = 30mA
a) Find initial current at each branch
b) Find initial value of dv/dt
c) Find v(t)
d) Sketch v(t)
c)
(Units of s1,s2 are inverse of time)
exponential terms are unitless
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
7
Example 1 continued..
For the circuit, v(0+) = 12 V, iL(0+) = 30mA
a) Find initial current at each branch
b) Find initial value of dv/dt
c) Find v(t)
d) Sketch v(t)
d)
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
8
Example 2
For the previous example, derive the branch currents:
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
9
Underdamped voltage response
Roots are complex
B1 = A 1 + A 2
B2 = j(A1 - A2)
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
10
Underdamped voltage response
B1 = A 1 + A 2
B2 = j(A1 - A2)
Asst. Prof. Onur Ferhanoğlu
The response is oscillatory (sin & cos)
wd : oscillation frequency
α: damping coefficient
(How fast the oscillations die)
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
11
Example 3
In the circuit V0 = 0, I0 = -12.25 mA
a) Calculate the roots of the
characteristic equation
b) Calculate v, dv/dt at t= 0+
c) Calculate v(t) for t > 0
d) Plot v(t)
a)
Underdamped
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
12
Example 3 continued…
In the circuit V0 = 0, I0 = -12.25 mA
a) Calculate the roots of the
characteristic equation
b) Calculate v, dv/dt at t= 0+
c) Calculate v(t) for t > 0
d) Plot v(t)
c) Across capacitor:
b) Across capacitor:
The current in resistive branch is 0 (V=I.R)
iC = -iL
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
13
Example 3 continued…
In the circuit V0 = 0, I0 = -12.25 mA
a) Calculate the roots of the
characteristic equation
b) Calculate v, dv/dt at t= 0+
c) Calculate v(t) for t > 0
d) Plot v(t)
d)
Underdamped response:
Damped & oscillatory
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
14
Critically damped voltage response
or
When roots are equal, the solution takes the
following form:
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
When two roots are equal,
this equation can not satisfy 1
constant (A0) with two independent
initial conditions (V0, I0)
15
Example 4
a) Find the value of R that results in
critically damped voltage
response
b) Calculate v(t)
c) Plot v(t)
b) Same as previous example
(given by the problem)
a)
(same as previous example)
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
16
Example 4 continued…
a) Find the value of R that results in
critically damped voltage
response
b) Calculate v(t)
c) Plot v(t)
c)
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
17
Step response of a parallel RLC circuit
Indirect approach
Express
in terms of v
KCL
differentiate
3 possible solutions
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
18
Step response of a parallel RLC circuit II
Direct approach
The solution for a 2nd order D.E with a
constant forcing function equals:
KCL
Final values of the response function
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
19
Example 5 - overdamped step response of parallel RLC
The initial stored energy is 0.
I = 24 mA. R = 400 Ω. Find
a) Initial value of iL
b) Initial value of diL/dt
c) Roots of the characteristic equation
d) iL(t)
c)
a) No energy stored for t< 0
iL(0-) = iL(0+) = 0
b) vC(0-) = vC(0+) = 0
= vC
d) Overdamped response is of the form:
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
20
Example 5 - continued..
The initial stored energy is 0.
I = 24 mA. R = 400 Ω. Find
a) Initial value of iL
b) Initial value of diL/dt
c) Roots of the characteristic equation
d) iL(t)
d)
Asst. Prof. Onur Ferhanoğlu
Inductor will become a short circuit, and
final value of its current If = I = 24 mA
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
21
Example 6 - underdamped step response of parallel RLC
The initial stored energy is 0.
I = 24 mA. R = 625 Ω. Find
=
Same as before, does not change with R
iL(0+) = 0 from prev. question
from prev. question
Complex roots, underdamped
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
22
Example 7 - critically damped step response of parallel RLC
The initial stored energy is 0.
I = 24 mA. R = 500 Ω. Find
Same as before, does not change with R
Critically damped
We know that both
equals 0 from prev.
question
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
23
COMPARISON
For all cases:
iL(0) = 0, if = 24 mA
• Underdamped overshoots the final value
• Overdamped & Critically damped always
stay below if
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
24
Natural response of series RLC
KVL
differentiate
rearrange
Parallel RLC equation
Assume similar solution
Similarly with the
parallel RLC circuit
Different from
parallel RLC
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
25
Step response of series RLC
KVL
Similar to parallel RLC
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
26
Example 8 – underdamped natural response of series RLC
Initial voltage on capacitor is 100V.
a) Find i(t)
b) Find vC(t)
underdamped
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
27
Example 8 – continued…
Initial voltage on capacitor is 100V.
a) Find i(t)
b) Find vC(t)
This one is easier to handle 
Asst. Prof. Onur Ferhanoğlu
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
28
Example 9 – underdamped step response of series RLC
No energy stored in the inductor or the capacitor.
Find vC(t)
underdamped
Asst. Prof. Onur Ferhanoğlu
Initially no energy stored
Initial current is 0
(proportional to voltage
derivative)
RLC Circuits/ BASICS OF ELECTRICAL CIRCUITS
29