Spring 2021
SEEE
Lecture #2
Natural and Step Responses
of RLC Circuits
Chapter #8
Text book:
Electric Circuits
James W. Nilsson & Susan A. Riedel
9th Edition.
link: http://blackboard.hcmiu.edu.vn/
to download materials
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Objectives
Be able to determine the natural response and the step response of
parallel RLC circuits.
Be able to determine the natural response and the step response of
series RLC circuits.
Outline
The natural and step response of a parallel RLC circuit
The natural and step response of a series RLC circuit
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General
The natural response:
The response that arise when stored energy in an inductor or
capacitor is suddenly released.
The step response:
The response that arise when energy is being acquired by an
inductor or capacitor due to sudden application of a dc voltage or
current source.
Second order circuits (RLC circuits):
Circuits where voltages and currents are described by second-order
differential equations.
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A second-order circuit is
characterized by a second-order
differential equation.
• A second order circuit consists of resistors and the
equivalent of two energy storage elements.
• Two initial conditions (boundary conditions) are
needed for solving a second-order differential
equation, and those initial conditions can often be
obtained via circuit analysis.
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Finding Initial Values
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Finding Initial Values (cont.)
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Finding Final Values
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Example 1
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Example 1 – Sol.
12
i (0 ) =
= 2 A,
4+2
i (0 + ) = i (0 − ) = 2 A
−
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v(0− ) = 2i(0 − ) = 4 V
v (0 + ) = v (0 − ) = 4 V
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Example 1 – Sol.
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Example 1 – Sol.
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Natural and step responses of RLC circuits
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Natural response
Step response
of a parallel RLC circuit.
of a parallel RLC circuit.
Natural response
Step response
of a series RLC circuit.
of a series RLC circuit.
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Natural responses of a parallel RLC circuit
Or the Source-Free Parallel RLC Circuit
vdt
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d 2i 1 di
i
+
+
=0
2
dt
RC dt LC
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Natural responses of a parallel RLC circuit
Characteristic equation
v = Aest
, where A and s are unknown constants
Characteristic
equation
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Natural responses of a parallel RLC circuit
Damping affects the way the voltage response reaches its final (steady-state)
value.
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Natural responses of a parallel RLC circuit
The over-damped response
Case of:
2
0
2
s1, s2, will be real & distinct
Then: v(t ) = A1e 1 + A2e
st
s2t
To solve for A1 & A2: v(0+) = A1 + A2
Besides,
dv(0+ )
and
= s1 A1 + s2 A2
dt
dv(0+ ) iC (0+ )
=
dt
C
iC (0 + ) = −
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V0
− I0
R
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Natural responses of a parallel RLC circuit
The over-damped response
Procedure for finding the over-damped response v(t):
(1) Find the roots of the characteristic equation, s1 and s2, using the
values of R, L, and C.
(2) Find v(0+) and dv(0+)/dt using circuit analysis.
(3) Find the values of A1 and A2 by solving simultaneously equations:
( )
dv(0 ) i (0 )
=
=s A +s
v 0 + = A1 + A2
+
+
C
dt
C
1
1
2
A2
(4) Substitute the values for s1, s2, A1, and A2 into equation
v = A1es1t + A2es2t, to determine the expression for v(t) for t ≥ 0.
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Natural responses of a parallel RLC circuit
The under-damped response
The under-damped response:
v(t ) = B1e − t cos
d
t + B2 e − t sin
d
t
The process for finding the under-damped
response is the same as that for the overdamped response, although the response
equations and the simultaneous equations used
to find the constants are slightly different.
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Natural responses of a parallel RLC circuit
The critically damped response
When a circuit is critically damped,
the response is on the verge of
oscillating.
It rarely happens in practical.
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under-damped = sinusoidal + exponential decay
over-damped = exponential decay
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The step response of a parallel RLC circuit
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i L + i R + iC = I
v
dv
iL + + C
=I
R
dt
d 2iL
L di L
iL +
+ LC 2 = I
R dt
dt
d 2iL
iL
1 di L
I
+
+
=
RC dt LC LC
dt 2
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If & Vf represent the final value of the response function.
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The natural response of a series RLC circuit
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d 2v R dv v
+
+
=0
2
dt
L dt LC
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The natural response of a series RLC circuit
The procedure for finding the natural or step responses of a series RLC
circuit are the same as those used to find the natural or step responses
of a parallel RLC circuit.
The neper frequency
Resonant radian frequency
Characteristic
equation
The roots
The current response will be overdamped, underdamped, or
critically damped according to whether
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The step response of a series RLC circuit
Applying Kirchhoff’s voltage law
Capacitor voltage step
response forms in
series RLC circuits
3 possible
solutions for vC
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Example 2
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Finding the Roots of the Characteristic
Equation of a Parallel RLC Circuit
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Example 2: Solution
The roots of the characteristic equation:
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Example 2: Solution
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Example 2: Solution
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Example 3
Finding the Overdamped Natural
Response of a Parallel RLC Circuit
For the circuit, v(0+) = 12V, iL(0+) = 30mA
a) Find the initial current in each branch of the circuit.
b) Find the initial value of dv/dt.
c) Find the expression for v(t).
d) Sketch v(t) in the interval 0 ≤ t ≤ 250 ms.
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Example 3: Solution
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Example 3: Solution
Because the roots are real and distinct, we know that the
response is overdamped. We find the co-efficients A1 & A2
➔The overdamped voltage response:
(ms)
d) plot of v(t) versus t over the interval
0 t 250 ms
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Example 4
Solution:
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Example 5
Finding the Underdamped Natural
Response of a Parallel RLC Circuit
For the circuit, V0 = 0 V, I0 = -12.25 mA
a) Calculate the roots of the characteristic equation.
b) Calculate v and dv/dt at t = 0+.
c) Calculate the voltage response for t ≥ 0.
d) Plot v(t) for the time interval 0
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t
11ms.
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Example 5: Solution
a) Because
Therefore, the response
is underdamped
complex frequencies
b) Because v is the voltage across the terminals of a capacitor, we have v(0) = v(0+) = V0 = 0.
Because v(0+) = 0, the current in the resistive branch is zero at t = 0+. Hence the current in
the capacitor at t = 0+ is the negative of the inductor current: iC(0+) = -(-12.25) = 12.25 mA.
Therefore the initial value of the derivative is
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Example 5: Solution
d) The figure shows the plot of versus t for the
first 11 ms after the stored energy is released. It
clearly indicates the damped oscillatory nature
of the underdamped response. The voltage
approaches its final value, alternating between
values that are greater than and less than the
final value. Furthermore, these swings about the
final value decrease exponentially with time.
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Example
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Example 6
Finding the Overdamped Step
Response of a Parallel RLC Circuit
The initial energy stored in the circuit in Fig. is zero. At t = 0
a dc current source of 24 mA is applied to the circuit. The
value of the resistor is 400 .
a) What is the initial value of iL?
b) What is the initial value of diL/dt?
c) What are the roots of the characteristic equation?
d) What is the numerical expression for iL(t) when t 0?
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Example 6 Solution
a) No energy is stored in the circuit prior to the application of the dc current source,
so the initial current in the inductor is zero. The inductor prohibits
an instantaneous change in inductor current; therefore iL(0) = 0 immediately after
the switch has been opened.
b) The initial voltage on the capacitor is zero before the switch has been
opened; therefore it will be zero immediately after. Now, because
v = LdiL/dt,
c) From the circuit elements, we obtain
roots of the
characteristic
equation are real and
distinct
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Example 6 Solution
d) Because the roots of the characteristic equation are real and distinct,
the inductor current response will be overdamped. Thus iL(t) takes the
form of
Inductor current in overdamped parallel
RLC circuit step response
Hence, from this solution, the two simultaneous equations that determine A’1and
A’2 are
The numerical solution for iL(t) is
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Example 7
Finding the Underdamped Step
Response of a Series RLC Circuit
No energy is stored in the 100 mH inductor or the 0.4 F
capacitor when the switch in the circuit shown in Fig. is
closed. Find vc(t) for t ≥ 0.
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Example 7: Solution
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Example 7: Solution
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Example 8
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Example 8: Sol.
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Example 8: Sol.
v(t) = -0.2083e-t + 5.208e-50t
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Example 8: Sol.
v(t) = (5 – 50t)e-10t
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Example 8: Sol.
v(t) = (5cos6t – 6.667sin6t)e-8t
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Example 8: Sol.
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