Fall 2018
SEEE
Lecture #1
Response of First-Order
RL and RC Circuits
Chapter #7
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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Fall 2018
SEEE
Survival Skills
Try to attend every lecture. Supplements might be
provided during class.
Raise your questions if you don’t understand.
It’s not that you just get a ‘pass’. Try to have fun while
learning and practicing.
Take pride in yourself. Never cheat (you don’t need that).
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Fall 2018
EE Courses Overview
Freshman
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Physical
Physical
Training 1
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Academic
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English 1
English 2
Chemistry for
Engineers
Critical
Thinking
Physics 1
Physics 2
Calculus 1
Calculus 2
Introduction
to Electrical
Engineering
22 credits
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Ho Chi Minh’s
Thought
Introduction
to Computer
for Engineers
20 credits
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RF Design
General
Elective
1 Antenna and Microwave Engineering
2 RF Circuit Design
3
4
5
6
Electronics & Embedded Systems
Note:
Each student has to
select six elective
courses, where two of
the elective courses are
compulsory for each
specialization.
1 Analog Electronics
2 Embedded Real-time Systems
3
4
5
Physics 3
Calculus 3
Applied Linear
Algebra
Processes
Principles of
Marxism
SEEE
Physics 4
6
Wireless Communications
1 Computer & Comm. Networks
2 Telecommunication Networks
3
4
5
6
Probability
and Random
Processes
Differential
Equations
Processes
Signal Processing
Co- requisite
Lab
Mandatory
Courses
1 Control Systems
2 Embedded Real-time Systems
3
4
5
6
Revolutionary
Lines of VCP
Programming
for Engineers
Electronics
Devices
Principles of
EE1
Principles
of EE2
Signals &
Systems
Digital Signal
Processing
Entrepreneurship
Digital Logic
Design
Electromagnetic Theory
processor
Systems
Principles of
Communications
Senior
20 credits
15 credits
17 credits
23 credits
Prerequisite
Summer
Internship
14 credits
Thesis
10 credits
Fall 2018
SEEE
Objectives
Be able to determine the natural response of both RL and RC
circuits.
Be able to determine the step response of both RL and RC circuits.
Know how to analyze circuits with sequential switching.
Outlines
The natural response of an RL circuit & an RC
circuit
The step response of RL & RC circuits
Sequential switching
Unbounded response
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General Concepts
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.
First order circuits (RL or RC circuits):
Circuits where voltages and currents are described by first-order
differential equations.
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Four possible first order circuits
L or C connected to a
Thevenin equivalent
L or C connected to a
Norton equivalent
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The natural response of an RL circuit
The switch is closed for a long time and opened at t = 0
di
dt
t≤0
0
v=0
(short circuit)
All the source current I0 appears in the inductive branch
t≥0
Apply KVL:
L
di
dt
Ri
0 (first order differential equation)
the current cannot change instantaneously in a inductor
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The natural response of an RL circuit
Since the current cannot change instantaneously in an inductor
i0
it
The voltage across the resistor using Ohm’s law
v
iR
v0
I 0 Re
R/L t
0
t
0
v0
i0
I 0e
I0
R/L t
t≥0
The energy delivered to the resistor
during any interval of time after the
switch has been opened
I0R
The power dissipated in the resistor
p
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iv
I 02 Re
2 R/L t
t
0
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The time constant ( )
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The time constant ( )
Interpretation of the time constant of the RL circuit
Summary:
1) Find the initial current, I0 , through the inductor
2) Find the time constant of the circuit,
3) Use I0eT.V.Su
/t
= L/R
, to generate i(t) from I0 and .
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Example 1
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Example 1 - Solution
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Example 1 – Solution (cont)
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Example 1 – Solution (cont)
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Example 2
a)
b)
c)
d)
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Find i1 , i2 and i3 .
Calculate the initial energy stored in the parallel inductors.
Calculate the energy stored in the inductor as t
∞
Show that the total energy delivered to the resistive network equals
to the difference between the result obtained in (b) and (c).
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Example 2 – Solution
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Example 2 – Solution (cont)
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Example 2 – Solution (cont)
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The natural response of an RC circuit
Assume the switch has been in position a for a long time:
dv
dt
t≤0
i=0
0
(open circuit)
vC = Vg
t≥0
vt
Apply node voltage technique:
V0 e
t/
t
0
the voltage cannot change instantaneously in a capacitor
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The natural response of an RC circuit
The current goes through the resistor
vt
R
it
V0
e
R
t/
0
t
The power dissipated in the resistor
V02
vi
e
R
p
2 t/
t
0
The energy delivered to the resistor
t
w
pdt
0
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V02
e
R
0
t
2 t/
dt
1
CV02 1 e
2
2 t/
t
0
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Example 3
Find:
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Example 3 (cont)
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Example 4
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Example 4 (cont)
b) Calculate the initial energy stored in the capacitor C1 and C2
c) Calculate how much energy is stored in the Capacitors as t
∞
d) Show that the total energy delivered to the 250 kΩ resistor is the difference
between the results obtained in (b) and (c)
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The step response of an RL circuit
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The step response of an RL circuit
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The step response of an RL circuit
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Example 5
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Example 5 (cont)
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The step response of an RC circuit
SEEE
Apply KCL:
C
vC
R
dv
dt
vC t
it
it
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Is
IsR
C
dvC
dt
Is
V0
C V0
V0
e
R
IsR e
IsR
t / RC
t / RC
1
e
RC
, t
,
t
0
t / RC
0
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Example 6
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Example 6 (cont)
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Example 6 (cont)
e
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A general solution for natural & step responses
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A general solution for natural & step responses
t t0
xt
xf
x t0
xf e
x(t) the unknown variable as a function of time
xf the final value of the variable
x(t0) the initial value of the variable
t0 time of switching
Τ time constant
Procedure:
1) Identify the variable of interest of the circuit. For RC circuits, it is best to
choose vC ; for RL circuit, it is best to choose iL.
2) Determine the initial value of the variable.(vc(t0) in case of RC circuit and
iL(t0) in case of RL circuits)
3) Calculate the final value of the variable (value at t = ∞)
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4) Calculate the time constant for the circuit.
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Fall 2018
Sequential Switching
SEEE
Switching occurs more than once in a circuit.
The time reference for switching cannot be t = 0.
Procedure for sequential switching problem
(1) Obtain the initial value x(t0)
(2) Apply the techniques described previously to find current and
voltage value.
(3) Redraw the circuit that pertains to each time interval and repeat
step (1).
Note: Since inductive current IL and capacitive voltage VC cannot
change instantaneously at the time of switching, these value
should be solved first for sequential switching problem.
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Example 8
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Example 8 (cont)
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Example 8 (cont)
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Example 8 (cont)
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Fall 2018
Unbounded Response
SEEE
A circuit response may grow, rather than decay, exponentially
with time.
This type of response is called an unbounded response.
It may happen when the circuit contains dependent source.
In this case, the Thevenin equivalent with respect to the
terminals of either an inductor or a capacitor may be
negative, which resulting in a negative time constant.
To solve the circuit which have unbounded response, we
need to derive the differential equation that describes the
circuit containing the negative Rth.
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Fall 2018
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Example 9
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