VGU 61ECE205 - Analog Signals & 21 & 28 September & 5 October 2023

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VIETNAMESE-GERMAN UNIVERSITY
Electrical and Computer Engineering Study Program

Analog Signals & Systems

Udo Klein

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.1
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Lecture Topic 1

Response of First-Order
RL and RC Circuits

Analog Signals & Systems

Udo Klein

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.2
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Course Contents
• Response of first-order RL and RC circuits
• Response of second-order RLC circuits
• Introduction to the Laplace transform
• Application of the Laplace transform in circuit analysis
• Frequency selective circuits
• Fourier Series
• Introduction to the Fourier transform
• …

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.3
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Objectives
After today’s class you will be able to
• determine the natural response of both RL and RC circuits.
• determine the step response of both RL and RC circuits.
• analyze circuits with sequential switching.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.4
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Outline

• the natural response of an RL circuit

• the natural response of an RC circuit

• the step response of RL and RC circuits

• sequential switching

• unbounded response

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.5
Autumn 2023 Chapters 7.1…7.6
21/09/2023

Graphics: © https://www.asicnorth.com/offerings/design-services/analog-circuit-design/

Definitions
• The natural response:
The response that arises when stored energy in
an inductor or capacitor is suddenly released.
• The step response:
The response that arises 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.
Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.6
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Four Possible First-Order Circuits

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.7
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Natural Response of an RL Circuit

• The switch is closed
for a long time and
opened at t = 0.

short circuit
• All the source current IS appears in the inductive branch.
(first-order
differential
Apply KVL equation)

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.8
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Natural Response of an RL Circuit

(cont.)
• Since the current can not change instantaneously in an inductor:

• The voltage across the

resistor using Ohm’s Law:

• The power dissipated in

the resistor:
Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.9
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Natural Response of an RL Circuit

(cont.)
• The energy delivered to the resistor during any interval of time
after the switch has been opened:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.10
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Time Constant (τ)

• The time constant is the coefficient of time (t)

One time constant after the

inductor had begun to release its
stored energy to the resistor, the
current has been reduced to e-1, or
approximately 0.37 of its initial
value.

Long time implies that five or more

time constants have elapsed.
→ Steady State Response
Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.11
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Time Constant (τ) (cont.)

Summary
• Find the initial current, I0, through the inductor.
• Find the time constant of the circuit, τ = L/R.
• Use I0e-t/τ to generate i(t) from I0 and τ.
Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.12
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 1
The switch in the circuit has been closed for a long time before it is
opened at t = 0. Find
a) iL(t) for t ≥ 0.
b) io(t) for t ≥ 0.
c) vo(t) for t ≥ 0.
d) the percentage of the total energy stored in the 2-H inductor
that is dissipated in the 10-Ω resistor.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.13
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 1 – Solution

a) The switch has been closed for a long time prior to t = 0, so we

know the voltage across the inductor

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.14
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 1 – Solution (cont.)

b) io(t):

c) vo(t):

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.15
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 1 – Solution (cont.)

d) p10 Ω:

initial energy stored in the 2-H inductor is

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.16
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 2

a) Find i1 , i2 and i3 .
b) Calculate the initial energy stored in the parallel inductors.
c) Calculate the energy stored in the inductor as t → ∞.
d) Show that the total energy delivered to the resistive network equals
to the difference between the result obtained in (b) and (c).

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.17
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 2 – Solution

a) The switch has been closed for a long time prior to t = 0, so we

know the voltage across the inductor.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.18
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 2 – Solution (cont.)

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.19
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 2 – Solution (cont.)

b) Calculate the initial energy stored in the parallel inductors.

c) Determine how much energy is stored in the inductor as t → ∞.

d) Show that the total energy delivered to the resistive network

equals to the difference between the result obtained in (b) and (c).

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.20
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 3

Find i(t).

When the switch is closed for a long time, the inductor is a short.
a) Calculate the initial value of i:

b) Calculate the initial energy stored in

the inductor:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.21
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 3 (cont.)

c) What is the time constant of the circuit for t > 0?

d) What is the numerical expression for i(t) for t ≥ 0?

e) What % of initial
energy stored has
been dissipated in
the 2-Ω resistor 5 ms
after the switch has
been opened?

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.22
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 4

a) Calculate vo for t ≥ 0.

When the switch is in position a for a long time, the inductor is a short.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.23
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 4 (cont.)

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.24
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 4 (cont.)

b) What % of the initial energy stored in the inductor is eventually

dissipated in the 4-Ω resistor?

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.25
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Natural Response of an RC Circuit

• The switch is in position a
for a long time and flipped
to position b at t = 0.

open circuit

(first-order
Apply KCL differential
equation)

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.26
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Natural Response of an RC Circuit

(cont.)

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.27
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 5

a) vC(t) for t ≥ 0.

When the switch is closed for a long

time, the capacitor is an open circuit.
for t ≥ 0:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.28
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 5 (cont.)

b) vo(t) for t ≥ 0+?

c) io(t) for t ≥ 0+?

d) The total energy dissipated in the 60-kΩ resistor.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.29
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 6

a) Find v1(t), v2(t), and v(t) for t ≥ 0.

For t ≥ 0:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.30
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 6 (cont.)
b) Calculate the initial energy stored in the capacitors 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).

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.31
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 7

a) The initial value of v(t)?

b) The time constant for t > 0?

c) v(t) for t ≥ 0?

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.32
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 7 (cont.)
b) The initial energy stored in the capacitor.

c) The length of time required to dissipate 75% of the initially stored

energy.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.33
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 8

a) Find vo(t) for t ≥ 0?

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.34
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 8 (cont.)
b) What percentage of the initial energy stored in the circuit has been
dissipated after the switch has been open for 60 ms?
initial energy stored:

energy at 60 ms:

energy dissipated:

% energy dissipated:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.35
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Step Response of an RL Circuit

• Energy stored in the inductor at the
time the switch is closed is given in
terms of nonzero initial current i(0).

Apply KVL

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.36
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Step Response of an RL Circuit (cont.)

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.37
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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The Step Response of an RL Circuit (cont.)

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.38
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 9

Make-before-break switch: the connection at position b is established

before the connection at position a is broken, so there is no interruption of
the current through the inductor.
a) Find the expression for i(t) for t > 0.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.39
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 9 (cont.)

b) What is the initial voltage across the inductor just after the switch has
been moved to position b?

c) How many milliseconds after

the switch has been moved
does the inductor voltage
equal to 24 V?

c) Plot i(t) and v(t).

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.40
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 10
Assume the switch in the circuit has been in position b for a long time and
at t = 0 it moves to position a.
a) Find i(0-).

b) Find v(0+).

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.41
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 10 (cont.)

Another technique to find v(0+):

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.42
Autumn 2023 Chapters 7.1…7.6
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The Step Response of an RC Circuit

KCL:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.43
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 11

a) Find vo(t) for t ≥ 0.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.44
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Example 11 (cont.)

Apply source transformation or Norton equivalent:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.45
Autumn 2023 Chapters 7.1…7.6
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Example 11 (cont.)

b) Find io(t) for t ≥ 0+.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.46
Autumn 2023 Chapters 7.1…7.6
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A General Solution
for Natural & Step Responses

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.47
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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A General Solution
for Natural & Step Responses

1. Identify the variable of interest of the circuit. For RC circuits, it

is best to choose vC, for RL circuits it is best to choose iL.
2. determine the initial value of the variable; vC(t0) in case of RC
and iL(t0) in case of RL.
3. Calculate the final value of the variable (value at t = ∞).
4. Calculate the time constant for the circuit.
Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.48
Autumn 2023 Chapters 7.1…7.6
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Example 12

a) The initial value of vC.

Switch at a:

b) The final value of vC.

Switch at b:

c) Time constant when switch at b:

d) vC(t)?

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.49
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 12 (cont.)

e) i(t)?

f) How long after the switch is in position b does the capacitor voltage
equal zero?

g) Plot vC(t) and i(t).

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.50
Autumn 2023 Chapters 7.1…7.6
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Example 13

a) i(t) for t ≥ 0+:

b) v(t) for t ≥ 0+:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.51
Autumn 2023 Chapters 7.1…7.6
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Example 14

a) v(t) for t ≥ 0+:

b) i(t) for t ≥ 0+:

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.52
Autumn 2023 Chapters 7.1…7.6
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Sequential Switching

• Switching occurs more than once in a circuit.

• The time reference for switching cannot be t = 0.

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.53
Autumn 2023 Chapters 7.1…7.6
21/09/2023

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Example 15

At t = 0 switch 1 is opened. Then, 35 ms

later, switch 2 is opened. Find
a) iL(t) for 0 ≤ t ≤ 35 ms.
Both switches are initially closed for
a long time.

switch 1 is opened

Udo Klein Nilsson/Riedel, Electric Circuits, 9th Ed. ©2011, Prentice Hall 61ECE205 – 1.54
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Example 15 (cont.)
switch 1 is open

b) iL(t) for t ≥ 35 ms.

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Example 15 (cont.)

c) What percentage of the initial energy stored in the 150-mH inductor is

dissipated in the 18-Ω resistor?

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Example 15 (cont.)
d) Repeat (c) for the 3-Ω resistor?

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Example 16

t ≤ 0: switch at (a)
t = 0: switch at (b)
t = 15 ms: switch at (c)
Find vC(t).

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Example 16 (cont.)
t ≤ 0: switch at (a)
t = 0: switch at (b)
t = 15 ms: switch at (c)

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Example 17

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Example 18
t ≤ 0: switch a open & switch b closed

0 ≤ t ≤ 1 s: switch a closed &

switch b still closed

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Example 18 (cont.)

t ≥ 1 s: switch a open & switch b open

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Unbounded Response

• When the RTh for a circuit is negative, and

accordingly the time constant is also negative.
• Only valid for dependent source based circuits.

• Commonly used in oscillator design.

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Example 19

KCL:

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Exercises
1. The switch in the circuit below has been closed since the
reunification of Vietnam. If the switch is opened at t = 0, find
a) iL the instant after the switch changes;
b) v the instant after the switch changes;
c) iL(t) for t ≥ 0.

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Exercises
2. The switch in the circuit below is a single-pole, double-throw
switch that is drawn to indicate that it closes one circuit before
opening the other (make-before-break switch). Assuming the
switch has been in the position drawn for a long time determine
the value of v and iL
a) the instant just prior to the switch changing;
b) the instant just after the switch changes;
c) for t ≥ 0.

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Exercises
3.
a) Find vC(t) for all time in the circuit below.
b) At what time is vC = 0.1vC(0)?

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Exercises
4. The circuit below contains two inductors in parallel, thus
providing the opportunity of a trapped current circulating around
the inductive loop. Let i1(0-) = 10 A and i2(0-) = 20 A.
a) Find i1(0+), i2(0+), and i(0+).
b) Determine the time constant τ for i(t).
c) Find i(t) and v(t) for t ≥ 0.
d) Find i1(t) and i2(t) from v(t) and the initial values.
e) Show that the stored
energy at t = 0 is equal to
the sum of the dissipated
and remaining stored
energy for t → ∞.

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Exercises
5. After being closed for a long time, the switch in the circuit below
is opened at t = 0.
a) Find vC(t) for t ≥ 0.
b) Calculate values for iA(-100 μs) and iA(100 μs).

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Exercises
6. Determine values for R0 and R1 in the circuit below so that vC =
50 V at t = 0.5 ms and vC = 25 V at t = 2 ms.

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Exercises
7. The values of iS in the circuit below is 1 mA for t < 0, and zero
for t > 0. Find vx(t) for
a) t < 0;
b) t > 0.

- +

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End of Lecture 1

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