Lecture notes of introduction to electronics

1. Dr. Rik Dey
ASSISTANT PROFESSOR,
ELECTRICAL ENGINEERING, IIT KANPUR
ESC201A INTRODUCTION TO ELECTRONICS CIRCUITS
2023-24 SEM-II
Introduction to Electronics
Part 2: Transient Analysis
L12 Part B: Phasor and Impedance
1

2. ESC201, 2023-24 Sem-II
Dr. Rik Dey
What will you learn today?
Phasor: Sinusoid and Complex Number
2

3. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Representation of Sinusoidal Signals (Sinusoids)
3
𝑣(𝑡) = 𝑉
𝑚 cos( 𝜔𝑡 + 𝜃)
Canonical Form:
𝑉
𝑚 = peak value 𝜃 = phase
f = frequency in Hertz (which is 1/sec)
ω = 2πf = angular frequency in rad/sec
T = 1/f = 2π/ω = time period in sec
Vm


4. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Representation of Sinusoids: Example
4
5 sin( 4𝜋𝑡 − 60𝑜) 𝑣(𝑡) = 𝑉
𝑚 cos( 𝜔𝑡 + 𝜃)
= 5 cos( 4𝜋𝑡 − 60𝑜
− 90𝑜
)
Amplitude = 5 ;
Phase = –150o
Phase in radians: 360𝑜 = 2𝜋
𝜃 =
−150
360
× 2𝜋 = −2.618 radians
𝜔 = 4𝜋 𝑟𝑎𝑑/𝑠
𝜔 =
2𝜋
𝑇
= 4𝜋 ⇒ 𝑇 = 0.5 𝑠 𝑓 =
1
𝑇
= 2 𝐻𝑧
sin( 180 ) sin
cos( 180 ) cos
sin( 90 ) cos
cos( 90 ) sin
o
o
o
o
t t
t t
t t
t t
 
 
 
 
 = −
 = −
 = 
 =
Frequency and Time Period:
All
sin 
cos 
tan 

5. ESC201, 2023-24 Sem-II
Dr. Rik Dey
2 2
( ) cos( )
m
v t v t

=
1 1
( ) cos( 60 )
o
m
v t v t

= +
Representation of Sinusoids: Phase Relationship
5
The peaks of 𝑣1 𝑡 occur 60o before the peaks of 𝑣2 𝑡 .
In other words, 𝑣1 𝑡 leads 𝑣2 𝑡 by 60o
.

6. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Representation of Sinusoids: Phase Relationship
6
270o
Voltage leads current by 90o or lags current by 270o ?
Phase difference is usually considered between –180o to 180o
Add or subtract 360o to bring the phase between –180o to 180o
6

7. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Representation of Sinusoids: Example
7
𝑖1 = 4 sin( 377𝑡 + 25𝑜
)
𝑖2 = −5 cos( 377𝑡 − 40𝑜)
Find the phase difference between the two currents
𝑥(𝑡) = 𝑥𝑚 cos( 𝜔𝑡 + 𝜃)
Canonical Form
1 4cos(377 25 90 )
o o
i t
= + − 2 5cos(377 40 180 )
o o
i t
= − +
2 140o
 =
1 65o
 =−
1 2 205o
 
− =−
Does 𝑖2 lead 𝑖1 ?
1 2 205o
 
− =−
1 2 205 360 155
o o o
 
− =− + =
𝑖1 leads 𝑖2 by 155o

8. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Complex Numbers
8
• Complex number can be represented as a
point in the complex plane (2D)
𝑎
• Polar representation 𝑧 = 𝑎∠𝜃

9. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Complex Numbers
9
𝑎
𝑧 = 𝑎∠𝜃
𝑧 = 𝑥 + 𝑗𝑦
Absolute value
𝑧 = 𝑎
𝑧 = 𝑥2 + 𝑦2
Phase (Angle)
Angle(𝑧) = 𝜃
tan 𝜃 =
𝑦
𝑥
𝑥 = 𝑎 cos 𝜃
𝑦 = 𝑎 sin 𝜃
𝑧 = 𝑥 + 𝑗𝑦
𝑧 = 𝑎 cos 𝜃 + 𝑗𝑎 sin 𝜃
𝑧 = 𝑎 cos 𝜃 + 𝑗 sin 𝜃
𝑒𝑗𝜃 = cos 𝜃 + 𝑗 sin 𝜃
Euler Identity
= 𝑎𝑒𝑗𝜃
𝑧 = 𝑎∠𝜃 = 𝑎𝑒𝑗𝜃 = 𝑎 cos 𝜃 + 𝑗 sin 𝜃
𝑒𝑗𝜃 = 1∠𝜃

10. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Examples
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11. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Complex Numbers: Addition and Subtraction
11
To add or subtract two complex numbers:
➢ Convert them first into rectangular form and then perform the operations
➢ Then represent the result in polar form whenever needed

12. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Example
12
𝑣1 𝑡 = 20 cos( 200𝑡 − 45𝑜)
𝑣2(𝑡) = 10 cos( 200𝑡 + 60𝑜)
𝑣𝑠(𝑡) = 𝑣1(𝑡) − 𝑣2(𝑡)

13. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Complex Numbers: Multiplication and Division
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14. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Complex Numbers Multiplication/Division
14
➢ To multiply and divide complex numbers, it is easier to use polar coordinates

15. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Representation of Sinusoidal Signals: Phasors
𝑣(𝑡) = 𝑉
𝑚 cos( 𝜔𝑡 + 𝜃)
Complex plane
𝑉
𝑚∠𝜃 = 𝑉
𝑚𝑒𝑗𝜃 Re( 𝑉
𝑚∠𝜔𝑡 + 𝜃)
𝑉
𝑚 ∠(𝜔𝑡 + 𝜃) 𝑉
𝑚 cos( 𝜔𝑡 + 𝜃)
Vm

➢ Euler’s Identity:

16. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Frequency Components in Sinusoid
16
• Phasor: related to frequency domain (Phase)
cos 𝜔𝑡 + 𝜃 =
1
2
𝑒𝑗 𝜔𝑡+𝜃
+ 𝑒−𝑗 𝜔𝑡+𝜃
=
1
2
𝑒𝑗𝜃
𝑒𝑗𝜔𝑡
+ 𝑒−𝑗𝜃
𝑒−𝑗𝜔𝑡
for 𝜃 = 0
1/2
1/2
𝑒𝑗𝜃
𝑒−𝑗𝜃

17. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Phasor Representation of Sinusoids: Advantage
17
Performing algebra on sinusoids by representing them as complex numbers
Sinusoidal
variables
Complex
variables
Perform Algebra on
Complex variables
Transform
complex
variable
back to sinusoid
𝑣(𝑡) = 𝑉
𝑚 cos( 𝜔𝑡 + 𝜃)
Complex plane
Vm


18. ESC201, 2023-24 Sem-II
Dr. Rik Dey
What will you learn today?
Impedance Model in Complex Plane
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19. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Phasor Representation of Sinusoids
• Let us try to understand the “complex world” in time domain only
• Sinusoid:
• Actual signal is the real part of the phasor.
• Complex signals will represent actual signal via their real part.
• If 𝜔𝑡 is not written then:
• We’ll see that sinusoid remain sinusoid of same frequency for linear circuits.
19
( ) cos( )
m
v t V t
 
= + Re( )
m
V t
 
 +
m
V 

( ) cos( )
m
v t V t
 
= +

20. ESC201, 2023-24 Sem-II
Dr. Rik Dey
𝑣(𝑡) = 𝑉𝑀 cos( 𝜔𝑡 + 𝜃)
𝑖𝑅(𝑡) =
𝑉𝑀
𝑅
cos( 𝜔𝑡 + 𝜃)
𝐼𝑅 =
𝑉𝑀
𝑅
∠𝜃
𝑉𝑅 = 𝑉𝑀∠𝜃
𝐼𝑅 =
𝑉𝑅
𝑅
Impedance Model of Resistor
20
R
v(t)
• Fix the input sinusoid frequency and consider steady state
• All currents and voltages are sinusoids of the same frequency
𝑣𝑅(𝑡) = 𝑉𝑀 cos( 𝜔𝑡 + 𝜃)

21. ESC201, 2023-24 Sem-II
Dr. Rik Dey
𝐼𝐶 = 𝜔𝐶∠90 × 𝑉𝑀∠𝜃
𝐼𝐶 = 𝑗𝜔𝐶𝑉𝐶
𝑉𝐶 = 𝐼𝐶𝑍𝐶
𝑍𝐶 =
1
𝑗𝜔𝐶
= −𝑗
1
𝜔𝐶
𝐼𝐶 = 𝜔𝐶𝑉𝑀∠𝜃 + 90
𝑉𝐶 = 𝑉𝑀∠𝜃
Impedance Model of Capacitor
𝑣𝐶(𝑡) = 𝑉𝑀 cos( 𝜔𝑡 + 𝜃)
𝑖𝐶(𝑡) = 𝐶
𝑑𝑣𝐶
𝑑𝑡
= −𝜔𝐶𝑉𝑀 sin( 𝜔𝑡 + 𝜃)
𝑖𝐶(𝑡) = 𝜔𝐶𝑉𝑀 cos( 𝜔𝑡 + 𝜃 + 90𝑜)
In a capacitor, current leads voltage by 900
Generalized Ohm’s Law for
Capacitors in terms of Phasors

22. ESC201, 2023-24 Sem-II
Dr. Rik Dey
𝑉𝐿 = 𝜔𝐿∠90 × 𝐼𝑀∠𝜃
𝑉𝐿 = 𝑗𝜔𝐿𝐼𝐿
𝑉𝐿 = 𝐼𝐿𝑍𝐿
𝑍𝐿 = 𝑗𝜔𝐿
𝑉𝐿 = 𝜔𝐿𝐼𝑀∠𝜃 + 90
𝐼𝐿 = 𝐼𝑀∠𝜃
Impedance Model of Inductor
𝑖𝐿(𝑡) = 𝐼𝑀 cos( 𝜔𝑡 + 𝜃)
𝑣𝐿(𝑡) = 𝐿
𝑑𝑖𝐿
𝑑𝑡
) = −𝜔𝐿𝐼𝑀 sin( 𝜔𝑡 + 𝜃)
𝑣𝐿(𝑡) = 𝜔𝐿𝐼𝑀 cos( 𝜔𝑡 + 𝜃 + 90𝑜)
In an inductor, current lags voltage by 900
Generalized Ohm’s Law for
Inductors in terms of Phasors

23. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Impedance Model
23
Generalized Ohm’s Law for C and L

24. ESC201, 2023-24 Sem-II
Dr. Rik Dey
Frequency Dependency of Impedance
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25. ESC201, 2023-24 Sem-II
Dr. Rik Dey
𝑣(𝑡) = 𝑉0 cos( 200𝑡 + 45𝑜)
Circuit Analysis under Sinusoidal Signals: Example
Given
Compute current in R
R
v(t)
𝑖(𝑡) =
𝑣 𝑡
𝑅
=
𝑉
𝑜
𝑅
cos( 200𝑡 + 45𝑜
)

26. ESC201, 2023-24 Sem-II
Dr. Rik Dey
v(t)
L=0.1H
i(t)
𝑣(𝑡) = 2 cos( 200𝑡 + 45°) V 𝜔 = 200 𝑟𝑎𝑑/𝑠
𝑉𝐿 = 2∠45° V
𝑉𝐿 = 𝐼𝐿𝑍𝐿
⇒ 𝐼𝐿 =
𝑉𝐿
𝑍𝐿
𝐼𝐿 =
2∠45° V
20∠90°Ω
= 0.1∠ − 45° A
𝑖 𝑡 = ?
Circuit Analysis under Sinusoidal Signals: Example
26
𝑖(𝑡) = 0.1 cos( 200𝑡 − 45°) A
v(t)
L=0.1H
𝑍𝐿 = 𝑗𝜔𝐿
= 𝑗20 Ω
= 20∠90°Ω
𝑍𝐿 = 𝑗20 Ω
Multiply/Division:
Do it in polar form.

27. ESC201, 2023-24 Sem-II
Dr. Rik Dey
v(t)
L=0.1H R=50
𝑣(𝑡) = 2 cos( 200𝑡 + 45°)
𝑣𝑅(𝑡) = 1.85 cos( 200𝑡 + 23.2°)
𝑣𝐿(𝑡) = 𝑣(𝑡) − 𝑣𝑅(𝑡)
= 2 cos( 200𝑡 + 45°) − 1.85 cos( 200𝑡 + 23.2°)
Solving such circuits requires us to add/subtract sinusoids! Do it in cartesian form.
Circuit Analysis under Sinusoidal Signals: Example
Given
Compute 𝑣𝐿(𝑡)
Note that 𝑣(𝑡) and 𝑣𝑅(𝑡) have the same frequency. Q.: How to find 𝑣𝑅(𝑡) if not given?