Study notes on Phasor analysis
Table of contents
Chapters
Page
Sinusoids
review
2
RMS voltage
3
Complex
numbers
review
4
Phasor
analysis
6
Example
8
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Phasor analysis of circuits, Study notes of Electrical Engineering
Study notes on the topic of phasor analysis of circuits in electrical engineering. Straightforward set of notes that achieved perfect score in exams. Introduction with reviews on complex numbers and on sinusoids, to understand easily phasor transformation. Trinity College Dublin 2nd year engineering
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2023/2024
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Study notes on Phasor analysis
Table of contents
Chapters Page
Sinusoids
review
RMS voltage 3
Complex
numbers
review
Phasor
analysis
Example 8
Sinusoids review
The angular frequency of a waveform is given by the formula:
where f is the frequency. The angular frequency is used to represent the cycles per
second of the wave, and one cycle is 2π.
The frequency f is:
the inverse of the period of the wave T. The period is the distance between two
peaks of the wave, or in other words the Ɵme it takes for the wave to complete a
full cycle.
The amplitude A represents the peak value of the waveform.
The phase 𝜙 is the offset angle of the waveform; different phase values shiŌ the
wave right or leŌ along the Ɵme axis.
The equaƟon that represents a sinusoidal funcƟon of Ɵme is:
𝐴 sin(𝜔𝑡 + 𝜙)
Which is also equal to
𝐴 cos(𝜔𝑡 + 𝜙 −
Complex numbers review
There are three forms of complex numbers:
Rectangular form
Polar form
ExponenƟal polar form
The rectangular form is:
where z is the complex number, a is the real part and b is the imaginary part.
The complex conjugate of z is:
Complex numbers can be represented on a graph, with the Real part being the x
axis, and the imaginary part being the y axis, as illustrated below.
The modulus of the complex number is
r and given by the formula 𝑟 =
ଶ
ଶ
The graph shows that the real part a is
𝑅𝑒(𝑧) = 𝑎 = 𝑟 cos(𝜃), and that the
imaginary part is 𝐼𝑚
𝑟 sin(𝜃)
Figure 1: Complex number represented on a graph [1]
Hence, the complex number z can be wriƩen as 𝑧 = 𝑟 cos(𝜃) + 𝑟 sin(𝜃)𝑗, or
more simply as:
𝑧 = 𝑟(cos 𝜃 + jsin 𝜃)
where 𝜗 = tan
ିଵ
This is the polar form of the complex number.
Euler’s formula describes the relaƟonship between exponenƟals and sinusoids as
follows:
ఏ
= cos 𝜃 + jsin 𝜃
which introduces the last form of complex number, the exponenƟal polar:
ఏ
Final recap:
Any complex number can be wriƩen in three forms that are equal to each other:
𝑧 = 𝑎 + 𝑏𝑗 = 𝑟(cos 𝜃 + jsin 𝜃) = 𝑟𝑒
ఏ
For example, if in the Ɵme domain the angular frequency is 0.5 and the capacitance
is 1 F, then the equivalent in the phasor domain would be:
1 ×
× 𝑗
The capacitor is described by the formula:
Inductor
The value of inductance in the Ɵme domain transforms according to the equaƟon:
For example, if in the Ɵme domain the angular frequency is 2 and the inductance is
1 H, then the equivalent in the phasor domain would be:
𝐿 = 2 × 1 × 𝑗 = 2𝑗
The inductor is described by the formula:
Impedance and admiƩance
The equivalent of resistance in the frequency domain is called impedance,
expressed by the symbol Z and described by the equaƟon:
Z is a complex number with the resistance as its real part and the reactance,
denoted by the symbol X, as its imaginary part.
The reciprocal of the impedance is the admiƩance Y:
Y is a complex number with conductance G as its real part and the susceptance B
as the imaginary part.
Example
Figure 2: RL circuit with AC voltage source
The image shows a circuit with an AC voltage source of 8 sin 10𝑡, a resistance of 4Ω
and an inductor with an inductance value of 2 H, with unknown inductor voltage.
Firstly, every element of the circuit has to be converted to phasor domain: