ELECTRICAL TECHNOLOGY
Complex Numbers Revision
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1.0 Introduction
1.0 Introduction
⚫
A complex number is of the form (a+jb) where a is a
real number and jb is an imaginary number.
⚫
Complex numbers are widely used in the analysis of
series, parallel and series-parallel electrical networks
supplied by alternating voltages.
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1.0 Introduction
1.0 Introduction
⚫
The advantage of the use of complex numbers is that
complex processes become simply algebraic processes
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1.0 Introduction
1.0 Introduction
⚫
A complex number can be represented pictorially on an
Argand diagram
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1.0 Introduction
1.0 Introduction
⚫
A complex number of the form a+ jb is called a Cartesian
or rectangular complex number
⚫ Application of the operator j to any number rotates it 90◦
anticlockwise on the Argand diagram, multiplying a
number by j2 rotates it 180◦ anticlockwise, multiplying a
number by j3 rotates it 270◦ anticlockwise and
multiplication by j4 rotates it 360◦ anticlockwise.
⚫ By similar reasoning, if a phasor is operated on by −j then
a phase shift of −90◦ (i.e. clockwise direction) occurs,
again without change of magnitude.
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1.0 Introduction
1.0 Introduction
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2.0 Operations involving Cartesian
complex numbers
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2.0 Operations involving Cartesian
complex numbers
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2.0 Operations involving Cartesian
complex numbers
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2.0 Operations involving Cartesian
complex numbers
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2.0 Operations involving Cartesian
complex numbers
Example: Given Z1 =3+j4 and Z2 =2−j5 determine in
Cartesian form correct to three decimal places:
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2.0 Operations involving Cartesian
complex numbers
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2.0 Operations involving Cartesian
complex numbers
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3.0 The polar form of a complex
number
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3.0 The polar form of a complex
number
• This latter form is usually abbreviated to Z=r∠θ, and is called
the polar form of a complex number.
• r is called the modulus (or magnitude of Z) and is written as
mod Z or |Z|.
• r is determined from Pythagoras’s theorem on triangle OAZ,
i.e.
θ is called the argument (or amplitude) of Z and is written
as arg Z. θ is also deduced from triangle OAZ: arg Z=θ
=tan−1y/x
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3.0 The polar form of a complex
number
For example:
The cartesian complex number (3+j4) is equal to r∠θ in
polar form.
(3+j4)=??
What about (-3+j4)?
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3.0 The polar form of a complex
number
For example:
The cartesian complex number (3+j4) is equal to r∠θ in
polar form.
(3+j4)=5∠53.13◦
(−3+j4)=5∠126.87◦
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3.0 The polar form of a complex
number
Multiplication and division using complex numbers in
polar form
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3.0 The polar form of a complex
number
Multiplication and division using complex numbers in
polar form
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3.0 The polar form of a complex
number
Conversion
Example:
Convert 5∠−132◦ into a+jb form correct to four significant
figures.
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3.0 The polar form of a complex
number
Conversion
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3.0 The polar form of a complex
number
De Moivre’s theorem — powers and roots of complex
numbers
This result is true for all positive, negative or fractional
values of n.
Example:
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3.0 The polar form of a complex
number
De Moivre’s theorem — powers and roots of complex
numbers
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3.0 The polar form of a complex
number
De Moivre’s theorem — powers and roots of complex
numbers
Example:
Determine
in polar and in Cartesian form.
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3.0 The polar form of a complex
number
De Moivre’s theorem — powers and roots of complex
numbers
Example:
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3.0 The polar form of a complex
number
De Moivre’s theorem — powers and roots of complex
numbers
Example:
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3.0 The polar form of a complex
number
De Moivre’s theorem — powers and roots of complex
numbers
Example:
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