Complex
Numbers
Ger O'Farrell – Distributed Electrical Systems
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Complex Number Review.
When dealing with electrical quantities. We tend to
think of all numbers as being ‘real’; however, it is
sometimes more convenient to think of some values
as being ‘imaginary’.
Consider the square root of – 9, its value is not
considered to be a ‘real’ number because, whenever
we ‘square’ the result, we get a positive value and not
– 9.
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Complex Number Review.
A universally adopted solution is to say that if
x2 = - 9
Then x = imaginary operator . 9
This imaginary operator is (-1)
x = j . 9 = j3
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Complex Number Review.
Consider a quadratic equation f(x) = ax2 + bx +c = 0. The solution to this
equation is given by :
-b ± (b2 - 4ac)
x = ------------------2a
A complex number will arise if b2 - 4ac < 0, i.e. we are trying to get the root
of a negative number.
As an example consider the equation x2 -2x + 5 = 0.
The solution to this equation is given by :
x = [2 ± (-16)]/2
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Complex Number Review.
The solution for x has two parts;
a Real part and an Imaginary part.
Real part : 2/2 = 1
Imaginary part ± [(-16)]/2
The (-1) is assigned the symbol or operator j. So (-16) = j 16 =
j4 or 4j.
Solution for x is given by 1 ± j4/2 = 1 ± j2; i.e. 1 + j2 and 1 - j2.
Note that 1 - j2 is a Complex Conjugate of 1 + j2, and vice versa.
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Complex Number Review.
A complex number can be represented graphically on an Argand
Diagram.
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Complex Number Review.
Rectangular coordinates and polar coordinates are two different ways of using
two numbers to locate a point on a plane.
Rectangular
coordinates are in
the form (x,y), where
'x' and 'y' are the
horizontal and
vertical distances
from the origin
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Complex Number Review.
Polar coordinates are in the form: (r, Ѳ), where 'r' is the
distance from the origin to the point, and ‘Ѳ' is the angle
measured from the positive 'x' axis to the point:
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Complex Number Review.
To convert between polar and rectangular coordinates,
we make a right triangle to the point (x,y), like this;
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Polar to Rectangular
From the diagram above, these formulas convert polar
coordinates to rectangular coordinates:
x = r cos Ѳ, y = r sin Ѳ
So the polar point: (r, Ѳ) can be converted to
rectangular coordinates like this:
( r cos Ѳ, r sin Ѳ ) = ( x, y )
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Polar to Rectangular
Example: A point has polar coordinates: (5, 30 ⁰).
Convert to rectangular coordinates.
Solution: (x,y) = (5cos30⁰, 5sin30 ⁰)
= (4.3301 + j2.5)
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Rectangular to Polar
Again, from the diagram above, these formulas
convert rectangular coordinates to polar coordinates:
By the rule of Pythagoras:
Tan Ѳ = y/x , so therefore:
Ѳ = tan-1( y/x )
So the rectangular point: (x,y) can be converted to
polar coordinates like this:
(√x2 + y2, tan-1( y/x ) ) = ( r , Ѳ )
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Rectangular to Polar
Example: A point has rectangular coordinates: (3, 4).
Convert to polar coordinates.
Solution:
r = square root of(3² + 4²) = 5,
Ѳ = tan-1(4/3) = 53.13 ⁰
so (r, Ѳ) = (5, 53.13⁰)
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Questions
What is the polar form for the rectangular complex
number (2+j3)?
r = (√22 + 32) = 3.606
Ѳ = tan-1( y/x ) ) = tan-1( 3/2 ) = 56.31 ⁰
= ( r , Ѳ ) = 3.606, 56.31 ⁰
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Multipling Complex numbers
Complex numbers can be multiplied together either if
they are in Polar form or in Rectangular form, although,
preferably they should be in polar form.
Division Complex numbers
Division of complex numbers can be carried out in Polar
form or in Rectangular form, although polar form is the
simplest method.
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