1.5 The significance of reactive components equations
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The polar representation makes life easier when manipulating equations and it also has the
advantage of being easily represented by simple exponential functions which have an imaginary
exponent. This is possible thanks to Euler formula which is shown below
( ) ( )
From Figure 1.5-6 and basic trigonometry, it is apparent that
( ) ( )
Combining (1.5-4) and (1.5-5) yields
So
the complex exponential gives a very compact notation for our complex number!
One of the main advantages of complex numbers is that they allow
phase shifts to be
easily represented. For instance, the angle of a complex number may be increased by
by simply
multiplying it by the complex constant . As an example, let us consider a complex number with
equal real and imaginary parts (just for simplicity) and express it in both polar and Cartesian form
√
√
And
let us plot it on a graph (Figure 1.5-7)
Figure 1.5-7 Graphical representation of
Now, what happens if we multiply our complex number by ?
[
]
7
(1.5-4)
(1.5-5)
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