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By: Carl H. Durney
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Complex Analysis
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Phasors
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Tutorial
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Tutor: THE
PHASOR TRANSFORM
All
voltages and currents in linear circuits with sinusoidal sources are described
by constant-coefficient linear differential equations of the form
(1) 
where f is a function of time,
the an are constants,
C is a constant, 
is the radian frequency of the sinusoidal source, and 
is the phase of the
sinusoidal source. In (1), f represents any voltage or current in the circuit.
A particular solution to (1)
can be found by an elegant procedure called the phasor transform method. This supplementary material outlines the
mathematical basis of the method. The phasor transform is defined by
(2) 
where 
is a
function of 
called
the phasor transform of f(t), and Re
means the real part of the quantity in the brackets. 
is
complex; it has a real and an imaginary part.
Two key
mathematical relationships are used in finding a particular solution
to (1). The first is
(3) 
where W is any complex number
and W* is the complex conjugate of W. Using (3) with (2) gives
(4) 
where f has been written for f(t) and F for 
for
brevity. Note that F is not a function of time. The second
relationship is
(5) 
which is called Euler's formula.
Substituting
(5) and (4) into (1), taking the derivatives with respect to time, and collecting
terms gives
(6) 
.
Now because 
and 
are linearly independent functions (see, for example, C. R. Wylie, Advanced Engineering Mathematics, 3rd
ed., New York: McGraw-Hill, 1966, p. 444), (6) can be true for all time only
if
(7) 
and
(8) 
.
Equations (7)
and (8) are identical because one is the complex conjugate of the other, so
only one is needed. An expression for F
from (7) is
(9) 
.
A particular solution to (1) can
now be obtained from (9) and (2):
(10) 
.
Symbolically,
the notation for a phasor transformation is
(11) 
where the bold P means "phasor transform
of". Thus, F is the phasor
transform of f. Taking the derivative of both sides of (2) gives
which corresponds to
.
Similarly,
(12) 
and
(13) 
because
(14) 
.
From the basic
relation in (2) it can also be shown that
(15) 
and
(16) 
where
and
and "a" is a constant.
The relation in (15) means that the phasor transform of a sum of functions can
be found by taking the transform of each function and adding the transforms.
Equations (11),
(12), (13), (15), and (16) describe phasor transforms. An inverse phasor
transform relation is written as
(17) 
.
Equations (11) and (17) are
called a transform pair. Equation
(11) states how to get F when f is
known; (17) how to get f when F is
known. Equation (2) is the inverse transform relation. The transform relation
is derived as follows. f(t) will always be a sinusoid, because it is a
particular solution to (1). Thus f(t) can be written as
(18) 
.
Substituting (18) into (2),
using Euler's formula and (3) gives
.
Collecting terms and using the linear
independence of 
and 
, as
before, gives
(19) 
so the phasor transform of 
is 
. The transform pairs are
thus
(20) 
and
(21) 
.
With the phasor
transform relations given in (12), (15, (16), (20), and (21), a particular
solution to (1) can be found without going through the detailed derivation
using (3) and linear independence. The phasor transform of (1) is taken
term-by-term using (12), (13), (15), and (16) to get (7), which is then solved
for F. Having found F, f is found by taking the inverse
transform according to (21).
Ex: Let's
find a particular solution to
(22) 
.
Taking the
phasor transform of this equation gives
.
Solving for F,
.
Converting F to polar form gives
and finding the inverse
transform gives
.
Comment: The
phasor transform method is powerful because it transforms a differential
equation (1) into an algebraic equation (7), which can be solved for the phasor
F, and then f can be found by taking
the inverse transform.
Phasor
voltages and currents satisfy Kirchhoff's laws, because of (15). Consequently,
circuits can be transformed into the frequency domain, eliminating the need to
write differential equations in the time domain and solve them by phasor
transforms. The procedure for analyzing and designing circuits by transforming
them into the frequency domain is summarized in the figure below. Note that
impedance is defined as the ratio of a phasor voltage to a phasor current.
Impedance is not defined in the time domain.
