1. Sinusoids and Phasors
In electrical engineering the forced sinusoidal component remaining
after the natural component disappears is called the sinusoidal
steady-state response. The sinusoidal steady-state response is
also called the ac response since the driving force is an alternating
current signal. The phasor concept is the foundation for the
analysis of linear circuits in the sinusoidal steady state. Simply put,
a phasor is a complex number representing the amplitude and
phase angle of a sinusoidal voltage or current. The connection
between sinewaves and complex numbers is provided by Euler's
relationship:
Eq.(8-1)
To develop the phasor concept, it is necessary to adopt the point of
view that the cosine and sine functions
Eq. (8-2)
and
Eq.(8-3)
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When Eq.(8-2) is applied to the
general sinusoid we obtain
Eq.(8-4)
In the last line of Eq.(8-4), moving the amplitude VA inside the real
part operation does not change the final result because it is a real
constant.
By definition, the quantity
in the last line of Eq.(8-4) is the
phasor representation of the sinusoid v(t). The phasor V is written
as
Eq.(8-5)
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Note that V is a complex number determined by the amplitude and
phase angle of the sinusoid. Fig. 8-1 shows a graphical
representation commonly called a phasor diagram.
Fig. 8-1: Phasor diagram
Two features of the phasor concept need emphasis:
1. Phasors are written in boldface type like V or I1 to distinguish
them from signal waveforms such as v(t) and i1(t).
2. A phasor is determined by amplitude and phase angle and does not
contain any information about the frequency of the sinusoid.
In summary, given a sinusoidal signal
corresponding phasor representation is
given the phasor
multiplying the phasor by
, the
. Conversely,
, the corresponding sinusoid is found by
and reversing the steps in Eq.(8-4)
as follows:
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Eq.(8-6)
The complex exponential is sometimes called a rotating phasor,
and the phasor V is viewed as a snapshot of the situation at t=0.
Fig. 8-2: Complex exponential
Properties of Phasors
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First, the additive
property states that the phasor representing a sum of
sinusoids of the same frequency is obtained by adding the phasor
representations of the component sinusoids. To establish this
property we write the expression
Eq.(8-7)
where V1(t), V2(t), ... and VN(t) are sinusoids of the same frequency
whose phasor representations are V1, V2... and VN. The real part
operation is additive, so the sum of real parts equals the real part of
the sum. Consequently, Eq.(8-7) can be written in the form
Eq.(8-8)
Comparing the last line in Eq.(8-8) with the definition of a phasor,
we conclude that the phasor V representing v(t) is
Eq.(8-9)
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The result in Eq.(8-9)
applies only if the
component sinusoids all have the same
frequency so that
can be factored out as
shown in the last line in Eq.(8-8).
The derivative property of phasors allows us easily to relate the
phasor representing a sinusoid to the phasor representing its
derivative.
Equation (8-6) relates a sinusoid function and its phasor
representation as
Differentiating this equation with respect time t yields
Eq.(8-10)
From the definition of a phasor, we see that the quantity (
V)
on the right side of this equation is the phasor representation of the
time derivative of the sinusoidal waveform. This phasor can be
written in the form
Eq.(8-11)
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which points out that differentiating a sinusoid changes its
amplitude by a multiplicative factor
and shifts the phase angle by
90 .
EXAMPLE 8-1
(a) Construct the phasors for the following signals:
(b) Use the additive property of phasors and the phasors found in
(a) to find v(t)=v1(t)+v2(t).
SOLUTION
(a) The phasor representations of v(t)=v1(t)+ v2(t) are
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(b) The two sinusoids have the same frequent so the additive
property of phasors can be used to obtain their sum:
The waveform corresponding to this phasor sum is
The phasor diagram in Fig. 8-3 shows that summing sinusoids can
be viewed geometrically in terms of phasors.
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Fig. 8-3
EXAMPLE 8-2
(a) Construct the phasors representing the following signals:
(b) Use the additive property of phasors and the phasors found in
(a) to find the sum of these waveforms.
SOLUTION:
(a) The phasor representation of the three sinusoidal currents are
(b) The currents have the same frequency, so the additive property
of phasors applies. The phasor representing the sum of these
current is
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Fig. 8-4
EXAMPLE 8-3
Use the derivative property of phasors to find the time derivative of
v(t)=15 cos(200t-30 ).
SOLUTION:
The phasor for the sinusoid is V=15
-30 . According to the
derivative property, the phasor representing the dv/dt is found by
multiplying V by j .
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The sinusoid corresponding to the phasor j
V is
Finding the derivative of a sinusoid is easily carried out in phasor
form, since it only involves manipulating complex numbers.
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