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more signals having the same frequency, and how to find the particular solu-
tion of an ODE with a sinusoidal driving function.
There are two key ideas behind the phasor representation of a signal:

1. A real, sinusoidal time-varying signal may be represented by a
complex time-varying signal.
2. This complex signal can be represented as the product of a complex
number that is independent of time and a complex signal that is
dependent on time.

Example 6.7
Decompose the signal V = A cos(ωt + φ) according to the above prescription.

Solution: This signal can, using the polar representation of complex num-
bers, also be written as:

jφ
V = Acos(ω t + )φ = Re[ Aexp( j(ω t + ))]φ = Re[ Ae e j tω ] (6.64)
where the phasor, denoted with a tilde on top of its corresponding signal
symbol, is given by:

˜
V = Ae jφ (6.65)
(Warning: Do not mix the tilde symbol that we use here, to indicate a phasor,
with the overbar that denotes complex conjugation.)
Having achieved the above goal of separating the time-independent part of
the complex number from its time-dependent part, we now learn how to
manipulate these objects. A lot of insight can be immediately gained if we
note that this form of the phasor is exactly in the polar form of a complex
number, with clear geometric interpretation for its magnitude and phase.

6.6.1 Phasor of Two Added Signals

The sum of two signals with common frequencies but different amplitudes
and phases is

V = A cos(ω t + φ ) = A cos(ω t + φ ) + A cos(ω t + φ ) (6.66)
tot. tot. tot. 1 1 2 2
To write the above result in phasor notation, note that the above sum can also
be written as follows:

V = Re[ A exp( j(ω t + φ )) + A exp( j(ω t + φ ))]
tot. 1 1 2 2
(6.67)
ω
φ
= Re[( Ae j 1 φ + A e j 2 e ) jt ]
1 2
© 2001 by CRC Press LLC