Signals and Systems - Electrical Engineering

0.4 Complex or Real? 25

thatf 0 is the frequency in cycles/sec or in Hertz (Hz) (in honor of H. R. Hertz^8 ). Given that the units
of 0 is rad/sec, then 0 thas as units (rad/sec)×(sec)=(rad), which coincides with the units of the
phaseψ, and permits the computation of the cosine. Ifψ=0, thenx(t)is a cosine, and ifψ=−π/2,
thenx(t)is a sine.

If one knows the frequency 0 (rad/sec) in Equation (0.24), the cosine is characterized by its
amplitude and phase. This permits us to definephasors^9 as complex numbers characterized by the
amplitude and the phase of a cosine signal of a certain frequency 0. That is, for a voltage signal
v(t)=Acos( 0 t+ψ)the corresponding phasor is

V=Aejψ=Acos(ψ)+jAsin(ψ)=A∠ψ (0.25)

and such that

v(t)=Re[Vej^0 t]=Re[Aej(^0 t+ψ)]=Acos( 0 t+ψ) (0.26)

One can thus think of the voltage signalv(t)as the projection of the phasorVonto the real axis and
turning counterclockwise at a rate of 0 rad/sec. At timet=0 the angle of the phasor isψ. Clearly
the phasor definition is true for only one frequency, in this case 0 , and it is always connected to a
cosine function.

Interestingly enough, the angleψcan be used to differentiate cosines and sines. For instance, when
ψ=0, the phasorVmoving around at a rate of 0 generates as a projection on the real axis the
voltage signalAcos( 0 t), while whenψ=−π/2, the phasorVmoving around again at a rate of
 0 generates a sinusoidAsin( 0 t)=Acos( 0 t−π/ 2 )as it is projected onto the real axis. This estab-
lishes the well-known fact that the sine lags the cosine byπ/2 radians or 90 degrees, or that the cosine
leads the sine byπ/2 radians or 90 degrees. Thus, the generation and relation of sines and cosines
can be easily obtained using the plot in Figure 0.12.

Phasors can be related to vectors. A current source, for instance,

i(t)=Acos( 0 t)+Bsin( 0 t)

FIGURE 0.12
Generation of sinusoids
from phasors of a
frequency 0.

−cos

cos

−sin

sin

Ω 0

(^8) Heinrich Rudolf Hertz was a German physicist known for being the first to demonstrate the existence of electromagnetic radiation in
1888.
(^9) In 1883, Charles Proteus Steinmetz (1885–1923), German-American mathematician and engineer, introduced the concept of phasors
for alternating current analysis. In 1902, Steinmetz became a professor of electrophysics at Union College in Schenectady, New York.