Phasor Form Of Periodic Signals
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Phasor Form Of Periodic Signals

Figure 8.25 shows the sum Vs(t) of the sinusoidal voltage signals V1(t) and V2(t).
Determine the phasor form of Vs given the following information:
V1(t): amplitude = 15; frequency = 377; phase angle = 45o.
V1(t): amplitude = 15; frequency = 377; phase angle = 30o.

The string:
S7P2A21 (Identity - Physical Properties).
The math:

Pj Problem of interest is of type identity.
The expression for a generalized sinusoid is:
Acos(ωt + θ) -------(1)
Where A is the amplitude; ω is the frequency; and θ is the phase angle.
We can relate the general form of the sinusoid to Euler's Identity (Leonhard Euler (1707 - 1783)).
Basically, Euler's identity defines the complex exponential, e as a point in the complex plane, which has both real and imaginary components as follows:

e = cos θ + jsin θ -------(2)
Where cos θ is the real component and jsin θ is the imaginary component.
So, Ae = A(cos θ + jsin θ)
Now, Aej(wt + θ) = Acos(wt + θ) + jAsin(wt + θ)----(3)
So, we see that the real component of equation (3) is the expression for a generalized sinusoid.
By definition, the complex phasor notation for Acos(ωt + θ) is:

Ae = A<θ
The complex phasor notation for Acos(ωt + θ) is simply a mathematical definition that resulted from the need for a simple method for analyzing sinusoidal signals. It is important to note that the complex form ejwt is implicit in the simplification.

Now in time-domain form:
V1(t) = 15cos(377t + π/4)
V2(t) = 15cos(377t + π/6)

In phasor form:
V1(jω) = 15<π/4
V2(jω) = 15<π/6
Convert phasor voltages from polar to rectangular form:
V1(jω) = 15(cos45) + j15(sin45) = 10.61 + 10.61j
V2(jω) = 15(cos30) + j15(sin30) = 12.99 + 7.5 j
So, Vs(jω) = V1(jω) + V2(jω) = 23.6 + 18.11j
So, A2 = [(23.6)2 + (18.11)2]
So, A = 29.75.
So, sin θ = 18.11/29.75 = 0.609
So, θ = 37.52o.
So, Vs(jω) 29.75<37.52.
So, Vs(t) = 29.75cos(377t + 37.52 π/180)

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