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Phasor Form Of Periodic Signals

Figure 8.25 shows the sum V_{s}(t) of the sinusoidal voltage signals V_{1}(t) and V_{2}(t).

Determine the phasor form of V_{s} given the following information:

V_{1}(t): amplitude = 15; frequency = 377; phase angle = 45^{o}.

V_{1}(t): amplitude = 15; frequency = 377; phase angle = 30^{o}.

**The string**:

S_{7}P_{2}A_{21} (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 ^{jθ}* as a point in the complex plane, which has both real and imaginary components as follows:

e

Where cos θ is the real component and jsin θ is the imaginary component.

So, Ae

Now, Ae

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

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

Now in time-domain form:

V

V

In phasor form:

V

V

Convert phasor voltages from polar to rectangular form:

V

V

So, V

So, A

So, A = 29.75.

So, sin θ = 18.11/29.75 = 0.609

So, θ = 37.52

So, V

So, V

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

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