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Average AC Power

Figure 8.7 shows a simple AC circuit. Figure 8.7(a) is the time domain circuit while figure 8.7(b) is its phasor form.

Given that the sinusoidal voltage and current of the circuit are as follows:

v(t) = Vcos(ωt);
i(t) = Icos(ωt - θ); Determine:

(a) The average power of the circuit in the time domain

(b) The average power of the circuit in the frequency domain.

**The string**:

S_{7}P_{3}A_{32} (Force - Push).
**The math**:

Pj Problem of interest is of type *force*. Power and energy problems are *force problems*. Electric power is a *force-push*.

(a) General expression for electric power is:

p(t) = v(t)i(t)

So, p(t) = VIcos(ωt)cos(ωt - θ)----(1)

Using the following identities of trigonometry:

2cos^{2}ωt - 1 = cos2(ωt) and cosωtsinωt = (sin2ωt)/2
.

Equation (1) reduces to:

p(t) = (VI/2)cos(θ) + (VI/2)cos(2ωt - θ) ----(2)

(a) The average power, *P _{av}* is obtained by integrating p(t) over one cycle of the sinusoidal signal and dividing by the period T, of the signal.

So,

So, after substituting the expression for p(t) in equation (2) and integrating, we have:

P

(b) In the frequency domain:

V(jω) = Ve

So, impedance, Z = (V/I)e

So, I = (V/|Z|)e

Hence, P

Where, 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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