Electrical Power is the rate at which electrical energy is converted to another form, such as motion, heat, or electromagnetic field. It is symbolized by the uppercase letter P, and is measured in watts, symbolized by the letter W. Power analysis is of paramount importance as it plays a big role in electric utilities, electronics, and communication systems, since these systems involve transmission of power from one point to another.
Instantaneous Power p(t) is the power at any instant of time. It is the rate at which an element absorbs energy and is the product of the instantaneous voltage v(t) across the element and the instantaneous current i(t) through it. It is also measured in watts W.
p(t) = v(t) · i(t)
v(t) = Vm · cos(ωt + θv)
i(t) = Im · cos(ωt + θi)
Vm and Im are the amplitudes (or peak values), while θv and θi are the phase angles of the voltage and current. We can integrate those three formulas above into this one:
p(t) = Vm · Im · cos(ωt + θv) · cos(ωt + θi)
And by applying a certain trigonometric identity, we can further express the equation as:
p(t) = ½ Vm Im cos(θv – θi) + ½ Vm Im cos(2ωt + θv + θi)
The first part is constant or time dependent because its value simply depends on the phase difference between the voltage and current. While the second part is a sinusoidal function whose frequency is 2ω, which is twice the angular frequency ω of the voltage or current.
Here is the flow of an instantaneous power p(t) entering a circuit:
We can observe from the illustration above that this power goes to the positive and negative for every cycle. This means that whenever p(t) is in the positive, power is absorbed by the circuit. While in the negative, power is absorbed by the source – meaning power is transferred from the circuit to the source. Sort of a give-and-take. This is because of the presence of capacitors and inductors in the circuit.
We conclude from this that instantaneous power changes with time making it difficult to measure. Now, let us go to the other kind of power that is more convenient to measure. They even say that the wattmeter, the instrument for measuring power, responds to this power.
Average Power P is the average of the instantaneous power over one period. Still measured in watts W.
Deriving the main formula for average power, you’ll eventually get:
P = ½ Vm Im cos(θv – θi)
Note that p(t) is time-varying while P, does not depend on time. That’s why we can simply solve for average power when the voltage and current are expressed in the time domain. What is important is the difference in the phases of the voltage and current,then we can say that it can either go cos(θv – θi) or cos(θi – θv). It would still be equal and the sign would not matter. I think.
If θv is equal to θi, this would imply that the voltage and current are in phase and that a purely resistive circuit or resistive load – resistors – absorbs average power at all times. If the difference between θv and θi is ±90º, it means the circuit is purely reactive. And purely reactive loads – inductors and capacitors – absorb no average power.
To solve for the maximum average power transfer delivered to a load, the circuit should be in its Thevenin equivalent.
–where (a) is the circuit with a load, and (b) is the circuit’s Thevenin equivalent. The formula goes like this:
Pmax = (|Vth|²)/8Rth
While the current through the load is:
I = Vth/(Zth + Zl)
Also, the load impedance Zl must be equal to the complex conjugate of the Thevenin impedance Zth. Where, in rectangular form, the Thevenin impedance Zth and the load impedance Zl are:
Zth = Rth + jXth
Zl = Rl + jXl
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