Alternating current and Average power

• LagrangeEuler
In summary, the period of the power is not always the same as the period of the current. However, for consistency, the fundamental period of the current is always used to calculate the average power, even if it is not the fundamental period of the power. This is because the period of the power is half the period of the current due to simple trigonometry.
LagrangeEuler
For a current
$$i(t)=I_0\sin(\omega t+\varphi_0)$$
period is ##T=\frac{2\pi}{\omega}##,
Power is defined as
$$p(t)=Ri^2(t)$$. So period of power is not any more ##T=\frac{2\pi}{\omega}##. Why then average power is
$$P=\frac{1}{T}\int^T_0p(t)d t$$.
Why are we using the period of current and not of the power ##p(t)##?

Counter example: a postive sawtooth (##\ 0...A\ ##) in T: Period doesn't change.

Ok. But if the period changes why we used the period of the current and not of the power?

LagrangeEuler said:
Ok. But if the period changes why we used the period of the current and not of the power?
For consistency. The fundamental period of the current is always a period of the power, even if it is not the fundamental period of the power.

vanhees71
LagrangeEuler said:
For a current
$$i(t)=I_0\sin(\omega t+\varphi_0)$$
period is ##T=\frac{2\pi}{\omega}##,
Power is defined as
$$p(t)=Ri^2(t)$$. So period of power is not any more ##T=\frac{2\pi}{\omega}##. Why then average power is
$$P=\frac{1}{T}\int^T_0p(t)d t$$.
Why are we using the period of current and not of the power ##p(t)##?
Who says we are not using the period of the power? Using simple trigonometry,
$$P(t)=I_0^2R\sin^2(\omega t)=\frac{1}{2}I_0^2R[1-\cos(\Omega t)]~~~~~(\Omega \equiv 2\omega)$$Thus, the period of the power ##\frac{2\pi}{\Omega}## is half the period of the current. The average power is $$\langle P \rangle=\frac{\frac{1}{2}I_0^2R \int_0^{\frac{2\pi}{\Omega}}[1-\cos(\Omega t)]dt}{\int_0^{\frac{2\pi}{\Omega}}dt}=\frac{1}{2}I_0^2R.$$Same result.

vanhees71

1. What is alternating current (AC)?

Alternating current is a type of electrical current that constantly changes direction. It is commonly used in homes and businesses to power appliances and electronics.

2. How does alternating current differ from direct current (DC)?

Unlike alternating current, direct current flows in only one direction. This type of current is commonly used in batteries and electronic devices.

3. What is the frequency of alternating current?

The frequency of alternating current is the number of complete cycles it makes per second. In the United States, the standard frequency for AC power is 60 Hz.

4. How is average power calculated in an AC circuit?

The average power in an AC circuit is calculated by taking the product of the RMS (root mean square) voltage and current. This takes into account the constantly changing direction of the current.

5. What are the advantages of using AC over DC?

One advantage of AC is that it is easier to change the voltage using a transformer, making it more efficient for long-distance power transmission. AC is also more suitable for powering large appliances and machinery.

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