Question

Define power factor in a.c. circuit. An alternating current is represented by $I=30\sin 314t$ ( in SI units). Find the maximum value of current and time period.

Answer 1

Hint:A battery consists of two plates of opposite charges. Here we consider the capacitance for calculating the energy as well as the charge.AC circuits which contain resistance and capacitance or resistance and inductance, or both, also contain real power and reactive power.

Complete step by step answer:
The power factor of an AC circuit is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit, and is a dimensionless number.Power factor defines the phase angle between the current and voltage waveforms. Note that it does not matter whether the phase angle is the difference of the current with respect to the voltage, or the voltage with respect to the current. The mathematical relationship is given as:
$$\begin{gathered} Powerfactor={\displaystyle \frac{VI\cos \varphi }{VI}} \\ Powerfactor=\cos \varphi \end{gathered}$$

Now in the ac circuit, we have the expression for current, $I=I_{max}\sin \omega t$.Here the term $\omega$ is defined as, $\omega ={\displaystyle \frac{2\pi }{T}}$, where $T$ is the time period. So we have the given equation $I=30\sin 314t$. Now equating this expression with the general formula $I=I_{max}\sin \omega t$, we have,
$$\begin{gathered} I_{max}=30 \\ \Rightarrow \omega =314 \end{gathered}$$
Also we have the expression, $\omega ={\displaystyle \frac{2\pi }{T}}$. Now substituting the value of $\omega$ , we can calculate the time period as,
$$\begin{gathered} \omega ={\displaystyle \frac{2\pi }{T}} \\ \Rightarrow 314={\displaystyle \frac{2\times 3.14}{T}} \\ \Rightarrow T={\displaystyle \frac{2\times 3.14}{314}} \\ \therefore T={\displaystyle \frac{1}{50}}s \end{gathered}$$
Hence maximum value of current is 314 and time period is $0.02s$.

Note:In an AC circuit, the voltage and current waveforms are sinusoidal so their amplitudes are constantly changing over time. Since we know that power is voltage times the current maximum power will occur when the two voltage and current waveforms are lined up with each other. That is, their peaks and zero crossover points occur at the same time. When this happens the two waveforms are said to be “in-phase”.