Q1- An AC voltage is given below: V(t)=10+20cos(314t) Please find the average and RMS values of V(t) ?

To find the average and RMS values of the given AC voltage $V(t)$, we can use the following formulas:<br /><br />1. Average value:<br />The average value of a periodic waveform over one complete cycle is given by the formula:<br /><br />$\text{Average value} = \frac{1}{T} \int_{0}^{T} V(t) dt$<br /><br />where $T$ is the period of the waveform.<br /><br />In this case, the period $T$ can be found from the angular frequency $\omega$ as:<br /><br />$T = \frac{2\pi}{\omega}$<br /><br />where $\omega = 314$ rad/s.<br /><br />Substituting the given expression for $V(t)$, we can calculate the average value as:<br /><br />$\text{Average value} = \frac{1}{T} \int_{0}^{T} (10 + 20\cos(314t)) dt$<br /><br />Solving the integral, we get:<br /><br />$\text{Average value} = \frac{1}{T} \left[ 10T + \frac{20}{314} \sin(314t) \right]_{0}^{T}$<br /><br />Since $\sin(314T) = \sin(2\pi) = 0$, the average value simplifies to:<br /><br />$\text{Average value} = \frac{1}{T} \cdot 10T = 10$ V<br /><br />2. RMS value:<br />The RMS (Root Mean Square) value of a periodic waveform is given by the formula:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} \int_{0}^{T} V(t)^2 dt}$<br /><br />Substituting the given expression for $V(t)$, we can calculate the RMS value as:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} \int_{0}^{T} (10 + 20\cos(314t))^2 dt}$<br /><br />Solving the integral, we get:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} \left[ 100T + \frac{400}{314} \sin(314t) \right]_{0}^{T}}$<br /><br />Since $\sin(314T) = \sin(2\pi) = 0$, the RMS value simplifies to:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} \cdot 100T} = \sqrt{100} = 10$ V<br /><br />Therefore, the average value of $V(t)$ is 10 V, and the RMS value of $V(t)$ is also 10 V.