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 by looking at the argument of the cosine function, which is $314t$. Since the cosine function has a period of $2\pi$, we can equate $314t = 2\pi$ to find the period:<br /><br />$314t = 2\pi$<br /><br />$t = \frac{2\pi}{314}$<br /><br />$T = 2\pi t$<br /><br />Now, we can calculate the average value:<br /><br />$\text{Average value} = \frac{1}{T} \int_{0}^{T} (10 + 20\cos(314t)) dt$<br /><br />Evaluating the integral, we get:<br /><br />$\text{Average value} = \frac{1}{T} (10T + 20 \cdot \frac{1}{314} \int_{0}^{T} \cos(314t) dt)$<br /><br />Since the integral of $\cos(314t)$ over one period is zero, we have:<br /><br />$\text{Average value} = \frac{1}{T} (10T)$<br /><br />$\text{Average value} = 10$<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 />In this case, we can calculate the RMS value:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} \int_{0}^{T} (10 + 20\cos(314t))^2 dt}$<br /><br />Expanding the square and evaluating the integral, we get:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} (100T + 400 \cdot \frac{1}{314} \int_{0}^{T} \cos(314t) dt + 400 \cdot \frac{1}{314^2} \int_{0}^{T} \cos^2(314t) dt)}$<br /><br />Since the integral of $\cos(314t)$ over one period is zero, we have:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} (100T + 400 \cdot \frac{1}{314} \cdot 0 + 400 \cdot \frac{1}{314^2} \cdot \int_{0}^{T} \cos^2(314t) dt)}$<br /><br />We can use the identity $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$ to simplify the integral:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} (100T + 400 \cdot \frac{1}{314^2} \cdot \frac{1}{2} \int_{0}^{T} (1 + \cos(2 \cdot 314t)) dt)}$<br /><br />Evaluating the integral, we get:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} (100T + 400 \cdot \frac{1}{314^2} \cdot \frac{1}{2} (T + \frac{1}{314} \int_{0}^{T} \cos(2 \cdot 314t) dt))}$<br /><br />Since the integral of $\cos(2 \cdot 314t)$ over one period is zero, we have:<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} (100T + 400 \cdot \frac{1}{314^2} \cdot \frac{1}{2} (T + 0))}$<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} (100T + 200 \cdot \frac{1}{314^2} \cdot T)}$<br /><br />$\text{RMS value} = \sqrt{\frac{1}{T} (100T + \frac{200T}{314^2})}$<br /><br />$\text{RMS value} = \sqrt{100 + \frac{200}{314^2}}$<br /><br />$\text{RMS value} = \sqrt{100 + \frac{200}{98596}}$<br /><br />$\text{RMS value} = \sqrt{100 + 0.002}$<br /><br />$\text{RMS value} = \sqrt{100.002}$<br /><br />$\text{RMS value