Question

An ac voltage is given as $v = 14 \sin (314t)\text{ V}$. The average and the effective value of the voltage (in V) over a cycle are respectively :

• A. $14$ and $7$
• B. $10$ and $14$
• C. $0$ and $10$
• D. $10$ and $0$

Hint:

Average AC voltage over a half-cycle is $\frac{2V_0}{\pi} \approx 0.637 V_0$, but over a full cycle it is always $0$. Effective (rms) value is always $\frac{V_0}{\sqrt{2}} \approx 0.707 V_0$.

Answer 1

The Correct Option is C

Concept:
• The average value of a sinusoidal alternating voltage over one complete full cycle is strictly zero due to positive and negative half-cycles canceling each other out.

• The effective or root-mean-square (rms) value of voltage is given by $V_{rms} = \frac{V_0}{\sqrt{2}}$, where $V_0$ is the peak voltage amplitude.

Step 1: Determine the average voltage
Given sinusoidal voltage $v = 14 \sin(314t)\text{ V}$.
Comparing with standard expression $v = V_0 \sin(\omega t)$, peak voltage $V_0 = 14\text{ V}$.
The average value of a symmetrical AC voltage over a full cycle is:
\[ V_{avg} = 0\text{ V} \]

Step 2: Calculate the effective (rms) voltage
The effective value is the rms voltage:
\[ V_{rms} = \frac{V_0}{\sqrt{2}} \]
Substitute $V_0 = 14\text{ V}$ and $\sqrt{2} \approx 1.414$:
\[ V_{rms} = \frac{14}{\sqrt{2}} = 14 \times 0.707 \approx 9.9\text{ V} \approx 10\text{ V} \]

Step 3: Conclusion
The average value over a full cycle is $0\text{ V}$ and effective value is $10\text{ V}$, which corresponds to option (C).