Question

A current in circuit is given by i = 3 + 4 sin ωt.

Then the effective value of current is :

• A. 5
• B. $\sqrt{7}$
• C. $\sqrt{17}$
• D. $\sqrt{10}$

Answer 1

Answer: (C)

The effective value of the current is $\sqrt{17}$ A.

Answer 2

The current in the circuit is given by the equation:

$i = 3 + 4 \sin \omega t$

This current consists of two components:

1. A DC component ($I_{DC}$)
2. An AC component ($I_{AC}$)

From the given equation:

The DC component, $I_{DC} = 3$ A. The AC component is $I_{AC} = 4 \sin \omega t$. This is a sinusoidal current with a peak value (amplitude) $I_m = 4$ A.

The effective value (or RMS value) of a current waveform containing both DC and AC components is given by the formula:

$I_{eff} = \sqrt{I_{DC}^2 + I_{rms(AC)}^2}$

First, we need to find the RMS value of the AC component. For a sinusoidal current with peak value $I_m$, its RMS value is:

$I_{rms(AC)} = \frac{I_m}{\sqrt{2}}$

Substituting $I_m = 4$ A:

$I_{rms(AC)} = \frac{4}{\sqrt{2}}$ A

Now, substitute the values of $I_{DC}$ and $I_{rms(AC)}$ into the formula for the effective current:

$I_{eff} = \sqrt{(3)^2 + \left(\frac{4}{\sqrt{2}}\right)^2}$

$I_{eff} = \sqrt{9 + \frac{16}{2}}$

$I_{eff} = \sqrt{9 + 8}$

$I_{eff} = \sqrt{17}$ A

Thus, the effective value of the current is $\sqrt{17}$ A.