Understanding RMS Value vs. Instantaneous Value of AC Current

In alternating current (AC) circuits, the current varies sinusoidally over time. We often talk about the RMS (Root Mean Square) value and the instantaneous value of the current.

The instantaneous value of a sinusoidal AC current can be represented by the equation:

i(t) = I_{max} \sin(\omega t)

Where:

• i(t) is the instantaneous current at time t.
• I_{max} is the peak or maximum value of the current.
• \omega is the angular frequency (\omega = 2\pi f, where f is the frequency).
• t is the time.
• \omega t represents the angle in radians.

The RMS value of a sinusoidal AC current is defined as the equivalent DC current that would produce the same amount of heat (power dissipation) in a resistive load. For a sinusoidal waveform, the RMS value is related to the maximum value by:

I_{rms} = \frac{I_{max}}{\sqrt{2}}

The question asks at what angle the instantaneous value of the current is numerically equal to its RMS value. We need to find the angle \theta = \omega t where i(t) = I_{rms}.

Calculating the Specific Angle

Let's set the instantaneous current equal to the RMS value:

i(t) = I_{rms}

Substitute the expressions for i(t) and I_{rms}:

I_{max} \sin(\omega t) = \frac{I_{max}}{\sqrt{2}}

Now, we can simplify the equation by dividing both sides by I_{max} (assuming I_{max} is not zero):

\sin(\omega t) = \frac{1}{\sqrt{2}}

To find the angle \omega t, we take the inverse sine (arcsin) of both sides:

\omega t = \arcsin\left(\frac{1}{\sqrt{2}}\right)

The angle whose sine is 1/\sqrt{2} is 45 degrees (or \pi/4 radians). This is the first positive angle where this condition is met.

Therefore, the instantaneous value of a sinusoidal AC current is numerically equal to its RMS value when the angle is 45 degrees.