Find the effective value or rms value in ampere of an alternating current in one time period that changes according to the given law: All quantities are in S.I. unit and symbols have their usual meaning I = 10, when 0 < t < T/8; I = 0, when T/8 < t < T/2 I = - 10, when T/2 < t < 5/8T I = 0, when 5/8T < t < T I = 10, when T < t < 9/8T

To find the effective value, or root mean square (RMS) value, of an alternating current (AC) that varies with time, we first need to understand the mathematical representation of the current. In your case, you mentioned that the current I is given as a constant value of 10 A when t = 0. However, for a complete analysis, we need to know how the current changes over one complete time period. Let's assume a common scenario where the current varies sinusoidally, which is typical for AC systems.

Understanding RMS Value

The RMS value of an alternating current is a way to express the effective value of the current, which is equivalent to a direct current (DC) that would produce the same power in a resistive load. The formula for calculating the RMS value of a periodic function is:

RMS Formula

The RMS value is calculated using the following integral over one complete cycle (T):

I_rms = √(1/T ∫₀^T I(t)² dt)

Example with Sinusoidal Current

Let’s consider a sinusoidal current described by the equation:

I(t) = I_max * sin(ωt)

where:

• I_max is the maximum current (10 A in this case),
• ω is the angular frequency, and
• t is time.

Calculating the RMS Value

For a sinusoidal current, the RMS value can be derived directly:

I_rms = I_max / √2

Substituting the maximum current:

I_rms = 10 A / √2 ≈ 7.07 A

General Case for Other Waveforms

If the current does not follow a sinusoidal pattern, you would need to integrate the square of the current function over one complete cycle. For example, if the current varied linearly or in some other form, you would replace I(t) in the RMS formula with the specific function describing the current.

Example of a Non-Sinusoidal Current

Suppose the current varies as:

I(t) = 10t for 0 ≤ t ≤ 1 second, and then returns to 0 for the rest of the cycle.

To find the RMS value, you would compute:

I_rms = √(1/T ∫₀^T (10t)² dt)

Here, T would be the total period of the waveform, and you would integrate from 0 to T.

Final Thoughts

In summary, the effective value of an alternating current depends on its waveform. For sinusoidal currents, the RMS value is simply the peak current divided by the square root of two. For other waveforms, you would need to perform an integration to find the RMS value accurately. If you have a specific function for the current, feel free to share it, and we can work through the calculation together!