To find the expression for the instantaneous current of an AC sinusoidal current with a given RMS value and frequency, we can start by recalling some fundamental concepts. The RMS (Root Mean Square) value is a way of expressing the effective value of an alternating current (AC). For a sinusoidal current, the relationship between the RMS value and the peak (maximum) value is given by the formula:

Understanding RMS and Peak Values

The RMS value (I_rms) is related to the peak value (I_max) by the equation:

• I_rms = I_max / √2

Given that the RMS value is 40 A, we can rearrange this formula to find the peak value:

• I_max = I_rms × √2 = 40 A × √2 ≈ 56.57 A

Formulating the Instantaneous Current

The instantaneous current (i(t)) for a sinusoidal waveform can be expressed as:

• i(t) = I_max × sin(ωt + φ)

Where:

• ω = 2πf (angular frequency)
• f = frequency in Hz
• φ = phase angle (which we can assume to be 0 for simplicity)

Given that the frequency (f) is 50 Hz, we can calculate the angular frequency:

• ω = 2π × 50 ≈ 314.16 rad/s

Expression for Instantaneous Current

Substituting the values we have into the instantaneous current equation, we get:

• i(t) = 56.57 × sin(314.16t)

Calculating the Current at a Specific Time

Now, we need to find the value of the instantaneous current after 0.002 seconds:

• i(0.002) = 56.57 × sin(314.16 × 0.002)

Calculating the argument of the sine function:

• 314.16 × 0.002 ≈ 0.62832 rad

Now, we can find the sine of this value:

• sin(0.62832) ≈ 0.588

Finally, substituting this back into the equation for i(t):

• i(0.002) ≈ 56.57 × 0.588 ≈ 33.24 A

Summary of Results

To summarize, the expression for the instantaneous current is:

• i(t) = 56.57 × sin(314.16t)

And the value of the instantaneous current after 0.002 seconds is approximately:

• i(0.002) ≈ 33.24 A

This calculation illustrates how sinusoidal currents behave over time and how we can derive specific values from their mathematical representation. If you have any further questions or need clarification on any part of this process, feel free to ask!