Question

Consider a power source with its terminal voltage and current given as:

vₛ(t) = 100 + 80sin(ωt − 100° + 70cos(2ωt+120°)+25sin3ωtV

iₛ(t) = 12 +10sin(ωt+25°) + 5sin(2ωt − 30∘) + 2cos3ωtA​

where ω is the fundamental angular frequency.

Calculate:

The rms value of iₛ(t) and vₛ(t)

Answer 1

The rms value of iₛ(t) is sqrt((12^2 + 10^2 + 5^2 + 2^2)) A, and the rms value of vₛ(t) is sqrt((100^2 + 80^2 + 70^2 + 25^2)) V.

Step-by-step explanation:

The rms value of a sinusoidal function is the square root of the mean square value of the function. To calculate the rms value of iₛ(t), we need to find the mean square value of the current function. In this case, the current function is given as iₛ(t) = 12 + 10sin(ωt+25°) + 5sin(2ωt − 30°) + 2cos3ωtA​​.

To find the mean square value, we need to square the function and then take the average over one period. The average value of a sinusoidal function over one period is zero. Therefore, the mean square value of iₛ(t) is equal to the sum of the squares of the amplitudes of the individual terms:

iₛ(t) = sqrt((12^2 + 10^2 + 5^2 + 2^2)) A

Similarly, to calculate the rms value of vₛ(t), we need to find the mean square value of the voltage function. Using the same method as before, we can find that the rms value of vₛ(t) is:

vₛ(t) = sqrt((100^2 + 80^2 + 70^2 + 25^2)) V