RMS Current With Harmonics

There are two ways to do this problem. The hard way is determining the time-domain representation of the signal and then integrating to get the RMS value. The easy way is to use Superposition and apply Parseval's Theorem . This sounds more complicated than it actually is.

We have two frequencies in the source; one signal with an RMS value $V_1 = 1$ and angular frequency $\omega_1 = 1$ , and the other with an RMS value $V_2 = 1$ and angular frequency $\omega_2 = 2$ . We can represent the circuit as a superposition of two circuits: one at each frequency.

The complex reactance of an inductor is $X_L = \omega L$ . Since $L=1$ , $X_L = \omega$ . The key is that at each frequency, the resistance remains constant (at least ideally) while the inductive reactance scales with frequency.

Circuit 1: $\omega = 1;\, R = 1;\, X_L = 1;\, Z = R + jX_L = 1 + j;\, |Z| = \sqrt{1^2 + 1^2} = \sqrt{2}$

Circuit 2: $\omega = 2;\, R = 1;\,X_L = 2;\,Z = R + jX_L = 1 + 2j;\, |Z| = \sqrt{1^2 + 2^2} = \sqrt{5}$

Then we can calculate the RMS current magnitudes in both circuits using Ohm's Law .

$I_1 = \frac{1}{\sqrt{2}} \\ I_2 = \frac{1}{\sqrt{5}}$

Finally, in this application, Parseval's Theorem tells us that the RMS value of the composite current is the square root of the sum of the squares of the RMS currents at each frequency (the Euclidean Norm).

$I_{RMS} = \sqrt{I1^2 + I2^2} = \sqrt{\frac{1}{2} + \frac{1}{5}} = \sqrt{\frac{7}{10}} \approx 0.837$

As one final note, I also did it the semi-hard way by running a small-time-step simulation in code and numerically integrating. That approach gives the same result as the method presented above.