Question

The input voltage to the series R=4 Ohm and L=1/π Henry,network is expresed as the following Fourier seris;
Vi=(4/π)(sinπt+1/3sin3πt+1/5sin5πt)
1.What is the RMS value of the input voltage ?
2.What is the peak amplitude of the third harmonic of the circuit current response ?

Answer 1

The RMS value is computed by squaring the input voltage, integrating over one complete cycle, in this case 2 seconds, and then taking the square root of that definite integral. It's a good idea to use some computer algebra system instead of doing this by hand. Otherwise you will get bogged down in a sea of calculations.

The third harmonic of the input voltage has a frequency of 2.5 Hz. Since we are given the circuit's resistance, inductance, and oscillation frequency, we can compute its impedance. Because part b deals only with the third harmonic of the input voltage, you might as well disregard the first two terms of the input voltage and pretend that the input voltage is just 4/(5π) sin(5πt). It will take me a bit of time to come up with some actual numbers, but when I do I will post them so you can compare against your own.

Answer 2

I used wxMaxima to crank out the numbers tor part a:
\[V _{rms} = \sqrt{4144}/(15\pi) \approx 1.366 V.\]

Answer 3

For part b, I use only the third term in the given input voltage and call it E for electromotive force:
\[E = (4 \pi/5) \sin(5 \pi t), X _{L} = \omega L = 5\Omega.\]

Answer 4

Thank you so much.

Answer 5

:^)

Answer 6

The impedance of the circuit with E as input voltage is \[Z = \sqrt{R ^{2} + X _{L}^{2}} \approx 6.403 \Omega.\]

Answer 7

O.K.

Answer 8

We need to get the phase angle for the current relative to E. That angle will be pi/2 radians, so the current lags the voltage by pi/2 radians. The amplitude for the current will be the amplitude of the input voltage divided by the circuit's impedance:
\[I(t) = 4/(5 \pi Z) \sin(5t - \pi /2).\]

Answer 9

The amplitude of the response current is about 39.8 mA.

Answer 10

How do these figures compare with yours?

Answer 11

Thanks for the medal, Alma :^)

Answer 12

I do have posible answers;
1.0A
2.1/15A
3.4/15pi A
4.4/sq.root of 17piA