# Fourier series of output voltage

I've been practicing some Fourier series questions and then verifying my answers by generating an equivalent graph on MATLAB and comparing it with the graph generated by PSpice in simulating the same circuit.

This is my working:

The Fourier series of the source current:

$i_s\left(t\right)=1+\frac{4}{\pi}\sum_{n=1}^{\infty}{\frac{1-\left(-1\right)^n}{n}\sin{n\pi t}}$

Then do a source transformation to simplify the circuit: $v_s\left(t\right)=1i_s\left(t\right)$

simulate this circuit – Schematic created using CircuitLab

Then working in the phasor domain and using voltage division:

$\omega_n=\pi n$

$V_{out}=\frac{Z_L}{Z_L+3}V_s$

$V_{out}=\frac{j\omega_n}{j\omega_n+1}V_s$

$V_s=I_s=\frac{4}{\pi n}\left(1-\left(-1\right)^n\right)e^{j\left(-90\right)}$

$V_{out}=\left(\frac{j\omega_n}{j\omega_n+1}\right)\left(\frac{4\left(1-\left(-1\right)^n\right)}{\pi n}\right)e^{j\left(-90\right)}$

$V_{out}=\left(\frac{4\left(1-\left(-1\right)^n\right)}{\pi n}\right)\left(\frac{w_ne^{j\left(90\right)}}{\sqrt{1+\omega_n^2}e^{j\left(\tan^{-1}{\pi n}\right)}}\right)e^{j\left(-90\right)}$

Taking only odd n terms since evens result in 0:

$V_{out}=\left(\frac{8}{\sqrt{1+\pi^2n^2}}\right)e^{j\left({-\tan}^{-1}{\pi n}\right)}$

and finally in the time domain:

$v_{out}\left(t\right)=\sum_{k=1}^{\infty}{\frac{8}{\sqrt{1+\pi^2n^2}}\cos{\left(\pi nt-\tan^{-1}{\pi n}\right)}}$

with n = 2k - 1 for odd terms

I plot my answer in MATLAB and it seems to be the negative of what the equivalent PSpice graph shows.

Can someone point out what is wrong please?

• I'm actually impressed by the detail and clear presentation of your question. – Arsenal Apr 9 '18 at 14:14
• A guess: Your simulated Is waveform is inverted. It starts from -1A, not from 3A. Congratulations for the ability to ask something else than "Calculate this for me, I need it now!" – user287001 Apr 9 '18 at 14:23
• Is your current source value correct on your PSpice circuit? It says, -1A. Should it be +1A? Also, welcome to EESE! As User mentioned before, it's been a while since a new user asked a proper first question that they have with us ;) – KingDuken Apr 9 '18 at 14:34
• Thanks guys. I don't think its inverted though, here is the graph with the current source added in. You can see it matches the graph in the question: i.imgur.com/Z7KiXDn.png – Lachlan Apr 9 '18 at 15:18
• When working with phasors you implicity assume steady state. Your spice simulation is showing the transient leading to that steady state. The waveform is slowing removing the DC content from the input to become symmetric. The 'inversion' you see is just due to the fact that your steady state solution starts from an arbitrary point. You should plot both input and output together to see that they are 'in phase' both in the transient (spice) and steady state (phasor) analysis. – Sredni Vashtar Apr 9 '18 at 18:21