Calculate LC Lowpass Filter Output Signal

[etf]

Hi!
Here is my task:
Lowpass filter circuit is given:

Input signal to this circuit is:

L=36.4mH, C=85.4nF.
Calculate and draw output signal.

My idea is to:
1) Represent input signal in term of Fourier series (complex form),
2) Calculate transfer function for given circuit (for n-th harmonic),
Expression for output signal would be product of Fn of input signal (complex Fourier coefficient) and transfer function of circuit.

1)
Here is input signal in term of Fourier series (complex form):

2)
Here is transfer function for n-th harmonic:
(L'=L/2)

Output signal is now:

First question: what is purpose of second inductor? We don't take into account when calculating transfer function.
Second question: If I plot output voltage I calculated, it doesn't match with voltage I got in simulation software? Why?
Does it have something with location of pole of transfer function? Pole of our transfer function is on imaginary axes.

 

[NascentOxygen]

The inductor on the right achieves nothing without there being a load of finite impedance. With an open-circuit load, as you show, the second inductance does nothing.

Are you sure there isn't supposed to be a load of some specified resistance?

 

[etf]

Hi, thanks for reply.
Task is to calculate cutoff frequency, equivalent resistance and output signal. I used formula fc=1/(pi*sqrt(L*C)) to calculate cutoff freq., R=sqrt(L/C) to calculate equivalent resistance. But I'm not sure about output signal. My calculations are related to steady state response, but this circuit never goes to steady state (I would say)

 

[NascentOxygen]

You didn't actually answer my question re load impedance.

Your input signal here, is it just these two pulses then nothing, or is this waveform going to be repetitive?

For the record, you haven't yet learned to use Laplace Transforms, I gather?

I have to leave this for others, I'm afraid I'm just too rusty with Transform application.

 

[gneill]

I'd be tempted to use the non-complex version of the Fourier series for the squarewave, expanding it as a sum of sines.

For a squarewave of unit amplitude (+/- 1) and a period of T:
$$f(t) = \frac{4}{\pi} \sum_{n = 1,3,5,...} \frac{1}{n} sin \left( \frac{n 2 \pi}{T} t \right)$$
Each term is a sinewave with a frequency that's an odd multiple of the fundamental. Standard circuit analysis methods can then be applied for the transfer function, treating the magnitudes of the sine terms as phasors.

 

[rude man]

Hi!
Here is my task:
Lowpass filter circuit is given:

View attachment 90784

Input signal to this circuit is:

View attachment 90785
Calculate and draw output signal.
My idea is to:
1) Represent input signal in term of Fourier series (complex form),
2) Calculate transfer function for given circuit (for n-th harmonic),
Expression for output signal would be product of Fn of input signal (complex Fourier coefficient) and transfer function of circuit.

Is the input a pulse, as shown, or does it repeat ad infinitum?
For a pulse you don't use a Fourier series. You can use the Fourier integral or the Laplace transform.

 

[etf]

Input signal is periodic...

 

[rude man]

Input signal is periodic...

So then I'd go with post 5 but you should understand why
(1) there are no cosine terms; and
(2) why only odd harmonics.
Because a general periodic function has sin and cos terms and all harmonics, odd and even.
Still, drawing it isn't all that easy. Best would be software, even something simple as excel.

 

[etf]

So then I'd go with post 5 but you should understand why
(1) there are no cosine terms; and
(2) why only odd harmonics.
Because a general periodic function has sin and cos terms and all harmonics, odd and even.
Still, drawing it isn't all that easy. Best would be software, even something simple as excel.

But it isn't about which "type" of Fourier series I use (complex like in my first post or noncomplex like in post #5). Whatever type I use, I will get response of circuit in steady state, which actually never appears in my case (ideal LC circuit), right?

 

[rude man]

But it isn't about which "type" of Fourier series I use (complex like in my first post or noncomplex like in post #5). Whatever type I use, I will get response of circuit in steady state, which actually never appears in my case (ideal LC circuit), right?

In Suze Orman's parlance, "here's the problem": your circuit has no dissipative elements like resistors. So no matter when the waveform was first applied to the network, there will always be two waveforms with separate Fourier series: the steady-state one you'd get if there was even a small amount of R in the circuit, dictated by the input waveform, and the other which is dictated by the L-C time constant only.

But wait, there's more! Since there's no R the network "remembers" what the exact state of the input, i.e. the phasing, was when it was first applied. So the steady-state solution depends on when exactly the waveform was applied. This makes your problem not only very difficult but actually undefined.

This problem IMO should never have been assigned. No network has zero dissipation.