# derive the expression for vo/vi in terms of the angular frequency of a square wave?intergreating amp

#### halfeclipse

Joined Feb 2, 2017
11
Relevant circuit is Attached.I'm not really sure where to begin here.been trying to find something that points the way but there's nada. I'm pretty sure I can do this if it was ac input, but a square wave not so much. about the only thing I can think of would be to do Vin=sgn(coskt) and then do something in the frequency domain like I would with ac. I can even get an expression for Vo/Vi, but I haven't the slightest clue if doing that would be proper.

#### WBahn

Joined Mar 31, 2012
26,398
If you were given an input that was just a step input, could you find the output?

A square wave is just a string of step inputs of equal magnitude but opposite sign.

A square wave is also a string of harmonically-related sinusoids.

#### halfeclipse

Joined Feb 2, 2017
11
yes but that approach gives me a 1st order DE that doesn't seem to get me any closer to expressing the ratio in terms of the angular frequency

obviously

that will involve a Fourier expansion? not expected at this point in the course and not been taught yet

Please do me a favour and don't do the guiding questions thing. it's immensely unhelpful. I can ask myself that sort of question just ifne, and if it was going to get me anywhere, it would have well before for I decided to ask here. This course is some bastard hybrid of two older courses with different pre-reqs and they did not rationalize the two very well. It's entirely probable this lab was written with the expectation that some subject had already been covered which is no longer the case (it would not be the first time). A run down on the mathematical process would be great. A reference text that covers the response of opamp circuits to non sinusoidal periodic wave forms would be helpful. a worked example of a similar problem would be fantastic.

Everything single thing I can find on the topic treats Vin as a piecewise function and avoids anything whatsoever to do with it's angular frequency. There is nothing what so ever in my textbook, the lecture notes, the lab notes,from a google search or any of the three other textbooks i've pulled out the the library that gives meaningful guidance on this. Beating my head on this problem like that any further will get me no where.

I know that this circuit should give me a vo with a triangular waveform. I know a triangular wave can be described by the integral of sgn[cos(x)] which implies that doing something with sgn(coskt). I know I can also describe that as a Fourier series, which implies I should be able to treat this as a sinusoidal function and and I think that implies I can do something with the frequency domain, but past that I have zero clue and likely lack the math to justify any further steps on my own

#### WBahn

Joined Mar 31, 2012
26,398
A sine wave has three parameters, right? An amplitude, a frequency, and phase (ignoring DC offset)

A square wave has the same three parameters, right?

Wouldn't a triangle wave have the same three parameters?

What would the parameters for a triangle wave for this circuit be in terms of the parameters for a square wave?

Is it possible that the question is looking for the ratio of the amplitudes of the output triangle wave to the input square wave?

#### WBahn

Joined Mar 31, 2012
26,398
Please do me a favour and don't do the guiding questions thing. it's immensely unhelpful. I can ask myself that sort of question just ifne, and if it was going to get me anywhere, it would have well before for I decided to ask here.
Oops. Sorry. Just ignore my unhelpful guiding questions, since you've obviously already considered them. I won't bother you with any more of them.

#### The Electrician

Joined Oct 9, 2007
2,866
Oops. Sorry. Just ignore my unhelpful guiding questions, since you've obviously already considered them. I won't bother you with any more of them.
LOL! #### halfeclipse

Joined Feb 2, 2017
11
It's specifically wants Vo/Vi expressed as a function of the angular frequency. It does not want anything but that. The only way I can think of to get a periodic voltage expressed as a function of the frequency to to convert the expression from the time domain to the frequency domain. If if do that with AC I can express Vi/Vo purely in terms of the impedances, which given the capacitor, means I can express Vi/Vo as a function of the angular frequency. However we have seen nothing whatever that justifies doing so for non sinusoidal functions. I know enough to know it ought be doable, however I don't have the mathematical tools to justify doing so with sufficient rigor to satisfy the rubric. "I'm pretty sure I can do this?" won't cut it.

I've tried looking for examples, but not found any presumably because any such example are best addressed *after* handling the fourier expansion stuff, and thus all the material treats that as expected knowledge. I also know the fourier expansion was taught far earlier in the old course that these labs were designed for.

For that matter vo/vi is the ratio of a forcing function to a forcing function and I have just enough diff calc to know there are tools to deal specifically with that, however i have zero clue how that works when going from a time-dependent function to a frequency dependant function (frankly i can barely deal with them as time dependent function). However given that such tools exist and that ratio like vo/vi seem pretty fundament and that frequency dependant is a much better way to deal with periodic functions than time dependant, I guarantee that there are analysis tools that deal with this directly.

Can I use a phasor to express a non sinusoidal function without any issue resulting from the non sinusoidal function being an infinite sum of sinusoidal functions. If so, what tool or theorem justifies thats, if not, what specifically do I need to look into in order to do so. I don't expect you do to the question for me, but something to the effect of "Look up X, you can find a proof Y to the effect of blah blah blah" would be extremely helpful, since that would prevent me from wading through literally 300 pages of the advanced analysis section of my textbook trying to find it (assuming it even is in this godawful textbook)

#### MrAl

Joined Jun 17, 2014
8,549
Hi,

This actually isnt that hard at all if you dont mind working in the frequency domain or the time domain.

When in the time domain, the frequency is just 1/T so you can always convert like that if you find a time domain solution which could come in the form of a series with a closed form solution, and this is true of an RC network so i would think it would be true here too.

When in the frequency domain it's even simpler as long as you know how to derive the Fourier Series for the input waveform, which in this case is a square wave. I'll wait to see if you want to do that yourself.
Once you have the input form, you would then derive the transfer function in the frequency domain. Then apply the input, one harmonic at a time, to the frequency domain transfer function and sum the results, possibly taking phase into account also. The result will be a series in the frequency domain which you can then even convert to the time domain if you choose to do so. You loose the exponential components of the true full form time domain solution, but in the steady state solution we are not after that information anyway.

I'll wait to see what you want to do yourself, but if you already did a triangle wave i dont see why you would not be able to do a square wave. It's just a different series, and amazingly almost the same.
I am assuming for now that you "did" the triangle wave by the series method and summed the results.
If you dont know the correct frequency domain form for the square wave, we can go over that or you can use the Fourier Series formulas or you can look it up on the web as this is a very commonly asked question.

#### halfeclipse

Joined Feb 2, 2017
11
nope, 2 am and done it without, I think. Or maybe this touches on that?

if I have some input and output signal vi(t) vo(t) then I can take the Laplace transform of both of those to get vi(s) and vo(s). if s is a complex variable then I have a+jb, and so the simple substitution gives me the frequency domain representation of the input and output. Also steady state, so a=0

the circuit is treated as ideal so linear time invariant, and there's a linear mapping such that vo(s)=f(s)vi(s) and so vo(s)/vi(s)=f(s). Nice thing about that is It doesn't matter *what* vo and vi are f(s)will be some function of the circuit itself. I'll knock out that tomorrow, but I rather expect it will be of the form -Zf/Zi, where Zf in this case is that resistor and capacitor in parallel. gett

#### MrAl

Joined Jun 17, 2014
8,549
nope, 2 am and done it without, I think. Or maybe this touches on that?

if I have some input and output signal vi(t) vo(t) then I can take the Laplace transform of both of those to get vi(s) and vo(s). if s is a complex variable then I have a+jb, and so the simple substitution gives me the frequency domain representation of the input and output. Also steady state, so a=0

the circuit is treated as ideal so linear time invariant, and there's a linear mapping such that vo(s)=f(s)vi(s) and so vo(s)/vi(s)=f(s). Nice thing about that is It doesn't matter *what* vo and vi are f(s)will be some function of the circuit itself. I'll knock out that tomorrow, but I rather expect it will be of the form -Zf/Zi, where Zf in this case is that resistor and capacitor in parallel. gett
Hi,

Yes that sounds reasonable. I didnt think you were into that form of the solution, but if that is acceptable then you should be good to go. Perhaps you can post your solution too so we can compare notes.
Yes the square wave has a Laplace form too.