I met this discussion
https://forum.allaboutcircuits.com/...using-laplace-transform-understanding.136340/

I, even though I am not a student, I would like you to help me to know through practice Laplace transform in electronics.
I have not found another better place for this discussion.
https://en.wikipedia.org/wiki/Laplace_transform

 

Hi,

So this is not really homework then?

The Laplace Transform and it's inverse help to analyze electronic circuits because of the fact that the transform takes one function and maps it into another, and after that the solutions are usually simpler, and then transforming back we get a solution in the original domain of interest.

The Laplace Transform (hereafter referred to as LT) can be evaluated with the integral definition for many functions, while the Inverse Laplace Transform (ILT) is harder to evaluate directly so various techniques come into play that help to do the inverse transform such as partial fractions.

A complete survey of the LT would start from the Fourier Transform, but it's simpler just to start with the LT itself and work backwards if you ever need the Fourier Transform.

To start in the field of electronics it would be good to find out how to transform individual components and analyze the circuits using those transforms in the purely AC domain. Much of circuit analysis can be done that way. Once you get familiar with AC analysis, if you are not already, you can move on to doing some inverse transforms using the ILT. This gets you into the time domain where you can graph the response and that is usually an end goal.

Because the LT is a mathematical entity, it can be applied in various ways. Another way is to apply it after we've written the differential equations, which are best in the form of ordinary differential equations (ODE's). Either way i think one of Laplace's goals was to be able to transform the DE or ODE's and then end up with a system that is much easier to solve, and that's what we do today too. To use this technique you have to know how to write the differential equations for the circuit.

Whatever way you proceed, it's not as difficult as it may look in the literature. Doing a few circuits (ok maybe more than that) gets you familiar with the ideas behind using these powerful techniques.

The start is to find the transform of components like the capacitor, inductor, resistor, and power sources in purely linear circuits.
The next step would be to learn some common transforms and inverse transforms.
After that learn some of the LT properties, which show how we can benefit from using such ideas.
After that start to apply the transforms and inverse transforms and properties.
After that introduce some non linear components and how to handle them.

Concurrently you can also study numerical methods, which are in use by every circuit simulator program under the sun except for special purpose ones. Numerical methods provide more insight into the operation of the circuits as well as give you a secondary method for testing your calculations. Without a second method it's hard to be sure you have the right result every time.

So maybe it would be best to start with a simple RC circuit, then RL circuit, then RLC circuit. That would be a typical order to follow anyway in the study of circuits using the LT and ILT.

For the benefit of readers who want to help you, you should provide some background information so everyone knows what kind of explanations would be best for you. This should include your math background as a minimum, such as algebra, trig, calculus, etc. Knowing that helps others in writing replies which you will be able to appreciate.
I will say that knowing algebra well will get you pretty far really.

 

Whatever way you proceed, it's not as difficult as it may look in the literature.

Thank you very much.

The start is to find the transform of components like the capacitor, inductor, resistor, and power sources in purely linear circuits.
.................
So maybe it would be best to start with a simple RC circuit, then RL circuit, then RLC circuit.

Eg.




I gave a random example.

 

We should derive to get rid of integrate:
http://www.wolframalpha.com/input/?i=derivate+10*(e^-x)*sin(10x+5)

To solve this I will use Wolfram:
https://www.wolframalpha.com/input/?i=L*y''(t)++R*y'(t)+(1/C)+y(t)+=100*(e^-t)*cos(10*t+5)-10*(e^-t)*sin(10*t+5)


For me, if verify in practice it is the result

Edit - I understand the Fourier transform.
These bigger formulas I use them with the computer and draw some graphics.

 

Hello again,

Are you saying that you want to solve this with Laplace Transforms:


Note that it looks like you may not have entered it properly for the Wolfram Alpha computation.

To solve that using LT's you would transform each term and then calculate, then perhaps transform back using ILT.
Do you know how to transform the terms in that equation?

 

The RLC case is actually real the case for the LC oscillator. In fact, there is no LC oscillator without a bandwidth.
No bandwidth can be calculated without resistance.

Let's make it simpler then:

V is a battery, a constant voltage source that by derivation gives 0.
I expect the circuit to enter in damped oscillation as the oscillation is not maintained.

with e ^ x solution


(eq.1)








Check the solution with Wolfram
https://www.wolframalpha.com/input/?i=L*y''(t)++R*y'(t)+(1/C)+y(t)+=0




The results are the same.
If R is sufiecient small (R*R<4L/C) the LRC circuit starts oscillating. If R is greater than this value, it has the form of charging/discharging.

Are you saying that you want to solve this with Laplace Transforms:
.........................
To solve that using LT's you would transform each term and then calculate, then perhaps transform back using ILT.
Do you know how to transform the terms in that equation?

No, I'm now looking for.

 

I will simplify the problem further and put R = 0.Now (eq.1) becomes (simpler):

So,

 

For example:

http://www.wolframalpha.com/input/?i=integrate+e^(-s*x)*3*x^2

Now,

It would become, but I did not understand why.

 

i''(t)+1/(LC)=0, become
s*L{i'(t)}-i'(0) + 1/(LC)=0 =>
s*(s*L{i(t)}-i(0))-i'(0) + 1/(LC)=0

I could say that at the initial moment for the circuit current is 0A.
i(t)=0
But how do I know i'(t)? (Its "increasing tendency")
I think it's more difficult so.

 

Motanache.......now tell us why the equation works. Does it matter to you why the equation works?

 

First of all, I'm a hobbyist.
Just as some collect stamps or coins.
When I see the formulas, I better imagine what's going on and I know what components to change for the intended purpose.

Now I'm doing as a hobby. No one used my devices except me.

 

Plus I burned a lot of components, that I could buy a new market device.
Lost time I do not count it.............

 

Pardon me. I thought your profile said student....from the paradox post.....so I censored my responses there.
But student is not on your profile now. I must have misread it.

 

The usual practice is not to begin with the circuit differential equations but rather to write the KVL or KCL equations directly in the complex frequency domain using the s-domain impedance of each R, L, & C component. So in this problem it is only necessary to find the Laplace transform of the driving point voltage L(V) to obtain the Laplace transform of the loop current i in the s-domain. The final step is to calculate the inverse transform of the current i to get the time domain expression for the loop current. Note how starting the excitation voltage at some arbitrary phase angle greatly complicates the expression for the time-domain current. (Math shown in Maple)

If one is interested in the voltage drop across any component, first multiply the s-domain current by the s-domain impedance of the component, then find the inverse transform of the result.

 

i''(t)+1/(LC)=0, become
s*L{i'(t)}-i'(0) + 1/(LC)=0 =>
s*(s*L{i(t)}-i(0))-i'(0) + 1/(LC)=0

I could say that at the initial moment for the circuit current is 0A.
i(t)=0
But how do I know i'(t)? (Its "increasing tendency")
I think it's more difficult so.

View attachment 132926

Hello again,

I think you are going to have to explain better what you really want to do in this thread, what you want to accomplish. I say this for several reasons but this last one really sticks out where you want the Laplace Transform of cos(x) which does not make any sense unless you are working in at least two dimensions where one is usually of distance.
I think what you really want may be the transform of cos(t) or cos(wt) etc., but as i said you really need to speak in more coherent statements and explain exactly what you want to do. This will help greatly to understand what you are trying to learn.

When i do the inverse transform i dont like to assume anything but look to at least two solutions one damped and one underdamped and see what the conditions are that change the form of the response from one to the other. The main parameter is the part that may appear inside cosh() or sinh() which if imaginary tells you right away that there is at least some sinusoid for at least some time, which for an oscillator would be the thing to look for.

It's very hard to know exactly what you want to do at this point though. it is just analyze an oscillator circuit perhaps?