Using Laplace to analyse circuits

[IanB]

Here is a reference that explains something about \$\zeta\$ (damping ratio) and \$\omega_0\$ (natural frequency) in second order systems:

https://ocw.mit.edu/courses/mechanical-engineering/2-003-modeling-dynamics-and-control-i-spring-2005/readings/notesinstalment2.pdf

That particular article is very mathematical (sorry), but once you know the keywords "second order system", "damping ratio" and "natural frequency" you will find more hits when searching.

[Simon]

Thank you, sadly the books I do have by John Bird don't go into any detail on the subject. I effectively reached the end of the book as the subject was broached briefly. I have ordered another book of his called understanding engineering mathematics which hopefully gives a slightly more practical approach to maths than the book I have which is good for getting the hard facts but is not exactly a light reading.

[IanB]

I little confused. As using Laplace takes the time and frequency out of the equation what is the difference between the voltage across the capacitor at a steady-state and during the step function. The net result of the step function is that the voltage is applied what is the difference between the result when the step function happens and in the steady-state later on? Or is it that by using Laplace I'm not having to calculate with time but once transformed back the results will take into account the timing?

The key concept here is that the Laplace transform method allows you to solve not for a single steady-state answer, but instead to solve for a complete time dependent function for the solution. What you are really doing is solving differential equations, where the solution has two components: a transient response that usually decays with time, and a steady state response which is what is left after the transient response is over. That result after you convert the Laplace solution back to the time domain is a mathematical function that has both transient and steady-state responses included in it.

[IanB]

Thank you, sadly the books I do have by John Bird don't go into any detail on the subject.

You are likely to find coverage of the subject of system responses and first and second order systems in books on control theory. That would be another section of the "bookshelf" to browse for useful books to look at.

[Benta]

You are likely to find coverage of the subject of system responses and first and second order systems in books on control theory.

Good point, those are well documented. 3rd order systems and higher is where you'll need Laplace.

[danadak]

If you go to archive.org and do searches many books, papers, ap notes,
vendor info on just about everything.


Regards, Dana.

[rstofer]

My new best friend, MATLAB, does all this stuff.

https://www.mathworks.com/help/control/ref/damp.html

It knows how to deal with all kinds of math, including Laplace Transforms and Inverse Laplace Transforms.

https://www.mathworks.com/help/symbolic/laplace.html
https://www.mathworks.com/help/symbolic/ilaplace.html

I realize you can't use MATLAB to take a test but sometimes it is nice to know if you are on the right track.

In real life, that arena outside academia, everything is open book and unlimited computing.  It's sad that colleges and universities insist on having students grind through math rather than take the time to learn what it means.

http://www.seas.upenn.edu/~ese216/handouts/Chpt13LaplaceTransformsMATLAB.pdf
http://www.egr.msu.edu/~aviyente/LT_matlab.pdf

I'm trying to ease my grandson into using things like MATLAB.  The other night we were having to do the ever popular 'keep reducing the length of the secant line until it approaches a tangent line' with unending button pushing on a calculator.  He soon came to the conclusion that we should just use MATLAB.  Outstanding!

[IanB]

My new best friend, MATLAB, does all this stuff.

I never really thought about MATLAB doing symbolic math, though I see that it does. I think we have used it at work for helping with symbolic differentiation of complex expressions to avoid human error in the algebra.

Can it pretty-print the algebra in math notation, or is it only text based?

Also, what about the cost? I have the impression MATLAB licenses are pretty expensive outside of academia?

[f5r5e5d]

http://cirlab.dinfo.unifi.it/Sapwin4/ does schematic to Laplace, exports in Matlab .m format, Octave is a free workalike for Matlab that is highly compatable

[Nitrousoxide]

Just going to add my 2 cents here.

From my personal experience I used the textbook "Control Systems Engineering" during a control systems subject, however, they do cover responses in the Laplace and the time domain fairly well (In chapters 2, 3 and 4 respectively). Just another book to add to a (digital) pile.

https://www.amazon.com/Control-Systems-Engineering-Norman-Nise/dp/0470917695

[KE5FX]

My new best friend, MATLAB, does all this stuff.

I never really thought about MATLAB doing symbolic math, though I see that it does. I think we have used it at work for helping with symbolic differentiation of complex expressions to avoid human error in the algebra.

Can it pretty-print the algebra in math notation, or is it only text based?

Also, what about the cost? I have the impression MATLAB licenses are pretty expensive outside of academia?

They now have a "home use" license option.  A couple hundred bucks gets you pretty much everything useful including the EE-related toolboxes at around $50/each. 

There are some unfortunate exclusions, like HDL generation, but it's still a darned good deal.

[mtdoc]

Don’t forget about Scilab which is a free open source Matlab alternative. I believe it does Laplace and has a control systems toolbox as well.

[IconicPCB]

Schaums is the bees knees in many areas.

Once you have mastered S domain... and in particular in computer control systems the dreaded Z domain steps into the picture....

[rstofer]

They now have a "home use" license option.  A couple hundred bucks gets you pretty much everything useful including the EE-related toolboxes at around $50/each. 

There are some unfortunate exclusions, like HDL generation, but it's still a darned good deal.

For those attending a degree issuing institution, the price is even better!  The Student Suite is $99 and contains MATLAB and Simulink.

https://www.mathworks.com/academia/student_version/class_use.html

Add on packages for the Home edition run around $49 so I suspect the Education version might be a little cheaper.

Do be aware that at the end of the year, Mathworks wants you to pay a maintenance fee and, for the packages I have, this runs around $200/yr.  I'm too old to be a student so I just have the Home version.

I don't know about 'pretty print', mostly I use the command line in the IDE or I write a script and the output is just text.  Then there is MuPad which claims to produce typeset output:

https://www.mathworks.com/help/symbolic/typeset-math-and-other-output-modes.html?searchHighlight=typeset%20output&s_tid=doc_srchtitle

https://www.mathworks.com/help/symbolic/getting-started-with-mupad.html

HDL Coder is available for the Student edition but not the Home edition.  They have a truly lame excuse that Home users without support would have a hard time using the package.

https://www.mathworks.com/matlabcentral/answers/332885-why-is-hdl-coder-not-available-for-home-license

[Kalvin]

I went to Youtube for Laplace circuit analysis videos. I happened to run into this nice recap of the Laplace circuit analysis video by rolinychupetin:



Here is a nice video for the second order circuit with some very useful math tricks involved when performing Laplace inverses:



He has other very good videos about circuit analysis, for example tutorial on Modified Nodal Analysis (MNA) which can be used also in Laplace-domain:



Best of all, he doesn't spend half of the video writing things to the whiteboard, but the video rolls on quite smoothly in comfortable pace.

[rstofer]

That Modified Nodal Analysis video (and approach) is terrific.  I need to do a few more problems to make sure I understand the idea of 'evil currents' and 'evil branches' but the author's example shows how to deal with them.

I graduated before the method was invented so I guess I don't feel too left out!

[sibeen]

I'll agree about the MNA video. MNA may have been invented before I graduated, but by less than 10 years so it certainly hadn't made its way anywhere near any text book that I sighted. I'll also be working through a few problems. In the initial video he had 6 unknowns and I immediately thought, "hey, there's a supernode in there, get rid of at least one. " He uses it as an evil current and so may be required, so another view of the video and a run around a few problems may inform me. 

[Wimberleytech]

Similar story...I learned the old way for my undergraduate.  Then I went into industry later to return to academia for my Ph.D.  My particular course of study led me to methods of doing circuit analysis using computer and that is where I was introduced to MNA.  This is a great video on MNA, but learned it in an even more regimented way.  You have to be more methodical if you are going to write a SPICE simulator--which is what I had to do.  The circuit parser just plugs each element into a nodal admittance matrix and then you send that to a sparse matrix solver...voila!  All of the work is in the parser!!

>>I found this on the internets...this is the way I learned it. 
https://www.swarthmore.edu/NatSci/echeeve1/Ref/mna/MNA2.html

[IanB]

Interesting. I watched that video on MNA and as I write simulators for a living it contained no surprises. I'm curious that it is something that might need to be "invented".  Isn't it just the logical application of basic circuit principles in a systematic way?

(I remember as an undergraduate my fellow students writing spice-like simulators using using similar principles round about 1980.)

[Wimberleytech]

Isn't it just the logical application of basic circuit principles in a systematic way?

Yes, I agree.  What has changed is ubiquitous computing power.  With computers, solving large matrices is trivial.    Now that we have huge computing power, systematic methodologies were developed that fed naturally into a computer solution.

[rstofer]

It's slightly different in that it solves for both voltage and current simultaneously.  Nodal analysis give node voltages, mesh analysis gives  branch currents and this modified nodal analysis seems to do both.  There are more equations and more unknowns in that there are KCL equations for every node, EVL equations for every 'evil current' and CTL equations for every controlling variable.  In the old days (slide rules), this wouldn't be considered an advantage.  It's easy to solve a 2x2, even a 3x3, but once you get to 4x4 things get out of hand.  Creating 6 equations in 6 unknowns is only useful when using a computer or high end calculator.

[rstofer]

For giggles, I fed the equations into MATLAB see attached screen shot

[rstofer]

While I was on a roll, I decided to port the equations to wxMaxima.  For folks looking for a terrific math tool that happens to be free, wxMaxima is worth the time to evaluate.

Attached is a PDF of the code and the output.  Note that the equations look an awful lot like the ones I used with MATLAB.  In fact, I copied and pasted and just changed a wee bit for syntactic reasons.

Everything below 'res : ' is just used to produce a sorted output list.  That very same code is used in all of my files - now that I figured out how to do it.

So, why do I get into this stuff?  Well, in a year or so, my grandson is going to take his very first circuit analysis course and I need a head start.

BTW, don't expect a flat learning curve with either of these tools.  It takes time.  Mostly it takes copy and paste!

[Hextejas]

   I take my hat off to you folks that have the patience to learn this "stuff".
I'm not sure how or if it would be used to  try and figure out why my ham radio stopped working.
Wowsers!

[rstofer]

  I take my hat off to you folks that have the patience to learn this "stuff".
I'm not sure how or if it would be used to  try and figure out why my ham radio stopped working.
Wowsers!

It probably won't help a bit!

However, you can bet that Laplace Transforms were used to design some of the more sophisticated filters.

Here is a Wiki on the Butterworth Filter that makes my head spin:

https://en.wikipedia.org/wiki/Butterworth_filter

To think there was a time, back 45 years ago, where I actually understood this stuff.  Today it would be a stretch!  The good news today is that MATLAB will solve all the Laplace and Inverse Laplace transforms.