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Commentator

Join Date: Apr 2008
Posts: 86

# Laplace Transform and Complex Variable "s"

02/02/2009 11:25 PM

In laplace transform, there is a complex variable "s".

I guess "s" contains frequency information or so...

s = a + bi,(where i is complex number).

What is a physical meaning of s in mechanical system...

it could be eigenvalues ... etc..

s(a+bi).. what does "a" / "b" mean, respectively ?? physically...

Let me know.. pls ~ !!

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Join Date: Jan 2008
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#1

### Re: complex variables "s"

02/03/2009 12:55 AM

The Laplace transform is not directly time to frequency domain conversion. That is the job of the Fourier transform. The Laplace transform is a mathematical operator. In s-space, s operating on a function denotes differentiation, and 1/s operating on a function denotes integration.

The best book I have seen on the subject of transforms in general and the Laplace transform in particular is Operational Mathematics by Churchill. It dates from the 1940s, when they were interested in solving real problems in heat conduction and the like. Google it and get many hits and you can buy it for under US\$10 used.

I find math and physics books from the 1940s very useful, very practical, with excellent intuitive explanations of why things work as they do.

This was the era when academia understood that its purpose was to serve society, as opposed to today, when the purpose of society is to fund academia. (Speaking here of the USA.)

Guru

Join Date: Mar 2007
Posts: 2556
#2

### Re: complex variables "s"

02/03/2009 2:15 AM

i don't know why an old timer should not GA you just for the last few lines (others are known facts)

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#3

### Re: Laplace Transform and Complex Variable "s"

02/03/2009 11:10 PM

The use of complex numbers, with the "s" operator, lets you manipulate amplitude, frequency and phase info for the system.

It's pretty dumb for emcc to say that the books from the 40's (probably when he went to school) are the best. Things have changed a fair bit since then (transistors, computers, control theory etc) and the subject has grown a bit.

This whole subject is of special interest to student elec. engineers, so if you've got any problems ask them or get them to recommend a suitable textbook..

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2
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#4

### Re: Laplace Transform and Complex Variable "s"

02/03/2009 11:23 PM

I'm actually quite a bit younger than that, just had the good fortune to have profs and mentors who went to school under the original GI bill or thereabouts and still used the older texts.

And I truly and sincerely appreciate ffej's characterization. I have an excellent and quite remunerative career as a self-employed consultant, and I have made my reputation cleaning up messes made by people who think that because technology changes, they can ignore basic physics.

Keep it coming ffej and all you young whippersnappers out there: it's music to my ears, and money in the bank.

Guru

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#5

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 1:14 AM

I was never aware that new inventions have really changed so much of basic mathematics and may be proved 2+2=4.8

And I never thought that Fourier or Laplace were 21st century products. We still use it in the same way only with computers (Viz FFT is it not a Fourier transformation? we used to do other way in our schools)

Ever heard about students making a blank face when the old timers like us tried to explain clock-wise and anti-clock wise rotation of phasors ?

May be now with auto cads and other mathtools they only talk in + and -

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Fantastic ideas for a Fantastic World, I make the illogical logical.They put me in cars,they put me in yer tv.They put me in stereos and those little radios you stick in your ears.They even put me in watches, they have teeny gremlins for your watches
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#7

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 4:57 AM

wow, strictly speaking, FFT is not a Fourier transform.

only a way of calculation in discrete system.

btw, what is phasor? is it vector?

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#6

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 1:21 AM

My apologies, that comment came out a bit ruder than intended.

Part of the difference in texts may be due to the different mathematical backgrounds students have. Most now know, for example, matrix & vector algebras which can completely change how equations are presented. Maxwell's eqn's using Div and Curl are an example. You're right though, the basic physics doesn't change.

By the way, thanks for calling me a young whippersnapper, that hasn't happened for a long while.

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#8

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 5:23 AM

haha, it sounds everywhere is fulfilling with fighting atmosphere.

lets be humourous.

why cannt basic physics change? it of cause can be. diffrent field, different basic application. its well known Newton and 'Einstan.

Ether is still existing?

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#12

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 2:22 PM

Like I said, not a problem. It's how I make my living - remembering things that others have forgotten, or never learned to use.

I may be misinterpreting your post, but it seems to me that you imply that "back in the day" they didn't have vector calculus (div, grad, curl) or matrix algebra as tools.

These tools have been around a very long time. Einstein used tensors (an n-dimensional generalization of a matrix) to develop the General Theory of Relativity in 1905. And I see the use of div, grad, and curl in the 1950 bible of antenna engineering, "Antennas," by Kraus. Also in the 1942 classic "Fundamentals of Electric Waves," by Skilling.

BTW, the Skilling book is a very slim volume, less than 200 pages, and it is a 6 x 9 inch format; way smaller than a modern textbook on electromagnetics.

Here is a snippet of wisdom from that book, a footnote from the chapter on antennas.

"A question that very commonly arises in reference to receiving antennas is: Is the antenna voltage produced by the electric field of the passing wave, or the magnetic field, or both? This is a natural question, but the answer is clear when it is considered that anywhere in space the electric field of a traveling wave is the result of a changing magnetic field. The electric field induced in an antenna is likewise the result of the changing magnetic field, and whether one wishes to consider the electromotive force as the integral of the electric field of the wave in space (which it is) or as produced by the change in magnetic field (which it also is) is immaterial. The above question is analogous to asking whether a cork rising on the crest of a water wave is lifted by increasing pressure or by the higher water level; in wave motion there cannot be one without the other."

The footnote goes on with some very important technical details of shielding magnetic loops and the nature of the near vs. far field: words which are central to much of what I do, but unnecessary detail for a general post to CR4.

Now I don't know how well the above quote works for you, but for me it was vastly more enlightening than a whole year of div/grad/curl/Stokes theorem mathematical manipulations in "Electromagnetic Fields and Waves," by Lorraine and Corson. And still better than an earlier but heavily theoretical treatment, "Classical Electromagnetic Radiation" by Marion (1965).

I could do the math, but I gained no "feel" for what I was doing - I was just solving math problems.

Long after graduation, i gained the insight into the physics of electromagnetics, but by then the math skills had atrophied from disuse.

A common lament.

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#13

### Re: Laplace Transform and Complex Variable "s"

02/05/2009 3:58 AM

I am not any more so young but I still use books edited 40 (or even more) years ago.

I noticed this reaction to reject old for new several times, in general it comes from persons with a lower level of understanding (although some enjoy a high level of education).

Even if products do evolute at a constantly higher rate the basics of physics and mathematics are very very stable!

Euler or Gauss whose equations an engineer meets very often lived centuries ago. The same is valid for Young or others.

It is a wrong approach to reject past for present or future even in life since past is equal to experience and future can be more economically build up if already made errors are not repeated.

I remember a very funny situation, visiting a conference on advanced hydraulics, a team of young engineers presented a "new" and "revolutionary" pump control totally mechanical with less sensitivity than the,, at that time already used, electronic (electrohydraulic) systems. This was the consequence of not looking back at evolution and also neglecting the bases of tribology or servo-control. By the way most of differential equations used in dynamic analysis of control loops are not from the last year!

Guru

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#9

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 5:43 AM

Hi, guy, you owned me a GA at the front issues. you should pay me again for my industrious work. haha.

neednt guess, operation S indeed includes frquency(angle) .no doubt.

where a is amplitude and b is argument of an osillation.(viberation)

it may not be eigenvalue of an object. but if it is , the object will be taken place a socalled resonance.

a/b = normalization. for convinient calculation.

we generally use sigma and oumiga instead of a,b

s= sigma + omga i = sigma* exp(omiga)

from this transformtion, we can get frequency domain from time domain.

Participant

Join Date: Dec 2008
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#14

### Re: Laplace Transform and Complex Variable "s"

02/05/2009 12:02 PM

not e exp(omiga) but e exp(i*omega)

Guru

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#15

### Re: Laplace Transform and Complex Variable "s"

02/05/2009 8:55 PM

Very good.

I found this erro two days ago when I sent a private massage and told to #11 writer.

you are the first to point out it after it.

In fact, the last part is all wrong part, should stroke out.

you can wite exp(d), instead of e exp(d)

--

ps

a/b could be served as a 3db point of crossover frequency.

Regards

Anonymous Poster
#10

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 12:09 PM

The Laplace transform just changes the description of a signal or the relationship between output and input (transfer function) into a form that is easier to manipulate mathematically. Because the derivative of an exponential (e^-st) is just -s*(e^-st), you can translate the linear differential equations that describe the system into algebraic equations (by dividing out the (e^-st).

A short answer to your question is: if s=a+bi, the "a" indicates growth or decay of the signal amplitude. The "b" denotes the frequency that it is oscillating. A larger "a" and the signal decays rapidly. If "a" is negative, the signal grows. A larger "b" and the signal oscillates at a higher frequency.

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#11

### Re: Laplace Transform and Complex Variable "s"

02/04/2009 12:14 PM

Hi;

Every body over complicates the description of the LaPlace Transform, especially the prof's in college!

First, the FFT is a "fast" way to compute the spectrum of any time series (waveform). It does this by characterizing the waveform with pure sine waves/cosine waves, and works great for pure tones such as rotating machinery makes.

But the LaPlace transform characterizes a wave with DAMPED SINEWAVES. It is therfore very useful for determining the MODES of any system, electrical or mechanical. At Tektronix we used it on the now obsolete 2642 Dynamic Analyzer to determine the poles and zeros of structures (e.g. automobiles) and of servo amplifiers.

The poles are complex; the real part is the Frequency of resonance, the imaginary is the damping of the resonance (describes how quickly the pole will damp to zero). This is often ploted in the "s" plane with F on one axis, damping on the other.

Two other parameters are associated with each pole; Magnitude and Phase. The phase is usually ZERO or 180 degrees for uncoupled poles. So, each pole of a system has these properties; Frequency, Damping, Magnitude, and Phase. For some special applications (e.g. speach processing) a Delay parameter is also added to describe "when" it begins in a specific time segment.

Regards,

Dan Woodward

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#16

### Re: Laplace Transform and Complex Variable "s"

07/25/2013 6:05 PM

Great explanation! One detail, though: isn't the real part (sigma) the damping/growth constant, while the imaginary part (omega) is the frequency?

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