Question about laplace transform function

[david753]

1.
In a system analysis, we can transfer a system to a Transfer function, H(W)
, the equation may be like that, H(s)=s/(s^2+3s+2)
Then, we can plot a frequency response (BODE PLOTS)by H(s) function.
and, "s" equals "w".

But, in Laplace Transform function, "s" equals "sigma+j*w".
Obviously, it is different.

How to explain it?

2.
laplace u(t) is equal U(S)/S.
What does it,U(S)/S mean in terms of physic ot math viewpoint?

 

[Dmitrij]

Re: Laplace Transform

1) First of all, remember that Laplace and Fourier transforms are defined for different types of signals.

In order to be Laplace-transformed, the signal should be given only for positive times, must satisfy Dirichle conditions and possess the increase extent less rapid than the exponential one.

For Fourier transform again Diricle conditions are actual but also the signal must be squared-integrable.

If the signal (or any other function) satisfies simultaneously all the enumerated conditions, it may be represented in s-plane and then s may be substituted for jw to obtain Fourier frequency representation.

2) The second correspondence between originals and their laplace-transforms is explained by the following:

s1(t) ---> S1(s)
int{0,t}(s1(t)) ---> S1(s)/s,

It may be lead out from the direct proof.

With respect,

Dmitrij

 

[aersoy]

Laplace Transform

using these kind of transformations let us to trasnfer the complicated differenetial equations to linear domain so we can analyze circuits or sysytems easily this is one of the aims

 

[electricpete]

Re: Laplace Transform

A linear system can be represented as H(s) or H(w) with ths substition s=sigma * j*w

There may be some conditions necessary as described above, but for many simple systems (for example constant coefficient differential equations which can be represented by ratio of polynomials), the substitution applies. So we have two different ways to express the same system and each has it's own benefit.

The H(s) for has more relevance for determining natural response. The frequency will be the frequency of the poles in the complex plane... decaying exponential sinusoid for poles in left half.... etc

The H(w) can have more relevance for determining forced response. A single frequency input at w0 gets multiplied by the complex factor H(0) (changes amplitude and phase). A more complex input function generally be decomposed into a sum of sinusoids and the response is the sum of the responses at each of those frequencies.

 

[Kral]

Re: Laplace Transform

david753,
A simplified way of looking at "s" is to consider the real part of the exponential as representing the time-varying portion of the response, and the complex part of the eponential as representing the steady-state portion of the response.
Regards,
Kral

 

[electricpete]

Re: Laplace Transform

I wouldn't say it that way

If we have a pole at s=sigma * j*w

The associated time function is something like:

A*exp(sigma*t) * sin(w*t-C)

The real portion (sigma) relates to the first factor carrying the exponential growth or decay characteristic.

The imaginary portion (w) relates to the second factor carrying the oscillatory frequency.