What is the significance of Laplace Transform in determining system stability?

[I_am_learning]

After solving a simple circuit with a inductor L in series with resistor R Driven by Voltage source V₀ by using Laplace Transform we get
I(s) = $$\frac{V}{s (R + Ls}$$
Why do we call this the frequency Domain response?
What does 's' represents?

 

[Bob S]

s = jω

Bob S

 

[I_am_learning]

s = jω
Bob S

What I was wondering is How come the Mathmatical abstract variable s come to represent the Circuit Frequency?
is s = $$\sigma$$ + jω or just jω ?

Please, try to expand, more. At least write longer than your short name this time!

 

[Bob S]

See

http://en.wikipedia.org/wiki/Laplace_transform

Bob S

 

[I_am_learning]

See

http://en.wikipedia.org/wiki/Laplace_transform

Bob S

Already did, (in fact, before starting this thread, in fact I always learn from wiki before starting thread), but either there is no explanation about my OP or that, I couldn't recognize it!

 

[bassplayer142]

S represents the plane made up from the complex (jw) and real (sigma) axis. When studying the frequency response we are only interested in the complex axis so we cancel out sigma so that s = jw.

 

[n.karthick]

Hai you can refer to this book
Circuits and filters handbook chapter 3 on Laplace transform.
https://www.amazon.com/dp/0849383412/?tag=pfamazon01-20

I had the similar doubt like you. After reading the above book, I could understand how Laplace transform (LT) works. It is better if you yourself read and understand.
In the above book, LT is explained some what differently and is easily comprehensible.
free preview of some pages is available in google books.

 

[paulfr]

Why do we call this the Frequency Domain response?
What does 's' represents?

The Laplace Transform gives you the Transient Frequency Response while
The Fourier Transform gives the Steady State [long term] Frequency Response

This is in contrast to the Time Domain Response which is what you see on an oscilliscope.

s is complex frequency
s = k + jw k the real part of the complex pole/zero and w the imaginary part

If you disturb a harmonic oscillator, it will oscillate and slowly die down until it stops.
Same is true for any system responding to an input.
Unless overdamped, you will see a sinusoidal oscillation dying down exponentially.

If you hit any system with a step function input [ abrupt change to a new value ]
it will go to the new commanded value but it will oscillate about that value before
settling down. Think of your car suspension system going over a bump.

If k is very small, the transient oscillation response takes a long time; if very large, response is over quickly.
If w is very small, the oscillation frequency is low; if very large, the frequency is very large.

 

[Vectorspace]

In systems it is used to figure out the stability of the system.

What I was wondering is How come the Mathmatical abstract variable s come to represent the Circuit Frequency?
is s = $$\sigma + j\omega$$ or just jω ?

$$s = \sigma + j\omega$$

But usually that $$\sigma = 0$$

 

[Vectorspace]

Isn't it that when $$\sigma \neq 0$$ means the poles do not lie on the unit circle in the pole zero plot?

 

[Vectorspace]

Is that right?

 

[paulfr]

Yes the Laplace Transform provides an analytic tool to determine system stability.
There are two kinds of stability a control systems engineer is concerned with.

First is Global Stability. That is if the system will "blow up" or oscillate until one of the components burns up, vibrates or fails for another reason. This occurs if the closed loop poles move into the right half plane of the complex Root Locus plane.

Second is Relative Stability. That is the characteristics of the transient response. The response to a disturbance or command signal. In general, relative stability falls into three categories;
1/ Overdamped; the losses are high and the system exponentially moves to the new value/state, but very slowly. Not a good design.
2/ Underdamped; the system oscillates about the new position and this oscillation decays exponentially to the new state/value.
3/ Critically damped; the system responds in the fastest possbile manner. This will be a small overshoot of about 2 to 4% to the new value and then settles down with only one or two periods of oscillation.

On the complex plane, if the poles lie on the line y = - x which is at 45degrees to the real axis, the response will be critical, the optimum response all systems shoot for.

If you are ever in an elevator that makes you feel a little impulse (jerk = first derivative of acceleration), either starting or stopping, the system is out of tune and needs adjustment to bring it back to critical response characteristics.

Cheers