a very basic question on frequency domain analysis[very urgent..plz help]

[Tanya Moitra]

why do we substitute s=jω instead of s=σ+jω in frequency domain analysis methods?...i came across sumthin tat...because we are considering steady state analysis we are to neglect σ.But then how is σ related to transient response...σ is supposed to denote the real axis and jω the imaginary axis...plz help..:sad:

 

[LvW]

I`ll try a short answer.
* Solving the 2nd order diff. equation in the time domain you calculate the transient behaviour resulting in an expression exp(sigma*t)*exp(jw*t)=exp(sigma*t+jw*t). The 2nd term gives the frequency and the 1st term gives the amplitude (decreasing for sigma<0). Thus you get a transient behaviour equal to a decaying sinusoidal wave. This approach leads to the definition of a complex frequency variable s=sigma+jw and exp(s*t).
* All parameters and properties of a system in the frequency domain are defined for steady-state conditions only. That means: constant amplitudes of the sinusoidal signal (assuming that all transients have disappeared).
*That means: sigma=0 and s=jw (remember: steady-state conditions with rising/falling amplitudes are not possible; it is a contradiction).
Hope this answers your question.

---------- Post added at 12:37 ---------- Previous post was at 12:29 ----------

Additional remark: On the other hand, it is common practice to use the complex variable s=sigma+jw also in the frequency domain - however, only for theoretical purposes (e.g. pole/zero locations, definition of pole parameters like Q and pole frequency). Thus, it is very easy and illustrative to describe, for example, filter properties. However, if you speak about the real frequency response, which can be measured or simulated, you always set sigma=0.

 

[Tanya Moitra]

but y r we assuming steady state conditions in freq domain analysis?is ther any specific reason?.....another thin..where can i refer this entire derivation for the solution of the second order differential equation in time domain which gives me the expression exp(sigma*t)exp(jw*t).....or if u cud plz tel me in short how to arrive at the xpression?....

 

[LvW]

but y r we assuming steady state conditions in freq domain analysis?is ther any specific reason?.....another thin..where can i refer this entire derivation for the solution of the second order differential equation in time domain which gives me the expression exp(sigma*t)exp(jw*t).....or if u cud plz tel me in short how to arrive at the xpression?....

* All known formulas for frequency dependent parts/systems are based on a sinusoidal signal shape (examples: 1/wC, wL). That means, the impedances/conductances are defined for sinus only - equivalent to steady-state system conditions (otherwise the signal has no sinusoidal shape).
* The solution of a 2nd order diff. equation can be found in textbooks, several internet contributions and also on wikipedia. Sorry, but you have to spend some effort and time to search respective information.

 

[Tanya Moitra]

thankz a lot for the help..