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tfahim@gmail.com electronics forum beginner
Joined: 18 Apr 2006
Posts: 1

Posted: Tue Apr 18, 2006 8:50 pm Post subject:
Modeling nonlinearities with LaPlace Tranfer functions



Dear All,
I'm working on modeling the behavior of an analog equalizer at high
frequency.
The way I look at the system is that it should include 2 main responses
: DC gain and AC gain. The DC gain is easily obtained by DC sweeping
the input and measuring the resultant DC output voltage. The resultant
DC response looks more like a tanh function with a linear operation
region and a saturation region at its extremes. As for the AC response,
based on smallsignal analysis, it's basically a transfer function with
a certain number of poles and zeros that I implement using laplace
transform function.
To obtain the overall gain, the DC response is multiplied by the AC
response, where the latter is first normalized to its DC gain to make
sure the DC gain is not included twice.
Results are as expected for the range of inputs that fall within the
linear region of the DC tanh function. On the other hand, when the
input amplitude exceeds the linear region, the gain values can not be
anticipated with hand analysis.
Did anybody encounter a similar problem ? Do you have any
recommendations on ways to include such "largesignal" effects into the
model.
Your contribution is highly appreciated.
Thanks,
Tamer. 

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bruce varley electronics forum beginner
Joined: 19 Jun 2005
Posts: 17

Posted: Wed Apr 19, 2006 10:47 am Post subject:
Re: Modeling nonlinearities with LaPlace Tranfer functions



<tfahim@gmail.com> wrote in message
news:1145393413.493888.119960@u72g2000cwu.googlegroups.com...
Quote:  Dear All,
I'm working on modeling the behavior of an analog equalizer at high
frequency.
The way I look at the system is that it should include 2 main responses
: DC gain and AC gain. The DC gain is easily obtained by DC sweeping
the input and measuring the resultant DC output voltage. The resultant
DC response looks more like a tanh function with a linear operation
region and a saturation region at its extremes. As for the AC response,
based on smallsignal analysis, it's basically a transfer function with
a certain number of poles and zeros that I implement using laplace
transform function.
To obtain the overall gain, the DC response is multiplied by the AC
response, where the latter is first normalized to its DC gain to make
sure the DC gain is not included twice.
Results are as expected for the range of inputs that fall within the
linear region of the DC tanh function. On the other hand, when the
input amplitude exceeds the linear region, the gain values can not be
anticipated with hand analysis.
Did anybody encounter a similar problem ? Do you have any
recommendations on ways to include such "largesignal" effects into the
model.
Your contribution is highly appreciated.
Thanks,
Tamer.

Unfortunately there's no simple answer to this question. As soon as a system
goes outside the linear region, all bets are off. There are heaps of
examples of apparently 'simple' nonlinear systems exhibiting massively
complex behaviour that can't be predicted with any formal method. If you
want an example, google on Mandelbrot set or Henon maps.
Depending on what you want to do, a couple of techniques that *might* prove
useful are piecewise linearisation, which is deriving a linear relationship
about a chosen operating point for small deviations, and describing
functions which extend frequency domain analysis into the nonlinear regime 
slightly.
A practical approach for dealing with nonlinear systems is rigorous system
modelling in the time domain, this requires experience, time and patience to
get results. 

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Rene Tschaggelar electronics forum Guru
Joined: 24 Mar 2005
Posts: 540

Posted: Wed Apr 19, 2006 11:14 am Post subject:
Re: Modeling nonlinearities with LaPlace Tranfer functions



tfahim@gmail.com wrote:
Quote:  Dear All,
I'm working on modeling the behavior of an analog equalizer at high
frequency.
The way I look at the system is that it should include 2 main responses
: DC gain and AC gain. The DC gain is easily obtained by DC sweeping
the input and measuring the resultant DC output voltage. The resultant
DC response looks more like a tanh function with a linear operation
region and a saturation region at its extremes. As for the AC response,
based on smallsignal analysis, it's basically a transfer function with
a certain number of poles and zeros that I implement using laplace
transform function.
To obtain the overall gain, the DC response is multiplied by the AC
response, where the latter is first normalized to its DC gain to make
sure the DC gain is not included twice.
Results are as expected for the range of inputs that fall within the
linear region of the DC tanh function. On the other hand, when the
input amplitude exceeds the linear region, the gain values can not be
anticipated with hand analysis.
Did anybody encounter a similar problem ? Do you have any
recommendations on ways to include such "largesignal" effects into the
model.
Your contribution is highly appreciated.

The base of the fourier/laplace transform is
linearity. So if youf system is not linear,
then you have to look very precise what
actually you're doing. A thorough understanding
of the involved theory is necessary.
Start with the Schwartz space, where the transform
lives in.
IMO, the ground to use the laplace transform becomes
soft, wobbly and slowly unuseable.
Rene

Ing.Buero R.Tschaggelar  http://www.ibrtses.com
& commercial newsgroups  http://www.talkto.net 

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Jon electronics forum addict
Joined: 12 Oct 2005
Posts: 96

Posted: Wed Apr 19, 2006 2:58 pm Post subject:
Re: Modeling nonlinearities with LaPlace Tranfer functions



On approach is to approximate the nonlinear response by the sum of
functions (non necessarily linear) for which the LaPlace transform
exists. 

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Rene Tschaggelar electronics forum Guru
Joined: 24 Mar 2005
Posts: 540

Posted: Wed Apr 19, 2006 8:31 pm Post subject:
Re: Modeling nonlinearities with LaPlace Tranfer functions



Jon wrote:
Quote:  On approach is to approximate the nonlinear response by the sum of
functions (non necessarily linear) for which the LaPlace transform
exists.

This is excactly what does not work.
A property of the Laplace transform L() :
L(a*A+b*B+c*C)=aL(A)+bL(B)+cL(C) //a,b,c : Constant
Now since the input of the Transform is not linear:
{ Be f the nonlinear function }
f(a*A+b*B) != af(A)+bf(B) //a,b,c : Constant
Then the property above is not true either.
Sorry.
Rene

Ing.Buero R.Tschaggelar  http://www.ibrtses.com
& commercial newsgroups  http://www.talkto.net 

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