Modeling non-linearities with LaPlace Tranfer functions

[[email protected]]

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 small-signal 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 "large-signal" effects into the
model.

Your contribution is highly appreciated.

Thanks,
Tamer.

 

[bruce varley]

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 small-signal 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 "large-signal" 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.

 

[Rene Tschaggelar]

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 small-signal 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 "large-signal" 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

 

[Jon]

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

 

[Rene Tschaggelar]

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

 

[Ban]

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
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 small-signal 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 "large-signal" effects into
the model.

Your contribution is highly appreciated.

That looks like a differential amplifier. You can develop the 1+tanh
function like this:
1 +Vd/2Vt - Vd^3/24Vt^3... which is enough. Now you can substitute the
Vd=differential voltage with Vd*sinwt and get Vt is the temperatur
voltage (26mV) the linear range is around +/- 2Vt
1 + Vd/2Vt *sinwt - Vd^3/96Vt^3 *(3sinWt - sin3wt)
this is the distortion which is rising with the squared amplitude. Vd= peak
amplitude

 

[Timo!]

I understand that this solution does include the nonlinearities in some
way but I don't see how this will include the Gain as a function of
frequency ?

Thanks for your feedback.

 

[Ban]

I understand that this solution does include the nonlinearities in
some way but I don't see how this will include the Gain as a function
of frequency ?

Thanks for your feedback.

Timo, this is adding harmonics depending on the signal level. So you take
your signal to be processed and apply the filter functtion and after each
filter step calculate the level and then add those harmonics depending on
the envelope and do the next filter stage with the addaed harmonics. those
3rd order ones are pretty awful, but you can use 2nd order ones, that rise
lineary with amplitude. It is a difficult operation and you have to go from
frequency domain to time domain and back.