I am glad you asked this question, my work on transmission-line
simulation started as an attempt to improve accuracy of the S-parameter
The conventional brute-force approach of taking FFT of S-parameters to
obtain impulse response and then using numerical convolution in the time
domain is very general. But it is not as accurate as one may think. It
has several substantial sources of error, I will name some of them.
First, a very small difference in S-parameters, say between S11=0.9999
and S11=0.9998, causes a several orders of magnitude difference in
corresponding admittance, and voltages and currents in the circuit
simulation. This is because the range of S-parameters is confined
between 0 and 1.
There is frequency-response discretization error as S-parameters for FFT
are sampled at discrete points. Typically, you may need 1024 frequency
points just to get moderate accuracy.
Then, there is the frequency-response truncation error.
There is also aliasing error of FFT. This error is huge at higher
frequencies, and the entire high-frequency half of the impulse response
has to be simply cut off to keep this error acceptable.
There is impulse-response truncation error.
Finally, there is discretization error of discrete-time convolution.
In my algorithm, I perform difference approximation of the frequency
response, and indirect numerical integration based on analytical
solution of the system differential equations in the time domain. Thus,
I eliminate completely all of the above errors. The only source of
error in my algorithm is the difference-approximation error, and it is
typically under 0.01% in the full frequency range.
So, my answer is that for transmission lines the accuracy of my
algorithm is much better than the FFT/convolution scheme can achieve
with any number of samples. As I explained in my previous email, the
difference between the ac and transient models, and multi-segment lines
is because of the frequency-response correction required by non-physical
f*Gd and Sqrt(f)*Rs loss equations, and not because of poor accuracy.
This correction would also be required if the FFT/convolution approach
was to be used with these loss equations.
However, the FFT/convolution algorithm is still the most general
approach that can handle arbitrary linear systems, and, presently, the
only way to use measured S parameters in a time-domain simulator with
reasonable accuracy. By the way, Hspice also has the FFT/convolution
model where you can specify an arbitrary frequency table. It has been
in Hspice for quite a while.
Dmitri Kuznetsov, Ph.D.
ViewLogic Systems, Inc. e-mail: email@example.com
1369 Del Norte Rd. Tel: (805)278-6824
Camarillo, CA 93010 Fax: (805)988-8259
> Dmitri Kuznetsov wrote:
> > Dear SIers,
> > I am the developer of the transmission-line simulation technology
> > used
> > by several popular simulators including Hspice's W element.
> > Recently, a
> > number of postings in this reflector pointed out a discrepancy
> > between
> > the ac and transient responses of W elements with
> > frequency-dependent
> > loss. Although I am no longer with Avant!, I would like to respond
> > in
> > defense of my algorithm.
> > The answer may surprise you... It is supposed to be this way. And
> > it
> > does not indicate accuracy problems with either ac or transient
> > model.
> > This is a quite interesting phenomenon, and I would like to explain
> > it
> > here.
> > It is caused by non-physical nature of the Sqrt(f)*Rs skin-effect
> > and
> > f*Gd dielectric-loss equations. They are lacking imaginary parts,
> > or
> > the corresponding frequency dependence of L and C. Real and
> > imaginary
> > parts of any analytic complex function cannot be arbitrary but are
> > uniquely related by Riemann-Cauchy equations. This is not obvious,
> > as
> > one may envision specifying arbitrary unrelated functions for the
> > real
> > and imaginary parts (and have done so in this case). But there are
> > laws
> > for everything.
> > As a result, transient responses of transmission lines with
> > Sqrt(f)*Rs
> > and f*Gd loss are non-causal, i.e. the response starts before the
> > excitation is applied. If you take FFT of the W-element's ac
> > waveforms,
> > you can observe signal traveling faster than the speed of light, but
> > it
> > is a mathematically accurate frequency-domain solution.
> > To assure correctness and accuracy of the transient solution, I
> > change
> > the frequency response as to restore the correct relationship
> > between
> > the real and imaginary parts. This is why the frequency responses
> > of ac
> > and transient models are different.
> > The corrective change depends on line length. This creates another
> > side
> > effect, a slight difference between transient responses of segmented
> > and
> > unsegmented lines. The difference is small, as the correction
> > affects
> > primarily higher frequencies at which both transmission-line
> > responses
> > and excitation spectrum are significantly attenuated.
> > The transient model is just as accurate for non-zero Rs and Gd as it
> > is
> > for constant loss, but with respect to the corrected frequency
> > response. In fact, I use the same frequency-dependent algorithm for
> > both cases. The accuracy is not improved by segmenting the line or
> > changing .option RISETIME from it's actual value. It is important
> > to
> > set this option, especially for longer lines with low loss.
> > Another popular skin-effect equation, Sqrt(j*2*f)*Rs, has correct
> > real/imaginary part relationship and does not require correction.
> > However, its inductive component is L(f)=Lo+Rs/(2*Pi*Sqrt(f)), which
> > produces infinite inductance at dc, and causes other interesting
> > phenomena for large Rs.
> > I do have the solution that eliminates above problems. But I
> > believe
> > that present Hspice's implementation of my algorithm is still good
> > as it
> > provides a robust way to achieve simulation results that are very
> > close
> > to measurements with minimum number of model parameters.
> > This was verified by many users, I would recommend downloading IMAPS
> > presentation by Jim Foppiano. It was discussed recently in this
> > reflector and contains comprehensive comparison of time-domain
> > measurements and W-element simulations with non-zero Rs and Gd.
> > I hope you found these comments useful. I have been working hard
> > for 6
> > years developing my simulation technology, and can say with
> > confidence
> > that it is by far the most accurate and general transmission-line
> > simulation method.
> > Regards,
> > Dr. Dmitri Kuznetsov
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