tesche - the fast laplace transform

8/13/2019 Tesche - The Fast Laplace Transform

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HIGH POWER ELECTROMAGNETICS:

ENVIRONMENTS, INTERACTION,

EFFECTS, AND HARDENING---

The Fast Laplace Transform

Dr. F. M. Tesche

HPE 201-2011September 18 - 24, 2011

Schloss Noer Germany


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The Fast Laplace TransformSlide 2 / 25

Overview

Fourier and Laplace transform theory form the basis of

system analysis: It permits us to understand system behavior in both the time

and frequency domains

It is widely taught in engineering and science curricula.

In the past, most of the results from transform theory have

relied on analytical transformations from time to frequencyor back.

The recent development of the Fast Fourier Transform (FFT)

and fast computers have made the numerical

implementations of transform theory possible. The FFT has problems with certain waveforms, however.

A solution to these difficulties lies in the Fast Laplace

Transform (FLT)

In this presentation we discuss the

theory and applications of the FLT


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The discrete Fourier transform (DFT)

Frequently, the Fourier integral and the waveform

reconstruction integral must be performed numerically

using a finite limit of integration, and discretely sampled

data points.

This leads to the discrete Fourier transform (DFT):

The fast Fourier transform (FFT) [1] is a rapid way of

evaluating the above sums, and requires thatN= 2m

.

0

( ) ( ) j tF f t e dtww

1

0

( ) ( ) ( 0 1)N

jk n t

n

F k t f n t e for k Nww

1

0

( ) ( ) ( 0 1)N

jk n t

k

f n t f F k e for n Nww

1

( ) ( )

2

j tf t F e dww w

1.


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Properties of the DFT

The DFT is important, since most practical computer

representations of data involve sampled data.

If the time domain functionf(t) is sampled discretely,

the DFT spectrum isperiodic.

Furthermore, if the DFT spectrum is also discretely

sampled, f(t) is alsoperiodic.


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Difficulties with the DFT

"Fold-thru" errors in the transient response, Aliasing errors in the frequency response, and

Insufficient resolution (t) for fast pulses

Improper Nyquist sampling.

There is also a difficulty if there is an accuracyrequirement for both early time andlate time responses.


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Problem waveforms for the DFT

The following waveforms all provide difficulties for the

DFT, or equivalently, for the FFT.

The problems all occur because the waveform has not

decayed to zero by the end of the data record (at t=

Tmax

)



Problem waveforms for the DFT (cont.)

A simple illustration of one problem in using the DFT is

provided by a simple time-shift operator of the Fourierspectrum:

which shifts the reconstructed time waveform by ts=

20ms.

Note the early-time fold-through of the late-time

waveform

( ) sj tH e

ww

Responses are

computed using the

FFT of the time-

shifted spectra.


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Development of the FLT (cont.)

Once the modified spectrum was processed (byfiltering, time-shifting, etc., such thatFm(w;o)

Fm1(w;o) ), the modified transient response fm1(t;o)

was then computed using the inverse Fourier

relationship:

This modified response waveform was post-processed

to amplify the late-time response, back to (hopefully)what it should have been before the artificial

attenuation:

1 1

1( ) ( ; )

2ot j t

m of t e F e d w

w w

1 1

1( ; ) ( ; )

2

j t

m o m of t F e dw

w w

Exponential

Growth

Correction




In thinking about this seemingly ad-hock method, Bertholet

realized that this nothing more than a numerical implementation ofthe Laplace transform:

with its inverse,

where s= + jwis the complex frequency in the Laplace transform

domain.

0

0

( ) ( )

( )

ot j t

st

F s f t e e dt

f t e dt

w

1( ) ( ; )

2

1( )

2

o

o

o

t j t

m o

j

st

j

f t F e e d

F s e ds

j

w

w w




The inverse Laplace is essentially an integration along a

contour s= o + jw with wvarying from .

jw

jwm= 2jfm

-jwm= -2jfm

0

LaplaceTransform

FourierTransform

Complex frequency plane

s = + jw

To use the FLT, an appropriate for

the damping parameter omust bechosen.

More will be said about this later.


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Summary of the FLT

Using the operator notation for the Fourier transform pair

defined earlier,

the Laplace transform pair can be expressed in terms of the

Fourier operators as:

If the required integrations for the Fourier operators Fand

F-1

are evaluated numerically using the FFT, thecalculation is very rapid,

and we refer to this method as theFast Laplace Transform.

1( ) ( ) and ( ) ( )Fourier FourierF f t f t Fw w F F

0( ; ) ( ) ( )tLaplace oF e f t f t w F L

0 1 1( ) ( ; ) ( )t Laplace o Laplacef t e F F s w F L


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Past work on the FLT

This development was done by Bertholet about 6 years ago and

documented internally in French at the HPE Laboratory.

An early paper in 1968 [2] has described the principle of this method

and called it the modified Fourier transform(MFT).

More recently, two papers in the power community [3, 4] have

described this computational procedure.

And this method is described in a recent text book [5]

3. Ramirez, A., et. al, Frequency Domain Analysis of Electromagnetic Transients through the Numerical

Laplace Transform,Proc. 2004 IEEE PES Meeting, 10 June 2004, pp 113639.

4. Gmez Zamorano, P. and F. A. Uribe Campos, On the Application of the Numerical Laplace Transform

for Accurate Electromagnetic Transient Analysis,Revista Mexicana de Fsica, 52(3), Junio 2007.

2. Day, S. J., et. Al., Developments in Obtaining Transient Response using Fourier Transforms, Part III:

Global Response,Int. J. Elect. Engng. Educ., Vol. 6, pp 259-265, 1968.

5. Cohen, A. M., Numerical Methods for Laplace Transform Inversion, Springer Science + Business

Media, New York, 2007.


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Past work on the FLT (cont.)

Notwithstanding this previous work, this FLT concept is

notwidely appreciated or used in the EM community. A survey of workers in the IEEE EMC and Antennas and

Propagation areas showed that virtually nobody had heard

of this technique.

Thus, we have written a monograph on this method and

showing the details of how is can be used in

processing measured HPEM spectral data.


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Examples of the use of the FLT

A previous slide has shown the difficulty with the FFT calculations

for time-shifting the rogue waveforms. Here we illustrate the use of the FLT for such a time shift

operation.

Note that in the Laplace domain, the time shift operator changes

from( ) to ( )s s

j t stH e H s e

ww

FLT Response FFT Response

Note that now there is no fold-

thru of the late time waveform

into the early time using the FLT.


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Examples of the use of the FLT (cont.)

The choice of the damping constant ois crucial for the

functioning of the FLT. This parameter must be determined by the analyst by

examining the results.

If ois too large, late-time noise occurs in the inverse

transform:Example of obeing

too large

A rule of thumb for

this parameter is o4/Tmax.


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FLT Example for a resonant line

Another example of the FLT is provided by a highly

resonant transmission line [6] illuminated by an incidentEM field at angle = 45

We wish to compute V2andI1

For the highly resonant line withZL1= 0 andZL2=

6. Tesche, F. M., et. al, EMC Analysis Methods and Computational Models, John Wiley and Sons, New

York, 1997.

Line geometry Incident E-field waveform

I1


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A resonant transmission line (cont.)

The BLT equation [6] for the line in question provides

the following Fourier domain transfer functions for thevoltage V2/E

incand the currentI1/Einc:

These peaks are

actually infinite.


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A resonant transmission line (cont.)

Multiplying the voltage transfer function by the

excitation spectrum and taking the inverse FFT provides

the following transient response for the voltage V2(t):

This response is

incorrect.

The response is

non-causal.

The amplitude is

unrealistic.

The waveform is

not what onewould expect from

physical

considerations.

What is the correct

solution ?


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One possible solution

One solution used in the past with the FFT is to

add a small amount of loss to the short circuit or to the line wires,

Extend the FFT time window so the waveform damps out, and

Increase the number of sampling points so that a good early-time

response results.

This response is computed

for ZL1= 1 , time window

Tmax= 6 s and 32,768

sample points.

Note the very slight decay of

the response due to the

additional (unrealistic) loss

added to this model.


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A more efficient solution is to use the FLT

In this method, the excitation function spectrum and the

computed transmission line transfer function must becomputed in the Laplace (s) domain, not in the Fourier (jw)

domain.

The excitation spectrum can be calculated numerically from the

excitation waveform, using the FLT.

The transfer function can be computed by modifying the BLTequation for complex frequencies, or by numerically continuing the

Fourier transfer function on the jwaxis into thes-plane

This numerical continuation of the Fourier spectrum can be

accomplished by the following sequence of the FFT andFLT transforms:

1( )Laplace FourierH s H w L F


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Illustration of the FLT and FFT transfer

functions

Analytically continued FLT transfer function Fourier transfer function


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FLT-computed voltage response

The FLT provides an easy solution for the voltage

response using o= 3.612 x 106

Early time Late time

This response is computed

for ZL1= 0 , time window

Tmax= 2 s and only 4096

sample points.

Causality is

observed

Correct

waveform and

amplitude.

No damping as requiredfor the lossless model

No late time

noise. (ois OK).


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tesche - the fast laplace transform

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