[SI-LIST] Re: Interconnect Simulations-V2


>
>
>If you work out the amplitude of the harmonics of an infinitely repeating
>trapesoidal waveform with equal rising and falling edges, you find that the
>harmonics on a log-log plot fall at 20dB per decade until you reach
>1/(pi*Tr) ~ 0.318/Tr . After that knee frequency, they fall at 40dB per
>decade. Most of the significant harmonics are present to the knee frequency.
>Clayton Paul's book on EMC has both measurements and the derivation for
>those who last did a fourier analysis in undergrad. Does anyone know where
                                                     ^^^^^^^^^^^^^^^^^^^^^
>the 0.35 number can be derived from?
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>
>Cheers,
>

I've seen a number of different derivations for the ~.35 in .35/Tr
expression. Each comes up with a slightly different #.

I won't go into all the numerical details here, but one derivation
is based on the fact that the impulse response of a Gausssian filter is
Gaussian. Also the unit step response is the integral of the impulse 
response. It can be shown that the for the case of 10% to 90% risetime
the number arrived at in the derivation is .34/Tr

Another way to come around to the number (as Kai described) is to take a fft
of a repetitive trapezoidal waveform. It will be seen that the spectra falls
off at about 20dB/decade up to a certain freq.  (when pi*f*Tr <1) and then at
40 dB/decade  starting at pi*f*Tr~1.  Solving for f you get f~=.32/Tr.


Both of these derivations are approximations based on some assumed
conditions (gaussian response in the first case, and trapezoidal
waveform in the second). As such they are 'rules-of-thumb' which are
useful, but not analytically rigourous and correct.

See Ron Poon's text "Computer Circuits electrical Design" (Prentice Hall
1995) pages 144-145 for the mathematical details.

-Ray
Sun Microsystems Inc.

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