Skin effect on transmission lines?

The partial differential equations typically used to describe the behavior
of thransmission lines are:
Diff(e,x)=-R*i-L*Diff(i,t),
Diff(i,x)=-G*e-C*Diff(e,t).
The first equation says that the voltage across the line drops according to
the resistive and inductive drops distributed along the line. The second
equation says that current decreases along the line according to the
conductive shunting and capacitive charging.
If you include the effect of skin effect, R and L along the line increase as
a function of frequency. For a particular frequency, it is easy, in
principle, to find the additional resistance and reactance from skin effect.
If you want to see what happens with pulses on such a transmission line, you
can find the fourier transform of the pulse, calculate the transfer function
for each frequency, and then add all frequencies together to produce a
solution.
MY QUESTION: Is there a way to add terms to the equations given above so as
to include the skin effect but without having to calculate for each
frequency? I am looking for a way to integrate the equations so as to
propagate a pulse down the line.
Bill
Reply to
Repeating Rifle
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| The partial differential equations typically used to describe the behavior | of thransmission lines are: | | Diff(e,x)=-R*i-L*Diff(i,t), | Diff(i,x)=-G*e-C*Diff(e,t). | | The first equation says that the voltage across the line drops according to | the resistive and inductive drops distributed along the line. The second | equation says that current decreases along the line according to the | conductive shunting and capacitive charging. | | If you include the effect of skin effect, R and L along the line increase as | a function of frequency. For a particular frequency, it is easy, in | principle, to find the additional resistance and reactance from skin effect. | | If you want to see what happens with pulses on such a transmission line, you | can find the fourier transform of the pulse, calculate the transfer function | for each frequency, and then add all frequencies together to produce a | solution. | | MY QUESTION: Is there a way to add terms to the equations given above so as | to include the skin effect but without having to calculate for each | frequency? I am looking for a way to integrate the equations so as to | propagate a pulse down the line.
Not knowing if such a thing has already been done and produced a standard formula, my first inclination would be to go ahead and model a single pulse of "infinitely" short duration (as short as you want to go to cover all the cases you might have), then use that result as a function to combine other pulses expressed as a series of "infinite" pulses added together. A wider pulse would simply be the addition of 2 or more of the very short pulses shifted over time. Just do the same to the original results of the fourier method (this time it's just as simple as adding an array of numbers since the fourier work is already done).
Reply to
phil-news-nospam
in article snipped-for-privacy@news1.newsguy.com, snipped-for-privacy@ipal.net at snipped-for-privacy@ipal.net wrote on 8/12/04 6:34 PM:
If I understand this suggestion correctly, it suggests determining the Green's function (response to a delta function) by using fourier transform techniques. where the added G and L as functions of frequency are used for each configuration of conductors.
Off hand, that seems like a good way to go. I'll think about it a bit. Thank you.
Bill
Reply to
Repeating Rifle
I should think that the variation of R and L with frequency introduces a non-linearity into the process - will Fourier transform techniques handle this correctly? What is the skin effect seen with a delta function? You can solve the T-line equations with skin effect taken into account but it would be messy-introducing non-linearities and needing computer solution. and It is probably less computationally intensive to deal with each frequency as you indicated (transform + summation).
However there may be some useful information in the work done by Hermann Dommel et al in modelling of surges on power lines as skin effect will affect attenuation of these surges. IEEE transactions (PAS) and EPRI or Bonneville Power may give you links. It may be that this was ignored in favour of worst case solutions.
Reply to
Don Kelly
in article VwXSc.92904$gE.16563@pd7tw3no, Don Kelly at snipped-for-privacy@peeshaw.ca wrote on 8/12/04 9:33 PM:
It does not introduce nonlinearity but it does get away from the typical differential equation (DE) that is the heart circuit theory. These DEs usually have constant coefficients corresponding to the self and mutual impedances in a linear circuit. The DEs including frequency dependent skin effect will still have the property that for inputs x, y, and x+y, the outputs will be H(x), H(y), and H(x+y)= H(x)+H(y) where H is a transfer function.
Nonlinear systems are those where the G or L in a circuit is dependent upon the current or voltage in a circuit. A diode or a lamp in a Wein bridge oscillator produces nonlinearty.
Time dependent linear DEs are often found in devices like parametric amplifiers. While such devices may do things like mix frequencies, the behavior still follows the schem given above. There is additional confusion because sometimes nonlinear devices are used to change the parameters. For example, the capacitance of a back biased diode is often the provider of time varying capacitance. The property H(x+y)= H(x)+H(y) is the key to linear behavior.
Bill
Reply to
Repeating Rifle
in article G5cTc.97940$J06.18124@pd7tw2no, Don Kelly at snipped-for-privacy@peeshaw.ca wrote on 8/13/04 4:25 PM:
See below.
I too seem to be rusty. I should not have called H a $transfer$ function! It is a $response$ function! See the quote demarked by double $$. This response function is merely the product of a transfer function F(jw) and the input signal signal amplitude component at frequency w. The response of each spectral component E(jw) is H(jw)=E(jw)*F(jw). For lumped circuits, F is a polynomial in (jw) and does not depend upon the input waveform.
Fourier transforms allow a signal in the time to be broken up into a sum of signal at each frequency. The inverse transform allows them to be summed back into a response in time. It is all linear. Introducing skin effect changes the form of F. F will no longer be a simple polynomial.
This does not make sense to me. What I am trying to say is that the response if you have add two signals e1 and e2 and feed the sum through a network (that could include a transmission line}, the output at any one frequency w is
(E1+E2)*F(jw) = E1*F(jw) + E2*F(jw)
For the component at frequency w. The overall response in time rather than frequency is found by summing over all frequencies. That is what inverse transform does.
Bill
Reply to
Repeating Rifle
in article HCWTc.121715$gE.70341@pd7tw3no, Don Kelly at snipped-for-privacy@peeshaw.ca wrote on 8/15/04 9:20 PM:
Bessel functions are fun!
Bill
Reply to
Repeating Rifle

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