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Re: [Phys-l] Reference needed

Sl. relatedly: I find the curious statement in the Berkeley Physics Lab Part B-6 (Periodic Structures): "A mechanical or electrical structure w/ a basic repeating unit shows a number of remarkable physical properties. First a gradual disturbance introduced at one end of the structure will propagate down the structure worth out reflection. The second is about the cutoff; no problem here.

Aside: the discussion begins w/ the mechanical and then the electrical analogy. This illustrates the age of the manual. Now it's in reverse order, nicht wahr?

The text discusses dispersion, phase/group speed, delay, and later the strip line. Being v. non-maths. I suggest the following argument for a DC impedance, note the caveat. If there is no discontinuity in the evolution of a lumped constant line to a continuous, then it's follows that a lumped line has a dc resistance (characteristic impedance at DC)

bc

p.s the text includes lotsa heavy (for bc) maths.




Jack Uretsky wrote:

Hi all-
Thanks, Brian, for your comments. This is all new to me, and I'm working through some of this for the first time.
My understanding essentially agrees, I think, with Brian's comments. Terminating the transmission line with a sqrt(L/C) resistor makes the line "look" infinite, as Feynman points out in his "lumped" model. I found the exercise as problem 1, Chapter IX of Slater & Frank's
old book <Electricity and Magnetism>. The problem asks for a proof that for the infinitely long case the ratio of applied voltage to current is just the quoted termination value, independent of frequency. The ratio would therefore hold in the DC case. Here L and C are the inductance and capacitance per unit length.
In the lumped case, with proper termination, the impedence is that of a low-pass filter. At frequencies below cut-off the impedance is just
sqrt(L/C) with a frequency dependant phase that ranges from 0 to pi/2.
This can all be worked out from the discussion in Feynman's book.
Regards,
Jack




On Sat, 19 Aug 2006, Brian Whatcott wrote:


At 10:50 AM 8/19/2006, you wrote:

.... the
correct matching termination for a lossless transmission circuit is a
resistance with value \sqrt{L/C}. Feynman derives this result in a model
of lumped L-C circuit elements (II-22-12), although he doesn't tell us
that he's really modeling a transmission line circuit. I'm looking for
the classical derivation of this result.
I will be discussing Feynman's derivation, which has generated
some unnecessary controversy at the Feynman Festival at U. Md, starting on
8/25. The controversy arises from disbelief that a circuit of purely
reactive elements can look resistive at low frequencies. The DC
resistance of the circuit is just the termination value referred to above.
It doesn't seem to be in Jackson's or Schwinger's texts. If
someone has a text on transmission lines, it might be there.
Regards,
Jack

Although not responsive to Jack's question, I want to assert that
a circuit of purely reactive elements does not look resistive at
low frequencies. It is certainly true that a long transmission line
has a ratio of voltage to current at intermediate points which
behaves like a resistance of that value (called the characteristic
impedance) for a time period equal to the transmission time
for traversing that line both ways. After that time - i.e. in the
low frequency limit, the line can look capacitive, inductive or
resistive at the near end, depending on the value of the
terminating resistor at the far end, if any.



Brian Whatcott Altus OK Eureka!


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