FLZapped said:
If I have picked up where I think you are talking about......
Charge? Who said anything about charge? Impedance John, think impedance......he is <i>defining</i> the characterisitcs of characteristic impedance. By definition, he is quite correct, the impedance is fixed without regard to unit length and a set of lumped components, in fact, COULD be used in it's place. Radio would be impossible without this fact.
Therefore, at DC and frequencies below the transistion point, the impedance of a cable does not fit this definition as its impedance is variable with unit length, whch is what he went on to show in the paper when he talks about the effect of frequenncy.
His actual measurements in figure 7 backed up his work(through fig 6), so I don't see where the problem is.
Clear as pumpkin soup?
-Bruce
Hi Bruce..
For a capacitor:
E<sub>joules</sub> = <sup>1</sup>/<sub>2</sub>*C*V<sup>2</sup>
For an inductor:
E<sub>joules</sub> = <sup>1</sup>/<sub>2</sub>*L*I<sup>2</sup>
For a transmission line to set up a non distorting, propagating signal along it's length, the energy stored within the inductance per unit length is equal to the energy stored in the capacitance per unit length..this is the interplay between the two energy storage mechanisms which is part and parcel of energy propagation through space..and, for incremental slices of space, dissipation within that increment is of no significance to the propagation impedance.
So: set equivalency
<sup>1</sup>/<sub>2</sub>*C*V<sup>2</sup> =<sup>1</sup>/<sub>2</sub>*L*I<sup>2</sup>
lose the 1/2 on both sides:
C*V<sup>2</sup> = L*I<sup>2</sup>
Re-arrange c and l on one side,, I and V on the other:
<sup>V<sup>2</sup></sup>/<sub>I<sup>2</sup></sub> = <sup>L</sup>/<sub>C</sub>
Since <sup>V<sup>2</sup></sup>/<sub>I<sup>2</sup></sub> = Z<sup>2</sup> , substitute..
Z<sup>2</sup> = <sup>L</sup>/<sub>C</sub>
Now, square root each side:
Z = (<sup>L</sup>/<sub>C</sub>)<sup>1/2</sup>
In other words, the actual impedance of the media is dependent only on the equivalency of energy between the inductive and capacitive storage mechanisms...this applies equally well to e/m waves propagating through space, through media, and through wires.
What is being lost in the Belden analysis is the fact that in order to accurately measure impedance, one relies on reflection coefficients, whereas it is rather impractical to do so at 10 hz..
They made substitutions in order to simplify the equations and the models, but that model loses the simplicity of e/m theory..
What they did is perfectly valid for their product line, but their model falls apart for low frequencies as a result of their simplifications..
If their test method did not match somewhat, their model, do you believe that would have been included? Their test method is not indicative of impedance for low frequencies...it treats the line as a lumped element as the frequency goes down..and the distributed resistance for the timebase involved not only prevents their method from arriving at the real value, but confounds the measurement.
Sorry I had to keep the math simple, these equations are a ##### to do using html code.
Some real time movies showing propagation would have helped a bit..
Oh, the lumped components model works great as a visualization technique, but when you try to use a lumped component termination, you lose the physical reality of energy travelling along the media. The storage mechanism is entirely different for the two...using lumps for a wire that is millionths of a wavelength is stretching the simplistic model waaaaay too far, and in this case, presents incorrect conclusions. This is the exact same kind of pitfall everyone falls into with skin theory, with it's assumption of planar TEM waves.
Cheers, John
PS..the real problem? They are unable to think four-dimensionally...just like Marty.
