Walter Maxwell wrote:
The equation he misunderstands is Eq 4.23 on Page 100 in Johnson, which is
explained and derived on Pages 98 and 99, to derive the voltage E of the
standing wave for any position along a mismatched transmission line. Steve's
misunderstanding is that he believes the equation expresses the value of the
forward voltage on the line. Consequently, his Eq 6 in Part 1 says that 'Vfwd =
the terms on the right-hand side of the equation copied from Johnson', while the
correct version is 'E = the terms on the righ-hand side'.

Johnson's equation contains all possible components - both forward and
reflected, so Johnson's equation predicts the total net standing-wave
voltage. (Note that Johnson's equation contains both positive and negative
exponents.)

Dr. Best's equation contains only the forward components of Johnson's
equation. Therefore, Dr. Best's equation predicts *only* the forward-
traveling voltage. (Note that Dr. Best's equation contains only negative
exponents and represents the E+ half of Johnson's equation while omitting
the E- half of Johnson's equation.)

In other words, the two equations do *not* predict the same quantity
and they are *not* supposed to be the same equations. As I said before,
Dr. Best made a lot of conceptual errors but his equations seem to be
valid. If you break Johnson's equation into two parts representing E+
and E-, the E+ part will be Dr. Best's Eq 6.

Please consider the following matched system with a 1/2 second long lossless
transmission line. The physical rho is 0.5. Assume VF1 is a constant 70.7V.

100W XMTR---50 ohm line---x---1/2 second long lossless 150 ohm line---50 ohm load
VF1=70.7V-- VF2--
--VR1 --VR2

V1 is equal to 70.7(1.5) = 106.06V and is a constant value
V2 starts out at zero and builds up to 35.35V
VF2=V1+V2 assuming they are in phase

Here is (conceptually simplified) how VF2 builds up to its steady-state value.
VR2(t+1) is 1/2 of VF2(t) and V2(t+1) is 1/4 of VF2(t)

Time in
Seconds V1 VR2 V2 VF2
0 106.06 0 0 106.06
1 106.06 53.03 26.5 132.56
2 106.06 66.26 33.13 139.19
3 106.06 69.58 34.79 140.85
4 106.06 70.4 35.2 141.26
5 106.06 70.63 35.32 141.38
6 106.06 70.69 35.34 141.39
7 106.06 70.7 35.35 141.4 very close to steady-state

My rounding is a little off but above is a close approximation to what happens.
Note that VF2 is equal to V1 + V2 and converges on the proper value of 141.4V.

Steady-state VF2 = 106.06 + 26.5 + 6.63 + 1.66 + 0.41 + ...
This is indeed of the form 106.06(1 + 1/4 + 1/16 + 1/64 + 1/256 + ...)
or VF2 = V1(1 + A + A^2 + A^3 + A^4 + ...)
--
73, Cecil http://www.qsl.net/w5dxp



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