On Mon, 07 Jun 2004 22:22:49 -0500, Cecil Moore wrote:

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.

Sorry to spoil your fun, Cecil, but 141.4 v is not the forward voltage, it's the
max voltage of the standing wave. This is the same mistake that Steve made. I
asked you to rethink what the forward voltage and you have come up wrong.

You are using circuit theory for superposition, as Steve did, when circuit
theory fails to apply in certain transmission line cases. This is one of those
cases. In circuit theory you can superpose the voltages from two sources and V1
+ V2 equals Vtotal. But the re-reflected voltage CANNOT be added to the source
voltage in the transmission line case to obtain Vfwd, because there is only ONE
source.

I'll say it again, Cecil, V1 + V2 = maxV of the standing wave--V1 + V2 does NOT
equal Vfwd because V2 is not a second source, it came fr om ONE source, the
transceiver, and therefore superposition of V1 and V2 does not apply to
establish Vfwd.

Therefore, it is true that maxV may be incident on the load if the relative
phase between the reflected and forward waves permits, The only time the forward
wave is incident on the load is when the load = Zo.

Please, Cecil, go back to the drawing board and come up with the correct
Vfwd--it is not equal in any way to the right-hand side of Johnson's Eq 4.23;.
Remember, I said earlier that when rho = 0.5 and the circuit is matched, Vfwd =
Vsource x 1.1547. When you discover where the 1.1547 came from when rho = 0.5
you will have discovered the source of Vfwd.

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 + ...)

Wrong, Cecil. Change VF2 to 'E', the max value of the standing wave, and the
values obtained from the sum of the terms in the geometric series will be
correct.

Walt