| But see my
article 13 years previously; "Any apparently steady field
is a combination of two energy currents travelling in opposite directions
at the speed of l;ight."
My
letter to the author.
@@@@@@@@
IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 3, NO. 3, MARCH 1993
A DC Voltage is Equivalent to Two Traveling Waves
on a Lossless, Nonuniform Transmission Line
Ching-Wen Hsue, Senior Member, IEEE
Abstract - A static dc voltage can be treated as two traveling
waves propagating in opposite directions of a lossless, nonuni-form
transmission line. The amplitudes of these two traveling waves are
a function of the characteristic impedance of signal line. The concept
of two traveling waves is applied to a time--domain-scatterIng-parameters
analysis in a lossless, nonuniform transmission line terminated
with nonlinear loads.
I. INTRODUCTON
W HEN A LOSSLESS transmission line is connected to a dc voltage
source at one end and an appropriate resistive load at the other
end,.a steady state dc voltage will finally be reached in the lossless
signal line. Such a dc voltage is often treated as a static signal.
In this paper, we treat such a static dc signal as two traveling
waves propagating in opposite directions of the lossless signal
line. In general, these two traveling waves have different signal
amplitudes; the summation of these two traveling waves is equal
to the static dc voltage. Such an approach will give us physical
insights regarding the interaction between the transmission line
and associated terminations in time domain analysis [1]-[4J.
II. TRAVELING WAVE SOLUONS
The time-space domain solutions of a lossless, uniforRl transmission
line are [1]
V(t,x) = V+(t - ~) + V_ (t + ~), (I a)
l(t,x) = ~[V+(t-;)-V_(t+~)], (lb)
where V represents voltage, I is the current. Z = (£/C)1/2
is the characteristic impedance. u = (LC)-1/2 is t1}e wave velocity,
Land C are inductance and capacitance of the signal line per unit
length. t is the time and x is the space variable. Note that V+(t
- x/u) and V_ (t + x/u) represent the waves traveling in the +x
(forward) and -x (backward) directions. respectively.
We assume that a uniform transmission line having a characteristic
impedance Z is loaded with a dc voltage Vs, source resistance Rs
at the left-hand side and a dc voltage V L, resistor load RL at
the right-hand side. For such a
Manuscript received November 17, 1992.
The author is with AT&T Bell Laboratories. P.O. Box 900. Princeton.
NJ 08540.
IEEE Log Number 9207600.
Footnote; 'The terminology "dc voltge" here and in the
title might indicate not a continuous de voltage but a long-period
pulse.
....
....
to the circuit configuration at the load (right) end. For the circuit
previously stated, the traveling wave Vx+(t, I) makes a contribution
to the total incident and reflected waves a,(t) .. h(t) at the load
(Ieft) end and the existence of Vx+(t, I) extends over the time
interval II < I < T + I •. Notice that the summation of existing
time of traveling waves Vr+._ at both ends of the line is 2T.
IV. CONCLUSION
We decomposed a dc voltage on a lossless. nonuniform transmission
line into two traveling waves propagating in opposite directions
of the signal line. The amplitudes of two traveling waves are symmetric
with respect to a horizontal line represenling half of the steady
state voltage. This approach provides us physical insights regarding
the interaction between transmission lines and associated loads
in time-domain considerations.
.
.
|