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. Wakefield
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
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.
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
II. TRAVELING WAVE SOLUONS
The time-space domain solutions of a lossless, uniforRl
transmission line are 
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
IEEE Log Number 9207600.
Footnote; 'The terminology "dc voltge"
here and in the title might indicate not a continuous de voltage but a
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.
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