**Historical background.**

In the early nineteenth century electromagnetic theory made advances, a cornerstone of the theory being the doctrine of conservation of charge q, which developed into the doctrine of continuity of electric current flow dq/dt = i.

In the middle of that century Maxwell struggled with the paradox of the capacitor, where electric current entered one plate and then flowed out of the other plate apparently without traversing the space between the plates. It seemed that electric charge was being trapped on the upper plate and on the lower plate. Maxwell cut the Gordian knot (Ref.17) by postulating a new type of current, the extra current, as flowing across the gap between the capacitor plates so as to save the principle of continuity of electric current.

This extra current, later called 'displacement current', was a result of his postulation of 'electric displacement'. Maxwell said that the total outward displacement across any closed surface is equal to the total charge inside the closed surface.

Displacement current is sometimes explained as being the distortion of molecules in the dielectric, so that one end of the molecule is more positive and the other end is more negative. A difficulty arises if the dielectric is a vacuum, and has no molecules which could distort. So there have always been problems with displacement current. However, these are not the subject being discussed here.

**The Transmission Line.**

In the 1870's the young Oliver Heaviside wanted to speed up digital (morse) signalling in a coaxial undersea cable between Newcastle and Denmark. He discovered that a Transverse Electromagnetic Wave travelled undistorted at the speed of light for the dielectric, between the inner and outer conductor. When such a voltage step reached half way to Denmark, a uniform closed circuit of currents was made up of electric currents down the conductors and an equal amount of displacement current across the front face of the advancing step. Thus, at every instant, Kirchhoff's First Law was obeyed.

We will discuss a new view of this combination of displacement current at the front face and electric current on the side of a TEM step travelling down a transmission line.

**The Transmission Line Transmission Line.**

In a uniform transmission line, the cross sectional shape of the two conductors
and the vacuum between them determines Zo, the characteristic impedance of the
line. Zo determines the ratio of voltage to current for any TEM signal delivered
from the left into the line. Signals travel at the speed of light *C*.

If we deliver a 10 volt TEM step down a 100 ohm transmission line into a four
way series junction (Fig.14), the signal breaks up into four signals travelling
away from the junction. The amplitude of the four signals obeys the well known
laws for a change in characteristic impedance (Fig.11)^{[1]}.

If the junction is of four identical transmission lines each with Zo=100 ohms, then the incident signal sees before it an impedance of 300ohms. The coefficient of reflection is

The result is that a half amplitude signal of 5v returns back to the left. Since at the junction we see both incident signal plus reflection across the input line, the total voltage at the junction is 15v. So a 5v signal must travel forward down each of the downstream transmission lines.

The incident power was V.1=10v x 100ma = 1 watt. The power in each of the four signals leaving the junction is 5v x 50ma = 250 mw. So energy is conserved.

Now let us consider the case where the top and bottom transmission lines are
changed to a very small Zo=0.01 ohm ^{[2]}. The reflection coefficient
becomes a negligible 0.02/200.02 = 0.0001 ohm, and a negligible 1mv reflects
back to the left. A total of 10.001v forward signal is shared between the three
downstream transmission lines. The big one receives 9.999v while each of the
other two receive 1mv.

Let us introduce a second similar branching downstream to the right. This time, the incident signal of 9.999v (increased by the new 1mv reflection) splits up into forward going signals 1mv, 9.998v and 1mv.

At further branches downstream, (there is a further tiny reflection,) a tiny signal enters each of the branches, while a slightly reduced signal continues to the right.

**The Dielectric Constant of Copper.**

Consider three capacitors in series, each with plate area a, dielectric thickness d, dielectric constants . The formula for the capacitance c of the three in series is given by

In each case, a term on the right becomes .

If the 'dielectric' in the middle capacitor is copper, we know that the second
term disappears, and
. This means that
. It follows that the dielectric constant for copper must be
. ^{[3]}

**The transmission line with resistive conductors.**

Let us consider a transmission line with vacuum for dielectric and with characteristic impedance Zo=100 ohms. Its unusual feature is that instead of two copper conductors, it has very thin resistive conductors, where the resistance of each 1cm section of each conductor is 10mohm.

A 100v step is launched down the transmission line, in the vacuum between the two (resistive) conductors. During the first 30psec, when it traverses the first 1cm of the line, the 100v signal splits three ways, in the ratio 0.01 : 100 : 0.01. This means that a 99.98v signal arrives at the end of the first 1cm section, and proceeds to the right, through the vacuum dielectric. A 10mv step stays across the first 1cm of the upper conductor. A 10mv step stays across the first 1cm of the lower conductor.

During the second 30psec, the surviving 99.98v signal traverses the next 1cm of line, again splitting three ways, into 10mv, 99.96v and 10mv signals. Also, due to the 2mohm mismatch, a very small step reflects backwards up the line.

During the third 30psec, two more 10mv steps remain behind, while 99.94v proceeds to the right at the speed of light.

**The transmission line with transmission line
conductors.**

The situation is much the same as before, except that, instead of having 1cm sections of resistive (10mohm) conductor, each 1cm of each conductor is replaced by the front end of a transmission line with characteristic impedance 10mohm.

This time, instead of each laggardly 10mv step lingering across its prescribed 1cm of resistive conductor, it advances at the appropriate speed (for the dielectric of the new, Zo=10 mohm transmission line) outwards, sideways from the direction of the main voltage step travelling through the vacuum.

The conductor which delineates the further face of the outwards transmission line for the first 1cm of line, and also the nearer face of the outwards transmission line for the second 1cm of line, is very thin. It turns out that the electric current down one face of the conductor is equal and opposite to the current down the other face. As the conductor's thickness is further reduced, these two currents merge, cancel, and losses drop to zero.

Velocity of propagation into this row of transission lines, each with Zo = 10 mohm, is lower if the dielectric in them has a higher dielectric constant, reaching zero if the dielectric constant is infinite.

^{[1]}

Voltage and current must correlate at all times, and energy must be conserved.
These requirements more or less prescribe the laws of reflection.

^{[2]
}This very low Zo might be achieved by inserting
a dielectric with very high dielectric constant, or by changing the function
of geometry f (discussed above).

^{[3]
}The reader may be amused by Carter's approach to
this subject, Ref.3c, p265; "Nothing
has been written in this book which would enable any meaning to be attached
to the permittivity, k, of a metal;
we must merely assert here that the value is not very different from unity."