**The capacitor**

In the early 1960's I pioneered the inter-connection of high speed (1nsec)
logic gates at Motorola, Phoenix, Arixona (ref.25).
One of the problems to be solved was the nature of the voltage decoupling at
a point given by two parallel voltage planes. I asked Bill Herndon about the
problem, and he gave me the answer: "It's a transmission line".^{[1]} Bill learnt this from Stopper,
whom I never met, who later worked for Burroughs (UNISYS) Corp. in Detroit.

The fact that parallel voltage planes, when entered at a point, present a resistive, not a reactive, impedance, was for me an important breakthrough. (It meant that as logic speeds increased, there would be no limitation presented by the problem of supplying +5v.) The reader should be able to grasp the reason why voltage plane decoupling is resistive by studying Figure 64, which shows the effect of a segment only of two planes as they are seen from a point.

During the next ten years, with the help of Dr. D. S. Walton, I gradually came
to appreciate that, since a conventional capacitor was made up of two parallel
voltage planes it also had a resistive, not a reactive (i.e. capacitive or inductive)
source impedance when used to decouple the +5v supply to logic. Since the
source impedance (= transmission line characteristic impedance) is well
below one ohm, the transient current demand of logic gates approaching infinite
speed can still be successfully satisfied with +5v from a standard capacitor
of any type^{[2]}.

The capacitor is an energy store, and when energy is injected, it enters the
capacitor *sideways* at the point where the two leads are joined to the
capacitor. Nothing ever traverses a capacitor from one plate to the other^{[3]}. This is clearly understood in the case of a transmission
line. By definition, when a TEM wave travels down a transmission line, Figure 5,
nothing travels sideways across the transmission line, or we would not have
a transverse electromagnetic wave.

**Comparison of the transmission line model
with the lumped model of a capacitor in an RC circuit.**

Consider a transmission line as shown in Figure 65 with

characteristic impedance Zo terminating in an open circuit. We will assume that R>>Zo.

When the switches are closed (at time t=0) a step of voltage is propagated down the line. This reflects from the open circuit at the right hand end to give a total voltage of . Reflection from the left hand end makes a further contribution of and so on. In general, after n two-way

passes the voltage is and;

_{ }(1)

In order to avoid a rather difficult integration it is possible to sum the series to n terms using the formula

(2)

where a is the first term of a geometrical progression and v the ratio between terms. (This formula is easily verified by induction.) Substituting in (2) the parameters for (1),

(3)

(4)

We obtain,

(5)

(6)

This is a correct description of what is happening as a capacitor charges. We can now go on to show that it is approximated by an exponential. We have

(7)

Consider the term,

.

.

If Zo/R<<1 this term is asymptotically equal to

.

Now define

.

Substitution gives

.

By definition, as we have,

.

and therefore:

.

Now, after time t,
, where *C *= velocity of propagation.

Thus,

.

For any transmission line it can be shown (p19) that

where c= capacitance per unit length and f is a geometrical factor in each case. The "total capacitance" of length l

= lc = C.

Hence,

and therefore

which is the standard result. This model does not require use of the concept
of charge. A graphical comparison of the results is shown in Figure
66. ^{[4]}

Ref 25: Catt I., et al., A High Speed Integrated Circuit Scratchpad Memory, Fall Joint Computer Conference, Nov. 1966.

^{[1]}ref.15, p40.

^{[2]}Ref.3b, p216, refutes the fashionable nonsense about "RF
capacitors".

^{[3]}Similarly, the battery, p13,
note 24, and the electrolyte.

^{[4]}Calculations were by my co-author
Dr. D.S. Walton. First published in Wireless World, dec78, p51.