### Section 8: Optional Section

This is an advanced section, and will not be appropriate for everyone.

Instead of considering the “step function” voltage wave in Figure 1.5.2, consider the “sine wave” voltage wave of Figure 1.8.1. We no longer can track a discrete bounce of a voltage wave, instead we will have a reflection that constantly varies over time. The simple Wave Bounce Diagram tracking voltage amplitudes no longer applies. Additionally, since the voltage never settles to a steady state, some amount of reflected energy is continuously being directed back to the source.

An RF signal is a constantly
varying signal, and so it is no longer possible to think of discrete
voltage waves bouncing along the transmission line. However, is is
possible to treat each discrete frequency as a single voltage signal
using complex numbers. Looking at Figure 1.8.1, a sine wave can be
seen traveling down the transmission line. We can observe that *at
this frequency*, the
transmission line is quarter of a wavelength long. A quarter of a
wavelength corresponds to 90º
(a half wavelength is 180º, a full wavelength is 360º).
At 1ns, when the sine wave hits the end of the transmission line,
the voltage wave will reflect, and travel another 90º
(for a total phase shift of 180º).

Here is the interesting thing... the voltage wave begins to reflect at 1ns. From 0.5ns to 1ns, the transmitted signal travels 90º down the transmission line, where it is reflected (let's say the end of the transmission line is an open: all of the voltage is reflected). From 1ns to 1.5ns, the reflected voltage travels another 90º, for a total phase shift of 180º. It took 1ns for the wave to travel from the beginning of the transmission line and reflect back to the starting point. In that same 1ns, the transmitted voltage did not stay still, though, it too has changed by 180º. If we sum the transmitted voltage with the reflected voltage, we see that they cancel- the voltage is zero at the start of the voltage line. It is as if the start of the transmission line is a short... even though the end has been left open. So, a quarter-wavelength line has the interesting property that it makes an open look like a short, and it also makes a short look like an open.

The effective length (in terms of wavelengths) of the transmission line changes over frequency. It is possible to track how the voltage waves will bounce along the transmission line, for a single frequency at a time, using the phase information of the line length, since:

Exactly the same type of voltage wave bounces will occur as described above

Because the voltage is constantly changing, just tracking the amplitudes isn't enough

Since the phase length changes with frequency, the Wave Bounce Diagram has to be solved for each frequency

Equations 1.7.1a Equations 1.7.1b Equations 1.7.1c Equations 1.7.1d Equations 1.7.1e Equations 1.7.1f |

Rather than trying to graphically solve the problem (which is useful only to introduce the concepts), we can use a numerical version of the wave bounce diagram (which is also really only useful to introduce concepts), using the diagram in Figure 1.8.2 and the equations above. As a practicing engineer, I can say that I have not used wave bounce diagrams between the time I learned “The Theory of Small Reflections” and the time I wrote this guide.

**Main Point**

**Main Point**

Because the waves are constantly varying, tracking how voltages are transmitted and reflected are much more difficult for RF analysis compared to what goes on in the steady state. Because the voltages are constantly varying, if there is a mismatch some amount of energy is constantly being reflected back to the source. In most cases, this is wasted power, however sometimes we can use the reflected energy to our advantage, like making an open circuit look like a short circuit (or more commonly useful, a short circuit look like an open circuit).