We will now try to reproduce the experiment we have performed at the outset with the mathematical model of the transmission line. The experiment consisted of an ideal battery with zero source resistance driving an open ended line. We will do the same except it will be in terms of mutual interaction of electrons as outlined in the preceding sections. The visual presentation has been designed to have an appearance similar to the transmission line illustration.
Actuate the simulation and see the two wires exposing rows of electrons which are free to move under the influence of accelerating fields produced by their neighbors. Above the line is shown a row of atoms in orange to serve as the reference for spacings d but in reality represents the neutralizing charges. Just like in the transmission line simulation we show two graphs one of which displays in green the ratio of electron velocity v to electron spacing s as they are read from the lower wire. The quantity v/s is proportional to the flow magnitude in the lower wire. In electrical terms this is proportional to the current. The other graph displays in red the inverse spacing of atoms 1/d minus the inverse spacing of electrons 1/s as also read from the lower wire. The difference 1/d-1/s is proportional to the net density or in electrical terms to the net charge which for a uniform transmission line is proportional to the voltage. We should therefore be able to compare the behaviour of this structure with the earlier experiment based on the mathematical model. The source in this simulation permits any velocity to pass through it but enforces the difference in spacings between its two terminals to produce the nominal voltage between them.
Click the left mouse button over the window which immediately turns on our battery-the ideal DC voltage source. Be ready to click it again as soon as the transient has propagated to near the middle of the line. If you miss it, use Reset and start over. When you are there, observe the bunching of electrons in the upper wire and spreading in the lower. The velocity v of electrons in the lower wire is taken as positive to the left. The green graph is constructed from numbers obtained, when dividing the velocity by spacing as found in various positions along the lower wire. To the right of the disturbance front, the velocity is zero. To the left of it, it has a positive value.
The red graph on the bottom shows a net positive density where the electron spacing s is larger than that of positive atoms d. Beyond the wave front it is zero because s is equal to d, indicating a neutral state. Click over the window again and be ready to stop the simulation when the wave front has reached the center of the wires this time after turning around. The deceleration of the last electron at the wire end has sent a deceleration disturbance backwards which triggers a chain reaction of one electron stopping the other. But during the disturbance propagation time the electrons have narrowed down the distance even more in the upper wire and widened it more in the lower. The flux goes to zero when the electrons stop but the net positive density in the lower wire goes up when s becomes larger. As a matter of fact, the spacings established by disturbances propagating over widened distances, produce exactly a double net density in the red graph. The same doubling of net densities but of opposite polarity results in the upper wire when the propagation delays over now shorter distances are taken into account.
Continue the simulation and be ready to stop it at the moment the transient has reached the source. If you have missed it, a "Reset" followed by another try will get you there. The spacings presented to the source are larger on the bottom and smaller on the top than what the source wants to maintain. We saw the target spacing at the very beginning of the simulation. The source now enforces this spacing and in the process pulls the top electrons down, i.e., further apart and pushes the bottom electrons away, i.e., closer together. But this results in a negative velocity and if you now continue the simulation and stop it somewhere near the middle of the line you will see the spacings imposed from the source to be the same as at the start except that the flow has reversed its direction. This is seen in the green graph as a negative flow.
Allow now the simulation to continue past the end of the wire and stop it again somewhere in the middle of it. At the end of the line the deceleration has reversed the propagation direction of the disturbance as we have seen before and like dominoes the electrons stop one another. But before the field reaches them they succeed to get closer to each other in the bottom wire and to separate out more in the upper. When properly accounting for the delays between electrons at the particular spacing and for the velocities at which they happen to interact, the new spacing turns out to be exactly d in both wires.
It may be of interest to point out that while the disturbance is moving towards an electron this is, in general, moving also. The exact timing of when the encounter takes place depends on the spacing at the time of emission but also on the magnitude and direction of velocity of the electron. This particular simulation looks for coincidences of disturbances with electrons and accounts for this complex interaction automatically on the fly for all electron pairs. By maintaining the disturbance speed at constant value we have enforced the relativity principle.
If you had stopped the simulation as suggested earlier you should be seeing the wave front somewhere near the center of the wires with the left half of the green graph being negative and the right half being zero. The RHS of the wire has experienced a restoration of the original neutral state with electrons standing still and spacings being equal to that of the atoms d. Both the flux and the net density are therefore zero as seen in the two graphs. We can at this time predict that the voltage source when faced with this state will behave no differently than at the outset and will repeat the whole transient. Let us continue the simulation and let it go uninterrupted through a few cycles. When you count the number of passes needed to repeat the cycle you get four. Because each pass represents one quarter of the cycle, we have reproduced the quarter wave resonance. Recall that our source of energy is a plain DC battery and that we have applied only the first principles of nature to arrive at this result. The appealing feature of this simulation is that no familiarity with partial differential equations is necessary to follow it. Furthermore, the student does not need to be fluent in the concepts of electric current and voltage since the displays are presented in terms of spacings and velocities, these being familiar concepts from everyday life.
The spacings of electrons do not always appear to be uniform which is the result of discrete nature of displays. Internally the computation is done with double precision. Some improvements in display can be obtained by enlarging the window since "Open line" is scalable and the standard way of stretching the window can be applied to it. When enlarged to occupy most of the screen the spacings will appear uniform.
Let us now take a look at the temporal relationship between the flux, represented by v/s and the net charge, represented by 1/d-1/s. The diagram depicts the two quantities when plotted as a time sequence of the four phases. You can convince yourself of that by viewing the simulation and the phases simultaneously. The smaller phase diagram may be more appropriate for this or you may resize both windows if you wish. Reset the simulation and after starting it compare the flux and net charge with the phase diagram every time the wave front reaches either the open end or the source. Note also that the flux and the net density plots have exactly the same shape if we subtract the DC bias from the 1/d-1/s graph but they are shifted in time by a quarter cycle. Click over the phase animation diagram and observe that the 1/d-1/s graph must be shifted to the left in order to match the flux graph. This means that the flux was trailing the net charge in time by a quarter of a cycle. Because the quarter cycle is equivalent to 90 degrees shift in phase we can conclude that in an open line the flux lags the net charge by 90 degrees, a characteristic known to exist in capacitors. The open line driven by a voltage source is therefore a good physical model for behaviour of capacitors.
We leave now the open line and commence the study of the physical model of a shorted line driven by a current source. Recall that this was our last example in experimenting with the mathematical model of the transmission line. Quit the open line simulation but leave the phase diagram standing for later comparison. Call the relevant text for the shorted line with the link Shorted line.