Shorted Wires Driven by a Current Source

We will now try a combination of a DC current source and a shorted line which was seen to produce resonant behaviour when we were studying the mathematical model of the transmission line. Again we will do the simulation in terms of mutual interaction of electrons rather than by solving the telegraph equation. The layout of is similar in appearance to the one encountered in the mathematical model and almost identical to that used in the "Open line" simulation. Therefore we will assume that the meaning of all icons and labels is familiar. The difference is that we have this time a DC current source driving the electrons around the loop. Therefore they can never reverse their direction of motion. The source in this simulation permits any electron spacing at its two terminals, but adjusts the electron velocities to produce its nominal current.

Click the left mouse button over the simulation window and be ready to click it again when the transient has propagated to near the middle of the line. After you do so, 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 found in various positions along the lower wire. To the right of the transient front the velocity is zero as shown in the graph. The red graph shows a net positive density because the electron spacing s in the lower wire is larger than that of positive atoms but beyond the wave front it is zero because d is equal to s indicating a neutral state. So far there is no difference between this case and the open wire which we have examined earlier. This is in agreement with our earlier statement that it is not possible to distinguish between a voltage and a current source from the initial transient.

Click over the window again and be ready to stop the simulation when the wave front has reached the short. If you are very lucky to stop the simulation at the exact moment when the compression from the top and the rarefaction from the bottom meet at the center of the short, you will understand why the electron between the two, experiences a push from the top and a pull from the bottom simultaneously. This results in an enhanced acceleration of that particular electron which launches a fold downwards and a gap upwards. These travel along the wires towards the source and in the process accelerate the electrons they encounter. As we have learned earlier the acceleration disturbance is being reproduced by each electron as it experiences a speed-up. Now the effects of speed-up on electron spacing are different in the two wires. In the upper wire it results in a widening of the spacings and in the lower wire in its reduction. The net result of the interplay of acceleration fields encountering the approaching electrons in the upper wire and catching up with the departing electrons in the lower wire produces in each wire a spacing exactly equal to d, the original neutral state spacing. When we say that a short circuit cannot support potential differences across itself, we never think of it in these terms, yet this is how the nature deals with it.

Click again over the window and let the wave front reach the center of the line. Observe how the speed-up takes place from the shorted end and produces a doubling of the v/s quantity. At the same time the net charge goes to zero as the electron spacings s match that of the neutralizing atoms d. Continue the simulation and stop it when the wave front reaches the source. The current v/s is twice the original value supplied by the source. It is to be expected that the current source will impose a velocity limit on electrons and force a spacing s such that v/s will be equal to what the current source can support. Such a slowdown of electrons will produce a rarefaction in the upper wire because the electrons ahead of them are still moving at high velocity. The deceleration disturbance emanating from the source end will eventually catch up with them and will slow them down also. Conversely, the slowdown of electrons by the source will produce a compression in the lower wire where the electrons are approaching at high speed. These too, will eventually be slowed down by the decelerating disturbance. Allow the simulation to go until the wave front reaches the middle of the wire again. Observe the effects of the described phenomena on the flux and on the net charge. The current source imposed its characteristic flow in the wires and in the process forced the upper wire to become positive and the lower one negative.

Continue the simulation and be ready to stop it at the moment the transient has reached the short. If you find this difficult to achieve slow down the simulation with the slider on the bottom of the window. At the short we have now the opposite state as the last time, i.e., the electron at the center of the short experiences a pull from the top and a push from the bottom. The net effect is a complete stoppage of that electron. This in turn triggers the chain reaction moving from the short towards the source stopping the electrons along the way. The interplay of deceleration fields with the approaching electrons in the upper wire and with departing electrons in the lower wire results in spacings exactly equal to d at the time of stoppage. Because d is also the spacing of neutralizing atoms the net charge is therefore zero again.

Continue the simulation until the transient reaches the current source. We have now zero velocity and zero net charge on both wires, a condition equal to the one we started out with at the beginning. The whole cycle must repeat now and if you let the simulation go through a few repeats you will notice that it takes four passes for each repeat. In other words we have another quarter-wave resonance at hand.

To find out if this resonance differs from the one we have experienced with open wires driven by a voltage source we use another time graph depicting all four passes of v/s and of 1/d-1/s in a time sequence. Because we have a current source this time, we note a flux offset in the vertical direction but otherwise the waveforms of the net charge and the flux are identical. They are shifted in time by one quarter of a period, i.e., one of our four cycles, which becomes apparent when you click on the diagram and observe how the alignment is achieved. Note that the net charge peak preceeded that of the flux and we had to shift it to a later time, which is opposite to what we had to do in the case of the open line. This means that the flux is lagging the net charge by one quarter of a cycle or 90 degrees in phase. Such relationship is observed in inductors and consequently the shorted line driven by a current source is a good physical model for the inductive behaviour.

You may now quit all animations and simulations that may still be on your screen and read the Conclusions and Acknowledgements.