Electromagnetics Waves by German Leon

What we do that defines us


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What happens when an EM wave impinges from a dielectric medium on a planar interface with air? The EM wave bends its propagation direction. The angle forms between the surface normal and the propagation direction is bigger in the air than in the dielectric medium (Snell’s Law). Increasing the incidence angle in the dielectric, it be achieved an angle of propagation in the air equal to 90º. In this case, the incidence angle is called “critical angle”. For incidence angles bigger than the critical angle, total reflection can be achieve, that is, no energy flows from the dielectric to the air. That can be seen clearly in these experiment, where the light (a EM wave, of course) goes from water to air, forming different incidence angles.

This total reflection is a key element to understand optical fibers: the light can be “catched” because the angle of incidence from the dielectic medium is bigger than the critical angle.


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A wavelength/4 transformer -or (2n+1) wavelength/4 –  is a classical circuit scheme which allow us to match a real impedance to a transmission line (TL). The transformer impedance is the geometric average of the real impedance and the TL impedance. What happens inside the transformer? In the transitory state, an imping wave flows to the transformer. Part of the energy flows to the load, and part goes back to the generator, that is a backward wave does exist inside the TL (t=6s). The wave inside the transformer arrives the load and part of it returns to the TL (t=14s). During this time, a quasi-standing wave is made in the TL. But at t=24s, the backward wave from the load “compensates” the reflection from the transformer, and a flowing wave is formed in the TL, that is, in the steady state, all the energy that comes from the generator goes to the transformer, and it will be consumed in the load.


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These three videos show the behavior of a standing wave.  In this first one we can see a perfect standing wave. This wave is produced as a sum of two waves of the same amplitude, flowing in the same direction but in opposite sense. You can observe that there are positions where the wave amplitude is always zero and also, that there are instants that the amplitude is zero everywhere.

A voltage standing wave can be generated when a transmission line ended in a pure imaginary load, as for example a short circuit:

It is easy to realize that it is a short circuit, because the voltage is equal to zero at the end.

What happens if these waves have different amplitude? This is the general case, when a transmission line is ended in an arbitrary load:

We can see that a wave is flowing towards the positive sense. Also, a backward wave with lower amplitude is flowing in the line. The sum of both waves produce a net flux of energy toward the positive sense and also standing wave.


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This simulation shows the behavior of a lossy transmission line (TL). A sinusoidal signal propagates through the TL and, due to the losses, the signal decays when it advances. If you stop the motion at t=8s and join the peaks with an imaginary line, a decreasing exponential can be sketched. Moreover, this TL is not matched. It is very difficult to see the backward wave because it has very low level. However, if you stop the motion at t=10 s, you can see that the waveform at the end of the line is different to the waveform at the beginning.


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This video represents the potential and the intensity of two different Transmission Lines (TLs) joined in cascade.   This simulation is made using the Finite Different Time Domain (FDTD) method, which allows us to visualize what happens when the time goes by.

What can we see? A step signal feeds the first TL and the line is excited by certain potential. At t=2s, the step arrives to the discontinuity between the two TLs. Two waves appear: one wave is generated in the second line to forward direction and another wave advances to the backward direction. The first TL has a bigger transmission speed than the second.  The wave in the second line arrives at t=8s  to a certain load and a backward wave is produced that come back to the discontinuity.

Let‘s look a little more inside. We can make some affirmations. Try to understand why:

–          The two lines are lossless.

–          The two lines have different transmission speed and different impedance.  The impedance of the first line is smaller than the second one.

–          The load at the end of the second TL has a real value smaller than the impedance of the second line

–          The generator is matched to the first TL.