Basic Wireless Communication for Microcontrollers
Chapter 1 - Electricity and Magnetism
Boundary Conditions
E&M problems involving interaction with matter can be greatly simplified if one realizes that matter imposes certain restrictions on what the E and B fields can be inside it and it its surface. These restrictions are called boundary conditions, and we here discuss the boundary conditions on E and B fields and the consequences of the conditions.
Electric Boundary Conditions
Inside a perfect conductor, the electric field must be zero. This is true because if the E field were nonzero, charge would immediately move to cancel it out. At the surface of a perfect conductor, the component of the E field parallel to the suface must also be zero, for similar reasons.
The perpendicular component, however, will not be zero. It will cause electrons to be displaced within the conductor, which will alter, but not cancel the external field. For conductors which extend much farther than the area of interest, one can consider that the surface of the conductor creates a mirror image of the charges.
For example, if we have a 1 coulomb positive charge at 1 meter above a perfectly conducting plane, the E field in all the space above the conductor will be just like that which would be created by the positive charge and an equal negative charge 1 meter below where the surface of the conductor is (figure x). This type of analysis is called the method if images.
Because the E field is zero inside a perfect conductor, the voltage will be the same at every point on the conductor (voltage differences only exist when there is an E field between the points in question to create a tendency for the electrons to move. In a perfect conductor, the electrons would move effortlessly so no E field is needed). In good but imperfect (real) conductors, there will be very small voltage differences between points on the conductor.
For dielectrics (insulators with Epsilon_r>1), the situation is different. As mentioned before, dielectrics reduce the strength of the E field, but they only do this to the component of the E field which points into the surface of the dielectric. The component which points parallel to the suface is left unchanged.
Magnetic Boundary Conditions
The B field must not change (dB/dt=0) inside a perfect conductor. If it did change, it would induce currents which (according to Lenz's law) would immediately cancel the changes. At the surface of the conductor, the component of the B field which points into the surface cannot change, for the same reason. The component parallel can change, however, since it creates no flux in any immaginary loop drawn in the plane of the surface, so it doesn't induce current in the conductor.
For Magnetically active materials (with a Ur <> 1) which are not good conductors, the component of the B field pointing into the surface remains unchanged as it crosses the surface. The component parallel to the surface is strengthened by a factor of Ur between the outside and the inside (stronger on the inside).
Sheet Conductors and Ground Planes
In an infinite sheet of perfectly conducting material, there are an infinite number of paths along with current can flow. If you pick any pair of points, the sheet is equivalent to an infinite number of inductors in parallel, with some capacitance also in parallel. An infinite number of inductors in parallel results in a net inductance of zero. The capacitance is shorted by the net inductance of zero, so the entire sheet exhibits no inductance or capacitance (or resistance, since it is perfectly conducting) between any two points.
If current needs to be directed from one location to another with a minimum of voltage drop, such a sheet is ideal. While there is no such thing as a perfectly conducting sheet, sheets of good conductors are a fairly good substitute. This is why ground planes are often used in high frequency situations, because they allow current to flow from one location to another with almost no inductive or resistive effects to impede it. Ground planes also act to create images of charged objects placed above them (as mentioned above).
Reflection and Refraction
The boundary conditions given above cause several phenomena to occur when an EM wave encounters matter which is either conductive or has an epsilon_r or Ur other than 1. In these cases, a portion of the wave will be reflected and a portion will be transmitted (travel into the material). Good conductors reflect almost all of the incident wave, allowing only a tiny amount to be transmitted. Dielectrics and magnetically active materials reflect some and transmit some depending on how different their Epsilon_r or Ur is from 1. The farther from 1, the more reflection and less transmission.
If the wave does not enter exactly perpendicular to the surface, the path of the transmitted part will be bent (refracted). The degree and direction of bending will depend, once again, on the values of Epsilon_r and Ur. This is the basis for lenses in optics.
Diffraction
We said before that an EM wave propagates because at every point, the varying E and B fields create E and B fields in the entire surrounding area. A consequence of this is that attempts to control the shape of an EM wave do not always have the intended or expected effect. For example, when an EM wave encounters an object which obscures part of the wave (figure x), instead of the wave continuing on the other side with half the extent, it will begin to spread out on the other side. At the edge of object, there is an area below where no E or B fields exist. Therefore, the changing E and B fields in the portion of the EM wave which makes it over the boundary will induce E and B fields in the area below the boundary. The net effect is that a portion of the beam is bent downward. This is the origin of the "knife-edge refraction" type of propagation.
Attenuation and the Skin Effect
When an EM wave propagates into a material which is conductive, the majority is reflected, as mentioned above. The small amount which does penetrate the surface is quickly absorbed by being converted into heat by the resistance of the material. The attenuation occurs at an exponential rate, that is, for every millimeter deeper into the material, the wave's power is reduced by an additional power of some constant attenuation factor.
One way to express this is to calculate how far the wave must go into the material before its power is reduced by a factor of e, the base of natual logarithms. e ~= 2.71828. This depth is caled the "Skin Depth" and as a result, the phenomenon of the attenuation in conductors is called the Skin effect.
A more common example of the skin effect is when wires are used to conduct AC current. It is often pointed out that as frequency increases, the AC current gets more and more concentrated toward the surface of the conductor, increasing losses. While it may not be immediately apparent, this is actually the same effect as that mentioned above.
In any circuit, whether it be an circuit on a PCB, or in a transmission line, or an antenna, there are two paths for current, a forward path and a return path. This means that any AC circuit involves transmission lines (which will be discussed further in project 0). In a perfectly conducting transmission line (or circuit), no E field exists inside the wires and it only exists between the wires, with the E field exactly perpendicular to the wire's surface. This is equivalent to an EM wave propagating along between the two wires.
If the wires are lossy, however, some field must exist inside to push the electrons along. This results in a tiny component of the field along the direction of propagation. The E field vectors are no longer perpendicular to the wire's surface, but are slanted slightly. This is equivalent to an EM wave propagating mostly down the wire pair, but also slightly INTO the wires themselves. It is this component which propagates into the wires that experiences the skin effect.
Attenuation can also occur in media which are not conductive. Pure water, for example, is not a good condutor (although even slightly impure water can conduct well). Nonetheless, raindrops, which are quite pure, do absorb and attenuate EM waves. This is due to losses which occur when the electron clouds around the water molecules are distored by the E field in the EM wave. This results in a complex dielectric constant (effectively, Epsilon_r has an immaginary part). This is the basis for cooking with a microwave oven. The water in the food absorbs a large amount of the indicdent EM radiation, turning it into heat.
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