Telegrapher's Equations Revisited

Oliver Heaviside presented his version of electricity to humanity as the vector calculus version of Maxwell's nine equations. Heaviside also introduced us to the telegrapher's equations. The telegrapher's model for conductors remains with us to this day. Series resistance added linearly to a series inductance with a shunt resistance and a shunt capacitance.

Let's look at what this means as far as I learned in school as an electrical engineering student. The series resistance is easy to explain. Any conductor, copper for example, is made up of a lattice. as electrons traverse the lattice they accelerate and then slam into nuclei that make up the lattice. Electrons must also, critically, interact with one another. Traditionally we will say that some of the electrons repel each other when they come close to each other. All this bumping and grinding causes resistive loses.

Inductance tradition would tell us comes from the fact that every current carrying conductor has a magnetic field around it in the shape of an ellipse according to Gauss and Maxwell's magnetic equation. The magnetic field must be induced when the current starts to flow hence induction. The opposite happens when the current stops flowing.

I was taught that capacitance refers to the capacity of an electric field to store energy. For every telegraph line there is a return path for electrons. Phone and telegraph circuits work best when the return circuit is a conductor like the signal path. Capacitance is the representation in the telegrapher's model of the electric field. The electric field must be set up and it will unload its electrical energy when the circuit is de-energized.

Shunt conductance is the most ignored of the four elements of the telegraphy model. I'm not sure it should be. The conductance represents straight leakage from the signal path to the return path. Sounds simple but we ignore it to our peril in understanding how electric circuits really work.

Inductance and capacitance should be understood in terms of the shunt conductance; at least as far as the telegrapher's equations go. These two elements of the model store energy and then give it back.

Inductance should be explained without the use of a mythical magnetic field. Electrons accelerated down a conductor are going to pop off the conductor and statistically will make their way through any dielectric cladding. Electrons also have a massive propensity to spin in the presence of molecules. The spin or curl of the electron field is what has fooled humanity into believing in a mythical magnetic field. The spinning electrons in an eddy field will store energy regardless of how we want to perceive Maxwell and Heaviside's equations.

That leaves us with another highly statistical behaviour of capacitance. Some electrons that are accelerated off the signal wire will spin while others won't. The electrons that don't spin form the capacitive part of the telegraphers model. When the line is de-energized the electrons come back to the signal wire completing the capacitive model.

To summarize, electrons pop off the wire when there is an abundance of these elementary particles. Some electrons make it to the return wire forming conductance. Some electrons spin and this is represented by the inductive part of the telegrapher model. Other electrons pop off the line and don't make it to the opposing voltage line. This is capacitance.