Understanding Electrical Conduction in Wires

by Zvonko Fazarinc, Hewlett Packard Laboratories


The topic at hand is studied in college courses on transmission lines and is commonly introduced by means of the four Maxwell's equations. These require considerable preparation in higher mathematics and in electromagnetics theory and are as such not a good starting point for majority of the population. We will try to explain the conduction phenomena in wires from first principles of Nature and thereby make them accessible to scientists, teachers, engineers, college students and high school students alike. In the process we will not sacrifice any scientific rigor or limit ourselves to trivial cases. The result should be a thorough comprehension of the concepts involved. As such it should become a reliable platform for teachers and students of these topics. It should serve as a firm foundation for practitioners and researchers who must deal with phenomena encountered in high frequency analog and digital circuits. For the layman the course should result in an understanding and appreciation of processes taking place inside the wires.

The visualization of underlying processes relies on a number of dynamic simulations and animations which are linked to the text. Their optimal presentation calls for some system properties that should be considered. The desire to reach the users of various operating systems has been addressed by means of the Java language. The wide range of performance levels is being faced with manual speed adjustment of dynamic displays. The variety of presentation screens is being met with a selection of display window sizes.

The course should be taken on-line with the textual and graphic components side by side. The animations and simulations are designed to appear in the left half of the screen except in Linux which forces you to position them after the call. It is recommended that you move your browser window to the far right and adjust its width to occupy about half of the screen. Depending on the resolution of your screen you should then select the dimensions of the dynamic display by means of the red trigger points encountered within the text. Choose  if you have 800 or more pixels of horizontal resolution and  if your resolution is 600 or less. Of course, you may choose whichever you wish and you may even want to adjust the display window size to your liking.

To be able to follow the materials presented, the reader must know something about the potentials and currents , about the existence of electric charges and must accept first principles of Nature at face value. You may acquire the necessary amount of this knowledge from the linked texts.

In order to introduce the topic we will first simulate the behaviour of electric signals in wires by using the electric current conduction model arising from Maxwell's equations. The mathematical model will be used only in the background and will in no way require its understanding. It will serve only as a substitute for an experimental laboratory. As such the model will address the question What if?, i.e. what happens if we impose certain conditions. After we will have identified some observational facts we will ask the question Why? and will try to gain an understanding of the physics of the observed phenomena from underlying mechanisms. We will look at the microscopic events taking place inside the wires and relate them to the externally observed behaviour. One behaviour specifically chosen to be illuminated is the quarter wave resonance which has been known to be the source of headaches for many young electrical engineers.

The reader who is unfamiliar with the transmission line theory and with the inductive and the capacitive quarter wave resonances on wires should start with the Mathematical Model of Conduction. Others may start with the Physical Model of Conduction . Once you have chosen your entry point you will be guided to sections and their subsections without the need for returning to this page.

Most sections contain one or more supportive dynamic presentations. Red labeled pushbuttons provide access to them. The simulations are embedded in explanatory text which guides you through experiments deemed helpful in visualizing the phenomenon studied. For the reader's convenience in subsequent visits to this course all sections, their subsections as well as simulations are accessible directly through the linked list of contents shown below.

Table of Contents

Mathematical Model of Conduction


Transmission Line Simulation


Current source
Voltage source


Characteristic Impedance Standing Waves



Physical Model of Conduction

Force Field of Single Electron

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Relativistic Principles Applied to a Moving Electron

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A Thinking Aid How is Acceleration Information Carried by the Field?
Why is Acceleration Reproduced at each Electron

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Some Numeric Values of Interest

Interaction of Multiple Electrons

Electrons at Barriers

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Terminated Wires

Characteristic Impedance Open Wires Driven by a Voltage Source

Shorted Wires Driven by a Current Source

Conclusions and Acknowledgements