# FHSST Physics/Electricity/Nonlinear Conduction

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# Nonlinear conduction

Bernhard Haisch, Astrophysicist

Ohm's Law is a simple and powerful mathematical tool for helping us analyze electric circuits, but it has limitations, and we must understand these limitations in order to properly apply it to real circuits. For most conductors, resistance is a rather stable property, largely unaffected by voltage or current. For this reason, we can regard the resistance of most circuit components as a constant, with voltage and current being inversely related to each other.

For instance, our previous circuit example with the 3 ${\displaystyle \Omega }$  lamp, we calculated current through the circuit by dividing voltage by resistance (I = E/R). With an 18 volt battery, our circuit current was 6 amperes. Doubling the battery voltage to 36 volts resulted in a doubled current of 12 amperes. All of this makes sense, of course, so long as the lamp continues to provide exactly the same amount of friction (resistance) to the flow of electrons through it: 3 Ω.

However, reality is not always this simple. One of the phenomena we need to mention is of conductor resistance changing with temperature. In an incandescent lamp (the kind employing the principle of electric current heating a thin filament of wire to the point that it glows white-hot), the resistance of the filament wire will increase dramatically as it warms from room temperature to operating temperature. If we were to increase the supply voltage in a real lamp circuit, the resulting increase in current would cause the filament to increase temperature, which would in turn increase its resistance, thus preventing further increases in current without further increases in battery voltage. Consequently, voltage and current do not follow the simple equation ${\displaystyle I={\frac {E}{R}}}$  (with R assumed to be equal to 3 ${\displaystyle \Omega }$ ) because an incandescent lamp's filament resistance does not remain stable for different currents.

The phenomenon of resistance changing with variations in temperature is one shared by almost all metals, of which most wires are made. For most applications, these changes in resistance are small enough to be ignored. In the application of metal lamp filaments, the change happens to be quite large.

A more realistic analysis of a lamp circuit, however, over several different values of battery voltage would generate a plot of this shape:

The plot is no longer a straight line. It rises sharply on the left, as voltage increases from zero to a low level. As it progresses to the right we see the line flattening out, the circuit requiring greater and greater increases in voltage to achieve equal increases in current.

If we try to apply Ohm's Law to find the resistance of this lamp circuit with the voltage and current values plotted above, we arrive at several different values. We could say that the resistance here is nonlinear, increasing with increasing current and voltage. The nonlinearity is caused by the effects of high temperature on the metal wire of the lamp filament.