Nonlinear conduction
"Advances are made by answering
questions. Discoveries are made by questioning answers."
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 many circuit components as a
constant, with voltage and current being directly related to
each other.
For instance, our previous circuit example
with the 3 Ω 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 amps. Doubling the
battery voltage to 36 volts resulted in a doubled current of
12 amps. 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 explored in a later chapter is that 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
"I=E/R" (with R assumed to be equal to 3 Ω) 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.
This is just one example of "nonlinearity"
in electric circuits. It is by no means the only example. A
"linear" function in mathematics is one that tracks a
straight line when plotted on a graph. The simplified
version of the lamp circuit with a constant filament
resistance of 3 Ω generates a plot like this:
The straight-line plot of current over
voltage indicates that resistance is a stable, unchanging
value for a wide range of circuit voltages and currents. In
an "ideal" situation, this is the case. Resistors, which are
manufactured to provide a definite, stable value of
resistance, behave very much like the plot of values seen
above. A mathematician would call their behavior "linear."
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.
Another example of nonlinear current
conduction is through gases such as air. At standard
temperatures and pressures, air is an effective insulator.
However, if the voltage between two conductors separated by
an air gap is increased greatly enough, the air molecules
between the gap will become "ionized," having their
electrons stripped off by the force of the high voltage
between the wires. Once ionized, air (and other gases)
become good conductors of electricity, allowing electron
flow where none could exist prior to ionization. If we were
to plot current over voltage on a graph as we did with the
lamp circuit, the effect of ionization would be clearly seen
as nonlinear:
The graph shown is approximate for a small
air gap (less than one inch). A larger air gap would yield a
higher ionization potential, but the shape of the I/E curve
would be very similar: practically no current until the
ionization potential was reached, then substantial
conduction after that.
Incidentally, this is the reason lightning
bolts exist as momentary surges rather than continuous flows
of electrons. The voltage built up between the earth and
clouds (or between different sets of clouds) must increase
to the point where it overcomes the ionization potential of
the air gap before the air ionizes enough to support a
substantial flow of electrons. Once it does, the current
will continue to conduct through the ionized air until the
static charge between the two points depletes. Once the
charge depletes enough so that the voltage falls below
another threshold point, the air de-ionizes and returns to
its normal state of extremely high resistance.
Many solid insulating materials exhibit
similar resistance properties: extremely high resistance to
electron flow below some critical threshold voltage, then a
much lower resistance at voltages beyond that threshold.
Once a solid insulating material has been compromised by
high-voltage breakdown, as it is called, it often
does not return to its former insulating state, unlike most
gases. It may insulate once again at low voltages, but its
breakdown threshold voltage will have been decreased to some
lower level, which may allow breakdown to occur more easily
in the future. This is a common mode of failure in
high-voltage wiring: insulation damage due to breakdown.
Such failures may be detected through the use of special
resistance meters employing high voltage (1000 volts or
more).
There are circuit components specifically
engineered to provide nonlinear resistance curves, one of
them being the varistor. Commonly manufactured from
compounds such as zinc oxide or silicon carbide, these
devices maintain high resistance across their terminals
until a certain "firing" or "breakdown" voltage (equivalent
to the "ionization potential" of an air gap) is reached, at
which point their resistance decreases dramatically. Unlike
the breakdown of an insulator, varistor breakdown is
repeatable: that is, it is designed to withstand repeated
breakdowns without failure. A picture of a varistor is shown
here:
There are also special gas-filled tubes
designed to do much the same thing, exploiting the very same
principle at work in the ionization of air by a lightning
bolt.
Other electrical components exhibit even
stranger current/voltage curves than this. Some devices
actually experience a decrease in current as the
applied voltage increases. Because the slope of the
current/voltage for this phenomenon is negative (angling
down instead of up as it progresses from left to right), it
is known as negative resistance.
Most notably, high-vacuum electron tubes
known as tetrodes and semiconductor diodes known as
Esaki or tunnel diodes exhibit negative
resistance for certain ranges of applied voltage.
Ohm's Law is not very useful for analyzing
the behavior of components like these where resistance is
varies with voltage and current. Some have even suggested
that "Ohm's Law" should be demoted from the status of a
"Law" because it is not universal. It might be more accurate
to call the equation (R=E/I) a definition of resistance,
befitting of a certain class of materials under a narrow
range of conditions.
For the benefit of the student, however, we
will assume that resistances specified in example circuits
are stable over a wide range of conditions unless
otherwise specified. I just wanted to expose you to a little
bit of the complexity of the real world, lest I give you the
false impression that the whole of electrical phenomena
could be summarized in a few simple equations.
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REVIEW:
-
The resistance of most conductive
materials is stable over a wide range of conditions, but
this is not true of all materials.
-
Any function that can be plotted on a
graph as a straight line is called a linear
function. For circuits with stable resistances, the plot
of current over voltage is linear (I=E/R).
-
In circuits where resistance varies with
changes in either voltage or current, the plot of current
over voltage will be nonlinear (not a straight
line).
-
A varistor is a component that
changes resistance with the amount of voltage impressed
across it. With little voltage across it, its resistance
is high. Then, at a certain "breakdown" or "firing"
voltage, its resistance decreases dramatically.
-
Negative resistance is where the
current through a component actually decreases as the
applied voltage across it is increased. Some electron
tubes and semiconductor diodes (most notably, the
tetrode tube and the Esaki, or tunnel
diode, respectively) exhibit negative resistance over a
certain range of voltages.
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