Many commonplace electrical components connect to the world through two
terminals. From left to right above we have diodes, resistors, fuses,
crystal oscillators,
light bulbs and batteries. If we establish a voltage difference
between the two wires, a current will usually flow. Generally the
current will flow into (i.e., a positive current) the higher-voltage
terminal and an equal current will flow out the low-voltage terminal.
(The alternative to "current-in equals current-out" is a build-up of
net charge inside the device. The electric force is simply too strong
to sustain this. Even devices like capacitors and batteries which you
may think of as "storing charge" in fact closely maintain charge
neutrality.) Thus the relationship, *I*(*V*),
between the applied voltage and the
resulting current totally defines the device.

(Exceptions: some
devices do depend on time: a discharged or charged battery, or
temperature: a warm or cool filament in a light bulb... Perhaps
temperature is the most common secondary parameter. Often a
temperature coefficient, or "tempco", is specified to determine how a
device behaves at different temperatures.) I say again: the
relationship between applied voltage and resulting current -- the
"*IV* characteristic curve" -- almost totally defines the
device. This relationship can sometimes be usefully described with a
formula; it can always be displayed as a graph of current (on the
*y*-axis) vs. voltage (on the *x*-axis).

The most common example of this is the resistor, a two-terminal device with a linear relationship between current and voltage:

*I* = *V/R*

where *R* is the resistance, *V* is the voltage
difference between the two terminals
(*V*_{2}-*V*_{1}), and *I* is the
current flowing into terminal 2. [Because it is potential
*difference* that counts, often we take *V*_{1}=0
("ground") so *V*_{2} is the voltage difference,
*V*. While the above may ease discussion, it in no way should
suggest that in a real circuit *V*_{1} must be at ground
or that measurements can be made on the device by connecting
terminal-1 to ground.] In a resistor *R* is a
"constant" (of course subject to some tempco; years and use may also
have some effect).

For any two-terminal device we can rearrange this relationship to define a "voltage dependent resistance":

*R* = *V/I*

although more often the "dynamic resistance"

*r* = *dV/dI*

is the useful concept. (Note that this dynamic resistance is the
inverse of the slope of the *IV* curve.) The dynamic resistance
is particularly useful if we seek small changes in current due to small
changes in voltage near a particular "quiescent point" voltage
*V*_{Q}. Taylor expanding the function
*I*(*V*) in the vicinity of *V*_{Q}
yields:

We can express this mathematical formula as a model equivalent circuit:

which simply displays that if the voltage drop across the
device is changed by *V*
the resulting change in current is exactly what would have been produced
by a resistance *r*.
Thus the dynamic resistance tells us how the device responds to changing
voltages.

You may be under the impression the Ohm's "Law" is TRUE. While we can take various epistemological approaches to this, the most important point is that usually in electronics we're not so much looking for the TRUE as the useful. Real materials do not exactly follow Ohm's law...so what! The fact that many materials under normal conditions do approximately follow Ohm's law is what makes this a "law of electricity". That is to say, simple "models" of how devices behave normally and approximately are more valued than much more complex stories that might be more accurate. What is perhaps most valued is a sequence of increasingly accurate models allowing you to choose the needed level of accuracy and resulting burden of complexity. (Einstein: "Everything should be as simple as possible, but not simpler".) Note that we humans value simplicity as an aid to human understanding -- but computers rarely complain at being forced to do an additional million calculations.

Another aspect of my emphasis on __simple__ models is an inversion
of the revolutionary idea of interchangeable parts: The pistons coming
off a GM engine assembly line are to a very high accuracy the same size
so any can fill the cylinder. Alternatively we could design the engine
so that more sloppily constructed pistons would still work. (Think
piston rings.) In the electronics industry, available devices have a
comparatively low level of reproducibility, so engineers have had to
learn to make flexible (robust) designs. Given that individual
electronic devices are far from identical, there is often no reason to
use highly accurate models of the devices.

The upshot of all of this is that we're going to be looking for simple
descriptions of our *IV* curves and we won't be put off
if our model's predictions are a few percent off from measurements on a
real device. While not the focus of these pages, additionally it is
important to understand the physical basis of these models, so that, if
possible, devices can be constructed with the characteristics we desire.

One of the first non-ohmic electronic devices invented was the diode.
With a resistor if you reverse the voltage difference the current also
reverses. (The two terminals on a resistor are interchangeable.)
With a diode, voltage differences of one sign produce quite
different results from the reversed sign. Perhaps the simplest model
would be if *V*_{2}>*V*_{1} the device
behaves like a resistor, whereas
*V*_{2}<*V*_{1} no current flows:

The first diodes constructed worked by having the two terminals at different temperatures. One terminal (the "cathode") was heated to so high a temperature that electrons were easily pulled from it, allowing an electron flow from the cathode to the other terminal (called the "anode"). The anode was kept at room temperature; so its electrons stayed put and no current would flow in the reversed direction. Thus our first model is easy current flow in one direction; zero current flow in the other.

Since we understand the physics of the device, we can make an improved model. For electron flow through a "vacuum", it turns out that the current is not proportional to the voltage, rather the 3/2 power of voltage (Child's law):

I won't bother to explain what all the constants are in this expression, but do note that the basis of this law is understood. Devices with different characteristics could be designed by changing the the geometric constants of the device.

Real manufactured tubes do not exactly follow Child's Law.
There is some current flow in the reverse direction ("cold emission") at large
reverse voltages (see insert, about 10^{-5} A reverse flow).
For large positive voltages
the measured current falls below that given by Child's law.
At
zero volts there is some positive (rather than
zero) current flow.
We could incorporate these additional complexities
in our model of the device, if we really needed to.

The above picture taken of the 3B24 vacuum tube while the above data was collected, shows the main problem: about 15 Watts of power required just to heat the cathode. The vacuum tube looks like an Edison light bulb (indeed, current flow in the vacuum tube is called the Edison Effect).

It was discovered you could make more efficient diodes (no heater
required) out of semiconductors like silicon and germanium
("crystals"). Again the 0^{th} model is easy current flow in
one direction, no current flow in the other. For silicon diodes the
current flow remains "small", unless the forward voltage is greater
than about 0.7 V. Thus our next model is easy current flow in one
direction for forward voltages more than 0.7 V, otherwise no current
flow. Again an understanding of how a semiconductor diode works
shows that in simple cases there is an exponential
relationship between voltage and current (Shockley's Law).

The aim of physics is to understand the basis of such a model, so
for example, we want to understand the cause of the parameters
*I*_{s} and *V*_{T} used
in our models. Again, I won't bother to explain what the symbols
mean in the below expression for *I*_{s},
but do note that the basis of Shockley's law is understood.

If you measure the *IV* curve for a real diode, you'll
discover
deviations from Shockley's law. Additional processes like
the recombination current and high-level injection modify
Shockley's law at low and high currents. Again, if we need
a more accurate model these factors they can be included. But
frankly the simplest model of a diode: that it is like a
wire for current flow in one direction, and like a disconnect
for current flow in the other, is what first comes to my mind
when I need think about a circuit with diodes.

- NPN Bipolar transistors
- PNP Bipolar transistors
- n-channel field effect transistors (FETs)
- p-channel field effect transistors (FETs)
- n-channel MOSFETs
- p-channel MOSFETs

Many commonplace electrical components connect to the world through
three terminals. Mostly I've displayed various types of transistors,
but I've also included some potentiometers ("pots"). As above, for
convenience we consider the voltage on one terminal to be zero
(*V*_{1}=0). We then have four variables, two of which
are independent: *V*_{2}, *I*_{2},
*V*_{3}, *I*_{3}. (Of course, device
neutrality guarantees that the current flowing in:
*I*_{2}+*I*_{3} must flow out the first
terminal.) For two terminal devices the current was just a function of
the voltage, and the defining property was the *IV*-curve. For
three terminals we have two independent quantities. Following the lead
of the two-terminal discussion, we will take one of these to be
*V*_{2}, but the other could reasonably be either
*V*_{3} or *I*_{3}.
*V*_{3} seems more natural, so let's begin there.

The
next problem is how to display functions of two variables, like
*I*_{2}(*V*_{2},*V*_{3}).
We could, of course, resort to a 3d perspective plot, with the height
above the *V*_{2}-*V*_{3} plane denoting
*I*_{2}.

While such a plot might be evocative, we reject it as:

- it would not be useful, because we can not easily extract numbers from such a perspective plot
- it erroneously
suggests some sort of equivalence between
*V*_{2}and*V*_{3}, whereas in fact for useful devices the two are designed to be quite different. (In future cases we'll use*I*_{3}as the second variable so the two axes in the base plane will not even have the same units.)

and displaying the slices together.

(You should have seen similar plots
displaying *PVT* relationships of gases in thermodynamics:
*PV* curves for various fixed *T* ("isotherms") and "topographic" maps
of mountains where the set of horizontal points at the same altitude
are displayed as a curve.)

Practically speaking such plots are easy to produce: you just fix a
value of *V*_{3} and then vary *V*_{2}
collecting data points (*V*_{2},*I*_{2})
to form your curve.

In the vicinity of a point Q, we can approximate changes in
*I*_{2} due to changes in *V*_{2} and
*V*_{3} using a 2d Taylor's expansion:

(In our 2-terminal Taylor's expansion we were approximating the characteristic curves with a tangent line. In our 3-terminal Taylor's expansion we are approximating the characteristic surface with a tangent plane.)

We can play exactly the same game to express
*I*_{3}(*V*_{2},*V*_{3}),
but I've never seen the results displayed a characteristic curves.
Instead the *y* (admittance) parameters are sometimes provided.

We can express these mathematical formulae as a model equivalent
circuit. *I*_{2} increases in proportion to
*V*_{2} (exactly like a resistor)
and there is an additional sink for current controlled by
*V*_{3}: a source sinking a current equal to
*y*_{23}*V*_{3}.
(The current source parameter *y*_{23} is often
called the transconductance and denoted by *g*.)

where

*r*_{out} = 1/*y*_{22}

*r*_{in} = 1/*y*_{33}

*g* = *y*_{23}

In typical devices like FETs, *y*_{32} is small
and is usually neglected:

Often even greater model simplicity can be achieved:
*r*_{in} (and sometimes *r*_{out})
are "large" enough to be neglected. In this simplest model, a FET is
just a current source controlled by *V*_{3}.

The whole game can be re-played using *I*_{3} as the
other independent variable:

We can express these mathematical formulae as a model equivalent
circuit. *I*_{2} increases in proportion to
*V*_{2} (exactly like a resistor)
and there is an additional sink for current controlled by
*I*_{3}: a source sinking a current equal to
*h*_{23}*I*_{3}.
(The current source parameter *h*_{23} is often
called the current gain and denoted by .)

where

*r*_{out} = 1/*h*_{22}

*r*_{in} = *h*_{33}

= *h*_{23}

In typical devices like bipolar transistors, *h*_{32} is small
(ca. 10^{-4})
and is usually neglected:

In this simplest model, a bipolar transistor is
just a "current multiplier" (i.e., a current source controlled by
*I*_{3}) with input and output impedances.

I left out these important players because I was
seeking *simple* models. Consideration
of alternating current (AC) immediately introduces
two additional parameters: frequency and phase shift.
Most any circuit operating above the audio range
(i.e., above 20 kHz) must include these additional complications.
Here I will only comment that "stray capacitance"
is the most important AC effect. More complex models of transistors
would include little capacitors connecting each pair of
terminals. At "low" frequencies the impedance of the capacitor:

*Z*_{C}=-*i*/*C*

is large and so the stray capacitors have little effect. However at higher frequencies these little capacitances are quite important.