for *I*_{B} = 10 µA, 20 µA, 30 µA, ... , 80 µA

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The behavior of an NPN bipolar transistor is largely controlled
by the current flowing *into* the base (i.e., a *positive* current).
For the usual collector-emitter voltage drops (i.e., the __active region__:
positive voltages from a fraction of a volt up to some breakdown voltage) the collector
current (*I*_{C}) is nearly independent of the collector-emitter voltage
(*V*_{CE}), and instead
depends on the base current (*I*_{B}).
(This is unusual behavior: usually more voltage
produces to more current, but here the current only increases
slightly with increasing *V*_{CE}.) The __current gain__,
i.e., the ratio of the collector current to the base current, is
often denoted by or *h*_{FE}:

= *h*_{FE} =
*I*_{C}/*I*_{B}

Thus in the simplest approximation the characteristic curves of a NPN are a set of flat, evenly spaced, lines:

Each (flat) curve shows that *I*_{C} doesn't change with changing
*V*_{CE}. The different levels show that *I*_{C}
does depend on *I*_{B}.

A slightly more complicated approximation takes into account
the sloping characteristic curves through the fiction of a constant Early
Voltage (*V*_{A}). Here we assume that the
characteristic curves all have a common *x*-axis intercept
at the large negative voltage -*V*_{A}. (The dashed curves
are far from the active region and in no way represent the actual behavior of
the transistor for negative *V*_{CE}. In fact, the transistor
is not designed to be operated with negative *V*_{CE}.)

(For the above measured 2N3904, the Early voltage ranges
from 100 to 150 V. That is the extrapolated characteristic curves do not
intersect at *a* point.)

The actual relationship between the collector current (*I*_{C})
and the controlling base current (*I*_{B}) and
collector-emitter voltage drop (*V*_{CE}) is some complicated
function which we can denote:

*I*_{C}(*I*_{B},*V*_{CE})

Like any function we can approximate it near a particular point using just the first terms of a Taylors expansion:

Clearly these *hybrid (h) parameters* are not constants.
For example *h*_{oe} is the slope of a characteristic
curve, which is nearly zero for small *I*_{B} and increases
for larger *I*_{B}. (Note that slope on an *I-V*
is basically the inverse of the resistance. Thus 1/*h*_{oe}
can be described as the output impedance. A typical value for
1/*h*_{oe} would be 100,000 .)

Even the defining parameter -- the current gain
or *h*_{FE} -- is not particularly
constant and depends on *I*_{B}. Here is the measured
relationship for the above 2N3904:

The characteristic curves focus on the output of the transistor, but we
can also consider the behavior of the input. In the active region the base
is a forward biased diode, and so *V*_{B} would be about .7 V,
typical for a conducting Si diode. Of course in greater detail the relationship
between *V*_{B} and *I*_{B} would be given by the
Shockley diode equation:

where *I*_{s} is a constant and the thermal voltage
*V*_{T} is given by:

Because of the exponential relationship between base current and base voltage,
the slope of this relationship (which could be called the
input conductance, or 1/*r*_{B}, or 1/*h*_{ie}) can be
approximated by:

where in the last equation (25 /*I*_{B})
the base current must be entered in mA.

A desirable characteristic of a transistor is that the outputs have little
effect on the inputs, but if we look in detail we find that *V*_{CE}
affects *V*_{B}. The actual functional relationship giving the
base voltage (*V*_{B}) from
the base current (*I*_{B}) and
collector-emitter voltage drop (*V*_{CE}) is some complicated
function which we can denote:

*V*_{B}(*I*_{B},*V*_{CE})

Like any function we can again approximate it near a particular point using just the first terms of Taylors expansion:

The small value of *h*_{re} shows that the input is
largely unaffected by the output.

The spec sheet reports the following values for the 2N3904:

Characteristics | Symbol | Min | Max | Unit |
---|---|---|---|---|

Input Impedance | h_{ie} |
1 | 10 | k Ohms |

Voltage Feedback Ratio | h_{re} |
0.5 | 8 | ×10^{-4} |

Small-Signal Current Gain | h_{fe} |
100 | 400 | |

Output Admittance | h_{oe} |
1.0 | 40 | µ mhos |