Andy Collinson
Introduction
The primary function of a "model" is to predict the behaviour of a device in a particular operating region. At DC the bipolar junction transistor
(BJT) works in either the cutoff or saturation regions (as a switch). See these articles:
In the AC domain (audio frequencies) operation is quite different and the transistor works in the linear operating region. The r_{e} model
reflects the operation of the BJT at midfrequencies and is sufficiently accurate. The r_{e} model is an equivalent circuit that can be
used to predict performance.
The r_{e} Model
Small r
_{e} is the resistance looking into the emitter terminal of a transistor. As there is a voltage on the base of a transistor and a
current flowing in the emitter, then from ohm's law r
_{e} = v/i, see diagram below.
If the BJT is working in the linear region of its characteristic curves and base emitter junction is forward biased,
then r
_{e} can be defined as:
The base emitter junction acts the same as a conducting diode and has an exponential relationship between the current and voltage in the forward
region. The following equation can now be used to find an approximate value for r
_{e}:

where:
K is Boltzman's constant 1.38 x 10^{23} joule/K
T is absolute temperature in Kelvin (K = 273 + °C)
q is electronic charge 1.602 x 10 ^{19} coulombs

At room temperature r
_{e} equates to 25 / I
_{E} at 20 °C and 26 / I
_{E} at 30 °C, see below:
As I
_{E} is approximately the same as I
_{C} some text books quote r
_{e} as 25 / I
_{C}.
It is important that I
_{E} is measured in milliamps and to use the appropriate ambient temperature to calculate
r
_{e}.
In any BJT, the collector current i
_{c}, is equal to the product of the base current, i
_{b} multiplied by the small signal forward
current gain, h
_{fe} or β of the transistor. Thus βi
_{b} can be thought of as a constant current generator. The equivalent
circuit is shown below:
This model is quite accurate provided the DC conditions are evaluated to find the quiescent point of the circuit. Just one parameter is required
which can be measured or taken from the manufacturers data sheet. Separating the above diagram and arranging in common emitter, the r
_{e}
model is drawn below:
Common Emitter r_{e} Model
The output equivalent circuit between terminals C and E is now a constant current
βi_{b} generator.
The input impedance is between terminals B and E and has a value of:
r_{e} ( β + 1 )
Common Base r_{e} Model
In common base the input signal is applied between B and E terminals and has the value:
r_{e}
Common Collector r_{e} Model
In common collector (emitter follower) the input impedance is:
r_{e} ( β + 1 )
The r
_{e} model can be used to quickly estimate input impedance, gain and operating conditions of transistor circuits.
Example Circuit
An example circuit using the r
_{e} model and a differential amplifier
can be found here in the Simulation
section.
Summary
The r
_{e} model is sufficiently accurate and only requires one parameter h
_{fe}. Input impedance is derived from just one parameter,
h
_{fe}. However the r
_{e} model does not have parameters for output admittance or reverse voltage ratio and contains no capacitance.
As such it is only suitable at dc and midfrequencies. For high frequency work the hybridpi model must be used.