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 cut-off 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 mid-frequencies 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 mid-frequencies. For high frequency work the hybrid-pi model must be used.