Transistor re  Model

Andy Collinson

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:

BJT Biasing
Transistor as a Switch

In the AC domain (audio frequencies) operation is quite different and the transistor works in the linear operating region. The re model reflects the operation of the BJT at mid-frequencies and is sufficiently accurate. The re model is an equivalent circuit that can be used to predict performance.

The re Model
Small re 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 re = 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 re 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 re:

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 re equates to 25 / IE at 20 °C and 26 / IE at 30 °C, see below:

As IE is approximately the same as IC some text books quote re as 25 / IC. It is important that IE is measured in milliamps and to use the appropriate ambient temperature to calculate re.

In any BJT, the collector current ic, is equal to the product of the base current, ib multiplied by the small signal forward current gain, hfe or β of the transistor. Thus βib 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 re model is drawn below:

Common Emitter re Model
The output equivalent circuit between terminals C and E is now a constant current βib generator.
The input impedance is between terminals B and E and has a value of: re ( β + 1 )

Common Base re Model
In common base the input signal is applied between B and E terminals and has the value: re

Common Collector re Model
In common collector (emitter follower) the input impedance is: re ( β + 1 )
The re model can be used to quickly estimate input impedance, gain and operating conditions of transistor circuits.

Example Circuit
An example circuit using the re model and a differential amplifier can be found here in the Simulation section.

The re model is sufficiently accurate and only requires one parameter hfe. Input impedance is derived from just one parameter, hfe. However the re 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.

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