#### Concept of Transistor Parameters and Ebers-Moll Equations

Introduction:

Some parameters can be defined for bipolar junction transistors that act as figures of merit in quantifying its performance. Most of such, however not all, are measured and specified for forward active mode of operation. A few parameters differentiate among electron and hole currents however such are not of particular interest and attention is confined to such that are based on the terminal currents of device. These can be assumed from the physical model shown in figure below that formed the basis of Ebers-Moll model.

Ebers-Moll Equations are as shown below:

IE = IES (eVBE/VT -1) – αR ICS (eVBC/VT -1)

IC = αF IES (eVBE/VT -1) – ICS (eVBC/VT -1)

IB = (1 - αF) IES (eVBE/VT -1) + (1 - αR) ICS (eVBC/VT -1)

Forward Transfer Ratio:

It is defined for the forward active mode of operation as the ratio of collector and emitter terminal currents. In forward active mode, the base-collector junction is reverse biased and hence eVBC/VT << 0 and therefore the currents approximate to:

IE ≈ IES eVBE/VT

IC ≈ αF IES eVBE/VT = αF IE

IB ≈ (1 - αF) IES eVBE/VT = (1 - αF) IE

Then,

Forward transfer ratio = IC/IE = (αF IES eVBE/VT)/(IES eVBE/VT) = αF

Figure: Model Showing Forward and Reverse Current Components

Figure: Minority Carrier Concentration in the Base Region

Forward Current Gain, βF:

Beneath steady-state conditions in forward active mode, the volume of minority charge stored in the base region is constant. The greater portion of this is charge in transit to the collector that forms the electronic component of collector current. A small percentage of minority charge continually rejoins with majority carriers that are replaced with holes supplied by an external base current. As the base region is very thin and is sandwiched among the two junctions, it is not surprising that the volume of minority charge exist in the base exercises a profound influence on the behavior of transistor.

However, the excess minority charge in the base region can be forced to modify by modulating the base current with an external signal. Atomic forces operating cause both the recombining charge and the charge in transit to collector to differ in sympathy. The junction voltages will adjust, in response to currents flowing, to conform to exponential laws governing them. Though, as the charge in transit to the collector is much bigger than the recombination charge, a big change in absolute charge terms is induced in the collector current than is present in the base current due to the signal. Therefore, the transistor is observed to amplify the input signal to the base as shown in figure below.

Figure: The Current Amplifying the Property of Bipolar Transistor

This current amplifying the property is explained by the current gain of the transistor in forward active mode that is taken simply as the ratio of the collector and base currents. Thus:

Forward current gain, βF = IC/IB = (αF IES eVBE/VT)/(1 - αF) IES eVBE/VT = αF/(1 - αF)

Alternatively,

βF = IC/IB = IC/(IE – IC) = (IC/IE)/[1 – (IC/IE)] = αF/(1 - αF)

Typical values of βF range from 40 to 100 for integrated transistors however can be as high as 300 to 500 for discrete devices.

Reverse Transfer Ratio and Current Gain, αR, βR

The parameters above can as well be defined for the reverse active mode if the B-E junction is reverse biased and the B-C junction is forward biased. Though such parameters are not of great interest as the transistor is much inefficient from an amplifying view point in this mode. Usually αR = 0.1 - 0.5 and βR = 0.1 – 1.0.

It is the average amount of time; a minority carrier can survive after being injected to the base region before rejoining with a majority carrier. It mainly depends on the physical properties of device, like the doping concentration and dimensions. Usually τB = 50 - 100ns.

Forward Transit Time, τF

It is the average time it takes a minority carrier injected to the base region to make the transition to the collector region whenever the transistor is operating in forward active mode. This can be shown as:

τF = (Wb2)/2Db = (1/2) (Wb/Lb)2 τb

In general τF = 0.1 – 0.5ns

The above parameters will be employed in the analysis of static and dynamic performance of transistor as a circuit switching element.

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