The ideal transistor model is based on the ideal p-n diode model and provides a first-order calculation of the dc parameters of a bipolar junction transistor. To further simplify this model, we will assume that all quasi-neutral regions in the device are much smaller than the minority-carrier diffusion lengths in these regions, so that the "short" diode expressions apply. The use of the ideal p-n diode model implies that no recombination within the depletion regions is taken into account. Such recombination current will be discussed in section 5. The discussion of the ideal transistor starts with a discussion of the forward active mode of operation, followed by a general description of the four different bias modes, the corresponding Ebers-Moll model and a calculation of the collector-emitter voltage when the device is biased in saturation.
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The ideal transistor model is based on the ideal p-n diode model and provides a first-order calculation of the dc parameters of a bipolar junction transistor. To further simplify this model, we will assume that all quasi-neutral regions in the device are much smaller than the minority-carrier diffusion lengths in these regions, so that the "short" diode expressions apply. The use of the ideal p-n diode model implies that no recombination within the depletion regions is taken into account.
Such recombination current will be discussed in section 5. The discussion of the ideal transistor starts with a discussion of the forward active mode of operation, followed by a general description of the four different bias modes, the corresponding Ebers-Moll model and a calculation of the collector-emitter voltage when the device is biased in saturation.
The forward active mode is obtained by forward-biasing the base-emitter junction. The minority-carrier distribution in the quasi-neutral regions of the bipolar transistor, as shown in Figure 5. The values of the minority carrier densities at the edges of the depletion regions are indicated on the Figure 5. The carrier densities vary linearly between the boundary values as expected when using the assumption that no significant recombination takes place in the quasi-neutral regions.
The minority carrier densities on both sides of the base-collector depletion region equal the thermal equilibrium values since V BC was set to zero. While this boundary condition is mathematically equivalent to that of an ideal contact, there is an important difference.
Instead, they drift through the base-collector depletion region and end up as majority carriers in the collector region. The emitter current due to electrons and holes are obtained using the "short" diode expressions derived in section 4.
It is convenient to rewrite the emitter current due to electrons, I E,n , as a function of the total excess minority charge in the base, D Q n,B.
This charge is proportional to the triangular area in the quasi-neutral base as shown in Figure 5. The emitter current therefore equals the excess minority carrier charge present in the base region, divided by the time this charge spends in the base.
This and other similar relations will be used to construct the charge control model of the bipolar junction transistor in section 5. A combination of equations 5. We now turn our attention to the recombination current in the quasi-neutral base and obtain it from the continuity equation 2. Next, we need to find the emitter efficiency and base transport factor.
The emitter efficiency defined by equation 5. It is typically the emitter efficiency, which limits the current gain in transistors made of silicon or germanium.
The long minority-carrier lifetime and the long diffusion lengths in those materials justify the exclusion of recombination in the base or the depletion layer. The resulting current gain, under such conditions, is:. From this equation, we conclude that the current gain can be larger than one if the emitter doping is much larger than the base doping.
A typical current gain for a silicon bipolar transistor is 50 - The base transport factor, as defined in equation 5. This base transport factor can also be expressed in function of the diffusion length in the base:. Calculate the emitter efficiency, the base transport factor, and the current gain of the transistor biased in the forward active mode. Assume there is no recombination in the depletion region. Solution The emitter efficiency is obtained from:. General bias modes of a bipolar transistor While the forward active mode of operation is the most useful bias mode when using a bipolar junction transistor as an amplifier, one cannot ignore the other bias modes especially when using the device as a digital switch.
All possible bias modes are illustrated with Figure 5. They are the forward active mode of operation, the reverse active mode of operation, the saturation mode and the cut-off mode.
Finally, there is the reverse active mode of operation. In the reverse active mode, we reverse the function of the emitter and the collector. In this mode, the transistor has an emitter efficiency and base transport factor as described by equations 5. Most transistors, however, have poor emitter efficiency under reverse active bias since the collector doping density is typically much less than the base doping density to ensure high base-collector breakdown voltages.
In addition, the collector-base area is typically larger than the emitter-base area, so that even fewer electrons make it from the collector into the emitter. Having described the forward active mode of operation, there remains the saturation mode, which needs further discussion. Cut-off requires little further analysis, while the reverse active mode of operation is analogous to the forward active mode with the added complication that the areas of the base-emitter and base-collector junction, A E and A C , differ.
The Ebers-Moll model describes all of these bias modes. The Ebers-Moll model is an ideal model for a bipolar transistor, which can be used, in the forward active mode of operation, in the reverse active mode, in saturation and in cut-off.
The model contains two diodes and two current sources as shown in Figure 5. The two diodes represent the base-emitter and base-collector diodes.
The current sources quantify the transport of minority carriers through the base region. These current sources depend on the current through each diode. The parameters I E,s , I C,s , a F and a R are the saturation currents of the base-emitter and base collector diode and the forward and reverse transport factors.
Using the parameters identified in Figure 5. This relation ship is also referred as the reciprocity relation and can be derived by examining the minority carrier current through the base. For the specific case where the base-emitter and base-collector voltage are the same and the base doping is uniform, there can be no minority carrier diffusion in the base so that:.
The forward- and reverse-bias transport factors are obtained by measuring the current gain in the forward active and reverse active mode of operation. The saturation currents I E,s and I C,s are obtained by measuring the base-emitter base-collector diode saturation current while shorting the base-collector base-emitter diode. Therefore, the base-collector junction is also forward biased.
Saturation also implies that a large amount of minority carrier charge is accumulated in the base region. As a transistor is switched from saturation to cut-off, this charge initially remains in the base and a collector current will remain until this charge is removed by recombination. This causes an additional delay before the transistor is turned off. Since the carrier lifetime can be significantly longer than the base transit time, the turn-off delay causes a large and undesirable asymmetry between turn-on and turn-off time.
Saturation is therefore avoided in high-speed bipolar logic circuits. Two techniques are used to reduce the turn-off delay: 1 adding a Schottky diode in parallel to the base-collector junction and 2 using an emitter-coupled circuit configuration.
Both approaches avoid biasing the transistor in the saturation mode. The Schottky diode clamps the base-collector voltage at a value, which is slightly lower than the turn-on voltage of the base-collector diode. An emitter-coupled circuit is biased with a current source, which can be designed such that the collector voltage cannot be less than the base voltage. Boulder, December Minority-carrier distribution in the quasi-neutral regions of a bipolar transistor a Forward active bias mode.
And the emitter current due to electrons, I E,n , simplifies to:. By applying it to the quasi-neutral base region and assuming steady state conditions:. Consider a pnp bipolar transistor with emitter doping of 10 18 cm -3 and base doping of 10 17 cm The quasi-neutral region width in the emitter is 1 m m and 0. The minority carrier lifetime in the base is 10 ns.
The emitter efficiency is obtained from: The base transport factor equals: The current gain then becomes: where the transport factor, a , was calculated as the product of the emitter efficiency and the base transport factor:. General bias modes of a bipolar transistor. While the forward active mode of operation is the most useful bias mode when using a bipolar junction transistor as an amplifier, one cannot ignore the other bias modes especially when using the device as a digital switch.
Calculate the saturation voltage of a bipolar transistor biased with a base current of 1 mA and a collector current of 10 mA. The saturation voltage equals:.
Ebers-moll model of transistor
If transistor circuits are to be of any use or amenable to diagnostic procedures, we must be able to model them. Transistors characteristically have multiple modes of conduction. We can view these phenomena in the two-diode model of a bipolar junction transistor BJT. Two diodes whose anodes join to form a center tap are analogous to an NPN transistor insofar as ohmmeter readings accurately represent the real device.
Chapter 5: Bipolar Junction Transistors
A bipolar junction transistor BJT is a type of transistor that uses both electrons and holes as charge carriers. Unipolar transistors, such as field-effect transistors , use only one kind of charge carrier. A bipolar transistor allows a small current injected at one of its terminals to control a much larger current flowing between two other terminals, making the device capable of amplification or switching. BJTs use two junctions between two semiconductor types, n-type and p-type, which are regions in a single crystal of material.