3.5: Ebers-Moll Model

A good, functional model of the BJT is the simplified Ebers-Moll model shown in Figure \(\PageIndex{1}\). This utilizes an ideal diode to model the base-emitter junction and a current-controlled current source located at the collector-base. This model is sufficient to achieve good analysis results with a variety of DC and low frequency circuits. It is important to remember, though, that \(\beta\) varies not only from device to device, but also varies with changes in temperature, collector current and collector-emitter voltage.

Figure \(\PageIndex{1}\): Ebers-Moll model of the NPN BJT.

We can put the Ebers-Moll model to use in basic DC biasing circuits. To properly bias the BJT we need to make the collector-base reverse-biased and the base-emitter forward-biased. In other words, \(V_C > V_B > V_E\). There are many ways to achieve this. One method places the emitter at ground, a modest DC source in the base-emitter loop, and a somewhat higher DC source at the collector. An example is shown in Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\): Simple base bias circuit.

The two resistors serve to limit the transistor's currents and voltages. Because the emitter is at ground, the common point, this circuit is classified as having a common emitter configuration. There are many possible common emitter circuits. We shall refer to this one specifically as base bias.

Now, let's replace the transistor with the Ebers-Moll model. The result, with added voltage polarities and current directions, is shown in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\): Base bias circuit with EbersMoll model.

Given values for the resistors, power supplies and \(\beta\), all of the currents and voltages may be determined using fundamental circuit analysis techniques. The basic idea is to create KVL equations for the two loops and then expand using Ohm's law. We begin with the base-emitter loop.

\[V_{BB} = V_{R_B} +V_{BE} \\ V_{BB} = I_B R_B+V_{BE} \\ I_B = \frac{V_{BB} - V_{BE}}{R_B} \label{4.3} \]

And for the collector-emitter loop:

\[V_{CC} = V_{R_C} +V_{CE} \\ V_{CC} = I_C R_C + V_{CE} \\ V_{CE} = V_{CC} − I_C R_C \label{4.4} \]

To obtain \(I_C\), recall that \(I_C = \beta I_B\).

Example \(\PageIndex{1}\)

Determine the circulating currents and device voltages for the circuit of Figure \(\PageIndex{2}\) if \(V_{BB} = 10\) V, \(V_{CC} = 15\) V, \(R_B = 200\) k\(\Omega\), \(R_C = 1\) k\(\Omega\) and \(\beta\) = 100. Assume that the transistor is silicon.

First, find the base current. KVL dictates that the voltage across \(R_B\) is 9.3 volts.

\[I_B = \frac{V_{BB} −V_{BE}}{R_B} \nonumber \]

\[I_B = \frac{10 V −0.7V}{200 k\Omega} \nonumber \]

\[I_B = 46.5\mu A \nonumber \]

Now find the collector current and follow with Ohm's law and KVL.

\[I_C = \beta I_B \nonumber \]

\[I_C = 100\times 46.5\mu A \nonumber \]

\[I_C = 4.65mA \nonumber \]

\[V_{CE} = V_{CC} −I_C R_C \nonumber \]

\[V_{CE} = 15V −4.65 mA\times 1k\Omega \nonumber \]

\[V_{CE} = 10.35 V \nonumber \]

For the sake of completeness, the voltage across \(R_C\) is 4.65 volts, \(V_{CB}\) is 9.65 volts and \(I_E\) is 4.6965 mA.

The preceding example illustrates that the place to start the analysis is in the baseemitter loop instead of the collector-emitter loop. This is because in the base-emitter loop we have the forward-biased base-emitter junction which has a known potential (approximately 0.7 volts). In contrast, the collector-emitter voltage is an unknown as it includes the reverse-biased collector-base junction. That voltage will depend on other circuit elements, most notably the collector resistor and associated supply.

An improvement on the circuit of Example \(\PageIndex{1}\) would be to redesign it for a single power supply rather than two supplies. This is easy to do. All that is needed is to keep the base current unchanged. If that remains at its original value then the collector current won't change and consequently nothing in the collector-emitter loop will change either. Using the 15 volt source for \(V_{BB}\) means that the voltage across \(R_B\) will increase to 14.3 volts. Ohm's law then indicates that \(R_B\) must be 14.3 volts divided by 46.5 \(\mu\)A, or 307.5 k\(\Omega\).

4.5.1: \(\beta\) Variation Issues

There is a major problem with the circuit of Figure \(\PageIndex{2}\), namely, it lacks stability of collector current and collector-emitter voltage. As we shall see in upcoming chapters, it is important to keep these parameters stable in order to ensure consistent performance for many kinds of circuits. As we noted from our inspection of the 2N3904 data sheet, the variation of \(\beta\) can be quite large at a given operating point. If we also add in the variance due to temperature and other factors, we may be looking at a 10:1 range. If we repeat Example \(\PageIndex{1}\) with a doubled \(\beta\) of 200, the base-emitter loop is unchanged but the collector current doubles to 9.3 mA. This increases the voltage drop across \(R_C\) to 9.3 volts which then forces \(V_{CE}\) to drop to 5.7 volts. Given a typical production run of transistors, this circuit might exhibit collector currents from less than 4 mA to more than 10 mA. In some applications this variation in current might be tolerable but not in all of them. For example, suppose an LED is placed in series with \(R_C\). Because the brightness of an LED depends on its current level, the brightness will now depend on the \(\beta\) of the specific BJT used. If this is one LED in a larger display made up of similar circuits, then the illumination will be uneven between them causing the entire display to appear off kilter.

In fact, if this circuit was built in the lab, it is quite likely that after turning on the power, you could watch \(I_C\) slowly rise on your ammeter. This is because the BJT will begin to warm up as it dissipates power. As noted from the data sheet, \(\beta\) increases with increasing temperature. Because \(I_B\) is a fixed value, any rise in \(\beta\) means that \(I_C\) must also rise. This increased current will tend to cause a further rise in power dissipation and temperature which causes a further increase in \(\beta\), and the process cycles. We have created an inadvertent thermal positive feedback loop. Left unchecked, devices could overheat and be destroyed. We will examine biasing circuits that achieve high stability in the next chapter.

There is another interesting aspect to this circuit. As noted, if we substituted the original BJT with another unit that had a higher \(\beta\), the collector current would rise. What if we continued this to higher and higher \(\beta\) values? For example, if we increased \(\beta\) to 400 (admittedly, rather high) the new collector current would seem to jump up to 46.5 \(\mu\)A \(\cdot\) 400, or 18.6 mA. There is a “small” problem with this value. Ohm's law indicates that this current would develop a drop of 18.6 volts across the 1 k\(\Omega\) \(R_C\) but that's impossible because \(V_{CC}\) is only 15 volts. The only way that “works” is if somehow the BJT is transformed magically into a 3.6 volt battery. No amount of prayer or letters to Santa will make that happen¹.

References

¹Both being equally effective.