Stabilisation
Since the collector current in a transistor changes rapidly with temperature, the transistor is replaced with another of the same type. This is due to the inherent variations of transistor parameters.
When the temperature changes or the transistor is replaced, the operating point (i.e., zero signal \(I_{C}\) and \(V_{CE}\)) also changes. However, for faithful amplification, it is essential that the operating point remains fixed. This necessitates making the operating point independent of these variations. This is known as stabilisation.
The process of making the operating point independent of temperature changes or variations in transistor parameters is known as stabilisation.
Once stabilisation is complete, the zero signal \(I_{C}\) and \(V_{CE}\) are independent of temperature variations or transistor replacement, i.e., the operating point is fixed. A good biasing circuit always ensures the stabilisation of the operating point.
Stabilisation of the operating point is necessary due to the following reasons :
Temperature dependence of collector current.
The collector current \(I_{C}\) for CE circuit is given by:
\[I_{C} = \beta I_{B} + I_{CEO} = \beta I_{B} + (\beta + 1) I_{CBO}\]
The collector leakage current \(I_{CBO}\) is greatly influenced (especially in a germanium transistor) by temperature changes. A rise of 10 °C doubles the collector leakage current, which may reach \(0.2 \,mA\) in low-powered germanium transistors. In such transistors, the biasing conditions are generally set so that the zero signal \(I_{C} = 1~mA \); therefore, the change in \(I_{C}\) due to temperature variations cannot be tolerated. This necessitates stabilizing the operating point, i.e., holding the \(I_{C}\) constant despite temperature variations.
Individual variations.
The values of \(\beta\) and \(V_{BE}\) are not exactly the same for any two transistors, even of the same type. Further, \(V_{BE}\) itself decreases as temperature increases. When a transistor is replaced by another of the same type, these variations change the operating point. This necessitates stabilizing the operating point, i.e., to hold \(I_{C}\) constant irrespective of individual variations in transistor parameters.
Thermal runaway.
The collector current for a CE configuration is given by :
\[I_{C} = \beta I_{B} + (\beta + 1) I_{CBO}\]
The collector leakage current, \(I_{CBO}\), is strongly temperature-dependent. The flow of collector current produces heat within the transistor. This raises the transistor temperature, and if no stabilisation is applied, the collector leakage current, \(I_{CBO}\), also increases. It is clear that if \(I_{CBO}\) increases, the collector current \(I_{C}\) increases by \((\beta + 1) I_{CBO}\). The increased \(I_{C}\) will raise the temperature of the transistor, which will increase \(I_{CBO}\). This effect is cumulative, and in a matter of seconds, the collector current may become very large, causing the transistor to burn out.
The self-destruction of an unstabilised transistor is known as thermal runaway.
To avoid thermal runaway and consequent destruction of the transistor, it is essential that the operating point be stabilised, i.e., the \(I_{C}\) is kept constant. In practice, this is done by automatically decreasing \(I_{B}\) with increasing temperature through circuit modification. Then, a decrease in \(\beta I_{B}\) will compensate for the increase in \((\beta + 1) I_{CBO}\), keeping \(I_{C}\) nearly constant. In fact, this is always the goal when designing a biasing circuit.
Stability Factor
It is desirable and necessary to keep \(I_{C}\) constant despite variations in \(I_{CBO}\) (sometimes represented as \(I_{CO}\)). The extent to which a biasing circuit is successful in achieving this goal is measured by the stability factor \(S\). It is defined as under:
The rate of change of collector current \(I_{C}\) with respect to the collector leakage current \(I_{CO}\) at constant \(\beta\), and \(I_{B}\) is called the stability factor, i.e., Stability factor,
\[S = \frac{dI_{C}}{dI_{CO}} \,\bigg|_{\text{at constant}\, I_B \,\text{and} \, \beta } \]
The stability factor indicates the change in collector current \(I_{C}\) resulting from a change in collector leakage current \(I_{CO}\). Thus, a stability factor of \(50\) of a circuit means that the \(I_{C}\) changes \(50\) times as much as any change in \(I_{CO}\).
To achieve greater thermal stability, it is desirable to have the lowest possible stability factor. The ideal value of \(S\) is \(1\), but it is never possible to achieve it in practice. Experience shows that values of \(S\) exceeding \(25\) result in unsatisfactory performance. The general expression of the stability factor for a C.E. configuration can be obtained as under:
\[I_{C} = \beta I_{B} + (\beta + 1) I_{CO}\]
Differentiating the above expression w.r.t. \(I_{C}\), assuming that \(\beta\) is independent of \(I_{C}\), we get,
\[1 = \beta \frac{dI_{B}}{dI_{C}} + (\beta + 1) \frac{dI_{CO}}{dI_{C}}\]
\[1 = \beta \frac{dI_{B}}{dI_{C}} + \frac{(\beta + 1)}{S}\]
\[\because \quad S = \frac{dI_{C}}{dI_{CO}}\]
\[S = \frac{(\beta + 1)}{1 - \beta \left(\frac{dI_{B}}{dI_{C}}\right)}\]
Stability factor for Fixed bias method
In the fixed-bias method, \(I_{B}\) is obtained as:
\[I_{B} = \frac{V_{CC}-V_{BE}}{R_{B}}\]
So, \(I_{B}\) is independent of \(I_{C}\), and \(dI_{B}/dI_{C} = 0\).
Putting the value of \(dI_{B} / dI_{C} = 0\) in the expression of Stability factor,
\[S = \frac{(\beta + 1)}{1 - \beta \left(\frac{dI_{B}}{dI_{C}}\right)}\]
We have a Stability factor,
\[S = \beta + 1 \]
Thus, the stability factor in a fixed bias is \((\beta + 1)\). This means that \(I_{C}\) changes \((\beta + 1)\) times as much as any change in \(I_{CO}\). For instance, if \(\beta = 100\), then \(S = 101\), which means that \(I_{C}\) increases \(101\) times faster than \(I_{CO}\). Due to the large value of \(S\) in a fixed bias, it has poor thermal stability.
Stability of Emitter bias.
The expression for collector current \(I_{C}\) for the emitter bias circuit is given by:
\[I_{E} \approx I_{C} = \frac{V_{CC} - V_{BE}}{R_{E} +\frac{R_{B}}{\beta}}\]
It is clear that \(I_{C}\) is dependent on \(V_{BE}\) and \(\beta\), both of which change with temperature. If \(R_{E} \gg \frac{R_{B}}{\beta}\), then expression for \(I_{C}\) becomes :
\[I_{C} = \frac{V_{CC} - V_{BE}}{R_{E}}\]
This condition makes \(I_{C}\) \((\approx I_{E})\) independent of \(\beta\).
If \(V_{CC} \gg V_{BE}\), then \(I_{C}\) becomes :
\[I_{C} = \frac{V_{CC}}{R_{E}}\]
This condition makes \(I_C (\approx I_E)\) independent of \(V_{BE}\).
If \(I_{C}\) \((\approx I_{E})\) is independent of \(\beta\) and \(V_{BE}\), the \(Q\)-point is not affected appreciably by the variations in these parameters. Thus, emitter bias can provide a stable \(Q\)-point if properly designed.
Stability Factor For Potential Divider Bias
By applying Kirchhoff’s voltage law to the base circuit, we have,
\[V_{2} = I_{B} R_{0} + V_{BE} + I_{E} R_{E}\]
where \(R_0 = \frac{R_1 R_2}{R_1 + R_2}\)
and
\[V_{2} = I_{B} R_{0} + V_{BE} + (I_{B}+I_{C}) R_{E}\]
\[\because \quad I_{E}=I_{B}+I_{C}\]
Hence,
\[V_{2} = I_{B} R_{0} + V_{BE} + I_{B} R_{E} + I_{C} R_{E}\]
Considering \(V_{BE}\) to be constant and differentiating the above equation w.r.t. \(I_{C}\), we have,
\[0 = R_{0} \frac{I_{B}}{I_{C}} + 0 + R_{E} \frac{I_{B}}{I_{C}} + R_{E} \]
or
\[0 = R_{E} + (R_{0} + R_{E}) \frac{I_{B}}{I_{C}} \]
\[\therefore \qquad \frac{I_{B}}{I_{C}} = \frac{-R_{E}}{(R_{0} + R_{E}) } \]
The general expression for the stability factor is Stability factor,
\[S = \frac{(\beta + 1)}{1 - \beta \left(\frac{dI_{B}}{dI_{C}}\right)}\]
Putting the value of \(dI_{B}/dI_{C}\) from the above equation into the expression for \(S\), we have,
\[S = \frac{(\beta + 1)}{1 - \beta \left(\frac{-R_{E}}{R_{0} + R_{E} }\right)}\]
\[S = \frac{(\beta + 1)}{1 + \left(\frac{\beta R_{E}}{R_{0} + R_{E} }\right)}\]
\[S = \frac{(\beta + 1)(R_{0} + R_{E})}{R_{0} + R_{E} + \beta R_{E}}\]
\[S = \frac{(\beta + 1)(R_{0} + R_{E})}{(\beta +1 )R_{E} +R_{0} }\]
\[S = (\beta + 1)\frac{(R_{0} + R_{E})}{(\beta +1 )R_{E} +R_{0} }\]
Dividing the numerator and denominator of R.H.S. of the above equation by RE, we have,
\[S = (\beta + 1)\frac{(1 + \frac{R_{0}}{R_{E}})}{(\beta +1 + \frac{R_{0}}{R_{E}} )}\]
This gives the formula for the stability factor \(S\) for the potential divider bias circuit.
For greater thermal stability, the value of \(S\) should be small. This can be achieved by making \(R_{0} /R_{E}\) small and can be neglected relative to \(1\).
\[\therefore \quad S = (\beta + 1) × \frac{1}{\beta + 1} = 1\]
This is the ideal value of S and leads to the maximum thermal stability.
The ratio \(R_{0} /R_{E}\) can be made very small by decreasing \(R_{0}\) and increasing \(R_{E}\). A low value of \(R_{0}\) can be obtained by making \(R_{2}\) very small. But with a low \(R_{2}\) value, the current drawn from \(V_{CC}\) will be high. This puts a restriction on the choice of \(R_{0}\). Increasing the value of \(R_{E}\) requires greater \(V_{CC}\) in order to maintain the same zero signal collector current. Due to these limitations, a compromise is made in selecting the values of \(R_{0}\) and \(R_{E}\). Generally, these values are so selected that \[\boxed{S \approx 10}\]
Design of Transistor Biasing Circuits
(For low-powered transistors)
In practice, the following steps are taken to design transistor biasing and stabilisation circuits:
Step 1.
It is a common practice to take \(R_{E} = 500 \,\Omega\) or \(1000 \,\Omega\). The greater the value of \(R_{E}\), the better the stabilisation. However, if \(R_{E}\) is very large, a higher voltage drop across it results in a lower voltage drop across the collector load. Consequently, the output is decreased. Therefore, a compromise must be made in selecting the value of \(R_{E}\).
Step 2.
The zero-signal current \(I_{C}\) is selected based on the signal swing. However, in the initial stages of most transistor amplifiers, a zero-signal \(I_{C} = 1~\text{mA} \) is sufficient.
The major advantages of selecting this value are :
- The output impedance of a transistor is very high at \(1~\text{mA}\). This increases the voltage gain.
- There is little danger of overheating as \(1~\text{mA}\) is quite a small collector current. It may be noted that operating the transistor below zero signal \(I_{C} = 1~\text{mA} \) is not advisable due to its strongly non-linear characteristics.
Step 3.
The values of resistances \(R_{1}\) and \(R_{2}\) are selected so that the current \(I_{1}\) flowing through \(R_{1}\) and \(R_{2}\) is at least \(10\) times \(I_{B}\), i.e., \(I_{1} \ge 10 I_{B}\). When this condition is satisfied, good stabilisation is achieved.
Step 4.
The zero signal \(I_{C}\) should be slightly more (e.g., 20 %) than the maximum collector current swing due to the signal. For example, if the collector current change is expected to be \(3~\text{mA}\) due to the signal, then select a zero signal \(I_{C} \approx 3.5 ~\text{mA}\). It is important to note this point. Selecting a zero-signal \(I_{C}\) below this value may cut off part of the negative half-cycle of the signal. On the other hand, selecting a value much above this (say \(15~\text{mA})\) may unnecessarily overheat the transistor, resulting in wasted battery power. Moreover, a higher zero signal \(I_{C}\) will reduce the value of \(R_{C}\) (for the same \(V_{CC}\)), resulting in reduced voltage gain.
