### Thermal Stability

As temperature increases Ic increases Ic increases biasing should be such that Ic does not change with temperature. The variation of Ic with temp is measured by stability factor. The stability factor should be low as possible lower the value of stability factor better is the thermal stability

• Ic independent of beta variation.

• Due to variation in manufacturing process two transistor never have the same value of beta thus as beta  changes  Ic will be different.

• Biasing  should be such that Ic can be made independent of Beta variation.

Biasing Circuit:

1. Fixed Biasing

2. Collector to base bias

3. Self biased

1. Fixed Biased:

Adding a resist of the emitter circuit stablizes the bias circuit, as show in figure

Stability Factor S For Fixed Bias Circuit

$\textbf{S}\boldsymbol\;\boldsymbol=\boldsymbol\;\frac{\boldsymbol\partial\mathbf{Ic}}{\boldsymbol\partial\mathbf{Ico}}$ Vbe, $\mathbf\beta$ constant

$\boldsymbol=\frac{\mathbf{\left({I\;+\;\beta}\right)}}{\mathbf1\boldsymbol\;\boldsymbol-\boldsymbol\;\mathbf\beta{\displaystyle\frac{\boldsymbol\partial\mathbf{Ib}}{\boldsymbol\partial\mathbf{Ic}}}}$

For the fixed Bias Circuit Ib = Vcc/Rb

$\boldsymbol\therefore\frac{\boldsymbol\partial\mathbf{Ib}}{\boldsymbol\partial\mathbf{Ic}}\boldsymbol=\boldsymbol\;\mathbf0$

$\boldsymbol\therefore\boldsymbol\;\boldsymbol\;\mathbf S\boldsymbol\;\boldsymbol=\boldsymbol\;\frac{\mathbf{\left({I+\beta}\right)}}{\mathbf1\boldsymbol-\mathbf\beta\mathbf{\left(0\right)}}$

$\boldsymbol\therefore\boldsymbol\;\boldsymbol\;\mathbf S\boldsymbol\;\boldsymbol=\boldsymbol\;\mathbf1\boldsymbol+\mathbf\beta$

1. Fixed Bias with Emitter Feedback:

1. Collector to base bias:

In this circuit as compare to previous circuit  stability doctor reduced and thermal stability increased.

Stability Factor S For Collector base bias

${\mathbf S\boldsymbol"\boldsymbol\;\boldsymbol=\boldsymbol\;\left.\frac{\boldsymbol\partial\mathbf{Ic}}{\boldsymbol\partial\mathbf\beta}\right|}_{\mathbf{Ico}\boldsymbol,\boldsymbol\;\mathbf{Vbe}\boldsymbol\;\mathbf{constant}}$

$\mathbf{Vcc}\boldsymbol=\mathbf{\left({\mathrm{Ib}+\mathrm{Ic}}\right)}\mathbf{Rc}\boldsymbol+\mathbf{IbRb}\boldsymbol+\mathbf{Vbe}$

$\mathbf{Vcc}\boldsymbol-\mathbf{Vbe}\boldsymbol=\mathbf{\left({\mathrm{Ib}+\mathrm{Ic}}\right)}\mathbf{Rc}\boldsymbol+\mathbf{IbRb}$

$\boldsymbol=\mathbf{Ib}\mathbf{\left[{\left(1+\beta\right)\mathrm{Rc}+\mathrm{Rb}}\right]}$

$\mathbf{Vcc}\boldsymbol-\mathbf{Vbe}$

$\boldsymbol\therefore\mathbf{Ib}\boldsymbol=\mathbf{\left({1+\beta}\right)}\boldsymbol\;\mathbf{Rc}\boldsymbol+\mathbf{Rb}$

$\boldsymbol\beta\mathbf{\left({Vcc-Vbe}\right)}$

$\boldsymbol\therefore\mathbf{Ic}\boldsymbol=\mathbf{\left({1+\beta}\right)}\boldsymbol\;\mathbf{Rc}\boldsymbol+\mathbf{Rb}$

1. Self biased circuit:

Self biased has lowest value of stability factor and hance best thermal stability among all circuit this improvement in thermal. Stability is due to negative feedback of emitter resistance. Here the Ic is independent on beta so thermal stability increases. Hence among the all fixed biased is best for good thermal stability.