The biasing in transistor circuits is done by using two DC sources V_BB and V_CC. It is economical to minimize the DC source to one supply instead of two which also makes the circuit simple.

The commonly used methods of transistor biasing are

Base Resistor method
Collector to Base bias
Biasing with Collector feedback resistor
Voltage-divider bias

All of these methods have the same basic principle of obtaining the required value of I_B and I_C from V_CC in the zero signal conditions.

Base Resistor Method

In this method, a resistor R_B of high resistance is connected in base, as the name implies. The required zero signal base current is provided by V_CC which flows through R_B. The base emitter junction is forward biased, as base is positive with respect to emitter.

The required value of zero signal base current and hence the collector current (as I_C = βI_B) can be made to flow by selecting the proper value of base resistor RB. Hence the value of R_B is to be known. The figure below shows how a base resistor method of biasing circuit looks like.

Let I_C be the required zero signal collector current. Therefore,

$$I_B = \frac{I_C}{\beta}$$

Considering the closed circuit from V_CC, base, emitter and ground, while applying the Kirchhoffs voltage law, we get,

$$V_{CC} = I_B R_B + V_{BE}$$

$$I_B R_B = V_{CC} - V_{BE}$$

Therefore

$$R_B = \frac{V_{CC} - V_{BE}}{I_B}$$

Since V_BE is generally quite small as compared to V_CC, the former can be neglected with little error. Then,

$$R_B = \frac{V_{CC}}{I_B}$$

We know that V_CC is a fixed known quantity and I_B is chosen at some suitable value. As R_B can be found directly, this method is called as fixed bias method.

Stability factor

$$S = \frac{\beta + 1}{1 - \beta \left ( \frac{d I_B}{d I_C} \right )}$$

In fixed-bias method of biasing, I_B is independent of I_C so that,

$$\frac{d I_B}{d I_C} = 0$$

Substituting the above value in the previous equation,

Stability factor, $S = \beta + 1$

Thus the stability factor in a fixed bias is (β+1) which means that I_C changes (β+1) times as much as any change in I_CO.

Advantages

The circuit is simple.
Only one resistor R_E is required.
Biasing conditions are set easily.
No loading effect as no resistor is present at base-emitter junction.

Disadvantages

The stabilization is poor as heat development cant be stopped.
The stability factor is very high. So, there are strong chances of thermal run away.

Hence, this method is rarely employed.

Collector to Base Bias

The collector to base bias circuit is same as base bias circuit except that the base resistor R_B is returned to collector, rather than to V_CC supply as shown in the figure below.

This circuit helps in improving the stability considerably. If the value of I_C increases, the voltage across R_L increases and hence the V_CE also increases. This in turn reduces the base current I_B. This action somewhat compensates the original increase.

The required value of R_B needed to give the zero signal collector current I_C can be calculated as follows.

Voltage drop across R_L will be

$$R_L = (I_C + I_B)R_L \cong I_C R_L$$

From the figure,

$$I_C R_L + I_B R_B + V_{BE} = V_{CC}$$

$$I_B R_B = V_{CC} - V_{BE} - I_C R_L$$

Therefore

$$R_B = \frac{V_{CC} - V_{BE} - I_C R_L}{I_B}$$

$$R_B = \frac{(V_{CC} - V_{BE} - I_C R_L)\beta}{I_C}$$

Applying KVL we have

$$(I_B + I_C)R_L + I_B R_B + V_{BE} = V_{CC}$$

$$I_B(R_L + R_B) + I_C R_L + V_{BE} = V_{CC}$$

Therefore

$$I_B = \frac{V_{CC} - V_{BE} - I_C R_L}{R_L + R_B}$$

Since V_BE is almost independent of collector current, we get

$$\frac{d I_B}{d I_C} = - \frac{R_L}{R_L + R_B}$$

We know that

$$S = \frac{1 + \beta}{1 - \beta (d I_B / d I_C)}$$

Therefore

$$S = \frac{1 + \beta}{1 + \beta \left ( \frac{R_L}{R_L + R_B} \right )}$$

This value is smaller than (1+β) which is obtained for fixed bias circuit. Thus there is an improvement in the stability.

This circuit provides a negative feedback which reduces the gain of the amplifier. So the increased stability of the collector to base bias circuit is obtained at the cost of AC voltage gain.

Biasing with Collector Feedback resistor

In this method, the base resistor R_B has its one end connected to base and the other to the collector as its name implies. In this circuit, the zero signal base current is determined by V_CB but not by V_CC.

It is clear that V_CB forward biases the base-emitter junction and hence base current I_B flows through R_B. This causes the zero signal collector current to flow in the circuit. The below figure shows the biasing with collector feedback resistor circuit.

The required value of R_B needed to give the zero signal current I_C can be determined as follows.

$$V_{CC} = I_C R_C + I_B R_B + V_{BE}$$

$$R_B = \frac{V_{CC} - V_{BE} - I_C R_C}{I_B}$$

$$= \frac{V_{CC} - V_{BE} - \beta I_B R_C}{I_B}$$

Since $I_C = \beta I_B$

Alternatively,

$$V_{CE} = V_{BE} + V_{CB}$$

$$V_{CB} = V_{CE} - V_{BE}$$

Since

$$R_B = \frac{V_{CB}}{I_B} = \frac{V_{CE} - V_{BE}}{I_B}$$

Where

$$I_B = \frac{I_C}{\beta}$$

Mathematically,

Stability factor, $S < (\beta + 1)$

Therefore, this method provides better thermal stability than the fixed bias.

The Q-point values for the circuit are shown as

$$I_C = \frac{V_{CC} - V_{BE}}{R_B/ \beta + R_C}$$

$$V_{CE} = V_{CC} - I_C R_C$$

Advantages

The circuit is simple as it needs only one resistor.
This circuit provides some stabilization, for lesser changes.

Disadvantages

The circuit doesnt provide good stabilization.
The circuit provides negative feedback.

Voltage Divider Bias Method

Among all the methods of providing biasing and stabilization, the voltage divider bias method is the most prominent one. Here, two resistors R₁ and R₂ are employed, which are connected to V_CC and provide biasing. The resistor R_E employed in the emitter provides stabilization.

The name voltage divider comes from the voltage divider formed by R₁ and R₂. The voltage drop across R₂ forward biases the base-emitter junction. This causes the base current and hence collector current flow in the zero signal conditions. The figure below shows the circuit of voltage divider bias method.

Suppose that the current flowing through resistance R₁ is I₁. As base current I_B is very small, therefore, it can be assumed with reasonable accuracy that current flowing through R₂ is also I₁.

Now let us try to derive the expressions for collector current and collector voltage.

Collector Current, I_C

From the circuit, it is evident that,

$$I_1 = \frac{V_{CC}}{R_1 + R_2}$$

Therefore, the voltage across resistance R₂ is

$$V_2 = \left ( \frac{V_{CC}}{R_1 + R_2}\right ) R_2$$

Applying Kirchhoffs voltage law to the base circuit,

$$V_2 = V_{BE} + V_E$$

$$V_2 = V_{BE} + I_E R_E$$

$$I_E = \frac{V_2 - V_{BE}}{R_E}$$

Since I_E ≈ I_C,

$$I_C = \frac{V_2 - V_{BE}}{R_E}$$

From the above expression, it is evident that I_C doesnt depend upon β. V_BE is very small that I_C doesnt get affected by V_BE at all. Thus I_C in this circuit is almost independent of transistor parameters and hence good stabilization is achieved.

Collector-Emitter Voltage, V_CE

Applying Kirchhoffs voltage law to the collector side,

$$V_{CC} = I_C R_C + V_{CE} + I_E R_E$$

Since I_E ≅ I_C

$$= I_C R_C + V_{CE} + I_C R_E$$

$$= I_C(R_C + R_E) + V_{CE}$$

Therefore,

$$V_{CE} = V_{CC} - I_C(R_C + R_E)$$

R_E provides excellent stabilization in this circuit.

$$V_2 = V_{BE} + I_C R_E$$

Suppose there is a rise in temperature, then the collector current I_C decreases, which causes the voltage drop across R_E to increase. As the voltage drop across R₂ is V₂, which is independent of I_C, the value of V_BE decreases. The reduced value of I_B tends to restore I_C to the original value.

Stability Factor

The equation for Stability factor of this circuit is obtained as

Stability Factor = $S = \frac{(\beta + 1) (R_0 + R_3)}{R_0 + R_E + \beta R_E}$

$$= (\beta + 1) \times \frac{1 + \frac{R_0}{R_E}}{\beta + 1 + \frac{R_0}{R_E}}$$

Where

$$R_0 = \frac{R_1 R_2}{R_1 + R_2}$$

If the ratio R₀/R_E is very small, then R0/RE can be neglected as compared to 1 and the stability factor becomes

Stability Factor = $S = (\beta + 1) \times \frac{1}{\beta + 1} = 1$

This is the smallest possible value of S and leads to the maximum possible thermal stability.

Base Resistor Method

Advantages

Disadvantages

Collector to Base Bias

Biasing with Collector Feedback resistor

Advantages

Disadvantages

Voltage Divider Bias Method

Collector Current, IC

Collector-Emitter Voltage, VCE

Stability Factor

Collector Current, I_C

Collector-Emitter Voltage, V_CE

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