# Biasing

As shown before, the DC operating point of a transistor amplification circuit needs to be set up properly (in the middle of the linear region) to avoid signal distortion. We now consider how the operating point is determined by the biasing circuit, in terms of , , and .

Fixed Biasing

By properly setting the voltage (not too low) and (not too large), the voltage can be approximated as a constant value of , as shown in the input characteristic plot:

The DC operating point in terms of and can be found in the following steps:

• Find the base current:

 (29)

• Find the collector current:

If the transistor is in linear region, then

 (30)

• Find the output voltage

 (31)

As both and depend on , which may differ for different transistors and change depending on the temperatures the operating point may be unstable and inconsistent.

Example 1

In the fixed biasing transistor circuit shown above, , , , find so that the DC operating point is in the middle of the linear region of the output characteristic plot, i.e., . We assume (may not be valid if is too large) and get

 (32)

For this DC operating point with to be in the middle of the load line, we need to have

 i.e. (33)

Alternatively, we can also require the short-circuit current to be twice of at the DC operating point:

 i.e. (34)

so that

 (35)

is indeed in the middle of the load line.

Example 2:

In the same circuit above, , , . Find the operating point for .

The load line is determined by these two points:

• Open-circuit voltage:
• Short-circuit current:
Same as before, we have

 (36)

We then find the folllowing for each of the three values:

 (37)

To minimize distortion, the DC operating point needs to be in the middle of the load line at . But in this case, we see that

• , too close to cutoff region.
• , in the middle of linear region as desired.
• , too close to the saturation region.

The DC operating point of this fixed biasing circuit is not completely determined by the parameters of the circuit such as the resistors, as it is also directly affected by factors such as value and temperature. This situation can be improved by introducing negative feedback into the circuit.

Self-Biasing

To correct the problem above, the self-biasing circuit shown below can be used to decrease the effect of changing by negative feed back due to the introduction of .

Qualitatively, an increased (caused by reasons such as increased due to temperature change) will cause the following to happen:

 (38)

This is a negative feedback loop that tends to stabilize the operating point.

Quantitatively, we can further carry out analysis of the circuit:

• If the resistances of and are small so that the current through is much larger than the base current i.e., , then the basis voltage can be approximated to be (voltage divider):

 (39)

Applying KVL to the base-emitter loop of the circuit, we get

 (40)

Solving for , we get

 (41)

Note that is independent of , as it is completely determined by , , and , as well as .

• If the condition is not satisfied, the method above is no longer valid. In this case, we can use Thevenin's theorem to replace the base circuit by an open circuit voltage (already found above), in series with the internal resistance :

 (42)

Applying KVL to the base loop we get

 (43)

Solving this equation for , we get:

 and (44)

If is not too small (strong negative feedback effect) and is not too large (approximately a voltage divider), so that (e.g., even for a small ), then can be approximated as

 (45)

which is the same as what we got previously. In this case, , and thereby and the DC operating point, is determined only by the resistors of the circuit, independent of the . Comparing this with fixed biasing with directly proportional to , the self-biasing circuit has a much more stable operating point.

For this approximation above to be valid, we desire to have smaller so that is less affected by , and large for stronger negative feedback. However, as the voltage gain of the circuit will be reduced due to the negative feedback, cannot be too large.

Example 3:

In the circuit of self-biasing, , , , , , Assume . The load line is determined by this equation:

 (46)

determined by these two points at:
• and (short-circuit current)
• and (open-circuit voltage)

To minimize distortion, the desired operating point should be in the middle of the load line at and .

• Based on the voltage divider approximation, we get the DC operating point independent of :
 (47)

• Based on the Thevenin theorem, we get more accurate results:
 (48)

The DC operating points corresponding to the three values can be found to be:

 (49)

We see that in all three cases, , , i.e., the DC operating point is always close to the middle of the load line.

Example 4

In a self-biasing transistor circuit, , , , , find so that the DC operating point is in the middle of the linear region of the output characteristic plot.

We first convert the base circuit into its Thevenin's equivalent voltage source composed of

 (50)

Then we get

 (51)

and

 (52)

To set this DC operating point to be in the middle of the load line, we need , and solving the equation we get .

Example 5

The circuit below shows yet another way to introduce feedback to stablize the DC operating point.

• The resistor connecting the collector to the base forms a feedback from the output to as well as providing the forward baising needed for the base-emitter PN junction:

 (53)

• The DC operating point can be found by applying KVL:

 (54)

Solving for we get

 (55)

and

 (56)

Note that if (strong negative feedback), then , independent of . Although typically is significantly greater than , the negative feedback still has the tendency to reduce the affect of varying and thereby stablize the DC operating opint.

• Given , , and a desired , find and so that the DC operating point is in the middle of the linear region.

We want the DC operating point to be at and , and get

 (57)

and

 (58)

 (59)

• Following the equations given above, we can find the DC operating pointt for each of the three values, which are all approximately in the middle of the linear region:

 (60)

As discussed above, to avoid distortion of the AC signal, we desire that the DC operating point is in the middle of the linear region of the output characterisc. For example, in the fixed biasing circuit, we want and . The corresponding power consumption is therefor , even when the AC signal is zero.