Assume in the circuit above
and the transistor is in
saturation region, i.e.,
, then we have

The second equation relates the output to the input , as shown by the red segment of the curve in the plot above. As the transistor is in saturation region,

which can be solved for to get:

It can be seen that when the transistor is in saturation mode the slope of the curve (red) indicates the ratio between input and output . And the voltage gain of the circuit is defined as:

**Example:** Assume , , , .
For the transistor to be in saturation region, we need

and the voltage gain is a function of the input :

This nonlinear equation can be represented by the table below:

0 | 1 | 1.4 | 1.5 | 1.8 | 1.9 | 2.0 | 2.1 | 2.2 | 2.3 | 2.32 | 2.35 | 2.4 | |

10 | 10 | 9.2 | 8.8 | 6.8 | 6.0 | 5.0 | 4.0 | 2.8 | 1.6 | 1.3 | 0.9 | 0.0 |

In summary, we see that

- When the transistor is in saturation mode, the circuit behaves as a voltage amplifier.
- The voltage gain is the slope of the tangent of the curve (red) as a function of .
- The value of the gain depends on the level of input . When , the gain is greater than one.
- The output voltage is out of phase with the input voltage (), as the slope of is negative.
- When , the transistor is cutoff. On the other hand, when , is more than one below , for example, , , the transistor is in triode region. In either of the two cases, the transistor has no amplification capability.

Next we consider a MOSFET circuit with sinusoidal input. Assume the drain resistor
is , , , and a sinusoidal input
. If the bias voltage is , the
input voltage varies between 1.4V and 1.6V. The output voltage can be
found to be:

In particular, corresponding to , the output voltage and the current are, respectively, , and , as shown in the figure below:

**Biasing:** In the example above, the DC offset of the input is at 1.5V, so that
the transistor is working in the saturation region when the magnitude of the AC input
is limited. However, if this offset is either too high or too low, the gate voltage
may go beyond the saturation region to enter either the triode or the cutoff region.
In either case, the output voltage will be severely distorted, as shown below:

It is therefore clear that the DC offset or biasing gate voltage has to be properly setup to make sure the dynamic range of the input signal is within the saturation region.

**Method 1:** One way to provide the desired DC offset is to use two resistors
and that form a voltage divider, as shown in the figure below (a). As
the input resistance of a MOSFET transistor is very high, therefore the gate of
the transistor does not draw any current, the DC offset voltage can simply obtained
as:

The input AC signal through the input capacitor is then superimposed on this DC offset.

**Method 2: ** Another way to set up the bias is the circuit shown in (b) above.
Assume , , , , and . The bias
voltage can be found to be
, and the voltage between gate and
source is
. The output voltage is

When , .

To determine the dynamic range of the input , recall the conditions for the transistor to be in saturation region:

- To avoid cutoff region:
. For this particular circuit,

Solving this we get with corresponding output . - To avoid triode region:
. For this particular circuit,

that is

i.e.,

Solving this for we get , with corresponding output

with the corresponding output range

and the overall voltage gain is about . Note that the output voltage is in phase with the input voltage.

**Source Follower: ** If the output is taken from the source, instead of the
drain of the transistor, the circuit is called a source follower.

Assume , and . To find the input and output voltages and the gain of the circuit, consider the current :

Plugging in the given values, we get

If , this equation becomes:

which can be solved to get or . We take the smaller voltage in order for the transistor to be outside the cutoff region:

Similarly, if , the equation becomes:

and we get . The voltage gain of the source follower is

To maximize the dynamic range for the input AC signal, the DC operation point in terms of the DC variables needs to be set around the middle point of the saturation region. If the AC signal around the DC operation point is small enough, the behavior of the circuit can be linearized (first term of Taylor expansion of the nonlinear relationship) to simplify the analysis.

Specifically, the nonlinear relationship between and can
be linearized around the DC operation point for small changes:

Here , called incremental transconductance, is the ratio between small change in and the small change in .