Lecture 10: Small Signal Model of the MOSFET Amplifier

What this lecture covers

The previous two lectures showed that a MOSFET amplifier operated in saturation gives amplification, but the response is nonlinear: a triangular input comes out looking like a distorted sinusoid. This lecture answers the question left open at the end of the last lecture — how to get a genuinely linear amplifier — by developing the small signal model. It reuses the "zoom in on a small piece of curve" idea from the earlier diode (expo dweeb) example, but works it out for the MOSFET's square-law transfer function directly, without Taylor series.

By the end, you can explain why biasing a small input signal on top of a DC operating point produces an approximately linear response, derive the small signal gain expression A = -K*RL*(VI - VT), and describe how gain, bias point, and valid input range interact when designing an amplifier.

Key ideas

• Bias point (operating point): the DC values VI, VO, IDS at which the amplifier sits before any small signal is applied; capital letters denote total or bias quantities, lowercase denote small signal (incremental) quantities.
• Small signal decomposition: total input vI = VI + vi and total output vO = VO + vo, where vi and vo are small variations around the bias point.
• Linearization by "zooming in": over a small enough region of the nonlinear transfer curve, the curve looks approximately like a straight line, so small variations map linearly.
• Small signal derivation: substituting VI + vi into vO = VS - (K/2)*RL*(vI - VT)^2, expanding, and discarding the vi^2 term (since vi is much smaller than VI - VT) leaves vo = -K*RL*(VI - VT)*vi.
• Transconductance gm: the constant K*(VI - VT), which combines with RL to give the small signal gain A = -gm*RL.
• Gain depends on the bias point: the gain A = -K*RL*(VI - VT) depends on where VI is set, not just on device and circuit constants, so changing the bias changes the amplification.
• Cascading amplifiers: connecting one amplifier's output to another's input couples their bias points, complicating design; a later lecture introduces a technique to decouple the stages.

Walkthrough

Review: saturation discipline and large signal analysis (3:17)

The lecture reviews the MOSFET amplifier pattern and the saturation discipline that let earlier lectures treat the MOSFET as a current source throughout analysis. It recaps large signal analysis as having two parts: deriving the nonlinear transfer function vO versus vI, and finding the valid input range for which the MOSFET stays in saturation. It also notes that in more advanced courses, without this discipline, you would first have to determine which region the MOSFET is in before choosing the right equations.

Setting up a bias point (16:00)

The lecture explains that to get a linear response, you operate the amplifier at a DC bias point VI, VO and superimpose a small signal vi on top of it. This bias, obtained by boosting the input with a DC offset, places the amplifier in a small region of the nonlinear transfer curve that looks approximately linear, similar to the small signal trick used earlier for the LED-like nonlinear device.

Triangular wave demo begins (18:00)

A live demo repeats the previous lecture's triangular-wave input into the amplifier. With a large-amplitude input, the output is visibly distorted (the earlier "corny" nonlinear response), motivating the need to shrink the signal amplitude while keeping the bias fixed.

Shrinking the input produces a linear response (20:00)

As the input amplitude is reduced while the DC bias stays fixed, the output triangular wave becomes visibly more linear — a small, clean triangular output tracking the small triangular input. Class discussion touches on why a small input is still useful (sensors may only produce small signals) and why cascading two smaller-gain amplifiers can be preferable to one large-gain amplifier, partly because high-gain amplifiers driven by large currents tend to be more sensitive to noise.

Total and bias variables at the output (32:00)

The lecture formalizes notation: the total output voltage is the sum of the DC bias output VO and the small signal output vo, mirroring the input decomposition. It notes in passing that cascading amplifiers couples one stage's output bias to the next stage's input bias, a complication that a later lecture resolves with a decoupling technique.

Deriving the small signal gain by expansion (38:00)

Substituting vI = VI + vi into the transfer function vO = VS - (K/2)*RL*(vI - VT)^2 and expanding the square produces a term in vi^2 alongside terms in (VI - VT) and (VI - VT)*vi. Because vi is assumed much smaller than VI - VT (the lecture gives a numeric example: 100 mV against a 5 V bias, roughly a 500x difference), the vi^2 term is dropped. Subtracting the bias-point relationship (which the DC terms alone satisfy) leaves vo = -K*RL*(VI - VT)*vi, a linear relationship. The lecture names K*(VI - VT) as gm, giving gain A = -gm*RL, and notes the same result follows more quickly by taking the derivative of the transfer function at the bias point and multiplying by vi.

Choosing gain and bias for a design (48:00)

The lecture closes by connecting the math to design decisions: gain is a key amplifier specification (illustrated with a cell-phone amplifier example needing very high amplification), and gain is proportional to both RL and VI - VT. The choice of bias point also determines how much distortion appears as the signal swings (too high or too low a bias causes clipping) and constrains the valid range of input excursions. These design tradeoffs are previewed as the subject of the next lecture.

Before you watch

• Review the MOSFET amplifier's saturation transfer function vO = VS - (K/2)*RL*(vI - VT)^2 from the two previous lectures (Lec 9 and Lec 9B).
• Recall the "Zen Method" / small signal linearization idea used earlier for the nonlinear diode-like device.
• Be comfortable with basic derivatives, since the lecture uses a slope-based shortcut to confirm the small signal result.

Check your understanding

1. Why is it valid to ignore the vi^2 term when deriving the small signal response, and what assumption makes this approximation accurate?
2. What is gm in terms of the circuit and device parameters, and how does it combine with RL to give the amplifier's gain?
3. Why does shrinking the amplitude of the triangular-wave input (while keeping the same bias) make the output look more linear?
4. According to the lecture, what problem arises when the output of one amplifier stage is connected directly to the input of another?
5. How does the choice of bias point VI affect both the gain and the risk of distortion in the amplifier's output?