The field-effect transistor as an amplifier: small and large signal
The FET parabola distorts only in the second harmonic, and by an exact amount: for the same input, almost eight times less than a bipolar. The price is amplifying less per milliampere.
01A square law instead of an exponential
The field-effect transistor —the MOSFET and the JFET— amplifies with the same idea as the bipolar: an input voltage controls an output current. The difference lies in the shape of that relationship. In saturation, the drain current follows a parabola of the overdrive voltage \( V_{ov} = V_{GS} - V_{th} \) (in the JFET, \( V_{GS} - V_P \)):
The comparison with the bipolar at equal current is the first design lesson: \( g_m / I_D = 2 / V_{ov} \) in the FET against \( 1 / V_T \) in the bipolar. With \( V_{ov} = 0.2 \) V, the FET gives 10 S/A and the bipolar 38.7: it amplifies almost four times less per milliampere. In exchange, its gate draws no current and, as you will see below, it is much more linear.
02The small-signal configurations
| Configuration | Voltage gain | Input resistance | Output resistance |
|---|---|---|---|
| Common source | \( -\dfrac{g_m R_D}{1 + g_m R_S} \) | Practically infinite (that of the biasing circuit) | \( \approx R_D \) |
| Common drain (source follower) | \( \dfrac{g_m R_S}{1 + g_m R_S} < 1 \) | Practically infinite | \( \approx 1/g_m \) |
| Common gate | \( g_m R_D \) | \( \approx 1/g_m \) | \( \approx R_D \) |
The source follower is less ideal than the emitter follower: since \( g_m \) is smaller, its gain is farther from 1 and its output resistance is higher. In an integrated circuit the body effect adds to this: the voltage between source and substrate raises the threshold and reduces the gain a little more.
03Large signal: the distortion of the parabola
With \( v_{gs} = \hat V \sin \omega t \) added to the bias, the current is \( \frac{k}{2} (V_{ov} + \hat V \sin \omega t)^2 \). Expanding the square gives a term in \( \sin^2 \omega t = (1 - \cos 2\omega t)/2 \): a second harmonic, and no other. The ratio is exact, not an approximation:
The comparison with the bipolar, \( HD_2 \approx \hat V / 4 V_T \), shows the other side of transconductance: with \( V_{ov} = 0.2 \) V the FET distorts almost eight times less for the same input. More \( V_{ov} \) means more linearity and less gain; less \( V_{ov} \), the opposite. Choosing \( V_{ov} \) is the central decision in FET design.
With larger signals the limits appear: if \( \hat V \) exceeds \( V_{ov} \), the transistor cuts off during part of the cycle; if the drain voltage falls too low, it enters the ohmic region. In both cases the output is clipped and all the harmonics appear.
A MOSFET in common source with a 12 V supply and a 1 V threshold. The current is calculated with the full model (saturation, ohmic region and cutoff) at each instant, without linearizing, and the harmonics of the output are analyzed.
04JFET, MOS and short channel
- The JFET follows the same parabola with \( V_P \) instead of \( V_{th} \) and has the advantage of low noise with high-impedance sources; its problem is parameter spread.
- The short-channel MOS departs from the parabola because of velocity saturation: its current grows almost linearly with \( V_{ov} \), which makes it more linear but with less \( g_m \).
- Below threshold, the MOS becomes exponential again, like a bipolar, with maximum \( g_m/I_D \): this is the regime of ultra-low-power circuits.
05Where it appears in electronics
| Case | Why a FET |
|---|---|
| Input of a high-impedance instrument | The gate does not load the source |
| Low-noise radio-frequency amplifier | GaAs and GaN transistors with very low noise at high frequencies |
| Analog CMOS integrated circuit | It is the only transistor available, and the design is done by choosing \( V_{ov} \) or \( g_m/I_D \) |
| Tube and FET audio amplifiers | Distortion that is mostly second harmonic has a characteristic sound |
06In the lab
Bias a 2N7000 at 2 mA with \( R_D = 2.2 \) kΩ, measure \( V_{GS} \) and estimate \( V_{ov} \). Measure the gain with 10 mV at the input and compare it with \( 2 I_D R_D / V_{ov} \).
With the oscilloscope's FFT, measure the second and third harmonics for three amplitudes. Verify that the second grows in proportion to the amplitude and that the third barely appears until clipping.
Build a common emitter with the same current and the same load resistance, and compare gain and distortion for the same input.
07Common mistakes
- Expecting the gain of the bipolar from the FET. At equal current, \( g_m \) is several times smaller.
- Confusing the saturation of the FET with that of the bipolar. In the FET it is the amplifying region.
- Forgetting that the parabola only holds in saturation. If the drain voltage falls, the transistor enters the ohmic region and clips.
- Applying \( \hat V / 4V_T \) to a FET. Its distortion scale is \( V_{ov} \), not \( V_T \).
08Self-assessment
With \( I_D = 1 \) mA and \( V_{ov} = 0.25 \) V, what is \( g_m \)?
\( 2 \times 1 / 0.25 = 8 \) mS.
What second harmonic does a common source with \( V_{ov} = 0.4 \) V and a 40 mV input produce?
\( 40 / (4 \times 400) = 2.5 \) %.
Why does a square law not generate a third harmonic?
Because squaring a sum produces only a term in \( \sin^2 \), which has a DC component and a second harmonic.
What gain does a source follower with \( g_m R_S = 4 \) have?
\( 4/5 = 0.8 \).
09Further reading
- Behzad Razavi. Design of Analog CMOS Integrated Circuits. 2nd ed., McGraw-Hill, 2017. The basic MOS configurations, with a physical intuition that is hard to find in any other text.
- Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. 7th ed., Oxford University Press, 2014. The MOS as an amplifier in parallel with the bipolar, for comparing them.
- Willy M. C. Sansen. Analog Design Essentials. Springer, 2006. The distortion of MOS stages and design with \( g_m/I_D \), in the form of annotated slides.