MOSFET biasing

The DC network around a MOSFET has one job: set a stable, well-defined Operating point (a specific $I_D$ and $V_{GS}$, device in saturation) so the small-signal amplifier on top of it works as designed. All small-signal analysis assumes a known bias point. Biasing is what makes that assumption true in a real, mass-produced circuit.

Why fixing $V_{GS}$ fails

The naive idea: apply a fixed gate-source voltage and read off the drain current from the MOSFET square-law $I_D=\frac{1}{2}{k}_n(V_{GS}-V_t{)}^2$. It fails badly in practice. Both the Threshold voltage $V_t$ and the MOSFET transconductance parameter $k_n$ vary widely between nominally identical transistors, with temperature, and across the manufacturing process. The current goes as the square of $(V_{GS}-V_t)$, so a small spread in $V_t$ or $k_n$ produces a huge spread in $I_D$. A fixed $V_{GS}$ giving $I_D=1\text{mA}$ in one transistor can give $100\mu \text{A}$ in another or $10\text{mA}$ in a third. The operating point is uncontrolled, and the amplifier built on it is worthless.

Fixing $V_{GS}$: the simplest and worst scheme. $V_t,k_n$ spread maps straight to $I_D$ spread.

The fix: source resistor + fixed gate reference

Insert a resistor $R_S$ between source and ground and hold the gate at a fixed reference $V_G$. The gate draws no current (Gate oxide), so KVL around gate → source → ground gives

$V_G=V_{GS}+I_D{R}_S$

Read it as feedback: if $I_D$ tries to rise (because $V_t$ happened to be low, say), the drop $I_D{R}_S$ across $R_S$ grows, which forces $V_{GS}$ down, which pulls $I_D$ back down. The source resistor closes a Negative feedback loop on the bias. If $V_G\gg V_{GS}$ the $V_{GS}$ term is a small part of the equation and

$I_D\approx \frac{V_G}{R_S}$

so the operating point is set almost entirely by stable external components ($V_G$ and $R_S$), with only weak dependence on the transistor’s wandering parameters. That’s the whole strategy: use negative feedback to make $I_D$ depend on resistors, not on $V_t$ and $k_n$.

Fix $V_G$, add $R_S$. $V_G=V_{GS}+I_D{R}_S$ stabilises the operating point.

Practical schemes

The fixed $V_G$ usually comes from a Voltage-divider bias (for a MOSFET the divider resistors can be enormous, since no gate current is drawn). A single-resistor alternative is Drain-to-gate feedback bias, which also guarantees saturation. Both rely on the same negative-feedback principle. The full board-level amplifier that puts these together with coupling and bypass capacitors is the Discrete-circuit MOSFET amplifier.