Bypass capacitor

A bypass capacitor is a capacitor placed across the source resistor $R_S$ of a MOSFET stage (or the emitter resistor $R_E$ of a BJT stage) to short it out for AC signals while leaving it in the circuit for DC. It resolves the conflict between wanting a source/emitter resistor for bias stability and not wanting it because it kills gain.

The conflict it solves

A source resistor is essential for DC: it provides the Negative feedback that stabilises the Operating point (see MOSFET biasing), making $I_D$ depend on resistors instead of the spread-prone $V_t$ and $k_n$. But the same resistor, present for AC, causes Source degeneration and divides the gain down by $(1+g_m{R}_S)$. You want it for DC and don’t want it for AC. A bypass capacitor lets you have both:

• At DC ($\omega \to 0$): the capacitor’s impedance $Z_C=1/(j\omega C)\to \infty$ — it is an open circuit. $R_S$ is fully present, so the bias stabilisation works normally.
• At signal frequencies: $C$ is chosen large enough that $|Z_C|=1/(\omega C)\ll R_S$ — it is effectively a short. $R_S$ is shorted out for AC, so the source sits at AC ground and the full undegenerated gain $-g_m{R}_D$ is restored.

So DC sees $R_S$; AC sees a wire. Best of both.

The low-frequency corner it creates

The capacitor is not a perfect short at all frequencies — it stops being a short as frequency drops. The bypass capacitor and the resistance it sees together form an RC highpass response with a corner frequency below which the gain rolls off back toward the degenerated value. Because the resistance looking into the source is low (roughly $1/g_m$ in parallel with $R_S$), this corner is often the highest of the low-frequency corners in the amplifier — i.e. the dominant one that sets where the low end of the passband begins. [Background from general knowledge, not the source PDF: the explicit identification of the bypass corner as the dominant low-frequency pole; the PDF states each cap forms an RC highpass and sets a low-frequency corner without ranking them.]

This is why the bypass capacitor must be sized generously: to pass audio down to 20 Hz, its corner has to sit well below 20 Hz. Contrast it with a Coupling capacitor, which sits in series in the signal path (blocking DC between stages) rather than across a bias resistor — different placement, different job, but both contribute corners to the Amplifier frequency response.