Miller effect

The Miller effect is the apparent multiplication of a feedback capacitance bridging the input and output of an inverting amplifier: a capacitor $C_f$ from input to output looks, from the input, like a much larger capacitor of value $C_f(1+|A_v|)$. In a Common-emitter amplifier (or Common-source amplifier) this turns the small base–collector capacitance into the dominant high-frequency limit.

BJT internal capacitances

Every real BJT has two parasitic junction capacitances that the ideal small-signal model ignores but that set the high-frequency response:

• $C_{\pi }$ — the base–emitter capacitance. The EBJ is forward-biased in active mode, so $C_{\pi }$ is dominated by the diffusion (charge-storage) capacitance of injected minority carriers, plus a junction term. It sits across $r_{\pi }$ in the BJT hybrid-pi model.
• $C_{\mu }$ — the base–collector capacitance. The CBJ is reverse-biased, so $C_{\mu }$ is a (smaller) depletion capacitance. It bridges base and collector, connecting the amplifier’s input to its output. That bridging is what makes it dangerous.

The MOSFET analogues are $C_{gs}$ and the bridging $C_{gd}$. For high-frequency analysis these capacitances are added to the small-signal model.

The multiplication, derived

Take an inverting amplifier with voltage gain $A_v$ (so $v_{out}=A_v{v}_{in}$, with $A_v$ negative and large). Put $C_f$ between input and output. The current the source must push into that capacitor at the input node is set by the voltage across it, $v_{in}-v_{out}$:

$i_{in}=j\omega C_f(v_{in}-v_{out})=j\omega C_f(v_{in}-A_v{v}_{in})=j\omega C_f(1-A_v)v_{in}$

From the input’s point of view this is the current drawn by an equivalent capacitor to ground of value

$C_{\text{Miller}}=C_f(1-A_v)=C_f(1+|A_v|)(\text{since }{A}_v<0)$

The far end of $C_f$ swings by $|A_v|$ times as much as the input, in the opposite direction, so charging it takes roughly $(1+|A_v|)$ times as much current as charging $C_f$ to ground would. The input sees a hugely magnified capacitor. With a CE gain of −100, a 1 pF $C_{\mu }$ presents ≈101 pF at the input.

This lumped equivalent is the Miller approximation: it assumes $C_f$ is the only input–output feedback path and uses the low-frequency gain $A_v$ as though it were constant. The gain itself rolls off with frequency, so the equivalent capacitance is accurate only at low and mid frequencies, enough to locate the dominant high-frequency pole but not a complete high-frequency model.

Consequences

That magnified input capacitance forms a low-pass filter with the source/input resistance, creating the dominant high-frequency pole that rolls the gain off. This is why:

• CE/CS amplifiers have limited bandwidth: their high gain multiplies $C_{\mu }$/$C_{gd}$ heavily.
• CB/CG amplifiers are fast. The base/gate is grounded, so $C_{\mu }$ is not bridged between a swinging input and output and is not Miller-multiplied. No dominant Miller pole.
• The Cascode amplifier uses this: the common-gate/base top transistor keeps the swing at the input transistor’s output node small, so the input transistor’s feedback capacitance is barely multiplied. High gain and wide bandwidth.

Emitter degeneration reduces the effective gain seen by $C_{\mu }$ at signal frequencies, part of why Emitter degeneration improves high-frequency response.