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 — i.e. it connects 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; see Amplifier frequency response.

The multiplication, derived

[Background from general knowledge, not the source PDF: the PDF states the result — $C_{\mu }$ multiplied by the gain — but does not derive it. The derivation below is standard.]

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