S4-1.3 Multistage & High-Frequency Amplifiers
Standard analog-electronics theory — written August 2026
What this is and why it exists
One stage rarely delivers enough gain, so real amplifiers are chains. Chaining multiplies the gains — and multiplies the frequency limits into each other, which is the actual subject of this unit. It also retires the h-parameter model: at high frequency the transistor's internal capacitances dominate, and the hybrid-pi model exists to say so honestly.
The vocabulary
- RC coupling — stages joined through a capacitor: cheap, blocks DC, loses the very low frequencies.
- Transformer coupling — stages joined magnetically: matches impedances well, bulky and expensive, poor at low frequency.
- Direct coupling — stages wired straight through: amplifies down to DC, but bias points interlock and drift compounds.
- Darlington pair — two transistors with the first's emitter feeding the second's base; behaves like one transistor with beta roughly the product of the two.
- Hybrid-pi model — the high-frequency small-signal model: transconductance, base resistance, and the capacitances the h-parameters pretend do not exist.
- f_T — the frequency at which the transistor's short-circuit current gain falls to one; the device's intrinsic speed limit.
The mental model
Stack panes of glass. Each pane (stage) is slightly tinted — it passes the middle frequencies well and dims the edges. Two panes stacked give the product of the tints: the middle gets darker slower than the edges, so the band that passes cleanly is NARROWER than either pane alone. That is the multistage law: overall gain multiplies, overall bandwidth shrinks. A two-stage amplifier built from identical stages has less bandwidth than one of its stages — the gain-bandwidth budget is being spent.
Why do the edges dim at all? At the high end, the hybrid-pi model answers: the base-collector capacitance, Miller-multiplied by the stage gain, shunts the input signal away as frequency rises. Following that mechanism to its end gives f_T, where the device's own current gain reaches one and amplification is over regardless of circuit cleverness. Datasheet f_T is the ceiling; Miller effect decides how far below the ceiling your circuit actually operates.
The named pairings are tools with characters: CE-CE for raw gain, CE-CB (the cascode) because the CB stage shields the CE stage from Miller multiplication and keeps bandwidth, and the Darlington (CC-CC) for enormous current gain and a gentle load on the source, paying for it with two base-emitter drops and slower switching.
What you should now be able to explain or do
Pick a coupling scheme for a stated job and name its cost. Compute overall gain and estimate overall bandwidth for cascaded identical stages. Explain what the cascode buys and what mechanism it defeats. Read f_T from a datasheet and say what it does and does not promise.
Check yourself
Two identical stages are cascaded. What happens to gain and to bandwidth?
Gain multiplies — twenty times twenty is four hundred. Bandwidth shrinks below a single stage's, because both stages' roll-offs act on the same signal.
Why does the cascode (CE-CB) keep more bandwidth than a plain CE stage of the same gain?
The CB stage holds the CE stage's collector at nearly constant voltage, so the CE stage sees almost no voltage gain across its base-collector capacitance — the Miller multiplication never happens.
What is f_T, and what happens beyond it?
The frequency where short-circuit current gain falls to one. Beyond it the transistor no longer amplifies current — no circuit arrangement recovers gain the device itself has run out of.
What does a Darlington pair trade for its huge current gain?
Two base-emitter drops of headroom, higher saturation voltage, and slower response — the second transistor cannot turn off quickly because nothing pulls its base down hard.
Which coupling scheme amplifies a signal that changes over seconds rather than milliseconds, and what does it cost?
Direct coupling — capacitors and transformers both block slow changes. The cost is interlocked bias design and drift that amplifies along with the signal.
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