Multistage Amplifiers: Cascading Gain Stages For students.

A multistage amplifier chains amplifier stages so the output of one feeds the next. A single transistor stage gives limited gain, but cascading multiplies gains: stages of 50 and 40 give 2,000 total. Between stages, a coupling network passes the AC signal while keeping each stage's DC bias undisturbed.

Real signals are tiny: a microphone makes millivolts, an antenna far less, yet speakers need volts. Bridging microvolts to volts takes a gain of thousands, which only cascades deliver. Nearly every audio system, radio and instrument is a chain of stages sharing the work.

Key Concepts

• Voltage gain: Av = Vout/Vin. Cascaded gains multiply: Av(total) = Av1 x Av2 x … x Avn; the sign shows overall phase.
• Decibels: 20 x log10(|Av|). In dB, cascaded gains add, keeping numbers manageable.
• RC coupling: A capacitor blocks DC and passes AC; cheap, standard for audio, but sets a low-frequency cutoff.
• Direct coupling: No capacitor; DC passes too. Used inside op-amps, but biases must be compatible.
• Transformer coupling: Older RF technique for impedance matching and isolation, now rare.
• Loading: Stage 2's input resistance loads stage 1: effective gain = -gm x (Rc1 in parallel with Rin2), less than -gm x Rc1 alone.
• Bandwidth shrinkage: Cascading multiplies each stage's high-frequency roll-off, so overall bandwidth narrows.

Formulas

• Total gain: Av(total) = Av1 x Av2 x … x Avn; in dB the gains add.
• Loaded gain: Av1(effective) = -gm x (Rc1 || Rin2). Ignoring loading overestimates total gain.
• Coupling cutoff: f = 1/(2 x pi x R x C); size C so this sits below your lowest frequency.
• Limits: Midband, small-signal, active-mode assumptions. Near band edges, phase shifts accumulate and simple multiplication breaks down.

Worked Example

Two stages: Av1 = -50, Av2 = -40 (each inverts phase). Step 1: total = (-50) x (-40) = 2,000, back in phase. Step 2: 2 mV input gives 2 mV x 2,000 = 4 V out. Step 3: in dB, 20log(50) = 34 dB and 20log(40) = 32 dB, totaling 66 dB, matching 20log(2000). Two modest stages turn a whisper into full-scale audio.

Common Misconceptions

• "Gains add." Voltage gains multiply; only decibel values add. 50 + 40 = 90 wildly understates 2,000.
• "More stages are always better." Each stage adds loading, noise and narrower bandwidth. Use as few as meet the spec.
• "Any coupling capacitor works." With surrounding resistances it forms a high-pass filter; too small a value kills bass. Keep f = 1/(2 x pi x R x C) below your lowest signal.

Real-World Applications

• Audio: Preamps boost millivolts; power stages drive speakers. Stereos and PAs are textbook cascades.
• Radio: RF, IF and audio stages each add gain at their own frequency.
• Instruments: Oscilloscope front ends amplify microvolt sensors through low-noise stages.
• Inside op-amps: Input, gain and output stages direct-coupled in one IC.

Multistage amplifiers win by cascading: gains multiply (add in dB), so 50 and 40 give 2,000, turning 2 mV into 4 V. RC capacitors couple audio stages while protecting each bias; direct coupling serves inside ICs. Always subtract loading loss, the next stage's input resistance steals gain, and remember bandwidth shrinks as stages pile up.