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A stadium microphone produces an electrical signal of barely 2 millivolts, but the loudspeakers demand over 20 volts to shake the room. How do audio engineers boost that signal by a factor of 10,000 without overloading a single circuit?
Instead of demanding impossible performance from one amplifier, engineers chain several simpler amplifiers together in a cascaded multi-stage amplifier system.

Cascading in Linear Gain

In a cascaded system, the output of the first stage feeds directly into the input of the second stage. Voltage gain (AvA_v) is the ratio of output voltage to input voltage: Av=VoutVinA_v = \frac{V_{\text{out}}}{V_{\text{in}}}.
Because each stage multiplies whatever voltage enters it, the overall voltage gain (Av,totalA_{v,\text{total}}) is the product of every individual stage's linear gain:
Av,total=Av1×Av2×Av3×⋯×AvnA_{v,\text{total}} = A_{v1} \times A_{v2} \times A_{v3} \times \dots \times A_{vn}

Why Linear Gains Multiply

For a two-stage amplifier, the first stage gives Vout1=Av1×Vin1V_{\text{out1}} = A_{v1} \times V_{\text{in1}}. Since the second stage receives this output, its input is Vin2=Vout1V_{\text{in2}} = V_{\text{out1}}.
The final output is therefore Vout2=Av2×Vin2=Av2×(Av1×Vin1)V_{\text{out2}} = A_{v2} \times V_{\text{in2}} = A_{v2} \times (A_{v1} \times V_{\text{in1}}). Dividing both sides by Vin1V_{\text{in1}} proves that the total gain Vout2Vin1\frac{V_{\text{out2}}}{V_{\text{in1}}} equals Av1×Av2A_{v1} \times A_{v2}.
Multiplying large numbers like 125×40×250125 \times 40 \times 250 in your head gets messy fast. Is there a way to turn all this multiplication into simple addition?

Cascading with Decibels

The decibel voltage gain (Av(dB)A_{v(\text{dB})}) converts linear gain into a logarithmic scale using the formula Av(dB)=20log⁡10(Av)A_{v(\text{dB})} = 20 \log_{10}(A_v).