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Signals & Wireless (RF)

Decibels explained

A decibel (dB) expresses a ratio, out ÷ in, on a logarithmic scale. For power it's 10 × log₁₀(Pout ÷ Pin); for voltage it's 20 × log₁₀(Vout ÷ Vin). Gain is positive, loss is negative, and 0 dB means no change. Because they're logarithms, decibels add where ratios multiply.

Two formulas, one number

dB = 10 × log₁₀(Pout ÷ Pin) = 20 × log₁₀(Vout ÷ Vin)

Why the 20? Power goes with voltage squared, and squaring a ratio doubles its logarithm. So measured across the same resistance, the power ratio and the voltage ratio give the same decibels. You rarely need a calculator, because a few values do most of the work:

Voltage ratioPower ratioDecibels
0.10.01−20 dB
0.50.25−6 dB
0.7070.5−3 dB
110 dB
24+6 dB
10100+20 dB

Live experiment

Build a chain of gains and losses

Put up to five blocks between a 1 V transmitter and Pip: −6, −14 and −20 dB pads and a +20 dB amplifier. Watch the voltage ratios multiply along the chain while the decibels simply add.

Try it yourself: free lesson

“Count the decibels” is lesson 3 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. Lessons unlock in order, so you’ll do the 2 before it first.

Multiply ratios, add decibels

Signals pass through chains: a cable, a connector, a filter, an amplifier. Each stage has a ratio, and the ratios multiply. On a log scale, multiplying becomes adding. A half (−6 dB) followed by a fifth (−14 dB) leaves a tenth: 0.5 × 0.2 = 0.1, and −6 − 14 = −20 dB. Add a ×10 amplifier (+20 dB) and you're back to 0 dB.

That's why radio engineers count in decibels: a whole link adds up in your head. A voltage ratio of a million becomes 120 dB.

A pad: decibels from resistors

The simplest attenuator is a resistor in front of a 50 Ω load, a voltage divider. The load gets 50 ÷ (R + 50) of the voltage. 50 Ω gives a half, −6 dB; 450 Ω gives a tenth, −20 dB. Resistors don't care about frequency, so a pad gives the same loss at 1 MHz as at 100 MHz. That assumes a stiff source: real RF pads use three resistors in a T or π, matched to 50 Ω at both ends, which is what lets their decibels simply add along a chain.

Decibels against a reference

Add a reference and a decibel becomes a level. dBm is decibels compared with 1 mW, so 0 dBm is 1 mW and +30 dBm is 1 W. dBV compares with 1 V. Sound levels are decibels above the quietest sound we can hear, where each +10 dB is ten times the sound power. And a filter's cutoff is its −3 dB point: half the power, 70.7% of the voltage.

Free lesson

Now build it yourself

Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.

Try it yourself: free lesson

“Count the decibels” is lesson 3 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. Lessons unlock in order, so you’ll do the 2 before it first.

Or start from the very first lesson, no account needed

Work it out

Practice questions

Three quick questions from the lessons. Pick an answer to check it.

  1. Question 1 of 3A loss of 40 dB leaves what fraction of the voltage?
    Show the answer

    Answer: A hundredth. Every 20 dB is a factor of ten in voltage: −40 dB is ten times ten, a hundredth.

    • Why not “A tenth”? A tenth of the voltage is −20 dB. A 40 dB loss is two lots of 20 dB, so it's a tenth of a tenth.
    • Why not “Half”? Half the voltage is only about −6 dB. Every 20 dB is a factor of ten in voltage, so 40 dB is far more than half.
    • Why not “A thousandth”? A thousandth of the voltage would be −60 dB, three factors of ten. A 40 dB loss is only two.
  2. Question 2 of 3Gains and losses in decibels…
    Show the answer

    Answer: Add up along a chain. That's the whole point of the log scale.

    • Why not “Can't be combined”? Combining stages is what decibels are for: add each stage's gain or loss, minus signs included, to get the whole chain's.
    • Why not “Cancel”? A gain and a loss only cancel when they're the same size, like +20 dB then −20 dB. In general they just add up, signs included.
    • Why not “Multiply”? It's the ratios that multiply. Decibels are their logarithms, so they add: ×10 then ×10 is 20 dB + 20 dB = 40 dB.
  3. Question 3 of 3A filter's output is half its input. In decibels, that's…
    Show the answer

    Answer: −6 dB. A ratio of 0.5 is −6 dB in the dB table. −3 dB is 70.7%, half the power, and −20 dB is a tenth.

    • Why not “−20 dB”? −20 dB is a tenth of the voltage, not half. For voltage, dB = 20 × log₁₀(ratio), and 20 × log₁₀(0.5) ≈ −6 dB.
    • Why not “−3 dB”? −3 dB is about 70.7% of the voltage, which is half the power. Half the voltage is a quarter of the power, −6 dB.
Keep practising: free lesson

“Count the decibels” is lesson 3 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. Lessons unlock in order, so you’ll do the 2 before it first.

Questions people ask

What is a decibel?

A way of writing a ratio, out ÷ in, on a logarithmic scale. Positive decibels are gain, negative are loss, and 0 dB means no change. The bel is named after Alexander Graham Bell; a decibel is a tenth of one.

Why is it 20 log for voltage but 10 log for power?

Power goes with voltage squared, and the log of a square is twice the log. So 10 × log₁₀ of the power ratio and 20 × log₁₀ of the voltage ratio give the same number of decibels, as long as both are measured across the same resistance.

What does −3 dB mean?

Half the power, which is about 70.7% of the voltage. That's why a filter's cutoff is called its −3 dB point.

What is dBm?

Decibels compared with 1 milliwatt. 0 dBm is 1 mW, +30 dBm is 1 W, and −30 dBm is 1 µW. Wi-Fi signal strength is measured in dBm: around −50 dBm is strong, −80 dBm is barely there.

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