Decibel

The decibel (dB) is a logarithmic unit for expressing ratios. For a unitless ratio $x$:

$x(\text{dB})=10{\log }_{10}x.$

So $x=100\to 20\text{dB}$, $x=1000\to 30\text{dB}$, $x=0.5\to -3\text{dB}$.

The factor of $10$ comes from the prefix “deci-”: one bel is ${\log }_{10}x$, and one decibel is $1/10$ of a bel, so $1\text{bel}=10\text{dB}$. The bel is named after Alexander Graham Bell.

Power ratios vs amplitude ratios

The formula $10{\log }_{10}$ is for power ratios. For amplitude ratios (voltage, current, or anything else whose square is power), use $20{\log }_{10}$:

$\text{Amplitude ratio in dB}=20{\log }_{10}\left(\frac{V_2}{V_1}\right).$

Because $P\propto V^2$, a power ratio $P_2/P_1$ corresponds to an amplitude ratio $(V_2/V_1{)}^2$, and $10{\log }_{10}((V_2/V_1{)}^2)=20{\log }_{10}(V_2/V_1)$.

Rule of thumb: $10\log$ for power, $20\log$ for amplitude. Confusing these is the single most common error in dB calculations.

In a Bode plot, $|H(j\omega )|$ is an amplitude ratio (voltage out over voltage in), so Bode magnitudes are plotted as $20{\log }_{10}|H(j\omega )|$, not $10{\log }_{10}$.

What can and cannot be added in dB

This is where dB trips people up.

Multiplications become additions in dB, from the log property $\log (ab)=\log a+\log b$.

• Cascaded gain: $G=G_1\cdot G_2$ (linear) $⟹G(\text{dB})=G_1(\text{dB})+G_2(\text{dB})$. Cascade gains add in dB.
• Power output of an amplifier: $P_2=P_1\cdot G$ (linear) $⟹P_2(\text{dBW})=P_1(\text{dBW})+G(\text{dB})$.

Additions of linear quantities do NOT work cleanly in dB.

• Power combining: $P_3=P_1+P_2$ (linear sum). You cannot write $P_3(\text{dBW})=P_1(\text{dBW})+P_2(\text{dBW})$; that would mean multiplying the linear powers. To combine powers in dB, convert each to linear, add, then convert back.

So: never add dBW values to combine powers, and never multiply dB values to combine gains.

Useful approximations

A few values worth knowing by heart:

• $3\text{dB}\approx 2\times$ in power, $\sqrt{2}\times$ in amplitude (half-power = $-3\text{dB}$).
• $6\text{dB}\approx 4\times$ in power, $2\times$ in amplitude.
• $10\text{dB}=10\times$ in power, $\sqrt{10}\approx 3.16\times$ in amplitude.
• $20\text{dB}=100\times$ in power, $10\times$ in amplitude.
• $30\text{dB}=1000\times$ in power, $\sqrt{1000}\approx 31.6\times$ in amplitude.

So a $40\text{dB}$ amplifier multiplies the signal voltage by 100; a $-60\text{dB}$ filter attenuates power by a factor of $10^6$.

Negative dB

Negative dB means the ratio is less than 1, i.e. attenuation. $-3\text{dB}$ = half power; $-20\text{dB}$ = 1/100 power; $-60\text{dB}$ = 1/1,000,000 power.

Why −3 dB defines a cutoff

Filters and amplifiers quote their band edge at "$-3\text{dB}$" straight from this scale. A power ratio of one-half is $10{\log }_{10}(1/2)\approx -3\text{dB}$, which in amplitude is a factor $1/\sqrt{2}\approx 0.707$ ($20{\log }_{10}(0.707)\approx -3\text{dB}$). So the half-power point and the $-3\text{dB}$ point are the same frequency, and that is what the Cutoff frequency of an RC filter or the bandwidth edge of an amplifier is defined to be.

See also

dBm and dBW for absolute power levels referenced to a fixed quantity.