# Half-power point

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The half-power point of an electronic amplifier stage is that frequency at which the output power has dropped to half of its mid-band value. That is a level of -3 dB. The half-power point is a commonly used specific definition of cutoff frequency, although not the only one.

This occurs when the output voltage has dropped by 1/√2 or 0.707 of the maximum output voltage (exact: ${\displaystyle 20\log _{10}\left({\tfrac {1}{\sqrt {2}}}\right)\approx -3.0103\,\mathrm {dB} }$) and the power has dropped by half (1/2 or 0.5) (exact: ${\displaystyle 10\log _{10}\left({\tfrac {1}{2}}\right)\approx -3.0103\,\mathrm {dB} }$). A bandpass amplifier will have two half-power points, whilst a low pass amplifier will have only one. A high pass amplifier stage will have only the lower half-power point.

The bandwidth of an amplifier is usually defined as the difference between the lower and upper half-power points. This is therefore also known as the 3 dB bandwidth.

## 3 dB

The half-power point is approximately 3 dB down, because ${\displaystyle \log _{10}2=.3010...\approx .3,}$ so ${\displaystyle 10\log _{10}2=3.010...\approx 3;}$ the decibel measure of a ratio r is defined as ${\displaystyle 10\log _{10}r.}$

Using 3 dB rather than the correct value of 3.010... yields a power factor of ${\displaystyle G=10^{\frac {3}{10}}\times 1\ =1.99526...\approx 2\,}$ which differs from a factor of 2 by about 0.24%. As logarithmic errors add, using 6 dB to approximate a factor of 4 difference yields an error of about 0.48%, and so forth.

This is the same mathematical coincidence as ${\displaystyle 2^{10}=1,024\approx 1,000=10^{3}}$, which is the source of the ambiguity of the prefixes kilo-, mega-, etc. in a computing context. Taking the logarithm of both sides of the equation ${\displaystyle 2^{10}\approx 10^{3}}$ yields ${\displaystyle 10\log _{10}2\approx 3}$.

## Antennas

The half-power point or 3 dB point of an antenna beam is the angle off boresight at which the antenna gain has fallen 3 dB below the peak.