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Half power point

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Title: Half power point  
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Subject: Full width at half maximum, Amplifier figures of merit, Decibel, Antennas, Feed horn
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Half power point

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: 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: 10\log_{10}\left(\tfrac{1}{2}\right) \approx -3.0103\, \mathrm{dB}). A bandpass amplifier will have 2 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 because \log_{10} 2 = .3010... \approx .3, so 10 \log_{10} 2 = 3.010... \approx 3; the decibel measure of a ratio r is defined as 10 \log_{10} r.

Using 3 dB rather than the correct value of 3.010... yields a power factor of 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 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 2^{10} \approx 10^3 yields 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.

See also

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