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# 3dB-point

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 Title: 3dB-point Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 3dB-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. Other names include the cutoff frequency.

This occurs when the output voltage has dropped by 1/√2 or 0.707 (exact: $20\log_\left\{10\right\}\left\left(\tfrac\left\{1\right\}\left\{\sqrt\left\{2\right\}\right\}\right\right) \approx -3.0103\, \mathrm\left\{dB\right\}$) and the power has dropped by half (1/2 or 0.5) (exact: $10\log_\left\{10\right\}\left\left(\tfrac\left\{1\right\}\left\{2\right\}\right\right) \approx -3.0103\, \mathrm\left\{dB\right\}$). 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 $\log_\left\{10\right\} 2 = .3010... \approx .3,$ so $10 \log_\left\{10\right\} 2 = 3.010... \approx 3;$ the decibel measure of a ratio r is defined as $10 \log_\left\{10\right\} r.$

This mathematical coincidence also means that $2^\left\{10\right\} \approx 10^3$ (precisely, $2^\left\{10\right\} = 1,024 \approx 1,000 = 10^3;$ taking the logarithm of this equation yields $10 \log_\left\{10\right\} 2 \approx 3$). This approximation is notably used in loose usage of the prefix kilo- (and associated mega-, giga-, tera-, etc.) in computing to refer either to $10^3$ or $2^\left\{10\right\}$ – see kilobyte.

Using 3 dB rather than the correct value of 3.010... yields a power factor of $G = 10^\frac\left\{3\right\}\left\{10\right\} \times 1\ = 1.99526... \approx 2 \,$ 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.

## 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.