continuousWave --> Radio-->dBm to Microvolt Conversion

## Conversion of Receiver Sensitivity

from micro-volts to dBm

### by James W. Hebert

*Conversion of signal levels from dBm measurement to voltage measurement is demonstrated.*

### Receiver Sensitivity Measurement

Communication receiver sensitivity is often stated in terms of the
radio-frequency voltage level at the input necessary
to produce a particular signal-to-noise ratio of the
desired signal at the audio-frequency output. Conversion
of this voltage level to a power level in decibels referenced
to one-milliwatt (dBm) can be useful when analyzing radio circuits.
The formula for circuit analysis predicts a certain received
power level. Conversion of rated receiver sensitivity in
micro-volts to a power level in dBm will help assess the
receiver performance for a particular predicted level of received
signal. We must also assume a particular resistance in the antenna, R,
normally about 50-ohms.

Power (P), voltage (E), and resistance (R) are related by Ohm's Law:

```
(1) P = E
```^{2} / R

If we express power in terms of dB relative to one milliwatt,
we get:

```
(2) dBm = 10 LOG ( E
```^{2} / R ) + 30

Separating the first term into components we have:

```
(3) dBm = 10 LOG ( E
```^{2} ) + 10 LOG ( R^{-1} ) + 30

Further simplifying we get:

```
(4) dBm = 20 LOG E - 10 LOG (R) + 30
```

Here E is in volts. If E_{µ} is the same voltage in micro-volts, then

```
(5) E = E
```_{µ} X 10^{-6}

Substituting into (4) we get

```
(6) dBm = 20 LOG (E
```_{µ} X 10^{-6}) - 10 LOG (R) + 30

Now simplifying we get

```
(7) dBm = 20 LOG E
```_{µ} - 20 LOG (10^{-6}) - 10 LOG (R) + 30

(8) dBm = 20 LOG E_{µ} -10 LOG (R) - 90

For the common situation where R=50, this simplifies to

```
(9) dBm = 20 LOG E
```_{µ} - 107

To convert the other way if voltage is in microvolts and resistance is 50-Ohm, we find

```
(10) (dBm + 107)/20 = LOG E
```_{µ}

(11) E_{µ} = 10^{(dBm + 107)/20}

DISCLAIMER: This information is believed to be accurate but there is no
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Copyright © 2005 by James W. Hebert. Unauthorized reproduction prohibited!

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Author: James W. Hebert
This article first appeared January 8, 2005.