# WirelessMathematics

## Interesting Mathematics

**Decibels (dBm) and milli-Watts (mW) Relationship:**

dBm = 10 log (mW)

**Typical loss per connector at 2.4Ghz:**

0.25 - 0.5 dBm

**Antenna Gain:**

Gain(dBi) = 10 log (n * ( (4*PI)/lambda^2 ) * A)

Where, A = Area of antenna

n = ?

**Approximate Antenna Gain (Yagi and Omni antennas):**

Gain(dBi) = 10 log (#elements)

**Wave Length:**

lambda(mm) = 300 / freq(Ghz)

(eg for 2.4Ghz, lambda = 123mm)

**Wave Length (in Coax):**

lambda(mm) = 300 / velocity(%) / freq(Ghz)

(eg LMR400 has velocity factor of 0.85, lambda = 123 * 0.85 = 104.5mm)

Note: For brass tubing (4/16") use a velocity factor of 0.95

**Path Loss, Attenuation Due to Air (signal loss in air):**

Loss (dB) = 10 log ( ( (4*PI*d) / lambda ) ^ 2 )

Where, d = distance (in meters)

lambda = wave length (in meters)

or for 2.4Ghz use

Loss (dB) = 40 + 20 log (d)

Note: Fade margin, usually an additional loss of 10dB is added to take into account things like frensal zones, multipath and other real world stuff

**EIRP (Effective Isotropic Radiated Power) Calculating:**

(Transmitter Gain:)

EIRP(dB) = Card Output(dBm) + Antenna Gain(dBi)

- Cable Loss(dBm) - Connector Losses(dBm)

Note: EIRP for 2.4 to 2.462Ghz ISM band must be <= 4W

**Receiver Gain:**

Gr = Antenna Gain(dBi) - Connector Losses(dBm) - Cable Loss(dBm)

**Link Budget:**

Link Budget = Path Loss + Rx Sensitivity + Gt + Gr

Adjusted Link Budget = Link Budget - 10dB (ie fade margin)

Note: Use -ve of Rx Sensitivity (ie value is positive, --84 => 84)

**Over-the-Air Data Rate:**

RF Data Rate = (Dpl + Dao + Dro) * (1 + rt) / time

Where, Dpl = payload data in bits

Dao = application overhead in bits/payload transmission

Dro = radio data overhead in bits

rt = retry percentage

**Receive Sensitivity (just read data sheet):**

Rx Sensitivity = Thermal Noise Floor + System Noise + 10 log (SymbolRate) + SNR

See also: PowerAndGainToDecibels

Version 1 (current) modified Mon, 26 Jul 2021 12:49:29 +0000

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