# Conversion Formulae:

## 1. Relation between “Microvolts per meter” and “Watts”

Watts are the units used to describe the amount of power generated by a transmitters. Microvolts per meter (uV/m) are the units used to describe the strength of an electric field created by the operation of a transmitter.

A particular transmitter that generates a constant level of power (Watts) can produce electric fields of different strengths (uV/m) depending on, among other things, the type of transmission line and antenna connected to it. Because it is the electric field that causes interference to authorized radio communications, and since a particular electric field strength does not directly correspond to a particular level of transmitter power, the emission limits of e.g. short range devices and broadcasting transmitters are specified in field strength.

Although the precise relation between power and field strength can depend on a number of additional factors, commonly-used equation to approximate their relationship is

```	(PG)/(4 π d2) = (E2)/(120 π)                  		(1)
```

where P is transmitter power in Watts, G is the numerical gain of the transmitting antenna relative to an isotropic source, d is the distance of the measuring point from the electrical center of the antenna in meters, and E is the field strength in Volts/meter.

4 π d2 is the surface area of the sphere centered at the radiating source whose surface is d meters from the radiating source.
120 π is the characteristic impedance of free space in Ohms.

Using this equation, and assuming a unity gain antenna (G = 1) and a measurement distance of 3 meters (d = 3), a formula for determining power given field strength can be developed:

```	P = 0.3 E2						(2)
```

where P is the transmitter power (EIRP) in Watts and E is the field strength in Volts/meter.

From this explanation the following simple expression relates power flux-density in dB(W/m2) with field strength in dB(uV/m):

```	E = (S + 145.8)						(3)
```

where E is field strength in dB(uV/m) and S is power flux-density in dB(W/m2)

Note: free-space propagation is assumed

## 2. Relation between “Microvolts per meter” and “Micro-ampere per meter”

Electromagnetic fields can be sub-divided into two components: the electric field E [measured in V/m] and the magnetic field H [measured in A/m]. The E-field and the H-field are mathematically interdependent in the far-field, that means only one component has to be measured.
For example, in free space if the H-field is measured in this region, it can be used to calculate the magnitude of the E-field and power density S [W/m2]:

E = H x Z,                                                                                                               (4)

S = H2 x Z0, knowing Z0 = 377 Ω                                                                          (5)

Consequential, the relation between dB(uV/m) and dB(uA/m) is as follows:

```	x   dB(uV/m) = (x - 51.5)   dB(uA/m)			(6)
```

## 3. Relation between field-strength and e.i.r.p.

The field strength emission limits are converted to an e.i.r.p. level in dBm using the following equation:

```	E.I.R.P. = E0 + 20 log10 D – 104.8			(7)
```

taken from section 2 of the NTIA document Assessment of Compatibility between Ultrawideband Devices and selected Federal systems, NTIA Special Publication 01-43, where

E.I.R.P. = e.i.r.p. corresponding with the electric field strength E0 (in dBm)
E0 = electric field strength (in dB(uV/m))
D = reference measurement distance (in meters).

Note: free-space propagation is assumed

## 4. Relation between isotropically transmitted power (in dB(W)) and field-strength (in dB(uV/m))

The field strength for a given isotropically transmitted power are related with each other as follows:

```	E = Pt - 20 log10 D + 74.8				(8)
```

where

E = electric field strength dB(uV/m)
Pt = isotropically transmitted power (dB(W))
D = radio path length (km)

Note: free-space propagation is assumed

## 5. Relation between field-strength (in dB(uV/m)) and isotropically received power (in dB(W)

The isotropically received power and the field strength are related with each other as follows:

```	Pr = E - 20 log10 f – 167.2				(9)
```

where

Pr = isotropically received power (dB(W))
E = electric field strength (dB(uV/m))
f = frequency (GHz)

Note: free-space propagation is assumed

## 6. Relation between power flux density (in dB(W/m2)) and e.i.r.p. (in dBm)

When the power flux density, pfd, is given in dB(W/m2) and the e.i.r.p., e, in dBm, the relation between these can be derived from relations (3) and (7) as

```	e = S + 20 log10 D + 41 					(10)
```

where

e = e.i.r.p. (dBm)
S = power flux density (dB(W/m2)
D = reference measurement distance (m)

Note: free-space propagation is assumed

## 7. Estimate of free-space transmission loss (in dB) for a given isotropically transmitted power (in dB(W)) and field strength (in dB(uV/m))

The free-space basic transmission loss, the isotropically transmitted power and electric field strength are related with each other as follows:

```	Lbf = Pt - E + 20 log10 f + 167.2			(11)
```

where

Lbf = free-space basic transmission loss (dB)
Pt = isotropically transmitted power (dB(W))
f = frequency (GHz)

Note: free-space propagation is assumed