##### To calculate free-space loss,

the equation 36.6 + 20LogD + 20LogF,

where D is the distance in miles and F is the frequency in MHz is commonly referenced.

You will note that if you subtract the free-space loss (in dB) from the transmitter ERP (in dBm) to get the receiver input power (in dBm) that when you use TAP™ to compute free space field, then use the Required Field utility to convert the field value (in dBu) to receiver input power (in dBm), the TAP™ value is about 4.3dB higher. What’s wrong?

Nothing is wrong. This is a case of both methods yielding the right answer, but expressed differently.

The equation referenced above is commonly used for computing free-space loss between isotropic antennas (see, for example, Engineering Considerations for Microwave Communications Systems, GTE Lenkurt, 1970, page 35; or Microwave System Path Design Considerations, TeleSciences, 1991, page 7, or Tech Note 101,equation 2.16).

## TAP™ programs

The TAP™ programs for field calculations (Broadcast, Carey, Bullington, Okumura, Longley-Rice) use antenna gain values referenced to a dipole (dBd), as is common in broadcast, land-mobile, and other industries. TAP™ field strength calculations involving Effective Radiated Power (ERP) generally assume the ERP is computed based on transmitter antenna gain above a dipole (dBd). (Note that microwave calculations are based on EIRP, or Effective Isotropic Radiated Power, which is more commonly used in microwave engineering.)

On the receiver end, the TAP™ conversion from received field strength (in dBu) to receiver input power (in dBm) allows you to specify the receiver gain referenced either to a dipole or to an isotropic radiator (dBi). The difference of gain values between a dipole reference and an isotropic reference is 2.15dB. (See other FAQs for a detailed discussion of antenna gain units, as well as the equations used for computing free-space field, and converting received field to received power.)

If you used the transmitter ERP value based on dBd gain (as assumed in TAP™), and if the field-to-power units conversion used a receive antenna gain of 0dBd (the default value, which is 2.15dBi), the net result would be 2.15dB “extra” gain on each end (“extra” meaning gain not considered by the free-space loss equation referenced above). Therefore, the net result computed by TAP™ will be 4.3dB higher.

Both methods are correct, but you must recognize the different assumptions for each. Since the free-space loss equation assumes isotropic antennas (0dBi), and TAP™ assumes gain values in dBd, you can adjust either calculation to correspond to the units of the other.

To adjust the values entered into TAP™ to correspond to the free-space loss equation, reduce your computed transmitter ERP entered into TAP™ by 2.15dB (so it is based on the gain of an isotropic antenna) and specify 0dBi in the TAP™ units conversion program (which will show as -2.15dBd). You will then get the same results as the free-space loss equation.

### The calculation of ERP based on EIRP is as follows:

ERP is the Transmitter Power Output (TPO) plus the gain: ERP = TPO + dBd

Since dBd = dBi - 2.15, therefore: ERP = TPO + (dBi - 2.15)

EIRP is the TPO plus the isotropic gain: EIRP = TPO + dBi

Substituting: ERP = EIRP - 2.15

As an alternative, to adjust the values in the equation to correspond to TAP™ free-space field calculations, subtract 4.3dB from the loss computed with the equation to account for the dBd values on both ends as assumed in TAP™ for Area Coverage, UHF/VHF Link Budgets, and the Single Point Field program. This would make the free-space loss equation (based on non-isotropic antennas)

32.3 + 20LogD + 20LogF.

The ERP minus this computed loss value will yield a received power value that agrees with the TAP™ calculations that assume dBd gain values.

## Relationship between computed free-space loss & computed free-space field

To demonstrate the relationship between computed free-space loss (based on isotropic antennas) and computed free-space field (based on dBd gain values), consider these equations:

For field strength in dBu:

(60-6) E(dbuV/m) = 106.92 + ERP(dBk) - 20LogD(km)

For received power:

(60-10) P(dBm) = E(dbuV/m) + Gr(dBi) - 20LogF(MHz) - 77.2

Substituting the received field:

P(dBm) = 106.92 + ERP(dBk) - 20LogD(km) + Gr(dBi) - 20LogF(MHz) - 77.2

For a 0dBi receiver antenna (Gr):

P(dBm) = 106.92 + ERP(dBk) - 20LogD(km) - 20LogF(MHz) - 77.2

For transmitter ERP in dBi instead of dBd, subtract 2.15dB:

P(dBm) = 106.92 + EIRP(dBk) - 2.15 - 20LogD(km) - 20LogF(MHz) - 77.2

For the distance in miles instead of kilometers:

P(dBm) = 106.92 + EIRP(dBk) - 2.15 - 20LogD(mi) - 4.13 - 20LogF(MHz) - 77.2

Convert transmitter output in dBk to dBm:

P(dBm) = 106.92 + EIRP(dBm) - 60 - 2.15 - 20LogD(mi) - 4.13] - 20LogF(MHz) - 77.2

Combine all the constants:

P(dBm) = EIRP(dBm) - 36.56 - 20LogD(mi) - 20LogF(MHz)

P(dBm) = EIRP(dBm) - [36.56 + 20LogD(mi) + 20LogF(MHz)]

So the free-space loss (the difference between the transmitter power and the received power for isotropic antennas) is:

36.56 + 20LogD(mi) + 20LogF(MHz)