Free Space Field and Loss Calculations

Q: To calculate free-space loss I use the equation 36.6 + 20LogD + 20LogF, where D is the distance in miles and F is the frequency in MHz. I subtract the free-space loss (in dB) from my transmitter ERP (in dBm) to get my receiver input power (in dBm). When I 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?

A: This is a case of both methods yielding the right answer, but expressed differently.

The equation you reference 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).

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 calculations in the Microwave Data Base Manager program 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 you used). 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 shown below:
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.

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)

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