The term Effective Radiated Power or ERP appears frequently in ham discussions, antenna literature and regulatory documents.

# The concepts

## ITU

(ITU 1990) defines Effective Radiated Power (erp) (in a given direction): [t]he product of the power supplied to the antenna and its gain relative to a half wave dipole in a given direction. (ITU 1990) defines gain relative to a half wave dipole as a half wave dipole isolated in space.

There is no suggestion that the Device Under Test is in free space, the implications are that the DUT is in its operating location.

## FCC

(FCC OET 2010) sets out a procedure and calculations relevant to determining ERP, and they define ERP by:

•  the expression erp = eirp/1.64 = (E x d)^2/(30 x 1.64) where: E = electric field strength in V/m, and d = measurement distance in meters (m), ; and
•  a definition in terms of measured field strength eirp =pt*gt=(E*d)^2/30 where: pt = transmitter output power in watts, gt = numeric gain of the transmitting antenna (unitless), E = electric field strength in V/m, d = measurement distance in meters (m) and erp=eirp/1.64.

The use of the factor 1.64 defines ERP relative to a half wave dipole in free space. They state [t]he gain of an ideal half-wave dipole antenna relative to an ideal isotropic antenna is 10log1.64 or 2.15 dBi.

There is no suggestion that the Device Under Test is in free space (which of course would be most impractical), the implication is that the DUT is in its operating location.

## Summary

The gain of a half wave dipole in free space relative to an isotropic radiator is 1.64 or 2.15dB.

ERP is the product of power into the Device Under Test and gain of that Device Under Test (as a dimensionless ratio) relative to a half wave dipole in free space.

# Examples

## 60m band somewhere in the world

If a licence condition states maximum effective radiated power (ERP) of 50 watts (W) peak envelope power (PEP), in the absence of other information, that could be taken to mean that in any direction, ERP must not exceed 50W PEP. If the antenna was a half wave dipole over real ground, and in the absence of actual measurements, an NEC model was used to indicate gain, maximum power to the antenna could be calculated. If that gain was for example 7dBi, that is 4.85dBd or 3.05 times that of a half wave dipole in free space, then the maximum power into the antenna would be limited to 50/3.05=16.4W. On the other hand, if a mobile antenna had a gain of -10dBi, that is -12.15dBd or 0.061 times that of a half wave dipole in free space, then the maximum power into the antenna would be limited to 50/0.061=819W. Note that the maximum field strength from both configurations would be similar at the same distance, albeit possibly in different directions.

(FCC 2010) states in Part 97.303(j)(2)(s):

Transmissions shall not exceed an effective radiated power (e.r.p) of 50 W PEP. For the purpose of computing e.r.p. the transmitter PEP will be multiplied with the antenna gain relative to a dipole or the equivalent calculation in decibels. A half wave dipole antenna will be presumed to have a gain of 0 dBd. Licensees using other antennas must maintain in their station records either manufacturer data on the antenna gain or calculations of the antenna gain.

This is loose drafting, the first mention of a dipole does not qualify it as a half wave dipole, and the mention of the gain of a half wave dipole does not state whether the dipole is in free space or in-situ. There appears to be a preference by hams to interpret this as in-situ where it is to their advantage, but in reality, using a half wave dipole in a variable situation (eg different heights, ground types etc) as a reference is plain nonsense and it is inconsistent with (FCC OET 2010).

Nevertheless, (ARRL nd) proposes this bodged up version of ERP where a dipole in situ, no matter what situation, is taken to have a gain of 0dBd, and all other antennas are adjusted: [t]he math is fairly straightforward. You must reduce your power by the number of decibels your antenna gain exceeds 0 dBd (0 dB relative to a half-wave dipole). Ok, so you come up with an estimate of the gain of your non half wave dipole in dBd, what is the actual gain in dBd of the reference half wave dipole over ground that you haven't got? This is again inconsistent with (FCC OET 2010) and sounds like it was written by lawyers rather than technicians, it lacks clarity, and as often the case, ambiguity in the law and subordinate regulations can be resolved... in a court of law.

(ARRL 2012) gives advice that [a]mateurs may transmit with an effective radiated power of 100 W or less, relative to a half-wave dipole, not making it clear that the method in (FCC OET 2010) is referenced to a half wave dipole in free space.

(Hendersen 2012) peddles the ARRL/QST view that the revised power limit on 60m limit is an effective radiated power of 100W or less, relative to a half-wave dipole and later lets assume that an ordinary half wavelength dipole antenna has a gain of 0dB without mentioning that in the real world, the gain of a half-wave dipole depends on its environment, it is not the fixed reference that a half-wave dipole in free space is, and in fact the gain of a half-wave dipole at low height over average ground is quite likely to be 6dB or more higher than a half-wave dipole in free space.

If you think this works by taking the gain of your own antenna in free space and comparing it to a dipole in free space, and making the ARRL's adjustment, what is the gain of the mobile whip example mentioned above in free space. It cannot work without some form of counterpoise, and if you transport the vehicle and natural ground to free space, you aren't in free space!

## 2200m band somewhere in the world

If a licence condition states that the licencee must not operate the station using a radiated power of more than 1 watt pX ERP (ACMA 2010),  and a particular antenna was taken to have a gain of -45dBd, then maximum power to the antenna could be calculated. If that gain was -45dBd, or 32e-6 times that of a half wave dipole in free space, then the maximum power into the antenna would be limited to 1/32e-6=31kW pX. The issue with practical antennas systems on the 2200m band is very low efficiency of the feed system, and the 1W pX ERP limit is less a constraint than the ability to dissipate the energy lost in the antenna system as heat.

(ACMA 2011) states that the licencee must not operate the station using a radiated power of more than 1 watt pX EIRP, which strictly speaking is internally in conflict as the ITU definition of pX (and PEP) is the average power supplied to the antenna transmission line by a transmitter during one radio frequency cycle at the crest of the modulation envelope taken under normal operating conditions. The gap being that transmission line loss is included in one and excluded from the other. It is obvious that what is intended is the combination of EIRP and the average power supplied to the antenna transmission line by a transmitter during one radio frequency cycle at the crest of the modulation envelope taken under normal operating conditions from the Peak Envelope Power definition. Perhaps the document should define a term EIRPEP as Effective Isotropic Radiated Peak Envelope Power (in a given direction) means the product of the average power supplied to the antenna by a transmitter during one radio frequency cycle at the crest of the modulation envelope taken under normal operating conditions and its gain (relative to an isotropic radiator). There is a sloppiness in the meaning of terms that is not usually found in the ACMA's regulatory instruments. The astute reader will have noticed that the ACMA's revision of this clause from ERP to EIRP is a 2dB reduction in permitted power level!

# Myths

It is often held that ERP is relative to a half wave dipole in the same position as the antenna being discussed. This has an appeal to simple minds, but it is not only impractical, it flies in the face of the definitions above. This position is sometimes justified by claiming that if there was a fixed relationship between dBd and dBi, then dBd is redundant. There is a fixed relationship between dBd and dBi, and if that makes dBd redundant, so be it.

Equally silly is the notion that the Device Under Test should be partially relocated into free space (eg a nominal radiator without its ground context), and the field strength measured or calculated relative to a half wave dipole also in free space.

In specifying ERP (or EIPR limits for that matter), regulators better constrain the maximum field strength due to a source at various distances, and can better manage the risk of interference to other users or services.

# Conclusions

• The gain of a half wave dipole in free space relative to an isotropic radiator is 1.64 (2.15dB).
• Antenna gain in dBd is referenced to a dipole in free space.
• Effective Radiated Power is the product of the power supplied to the antenna and its gain relative to a half wave dipole in free space.
• Though there is a clear definition of ERP in the ITU Radio Regulations, a lot of uses of the term, including by people who should know better, are wrong and for clarity, one is better using EIRP as it is less misused.