Effective radiated power

== Definitions ==
Effective radiated power and effective isotropic radiated power both measure the power density a radio transmitter and antenna (or other source of electromagnetic waves) radiate in a specific direction: in the direction of maximum signal strength (the "[[main lobe]]") of its radiation pattern.<ref name="NAB">{{cite book
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 | first1 = Graham A.
 | last2  = Layer
 | first2 =  David H.
 | last3  = Osenkowsky
 | first3 = Thomas G.
 | title  = National Association of Broadcasters Engineering Handbook, 10th Ed.
 | publisher = Elsevier
 | date   = 2007
 | pages  = 1632
 | url    = https://books.google.com/books?id=K9N1TVhf82YC&q=%22Effective+radiated+power%22+%22effective+isotropic+radiated+power%22&pg=PA1632
 | isbn   = 978-1136034107
 }}</ref><ref name="Huang">{{cite book
 | last1  = Huang
 | first1 = Yi
 | last2  = Boyle
 | first2 = Kevin
 | title  = Antennas: From Theory to Practice
 | publisher = John Wiley and Sons 
 | date   = 2008
 | pages  = 117–118
 | url    = https://books.google.com/books?id=MoI1T9fOxdIC&q=%22Effective+radiated+power%22+%22effective+isotropic+radiated+power%22&pg=PR13
 | isbn   = 978-0470772928
 }}</ref><ref name="Seybold">{{cite book
 | last1  = Seybold
 | first1 = John S. 
 | title  = Introduction to RF Propagation
 | publisher = John Wiley and Sons 
 | date   = 2005
 | pages  = 292
 | url    = https://books.google.com/books?id=4LtmjGNwOPIC&q=%22Effective+radiated+power%22+%22effective+isotropic+radiated+power%22
 | isbn   = 0471743682
 }}</ref><ref name="Weik">{{cite book
 | last1  = Weik
 | first1 = Martin H. 
 | title  = Communications Standard Dictionary
 | publisher = Springer Science and Business Media
 | date   = 2012
 | pages  = 327
 | url    = https://books.google.com/books?id=lv3xBwAAQBAJ&q=%22Effective+radiated+power%22+%22effective+isotropic+radiated+power%22&pg=PA327
 | isbn   = 978-1461566724
 }}</ref>  This apparent power is dependent on two factors: the total power output and the [[radiation pattern]] of the antenna – how much of that power is radiated in the direction of maximal intensity. The latter factor is quantified by the [[antenna gain]], which is the ratio of the signal strength radiated by an antenna in its direction of maximum radiation to that radiated by a standard antenna.  For example, a 1,000-watt transmitter feeding an antenna with a gain of 4 (6 dBi) will have the same signal strength in the direction of its main lobe, and thus the same ERP and EIRP, as a 4,000-watt transmitter feeding an antenna with a gain of 1 (0 dBi).  So ERP and EIRP are measures of radiated power that can compare different combinations of transmitters and antennas on an equal basis.

In spite of the names, ERP and EIRP do not measure transmitter power, or total power radiated by the antenna, they are just a measure of signal strength along the main lobe.  They give no information about power radiated in other directions, or total power.  ERP and EIRP are always greater than the actual total power radiated by the antenna.

The difference between ERP and EIRP is that antenna gain has traditionally been measured in two different units, comparing the antenna to two different standard antennas; an [[isotropic antenna]] and a [[half-wave dipole]] antenna:
*''Isotropic gain'' is the ratio of the power density <math>S_\text{max}</math> (signal strength in watts per square meter) received at a point far from the antenna (in the [[far field]]) in the direction of its maximum radiation (main lobe), to the power <math>S_\text{max,iso}</math> received at the same point from a hypothetical lossless [[isotropic antenna]], which radiates equal power in all directions <math display="block">\mathrm{G}_\text{i} = {S_\text{max} \over S_\text{max,iso}}</math> Gain is often expressed in logarithmic units of [[decibel]]s (dB).  The decibel gain relative to an isotropic antenna (dBi) is given by <math display="block">\mathrm{G}\text{(dBi)} = 10\log{S_\text{max}  \over S_\text{max,iso}}</math>
*''Dipole gain'' is the ratio of the power density received from the antenna in the direction of its maximum radiation to the power density <math>S_\text{max,dipole}</math> received from a lossless [[half-wave dipole]] antenna in the direction of its maximum radiation <math display="block">\mathrm{G}_\text{d} = {S_\text{max}  \over S_\text{max,dipole}}</math> The decibel gain relative to a dipole (dBd) is given by <math display="block">\mathrm{G}\text{(dBd)} = 10\log{S_\text{max}  \over S_\text{max,dipole}}</math>
In contrast to an isotropic antenna, the dipole has a "donut-shaped" radiation pattern, its radiated power is maximum in directions perpendicular to the antenna, declining to zero on the antenna axis.  Since the radiation of the dipole is concentrated in horizontal directions,  the gain of a half-wave dipole is greater than that of an isotropic antenna.  The isotropic gain of a half-wave dipole is 1.64, or in decibels 10 log 1.64 = 2.15 dBi, so
<math display="block">G_\text{i} = 1.64G_\text{d}</math>
In decibels
<math display="block">G\text{(dBi)} = G\text{(dBd)} + 2.15</math>

The two measures EIRP and ERP are based on the two different standard antennas above:<ref name="NAB" /><ref name="Seybold" /><ref name="Huang" /><ref name="Weik" />
*EIRP is defined as the RMS power input in watts required to a lossless [[isotropic antenna]]  to give the same maximum power density far from the antenna as the actual transmitter. It is equal to the power input to the transmitter's antenna multiplied by the isotropic antenna gain <math display="block">\mathrm{EIRP} = G_\text{i} P_\text{in}</math> The ERP and EIRP are also often expressed in [[decibel]]s (dB).  The input power in decibels is usually calculated with comparison to a reference level of one [[watt (unit)|watt]] (W): <math>P_\text{in}\mathrm{(dBW)} = 10 \log P_\text{in}</math>.  Since multiplication of two factors is equivalent to addition of their decibel values <math display="block">\mathrm{EIRP(dBW)} = G\text{(dBi)} + P_\text{in}\mathrm{(dBW)}</math>
*ERP is defined as the RMS power input in watts required to a lossless [[half-wave dipole]] antenna to give the same maximum power density far from the antenna as the actual transmitter. It is equal to the power input to the transmitter's antenna multiplied by the antenna gain relative to a half-wave dipole <math display="block">\mathrm{ERP} = G_\text{d} P_\text{in}</math> In decibels <math display="block">\mathrm{ERP(dBW)} = G\text{(dBd)} + P_\text{in}\mathrm{(dBW)}</math>

Since the two definitions of gain only differ by a constant factor, so do ERP and EIRP
<math display="block">\mathrm{EIRP(W)} = 1.64 \cdot \mathrm{ERP(W)}</math>
In decibels
<math display="block">\mathrm{EIRP(dBW)} =\mathrm{ERP}\text{(dBW)} + 2.15</math>