NTIA Technical Memorandum TM-10-469

Derivations of Relationships 

among Field Strength, Power in 

Transmitter-Receiver Circuits 

and Radiation Hazard Limits

Frank H. Sanders

NTIA Technical Memorandum TM-10-469

Derivations of Relationships 

among Field Strength, Power in 

Transmitter-Receiver Circuits 

and Radiation Hazard Limits

Frank H. Sanders

U.S. DEPARTMENT OF COMMERCE

June 2010
ii
iii

CONTENTS

Page

1 DIRECTIVITY, GAIN, EFFECTIVE ANTENNA APERTURE AND

ANTENNA CORRECTION FACTOR DERIVATIONS .............................................1

1.1 Introduction ..............................................................................................................1

1.2 Directivity and Gain of Power in Space ..................................................................2

1.3 Effective Aperture of Antennas ...............................................................................2

1.4 Relationship Between Power in a Circuit and Power Density in Space ..................3

1.5 Antenna Correction Factor and its Relationship to Gain .........................................3

2 FREE SPACE FIELD STRENGTH CONVERSION DERIVATIONS .......................5

2.1 Introduction ..............................................................................................................5

2.2 Incident Field Strength Related to Power in a Circuit .............................................5

2.3 Incident Power Per Unit Area Related to Field Strength .........................................7

2.4 Field Strength Units: Volts per Meter Related to Amperes per Meter ....................8

2.5 Field Strength Units: Decibel Volts per Meter and Amperes per Meter .................9

2.6 Field Strength Units: Amperes per Meter, Magnetic Field Intensity and 

Flux Density ...........................................................................................................10

3 EFFECTIVE ISOTROPIC RADIATED POWER CONVERSION 

DERIVATIONS ...........................................................................................................11

3.1 Introduction ............................................................................................................11

3.2 Power in a Circuit Related to Free-Space Loss and Transmitted EIRP .................11

3.3 Received Power as a Function of EIRP for Convenient Units of PDistance .........12

3.4 EIRP Conversions Between Convenient Engineering Units of Power ..................13

4 CONVERSION DERIVATIONS FOR RADIO RADIATION HAZARD 

LIMITS ........................................................................................................................14

4.1 Introduction ............................................................................................................14

4.2 Relationship Between EIRP and the Critical Separation Distance from a 

Transmitter Required to Comply with Radiation Hazard Limits...........................14

4.3 Determining the Measured Power in a Circuit that Corresponds to a 

Radiation Hazard Limit in Space ...........................................................................16

4.4 Limiting EIRP of a Transmitter Based on Measured Power and a Critical

Distance where a Radiation Hazard Limit is Reached ...........................................17

4.5 Maximum EIRP Related to Radiation Hazard Field Strength Limits ...................18

5 REFERENCES ............................................................................................................19
iv

ABBREVIATIONS/ACRONYMS

ACF antenna correction factor

ANSI American National Standards Institute

EIRP effective isotropic radiated power 

FS field strength

HERF hazards of electromagnetic radiation to fuel 

HERO hazards of electromagnetic radiation to ordnance 

HERP hazards of electromagnetic radiation to personnel

IEEE Institute of Electrical and Electronic Engineers

IF intermediate frequency

MILSTD military standard

RADHAZ radiation hazard

RF radiofrequency 
DERIVATIONS OF RELATIONSHIPS AMONG FIELD STRENGTH, POWER 

IN TRANSMITTER-RECEIVER CIRCUITS AND RADIATION HAZARD

LIMITS

Frank H. Sanders1

This NTIA Technical Memorandum provides a single, comprehensive set of 

derivations of the mathematical relationships among power in a radio 

transmitter or measurement circuit, the associated incident field strength in 

space, transmitter-receiver antenna characteristics, free space propagation 

equation variables and radiation hazard limits. 

Key words: antenna correction factor; effective isotropic radiated power; field strength; 

hazards of electromagnetic radiation to fuel; hazards of electromagnetic radiation to 

ordnance; hazards of electromagnetic radiation to personnel; measured power; military 

standard 461; power in circuits; radiation hazard; RF measurement 

1 DIRECTIVITY, GAIN, EFFECTIVE ANTENNA APERTURE AND 

ANTENNA CORRECTION FACTOR DERIVATIONS

1.1 Introduction 

Power in space couples into circuits, and vice versa, via antennas which couple energy 

between these two media by matching their (generally different) impedances. The 

relationships among the power in a circuit, the incident or radiated power in space around 

the circuit, and the characteristics of an antenna that couples the two are completely 

deterministic and are well-understood. Unfortunately, these relationships are often 

presented in engineering texts, data reports, data analyses and automated conversion 

calculators merely as formulae without references to sources of their derivations. While 

the use of formulae is an expedient way to perform calculations, a formula-based 

approach to an analysis can lead to calculation errors when the numerical inputs may not 

be legitimately used in that formula. This problem occurs, for example, when the origins 

of data inserted into a formula do not correlate to the assumptions behind the derivation 

of the formula. 

This publication seeks to address the lack of a published comprehensive set of derivations 

of the relationships among power in space, power in circuits, and the characteristics of 

antennas that couple between them. It provides, in a form that may be easily referenced, 

derivations of the relationships between these physical quantities. 

 

1 The author is with the Institute for Telecommunication Sciences, National Telecommunications 

and Information Administration, U.S. Department of Commerce, Boulder, Colorado 80305.
2

1.2 Directivity and Gain of Power in Space 

The derivations begin with directivity and gain, which are measures of the directive 

concentration of power in space by an antenna. Directivity, d, is the ratio of power 

density in some direction, pden, to the power density that would be produced if the power 

were radiated isotropically at pden_iso:

2

d = pden

pden _ iso

. (1)

Directivity makes reference only to power in space around an antenna; it is unrelated to 

power in a circuit that is connected to the antenna terminals. There is loss between the 

terminals and free space. Gain, g, includes these antenna losses: 

g = 4πr

2 ⋅ pden

pin _ out

, (2)

where:

pden = power density in space;

pin_out = power at the circuit input-output of the antenna; 

r = distance from the antenna.

Gain and directivity are related by efficiency, ε: 

g = d ⋅ ε (3)

1.3 Effective Aperture of Antennas

Effective antenna aperture, ae, is unrelated to the physical aperture of an antenna. It is 

defined as:

ae = λ2

g

4π

 (4)

where λ is the free-space wavelength of the radiation. Note that ae has the units of area. 

 

2 In this document, linear quantities are written in lower case italic (except for voltage, V) and 

decibel quantities (defined as decibels=10⋅log10(p2/p1), where p1 and p2 are linear power values) 

are written in upper case italic. For example, P(dBW) = 10log10⋅(p, watts).
3

1.4 Relationship Between Power in a Circuit and Power Density in Space

For an antenna matched to a load, the power in the load, pload, is related to the free-space 

power density by ae: 

pload = pden ⋅ ae (5)

If pload is in a 50-ohm circuit and pden is in a free-space impedance of 377 ohms, then the 

following relations apply: 

pload = Vload

2

50 (6a)

and

pden = Vspace

2

377 . (6b) 

Rewriting Eq. (5) using Eqs. (6a) and (6b) gives:

Vload

2

50 = ae ⋅ Vspace

2

377











 . (7)

Note that the voltage in the circuit, V2

load, is in units of volts, but that the free-space field 

strength, V2

space,, is in units of volts/m. The effective aperture, in units of m2

, converts the 

free-space power density on the right side to the power in a circuit on the left side. 

1.5 Antenna Correction Factor and its Relationship to Gain 

At this point, we introduce the antenna correction factor, acf, which is defined as: 

acf = Vspace

2

Vload

2 . (8)

Note that acf has units of m-2. Rewriting Eq. (7) with the substitution of Eq. (8) gives: 

acf = 377

50 ⋅

1

ae

. (9)

Because ae is dependent upon both gain and frequency, so is acf. Substituting Eq. (4) into 

Eq. (9) gives: 
4

acf = 377

50





 



 4π

λ2

g











 . (10) 

If the frequency, f, is in Megahertz, and the speed of light, c, is taken to be 3⋅108 m/s, then 

the substitution λ = c/(f⋅106) = 3⋅108/( f⋅106) gives:

acf = 377

50





 



( ) 4π

106

3⋅108













2

( ) f , MHz 2

⋅ 1

g





























 (11) 

which, calculating the constant term values, gives:

acf = 1.05 ⋅10−3 ( )⋅( ) f , MHz 2

⋅ 1

g











 . (12) 

Expressing acf in decibel terms (ACF = 10⋅log(acf)) gives3

: 

( ) ACF,dB = −29.8 + 20 log( ) f , MHz − 10 log( )g . (13) 

The quantity 10⋅log(g) is the directional gain, G, of an antenna in decibels relative to 

isotropic (dBi): 

( ) G,dBi = ( ) 10 log( )g = (20 log( ) f , MHz − 29.8 − ACF). (14) 

 

3 All log operations in this document are to the base 10. 
5

2 FREE SPACE FIELD STRENGTH CONVERSION DERIVATIONS

2.1 Introduction 

Signals are commonly measured in circuits as voltages either linear or log-detected, 

proportional to input power. The voltage induced in a measurement circuit by an input 

signal is usually converted to equivalent power by the circuit impedance. This conversion 

is usually accomplished automatically within the measurement device (e.g., a spectrum 

analyzer) after which the displayed power in a circuit must be converted into the field 

strength in free space that generated the voltage.

2.2 Incident Field Strength Related to Power in a Circuit

This power induced into a circuit, pload, is related to incident field strength through either 

the antenna correction factor or, equivalently, the antenna gain relative to isotropic; this is 

done as follows. Writing Eq. (5) with a substitution for ae from Eq. (4) gives: 

2 2

4 377

space

load

gV

p

λ

π

 

=   ⋅  

 (15) 

and substituting λ = c/f gives: 

pload = 3⋅108 ( )2

gVspace

2

( ) f , MHz 2

⋅ 106 ( )2

⋅ 4π ⋅ 377

















. (16) 

For power in milliwatts and field strength in microvolts/meter, the conversions 

(power, mW) = 1000⋅(W) and (field strength, V/m) = 10-6⋅(μV/m) are used:

pload ( ) ,mW = 1000 ⋅ 3⋅108 ( )2

⋅ g ⋅ 10−6 ( )2Vspace ( ) ,μV / m

2

( ) f , MHz 2

⋅ 106 ( )2⋅ 4π ⋅ 377

















 (17) 

which, upon combining the constant terms, becomes: 

pload ( ) ,mW = 1.90 ⋅10 −8

⋅g ⋅ Vspace ( ) ,μV / m

2

( ) f , MHz 2

















. (18) 

Taking 10log of both sides gives (where 10log(pload, mW) = (Pload, dBm)):
6

Pload ( ) ,dBm = −77.2 + G + 20 log Vspace ( ) ,μV / m − 20 log( ) f , MHz . (19) 

Rearrangement of terms yields:

( ) FS,dBμV / m = Pload ( ) ,dBm + 77.2 − G + 20 log( ) f , MHz , (20) 

where FS rather than Vspace is now used to denote incident field strength, still in units of 

dBμV/m. Note that Pload is related to the power measured within a circuit (e.g., a 

spectrum analyzer) by the total decibel correction, C, for all gains and losses between the 

antenna and the analyzer: Pload = Pmeas - C. This changes Eq. (20) to: 

( ) FS,dBμV / m = Pmeas ( ) ,dBm − C + 77.2 − G + 20 log( ) f , MHz (21) 

Eq. (21) is key for the conversion of measured power in a circuit into incident field 

strength in units of dBμV/m. For example, suppose that an antenna has gain G = 

16.9 dBi, at a frequency of 2300 MHz, with 31 dB of amplifier gain and 3 dB of loss 

between the measurement antenna terminal and the spectrum analyzer input. The 

measured power is taken to be -12.1 dBm on the spectrum analyzer display. Then the 

field strength at the measurement antenna must have been:

FS = (-12.1 dBm) – (31-3) + 77.2 – 16.9 + 20log(2300) = +87.4 dBμV/m.

If an equation is required to convert field strength in dBμV/m to the ACF, then Eqs. (7) 

and (9) are used to convert acf to voltage in a circuit and free-space field strength:

Vload

2

50 = ( ) pload = Vspace

2

50 ⋅ acf . (22) 

Converting power in watts to power in milliwatts, and converting field strength in 

volts/meter to microvolts/meter, gives

pload ( ) ,mW = 1000 ⋅ 10−6 ( )2

⋅( ) FS,μV / m

2

50 ⋅ acf

















 (23) 

which, combining the constants, gives: 

pload ( ) ,mW = 2 ⋅10−11 ⋅( ) FS,μV / m

2

acf . (24) 

Taking 10log of both sides, the incident field strength and the measurement antenna’s 

ACF is converted to the power coupled into the measurement circuit: 
7

Pload ( ) ,dBm = −107 + ( ) FS,dBμV / m − ACF . (25) 

Rearranging terms and taking into account the gains and losses, C, between the 

measurement antenna terminals and the final power measurement point, the measured 

power and the decibel antenna correction factor is converted to the field strength: 

( ) FS,dBμV / m = Pmeas − C + 107 + ACF . (26) 

For example, suppose a measurement of -12.1 dBm is taken on a spectrum analyzer, with 

28 dB of net gain in the path between the antenna and the analyzer, with a measurement 

antenna acf of 113 (corresponding to G = 16.9 dBi at 2300 MHz, as in the example 

above). Then ACF = 20.5 dB, and the corresponding free-space field strength is again 

computed using Eq. (26) to be +87.4 dBμV/m.

2.3 Incident Power Per Unit Area Related to Field Strength

Next, the conversion between field strength and incident power per unit area is 

considered. This incident power density, pden, in W/m2

, is equal to the incident field 

strength squared (units of (V/m)

2

), divided by the impedance of free space (377 ohms): 

pden , W

m

2







 =

fs, V

m









2

377 (27a) 

and

fs, V

m







 = 377 ⋅ pden , W

m

2







 = 19.4 ⋅ pden , W

m

2







 (27b) 

where fs is the field strength in linear units of (V/m). If field strength is expressed in 

μV/m (where 1 V/m = 106 μV/m), then

pden , W

m

2







 =

fs, μV

m









2

377 ⋅1012 = 2.65 ⋅10−15 ⋅ fs, μV

m









2

. (28) 

Using more common units for field strength (dBμV/m) and incident power (μW/cm

2

), 

where 1 W/m2 = [(106 μW)/(104 cm2)] =102μW/cm2, the relation for incident power 

density is: 

pden , μW

cm

2







 = 1

377 ⋅1010 ⋅ fs, μV

m









2

= 2.65 ⋅10−13 ⋅ fs, μV

m









2

 (29) 
8

and the conversion to mW/cm2 changes the conversion by another factor of 103: 

pden , mW

cm

2







 = 1

377 ⋅1013 ⋅ fs, μV

m









2

= 2.65 ⋅10−16 ⋅ fs, μV

m









2

. (30) 

2.4 Field Strength Units: Volts per Meter Related to Amperes per Meter

Returning to Eq. (27a), the power density in W/m2 may also be related to field strength 

expressed as amperes/meter. The basis of the conversion is the equivalence of a watt to a 

volt-amp. This can be used as a substitution in Eq. (27a) to yield: 

pden , W

m

2







 = pden , ( ) V ⋅ A

m

2





 



 =

fs, V

m









2

377 , (31a) 

which reduces to: 

( ) V ⋅ A = V2

377

→ ( ) V = 377 ⋅ A →

fs, V

m









2

377 = 377 fs, A

m









2

, (31b) 

so that 

pden , W

m

2







 = 377 fs, A

m









2

 (31c) 

and reciprocally

fs, A

m







 =

pden , W

m

2









377 = 5.15 ⋅10−2 ⋅ pden , W

m

2







 . (31d) 

Eq. (31c) may be combined with Eq. (27a) to relate field strength in V/m to field strength 

in A/m:

pden , W

m

2







 =

fs, V

m









2

377 = 377 fs, A

m









2

 (32a) 

which reduces to: 
9

fs, A

m









2

=

fs, V

m









2

3772 . (32b) 

Field strength is correctly rendered in unit terms of (V/m)2 or (A/m)2 but in reality, since 

all of the terms in Eq. (32b) are squared, field strength is sometimes conveniently but 

incorrectly rendered in un-squared units as “V/m” or “A/m”. If this technically incorrect 

but widely applied convention is used, the proportionality constant that must be used 

between V/m and A/m should be 377 rather than 3772

.

2.5 Field Strength Units: Decibel Volts per Meter and Amperes per Meter

Just as field strength may be expressed in dBμV/m, so it may also be expressed in 

dBμA/m by rendering Eq. (32b) in decibel terms: 

( ) FS,dBμA / m = ( ) FS,dBμV / m − 20 log 377 ( )= ( ) FS,dBμV / m − 51.5 , (33a) 

where, as always,

( ) FS,dBμV / m = 10 log fs,

μV

m





 





2

= 20 log fs,

μV

m





 



 (33b) 

and

( ) FS,dBμA / m = 10 log fs,

μA

m





 





2

= 20 log fs,

μA

m





 



 . (33c) 

(Note that, because field strength goes as (A/m)2, 1 dBA/m = 120 dBμA/m, just as 

1 dBV/m = 120 dBμV/m.) Incident power density in W/m2 is related to field strength in 

μA/m via Eq. (31c): 

pden , W

m

2







 = 377

fs, μA

m









2

1012 = 3.77 ⋅10−10 ⋅ fs, μA

m









2

. (34a) 

and thus 

fs, μA

m









2

= 1012

377 pden , W

m

2







 = 2.65 ⋅109 ⋅ pden , W

m

2







 . (34b) 
10

Converting Eqs. (34a) and (34b) into field strength in decibel terms (dBμA/m):

10 log pden , W

m

2







 = 10 log 1.94 ⋅10−5 ( )2

⋅ fs, μA

m









2 









 = ( ) FS,dBμA / m − 94.2

 (35a) 

and reciprocally

( ) FS,dBμA / m = 94.2 + 10 log pden , W

m

2







 = 94.2 + Pden ,dBW / m

2 ( ). (35b) 

2.6 Field Strength Units: Amperes per Meter, Magnetic Field Intensity and Flux 

Density

Sometimes there is a requirement to convert between a magnetic field intensity (H-field, 

in units of A/m) and a magnetic flux density (B-field, in units of Tesla (T) or Gauss (G), 

where 1 T = 104 G). In free space, B = μ0H, where μ0, the permeability of free space, is 

defined as 4π⋅10-7 N/A2, where force is expressed in MKS units of Newtons (kg⋅m/s2). 

Thus, 

( ) mfd,T = 4π ⋅10−7 ⋅( ) fs,A / m = 1.257 ⋅10−6 ⋅( ) mfi,A / m (36) 

where mfd and mfi are magnetic flux density and magnetic field intensity, respectively.
11

3 EFFECTIVE ISOTROPIC RADIATED POWER CONVERSION 

DERIVATIONS

3.1 Introduction 

It may be necessary to know the effective isotropic radiated power (eirp linear; EIRP

decibels) that a device transmits. The conversion from measured power in a circuit to 

EIRP is described in this section. 

3.2 Power in a Circuit Related to Free-Space Loss and Transmitted EIRP

First, free space loss must be determined. From transmit power (pt in watts), transmit 

antenna gain relative to isotropic (gt), receive antenna gain relative to isotropic (gr), 

receive power (pr in watts) and receive antenna effective aperture (ae) the effective 

isotropic radiated power is: 

eirp = pt ⋅ gt (37) 

and

pr = ae ⋅ eirp

4πr

2





 



 (38) 

with r being the distance between the transmit and receive antennas.

The effective aperture (Eq. (4)) of an antenna is the effective aperture of a (theoretical) 

isotropic antenna multiplied by the (actual) antenna gain (here a receiving antenna, gr) 

over isotropic:4

ae = λ2

4π











 ⋅ gr . (39) 

A change to decibel units makes Eq. (38): 

Pr = EIRP + Gr + 20 log( ) λ − 20 log 4( ) π − 20 log( )r . (40) 

In Eq. (40) and the following equations the term EIRP can also be written, from Eq. (37), 

as the decibel sum of the transmitted power, Pt, and the gain of the transmitter antenna, 

Gt, relative to isotropic:

EIRP = Pt + Gt . (41) 

 

4 By symmetry, the same relationships will hold for transmitting antenna gain, gt. 
12

Substituting c/f for λ, 

Pr = EIRP + Gr + 20 log( )c − 20 log( )f − 20 log 4( ) π − 20 log( )r . (42a) 

Gathering the constants, this becomes: 

Pr = EIRP + Gr + 20 log c

4π f











 − 20 log( )r , (42b) 

which can be written in terms of propagation loss, Lp, in space between the transmitter 

and receiver antennas:

Pr = EIRP − Lp + Gr , (42c) 

where

Lp = 20 log( )r − 20 log c

4π f











 . (42d) 

Note that the frequency dependence of the term Lp for so-called propagation path loss 

does not arise from any frequency-dependent loss of radio power through space. All 

frequencies of electromagnetic radiation propagate through free space equally, with only 

a geometric 1/r2 decrease in power density occurring as the distance increases from the 

point of transmission. The frequency dependence in the path-loss term is due to the 

wavelength dependence (and therefore frequency dependence) in the definition of 

effective aperture, ae, of an isotropic antenna (Eq. (4)). For algebraic convenience, when 

the gain values, gt and gr, of the antennas relative to the gain of an isotropic antenna are 

converted to the decibel variables Gt and Gr, the wavelength term in the effective aperture 

of an isotropic antenna is separated from the gain values and is gathered with other 

constants into the propagation path-loss term. While this gathering of multiple unrelated 

constants into the path-loss term provides convenience and efficiency for performing 

calculations, it contributes to a common misconception that propagation path-loss in free 

space is somehow frequency-dependent; it is not. 

3.3 Received Power as a Function of EIRP for Convenient Units of Distance 

If frequency and distance are expressed in megahertz and meters, then Eq. (42a) becomes 

Pr = EIRP + Gr + 20 log 3⋅108 ( )− 20 log f , MHz⋅106 ( )− 20 log 4( ) π − 20 log( )r , (43) 

which yields
13

Pr = EIRP + Gr + 169.5 − 20 log( ) f , MHz − 120 − 22 − 20 log( ) r,meters (44a) 

or 

Pr = EIRP + Gr + 27.5 − 20 log( ) f , MHz − 20 log( ) r,meters . (44b) 

Similarly, for r in kilometers, Eq. (44b) becomes: 

Pr = EIRP + Gr − 32.5 − 20log( ) f , MHz − 20log( ) r,km , (45) 

for r in statute miles:

Pr = EIRP + Gr − 36.5 − 20 log( ) f , MHz − 20 log( ) r,statute_ miles (46) 

and for r in nautical miles:

Pr = EIRP + Gr − 37.7 − 20 log( ) f , MHz − 20 log(r,nautical _ miles). (47) 

3.4 EIRP Conversions Between Convenient Engineering Units of Power 

For measurements of transmitters that are regulated by Part 15 or Part 18 of Federal 

Communication Commission (FCC) rules, Eq. (44b) is convenient. If the gain of the 

receive antenna is known and the received power has been measured at a known distance 

from the emitter, then Eq. (44b) can be rearranged to yield EIRP in decibels relative to a 

watt (dBW):

( ) EIRP,dBW = Pr ( ) ,dBW − Gr − 27.5 + 20 log( ) f , MHz + 20 log(r,meters). (48) 

If the received power is measured in dBm rather than dBW, Eq. (48) becomes 

( ) EIRP,dBW = Pr ( ) ,dBm − Gr − 57.5 + 20 log( ) f , MHz + 20 log( ) r,meters . (49) 

If EIRP in decibels relative to a picowatt (dBpW) is required, then Eq. (48) becomes: 

( ) EIRP,dBpW = Pr ( ) ,dBm − Gr + 62.5 + 20 log( ) f , MHz + 20 log( ) r,meters . (50) 

For example, if -10 dBm of received power is measured at a frequency of 2450 MHz, 

with an antenna gain of +16.9 dBi at a distance of 3 meters, then EIRP = +113 dBpW. 

Effective radiated power relative to a dipole (ERPdipole) is sometimes required. The 

conversion is accomplished by applying the fact that EIRP is 2.1 dB higher than 

ERPdipole, since by definition and theory the gain of a dipole antenna is 2.1 dBi. 
14

4 CONVERSION DERIVATIONS FOR RADIO RADIATION HAZARD 

LIMITS

4.1 Introduction 

Radiation hazard (radhaz) limits have been developed for non-ionizing (radio, or 

radiofrequency (RF)) energy by the American National Standards Institute (ANSI) and 

the Institute of Electrical and Electronic Engineers (IEEE) [1]. Another source for radhaz 

limits is the U.S. Department of Defense (DoD) Military Standard 461 (MILSTD-461) 

[2], which includes limits for hazards of electromagnetic radiation to fuel (HERF), 

ordnance (HERO), and personnel (HERP). 

All of these limits are generally expressed in units of average incident power per unit area 

for any given frequency. For example, a hypothetical limit might be specified as 

10 mW/cm2 for frequencies between 1-10 GHz.5

This section addresses the problem of computing the maximum permissible power level 

that a transmitter may emit in order to comply with a stated radhaz limit. It is assumed in 

this derivation that all transmitted and incident power levels are averaged in accordance 

with the definitions of [1]. This is a necessary step to take prior to operating any 

transmitter that may potentially produce a power level in excess of a radhaz limit at some 

point in space. 

4.2 Relationship Between EIRP and the Critical Separation Distance from a 

Transmitter Required to Comply with Radiation Hazard Limits

Radhaz limits are not limits on transmitter power; they are limits on the incident power 

density (power per unit area) at a point in space. Ordinarily, they are not limits on the 

peak power at a location, but rather on the average incident power, with the averaging 

interval being defined in existing documentation [1, 2].6

 Therefore, the designer of a 

transmitter system needs to be able to determine the critical distance, rcrit, at which the 

power from a transmitter with a given eirp will generate an incident power level that is 

equal to the critical radhaz limit. When the mathematical relationship between the eirp

and rcrit established, then either parameter may be used to control the other. 

The derivation assumes that radio power radiates from an antenna that behaves as a point 

source and that the power density in space will be assessed in the far field of the radiating 

antenna. This means that the power passes through a location of interest as a plane wave, 

 

5 Units of incident power per unit area are commonly used because they may be related to heating 

effects of radiofrequency energy.

6 In practice, this is equivalent to verifying the level of the power at a critical point in space by 

measuring it with a spectrum analyzer or RF power meter in a root-mean-square (RMS) detection 

mode, using measurement intermediate frequency (IF) and video bandwidths that are each equal 

to or greater than the 10-dB bandwidth of the transmitter’s emission spectrum.
15

and that nominal far-field antenna gain factors may be used in all calculations of power 

levels in space.

Since surface area multiplied by the power density per unit area yields incident power, 

the fundamental relationship between eirp, power density ρ at a point in space, and the 

distance r between the radiating antenna and the point in space is: 

eirp = ρ 4πr

2

( ). (51) 

Eq. (51) assumes that power radiating from a point in space will be distributed on a 

spherically shaped front (or a portion of such a front) of radius r. The density of power on 

the surface (or a portion of the surface) multiplied by the total area of the hypothetical 

spherical surface (4πr

2

) gives the total equivalent power that would be emitted from an 

isotropic source to produce that power density over an entire sphere of radius r.

7

A radiation hazard power density is a maximum specified power density, ρmax. Eq. (51) is 

converted to a radiation hazard relationship by giving subscripts to the variables to 

indicate that they have become critical:

eirpcrit = ρ max 4πrcrit

2

( ). (52) 

Since eirp is the product of transmitter power, pt, and transmitter antenna gain, gt, 

Eq. (52) can be written as:

ptg ( )t crit = ρ max 4πrcrit

2

( ). (53) 

Suppose, for example, that a clear distance of 12 m needs to be maintained from a 

transmitter that has a linear gain of 10, and that the critical radhaz power density limit is 

30 W/m2

. Then Eq. (53) becomes:

pt ( ) ⋅10 crit = 30 W

m2 ⋅ 4π ⋅( ) 12meters

2

( )= 54287watts . 

Dividing out the gain factor, gt, of 10 gives pt = 5429 W = 5.43 kW. 

Consider another example in which the transmitter power and antenna gain are already 

fixed and the minimum safety distance must be adjusted accordingly. If a power density 

of, for example, 10 W/m2 is not to be exceeded, and the transmitter in question is going to 

produce 10,000 W that will be radiated from an antenna with a linear gain factor of 2, 

then Eq. (53) becomes: 

((10,000 W)*2) = (10 W/m2

)*(4π(rcrit)

2

),

 

7 This argument is true even if the radiated power is concentrated in some direction; that is why 

the dependent variable has the word equivalent in its name.
16

or 

( ) ( ) 2

rcrit = × × = = 10,000 2 / 10 4 159 m 12.6 m π . 

4.3 Determining the Measured Power in a Circuit that Corresponds to a Radiation 

Hazard Limit in Space

The next problem to be considered is that of verifying through a direct measurement that 

the critical power limit is not being exceeded for an existing emitter. The average (RMS) 

power level that should not be exceeded can be written as (from Eq. (38)): 

pmeas_ crit = ae

eirpcrit

4πrcrit

2

 

  

 

  , (54) 

where

pmeas crit = average power measured at the critical radhaz distance limit;

ae = effective aperture of the measurement antenna.

The quantity inside the brackets in Eq. (54) is the total eirp divided by the surface area of 

a sphere of radius rcrit, and thus is the critical power density at distance rcrit from the 

transmitter. Multiplied by the effective aperture (which has units of area) of the 

measurement antenna, this yields the power that is coupled into the measurement circuit 

at rcrit. 

The effective aperture of the measurement antenna is given by Eq. (39). Substituting the 

terms of Eq. (52) into Eq. (54) yields: 

pmeas _ crit = λ2

4π

gr











 ⋅ ρ max 4πrcrit

2 ( )

4πrcrit

2











 = λ2

4π

grρ max . (55) 

At this point it is useful to convert the expression into decibel units by taking 10log10 of 

both sides: 

10log(pmeas_ crit) = 20log(λ) −10log(4π) +10log(gr) +10log(ρ max ). (56a) 

Writing decibel quantities in upper case and solving for the constant term we obtain: 

Pmeas_ crit = 20log(λ) −11+ Gr +10log(ρ max ). (56b) 

It is convenient to use engineering units of dBm for Pmeas_crit and MHz for frequency 

instead of the wavelength λ. The unit conversions are performed as follows: 
17

( )

8

6

_

max 2

2.99 10 / 20log 20log( , ) 10 / ,

11 10log , 30

meas crit

r

m s f MHz Hz MHz P dBm

W G

m

ρ

 ⋅    −

  =

  − + +   +  

 (57) 

where the quantity of 30 dB is added to adjust the equation units from dBW to dBm. 

Simplifying:

Pmeas_ crit ( ) ,dBm = 49.5 − 20log( f ,MHz) −11+ Gr +10log ρ max ,

W

m

2

 

   

  + 30 (58a) 

and combining constants and rendering the radiation density in the linear units that it 

would have in a look-up table such as is found in [1] result in: 

Pmeas_ crit ( ) ,dBm = 68.5 − 20log( f ,MHz) + Gr +10log ρ max ,

W

m

2

 

   

  . (58b) 

For example, [1] gives a radhaz limit for members of the general public of 10 W/m2 for 

frequencies between 2000 MHz to 100 GHz. If we want to verify that this limit is being 

met using a measurement antenna with a gain of 3 dBi at a frequency of 5700 MHz, then 

the critical power level that we would measure with that antenna connected to a spectrum 

analyzer would be: 

Pmeas crit = 68.5 – 75.1 + 3 + 10log(10 W/m2

) = +6.4 dBm. 

4.4 Limiting EIRP of a Transmitter Based on Measured Power and a Critical 

Distance where a Radiation Hazard Limit is Reached

How would this critical measured power limit relate to the eirp of the transmitter and 

some distance, rcrit, in meters from the transmitter? For this, Eq. (49) is used: 

EIRPcrit ( ) ,dBW = Pmeas_ crit ( ) ,dBm − 57.5 + 20log( f ,MHz) + 20log r( ) crit − Gr. 

To continue the example above, if the critical power value that must be measured is 

+6.4 dBm and rcrit = 3 meters, with a frequency of 5700 MHz and a measurement antenna 

gain of +3 dBi, then the maximum EIRPcrit power level of the transmitter and antenna 

that is connected to it must not exceed: 

EIRPcrit = (+6.4 dBm -57.5 + 75.1 + 9.5 – 3) = +30.5 dBW = 1133 W = 1.13 kW. 
18

4.5 Maximum EIRP Related to Radiation Hazard Field Strength Limits 

In addition to the physical criteria that have been discussed so far (power density in 

space, eirp from a transmitter, and power measured in a circuit), there is one more factor 

that is sometimes used in incident radiation criteria and measurements, and that is field 

strength. For this purpose, we adapt Eq. (21): 

(Pmeas_ crit,dBm) = FScrit − 77.2 − 20log( f ,MHz) + Gr

where FScrit is the incident field strength, in dBμV/m, at the radhaz limit. Substituting this 

expression for Pmeas_crit into Eq. (49) gives the connection between maximum allowable 

EIRP and the maximum allowable field strength at a point in space:

(EIRPcrit,dBW ) = FScrit + 20log(r,meters) −134.7. (59) 
19

5 REFERENCES

[1] ANSI/IEEE, “IEEE Standard for Safety Levels with Respect to Human Exposure to 

Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz,” ANSI/IEEE document 

C95.1, 1995. See in particular Tables 8 and 9 for recommended exposure limits. 

[2] Military Standard 461, “Electromagnetic emission and susceptibility requirements 

for the control of electromagnetic interference,” MIL-STD-461C, U.S. Department 

of Defense, August 1986. 
FORM NTIA-29 U.S. DEPARTMENT OF COMMERCE

(4-80) NATIONAL. TELECOMMUNICATIONS AND INFORMATION ADMINISTRATION

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION NO.

TM-10-469

2. Government Accession No. 3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Derivations of Relationships among Field Strength, Power in TransmitterReceiver Circuits and Radiation Hazard Limits

5. Publication Date

June 2010

7. AUTHOR(S)

Frank H. Sanders

9. Project/Task/Work Unit 

No.

6454000-200

8. PERFORMING ORGANIZATION NAME AND ADDRESS

Institute for Telecommunication Sciences

National Telecommunications & Information Administration

U.S. Department of Commerce

325 Broadway

Boulder, CO 80305

10. Contract/Grant No.

11. Sponsoring Organization Name and Address

National Telecommunications & Information Administration

Herbert C. Hoover Building, 14th & Constitution Ave., NW

Washington, DC 20230

12. Type of Report and 

Period Covered

Tech Memorandum, FY10

14. SUPPLEMENTARY NOTES

15. ABSTRACT

This NTIA Technical Memorandum provides a single, comprehensive set of 

derivations of the mathematical relationships among power in a radio transmitter or 

measurement circuit, the associated incident field strength in space, transmitterreceiver antenna characteristics, free space propagation equation variables and 

radiation hazard limits.

16. Key Words (Alphabetical order, separated by semicolons)

antenna correction factor; effective isotropic radiated power; field strength; hazards of electromagnetic 

radiation to fuel; hazards of electromagnetic radiation to ordnance; hazards of electromagnetic radiation to 

personnel; measured power; military standard 461; power in circuits; radiation hazard; RF measurement

17. AVAILABILITY STATEMENT

  UNLIMITED.

18. Security Class. (This report)

Unclassified

20. Number of pages

 19

19. Security Class. (This page)

Unclassified

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