TN1200.04 Calculating Radiated Power and Field Strength

Calculating Radiated Power and Field Strength
for Conducted Power Measurements
TN1200.04
Calculating Radiated Power and Field Strength
for Conducted Power Measurements
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Calculating Radiated Power and Field Strength
for Conducted Power Measurements
1 Table of Contents
1
2
3
4
5
6
7
8
Table of Contents............................................................................................................................. 2
1.1
Index of Figures ..................................................................................................................... 2
Introduction ...................................................................................................................................... 3
Antennas and Power Density .......................................................................................................... 3
3.1
Isotropic Antennas ................................................................................................................. 3
Free Space Path Loss ..................................................................................................................... 5
Antenna Coefficients........................................................................................................................ 6
5.1
Antenna Correction Factor..................................................................................................... 6
5.2
Antenna Factor ...................................................................................................................... 7
Field Strength Conversions ............................................................................................................. 7
Effective Isotropic Radiated Power Conversion .............................................................................. 8
References....................................................................................................................................... 9
1.1 Index of Figures
Figure 1: Isotropic Radiator..................................................................................................................... 3
Figure 2: One-Way Communications Link .............................................................................................. 5
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2 Introduction
The Federal Communications Commission (FCC) Regulations for Radio Frequency Devices defines
power and radiated limits in terms of electric field strength measured in volts/meter (typically at a
distance of 3 meters) from the source.
Unless the engineer has access to a RF anechoic chamber, GigaHertz transverse electromagnetic
(GTEM) cell, or open area test site (OATS), most measurements will be made as conducted power
measurements into the 50 Ω load presented by the test equipment.
This application note will explain the relationship between conducted power levels and radiated power
field strengths.
3 Antennas and Power Density
3.1 Isotropic Antennas
Since we are concerned with the radiated field generated from the antenna, we will consider the point
source radiator or isotropic antenna. The performance of most practical antennas is often referenced
in terms of this basic radiator.
The isotropic antenna radiates energy equally in all directions; hence the radiation pattern in any
plane is circular, as illustrated below in Figure 1.
Q
r
O
S
Figure 1: Isotropic Radiator
In the case of the isotropic radiator at point O which is fed with a power P watts. The power flows
outwards from the origin and must flow through the spherical surface, S, of radius, r.
We can define the power density, Pd, at the point Q as:
Pd =
P
4πr 2
Watts/m
2
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Poynting’s theorem defines the relationship between the power density to the E-field and H-field
vectors as defined below:
Pd = E × H
Watts/m2
The magnitude of the power density is thus:
E2
, where 120π is the impedance of free space or approximately 377Ω.
120π
E
E
P
From the above, we can see that
= 120π , and thus,
=
H
120π 4πr 2
| Pd |= EH =
Hence, the rms value of the E (or Electric) field can be calculated from:
E=
30 P
r
V/m
(2)
The isotropic antenna is considered to have a power gain, Gi = 1. Referring to Figure 1, above, if an
antenna with gain GT were placed at point O, the power received at point Q would be increased by
GT. Hence, the field strength at point Q will be increased:
E=
30 PT GT
V/m
r
(3)
Where PT is the transmitter power, GT is the antenna gain.
Similarly, for an antenna with gain GT, we can rewrite equation (1) as:
Pd =
PT GT
4πr 2
Watts/m2
(4)
If GT = 1 (e.g. the gain of an Isotropic antenna with gain, Gi), the radiated power calculated in equation
(4) is referred to as the Effective Isotropic Radiated Power (EIRP) and is the power that would be
radiated from a transmitter of Power PT from an ideal omnidirectional source.
Re-arranging (3), and with GT = 1, we can determine the EIRP in terms of the field strength:
PT =
E 2r 2
30
(5)
As an example, FCC Part 15.249(a) states that the maximum field strength for ISM band devices
operating in the 902 – 928 MHz band is 50 mV/m. Since field strength is typically measured at 3 m,
we can rewrite (5) to calculate the EIRP as follows:
PT = 0.3E 2
PT = 0.75mW = −1.25dBm
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Calculating Radiated Power and Field Strength
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4 Free Space Path Loss
Consider the one-way (or simplex) communications link illustrated below in Figure 2:
Q
r
T
Figure 2: One-Way Communications Link
Assume that a transmitter at T with transmitter power PT and antenna gain GT. The power density, Pd,
at the receiver, Q is as previously defined in equation (4).
Pd =
PT GT
4πr 2
Watts/m
2
The radiated power received at the antenna port is a function of the effective aperture of the antenna:
Ae =
Gant λ2
4π
m
2
(6)
If the gain of the receiver antenna, Gant = GR, we can calculate the power received as:
PR = Pd Ae
PR =
(7)
PT GT GR λ2
(4πr ) 2
(8)
Equation (8) is often referred to as the free space loss equation.
Whilst we can re-arrange (8) to determine the theoretical transmission range, it should be noticed that
in the real world situation, factors such as reflections, scattering, shadowing, terrain and environment
will impact upon the transmission range.
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5 Antenna Coefficients
5.1 Antenna Correction Factor
Typically laboratory based power measurements are made with test equipment that has a
characteristic impedance of 50Ω. As has been defined above, the characteristic impedance of free
space is (120π)Ω or 377Ω.
For an antenna matched to a load, we can re-write equation (7) as
PL = Pd Ae
(9)
If PL is measured in a 50Ω load and Pd in a free space impedance of 377Ω, then the following
relationships apply:
2
PL =
VL
E2
and Pd =
50
377
(10)
Hence, we can re-write equation (9):
2
 E2 
VL
 Ae
= 
50  377 
(11)
Note that VL2 is in units of volts, whilst E2 is in volts/meter. The effective aperture of the antenna, with
2
units of m , converts free space power density to circuit power.
The relationship between radiated and conducted power can be made clearer by defining an antenna
correction factor (ACF), which is introduced in equation (12):
ACF =
E 2 -2
m
2
VL
(12)
Re-writing (11) and substitute for ACF:
ACF =
377  1 
 
50  A e 
By substituting equation (6) into the above, the antenna correction factor can be defined as:
 377  4π  -2
ACF = 
m

2
 50  Gant λ 
(13)
If the frequency, f, is in MHz, then by substituting λ=c/f:
12
 377   10
ACF = 
4π 
 50   3 × 108


( f , MHz )2  1  



 Gant  


2 1
ACF = (1.03 × 103 )( f , MHz ) 

G
ant


(
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Calculating Radiated Power and Field Strength
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Taking 10(log) of both sides allows the ACF to be calculated in dB:
ACF ( dB ) = −29.78dB + 20 log( f , MHz ) − 10 log(G )
(14)
5.2 Antenna Factor
2
From the above, we see that the Antenna Correction Factor links power (in terms of VL /R) and power
density. Most antennas used for field strength and compliance measurements use a coefficient to link
field strength E to receiver voltage V directly, since VL can be measured, whilst E is the quantity that is
required by the FCC.
This coefficient is a called Antenna Factor (AF) and is generally quoted as a function of frequency (or
1/λ).
AF =
E
E2
=
= ACF
2
VL
VL
9.73
 377  4π 
m-1
AF = 
=

2 
50
G
λ
λ
G

 ant 
ant
(16)
6 Field Strength Conversions
From equation (9) the power in the load is related to power density by the effective antenna aperture,
Ae. Substituting equation (6) into this equation, we can relate load power directly to E as defined in
equation 10, as follows:
PL =
λ2 Gant E 2
4π 377
If the frequency, f, is in MHz and we substitute for λ=c/f:


(3 ×108 ) 2 Gant E 2
PL = 

2
6 2
 ( f , MHz ) (10 ) 4π 377 
For power in mW and field strength in µV/m, and calculating the constant terms:
1.90 × 10 −8 Gant ( E , µV / m) 2 
PL (mW ) = 

( f , MHz ) 2


(17)
Taking 10(log) of both sides:
10 log( PL , mW ) = −77.2dB + 10 log(Gant ) + 20 log( E , µV / m) − 20 log( f , MHz )
By rearranging the above:
E (dBµV / m) = PL (dBm) + 77.2dB + 20 log( f , MHz ) − Gant (dB )
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Since PL is related to power measured in a circuit (by using a spectrum analyzer) by the terms for any
path gains between the antenna and the analyzer, we can rewrite PL = Pmeas – Pgain. Equation (18)
becomes:
E ( dBµV / m) = Pmeas ( dBm) − Pgain ( dB ) + 77.2dB + 20 log( f , MHz ) − Gant ( dB )
(19)
Equation 19 is the key equation for converting power measured in a circuit (in dBm) into incident field
strength in dBµV/m.
7 Effective Isotropic Radiated Power Conversion
To calculate the effective EIRP from the power measured in a circuit, from equation (7), the power in
the receiver is related to the power density and the effective aperture of the antenna:
PG 
 | EIRP | 
PR = Pd Ae = Ae  T T2  = Ae 
2 
 4πr 
 4πr 
(20)
From equation (6), the effective aperture of the antenna is the effective aperture of an isotropic
antenna multiplied by the gain of the antenna (Gant = GR):
Ae =
GR λ2
4π
Taking log(10) of both sides of equation (20), and if the frequency, f, is in MHz, distance, r, is in
meters, and substituting for λ=c/f:
PR = EIRP + GR ( dB ) + 27.5 − 20 log( f , MHz ) − 20 log(r , meters )
(21)
For the measurements required by FCC Part 15, equation (21) is a convenient method of calculating
the power in the receiver.
Since measurements are typically made at a distance of 3 meters, we can rewrite equation (21):
PR = EIRP + GR ( dB ) + 17.96 − 20 log( f , MHz )
(22)
If the gain of the receiver antenna is known and the received power has been measured at a distance
of r meters from the source, we can rearrange equations (21) and (22) to calculate the EIRP:
EIRP(dBW ) = PR (dBm) − GR − 57.5 − 20 log( f , MHz) − 20 log(r , meters)
(23)
If EIRP relative to 1 mW is required:
EIRP(dBm) = PR (dBm) − GR − 27.5 − 20 log( f , MHz) − 20 log(r , meters)
(24)
And assuming that the distance r is 3 m as in the case defined for equation (22):
EIRP(dBm) = PR (dBm) − GR − 37 − 20 log( f , MHz )
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8 References
•
•
•
F.R. Connor; “Antennas” Edward Arnold, 1972
Softwright LLC; http://www.softwright.com/faq/faq_engineering.html)
F. Sanders; “Conversion of Power Measured in a Circuit to Incident Field Strength and
Incident Power Density, and Corrections to Measured Emission Spectra for Non-Constant
Aperture Measurement Antennas” (Appendix C of NTIA Report 01-383, Jan 2001)
END OF DOCUMENT
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