FIELD INTENSITY and POWER DENSITY

Sometimes it is necessary to know the actual field intensity or power density at a given distance from a transmitter instead of the signal strength received by an antenna. Field intensity or power density calculations are necessary when estimating electromagnetic interference (EMI) effects, when determining potential radiation hazards (personnel safety), or in determining or verifying specifications.

Field intensity (field strength) is a general term that usually means the magnitude of the electric field vector, commonly expressed in volts per meter. At frequencies above 100 MHz, and particularly above one GHz, power density (PD) terminology is more often used than field strength.

Power density and field intensity are related by equation [1]:

where PD is in W/m2, E is the RMS value of the field in volts/meter and 377 ohms is the characteristic impedance of free space. When the units of PD are in mW/cm2, then PD (mW/cm2) = E2/3770.

Conversions between field strength and power density when the impedance is 377 ohms, can be obtained from Table 1. It should be noted that to convert dBm/m2 to dBV/m add 115.76 dB. Sample calculations for both field intensity and power density in the far field of a transmitting antenna are in the Power Density Section which follows. Refer to the sections on antennas for the definitions of near field and far field.

Note that the "/" term before m, m2, and cm2 in Table 1 mean "per", i.e. dBm per m2, not to be confused with the division sign which is valid for the Table 1 equation P=E2/Zo. Remember that in order to obtain dBm from dBm/m2 given a certain area, you must add the logarithm of the area, not multiply. The values in the table are rounded to the nearest dBW, dBm, etc. per m2 so the results are less precise than a typical handheld calculator and may be up to dB off.

Table 1. Conversion Table - Field Intensity and Power Density PD = E2/Z0
( Related by free space impedance = 377 ohms )

E
(Volts/m)
20 log106 (E)
(dBµV/m) 		PD
(watts/m2) 		10 log PD
(dBW/m2)
Watts/cm2
dBW/cm2
mW/cm2
dBm/cm2
dBm/m2
7,000
5,000
3,000
4,000
1,000 		197
194
190
186
180 		130,000
66,300
23,900
10,600
2,650 		+51
+48
+44
+40
+34 		13
6.6
2.4
1.1
.27 		+11
+8
+4
0
-6 		13,000
6,630
2,390
1,060
265 		+41
+38
+34
+30
+24 		+81
+78
+74
+70
+64
700
500
300
200
100 		177
174
170
166
160 		1,300
663
239
106
27 		+31
+28
+24
+20
+14 		.13
.066
.024
.011
.0027 		-9
-12
-16
-20
-26 		130
66
24
11
2.7 		+21
+18
+14
+10
+4 		+61
+58
+54
+50
+44
70
50
30
20
10 		157
154
150
146
140 		13
6.6
2.4
1.1
.27 		+11
+8
+4
+0
-6 		1.3x10-3
6.6x10-4
2.4x10-4
1.1x10-4
2.7x10-5 		-29
-32
-36
-40
-46 		1.3
.66
.24
.11
.027 		+1
-2
-6
-10
-16 		+41
+38
+34
+30
+24
7
5
3
2
1 		137
134
130
126
120 		.13
.066
.024
.011
.0027 		-9
-12
-16
-20
-26 		1.3x10-5
6.6x10-6
2.4x10-6
1.1x10-6
2.7x10-7 		-49
-52
-56
-60
-66 		.013
66x10-4
24x10-4
11x10-4
2.7x10-4 		-19
-22
-26
-30
-36 		+21
+18
+14
+10
+4
0.7
0.5
0.3
0.2
0.1 		117
114
110
106
100 		1.3x10-3
6.6x10-4
2.4x10-4
1.1x10-4
2.7x10-5 		-29
-32
-36
-40
-46 		1.3x10-7
6.6x10-8
2.4x10-8
1.1x10-8
2.7x10-9 		-69
-72
-76
-80
-86 		1.3x10-4
66x10-4
24x10-4
11x10-4
2.7x10-6 		-39
-42
-46
-50
-56 		+1
-2
-6
-10
-16
70x10-3
50x10-3
30x10-3
20x10-3
10x10-3 		97
94
90
86
80 		1.3x10-5
6.6x10-6
2.4x10-6
1.1x10-6
2.7x10-7 		-49
-52
-56
-60
-66 		1.3x10-9
6.6x10-10
2.4x10-10
1.1x10-10
2.7x10-11 		-89
-92
-96
-100
-106 		1.3x10-6
66x10-8
24x10-8
11x10-8
2.7x10-8 		-59
-62
-66
-70
-76 		-19
-22
-26
-30
-36
7x10-3
5x10-3
3x10-3
2x10-3
1x10-3 		77
74
70
66
60 		1.3x10-7
6.6x10-8
2.4x10-8
1.1x10-8
2.7x10-9 		-69
-72
-76
-80
-86 		1.3x10-11
6.6x10-12
2.4x10-12
1.1x10-12
2.7x10-13 		-109
-112
-116
-120
-126 		1.3x10-8
66x10-10
24x10-10
11x10-10
2.7x10-10 		-79
-82
-86
-90
-96 		-39
-42
-46
-50
-56
7x10-4
5x10-4
3x10-4
2x10-4
1x10-4 		57
54
50
46
40 		1.3x10-9
6.6x10-10
2.4x10-10
1.1x10-10
2.7x10-11 		-89
-92
-96
-100
-106 		1.3x10-13
6.6x10-14
2.4x10-14
1.1x10-14
2.7x10-15 		-129
-132
-136
-140
-146 		1.3x10-10
66x10-12
24x10-12
11x10-12
2.7x10-12 		-99
-102
-106
-110
-116 		-59
-62
-66
-70
-76
7x10-5
5x10-5
3x10-5
2x10-5
1x10-5 		37
34
30
26
20 		1.3x10-11
6.6x10-12
2.4x10-12
1.1x10-12
2.7x10-13 		-109
-112
-116
-120
-126 		1.3x10-15
6.6x10-16
2.4x10-16
1.1x10-16
2.7x10-17 		-149
-152
-156
-160
-166 		1.3x10-12
66x10-14
24x10-14
11x10-14
2.7x10-14 		-119
-122
-126
-130
-136 		-79
-82
-86
-90
-96
7x10-6
5x10-6
3x10-6
2x10-6
1x10-6 		17
14
10
6
0 		1.3x10-13
6.6x10-14
2.4x10-14
1.1x10-14
2.7x10-15 		-129
-132
-136
-140
-146 		1.3x10-17
6.6x10-18
2.4x10-18
1.1x10-18
2.7x10-19 		-169
-172
-176
-180
-186 		1.3x10-14
66x10-16
24x10-16
11x10-16
2.7x10-16 		-139
-142
-146
-150
-156 		-99
-102
-106
-110
-116


NOTE: Numbers in table rounded off

VOLTAGE MEASUREMENTS

Coaxial cabling typically has input impedances of 50, 75, and 93, (± 2) with 50 ohm being the most common. Other types of cabling include the following: TV cable is 75 ohm (coaxial) or 300 ohm (twin-lead), audio public address (PA) is 600 ohm, audio speakers are 3.2(4), 8, or 16 ohm.

In the 50 ohm case, power and voltage are related by:

Conversions between measured power, voltage, and current where the typical impedance is 50 ohms can be obtained from Table 2. The dBµA current values are given because frequently a current probe is used during laboratory tests to determine the powerline input current to the system .

MATCHING CABLING IMPEDANCE

In performing measurements, we must take into account an impedance mismatch between measurement devices (typically 50 ohms) and free space (377 ohms).

FIELD STRENGTH APPROACH

To account for the impedance difference, the antenna factor (AF) is defined as: AF=E/V, where E is field intensity which can be expressed in terms taking 377 ohms into account and V is measured voltage which can be expressed in terms taking 50 ohms into account. Details are provided in the Emcon section.

POWER DENSITY APPROACH

To account for the impedance difference , the antenna's effective capture area term, Ae relates free space power density PD with received power, Pr , i.e. Pr = PD Ae. Ae is a function of frequency and antenna gain and is related to AF as shown in the Emcon section.

SAMPLE CALCULATIONS

The Power Density Section provides sample calculations using power density and power terms from Table 1 and Table 2, whereas the Emcon Section uses these terms plus field intensity and voltage terms from Table 1 and Table 2. Refer the examples in the Emcon Section for usage of the conversions while converting free space values of power density to actual measurements with a spectrum analyzer attached by coaxial cable to a receiving antenna.

Table 2. Conversion Table - Volts to Watts and dBµA
(Px = Vx2/Z - Related by line impedance of 50 ohms)

Volts dBV dBµV Watts dBW dBm dBµA
700
500
300
200
100 		56.0
53.9
49.5
46.0
40.0 		176.0
173.9
169.5
166.0
160.0 		9800
5000
1800
800
200 		39.9
37.0
32.5
29.0
23.0 		69.9
67.0
62.5
59.0
53.0 		142.9
140.0
135.5
132.0
126.0
70
50
30
20
10 		36.9
34.0
29.5
26.0
20.0 		156.9
154.0
149.5
146.0
140.0 		98
50
18
8
2 		19.9
17.0
12.5
9.0
3.0 		49.9
47.0
42.5
39.0
33.0 		122.9
120.0
115.5
112.0
106.0
7
5
3
2
1 		16.9
14.0
9.5
6.0
0 		136.9
134.0
129.5
126.0
120.0 		0.8
0.5
0.18
0.08
0.02 		0
-3.0
-7.4
-11.0
-17.0 		29.9
27.0
22.5
19.0
13.0 		102.9
100.0
95.6
92.0
86.0
0.7
0.5
0.3
0.2
0.1 		-3.1
-6.0
-10.5
-14.0
-20.0 		116.9
114.0
109.5
106.0
100.0 		9.8 x 10-3
5.0 x 10-3
1.8 x 10-3
8.0 x 10-4
2.0 x 10-4 		-20.1
-23.0
-27.4
-31.0
-37.0 		9.9
7.0
2.6
-1.0
-7.0 		82.9
80.0
75.6
72.0
66.0
.07
.05
.03
.02
.01 		-23.1
-26.0
-30.5
-34.0
-40.0 		96.9
94.0
89.5
86.0
80.0 		9.8 x 10-5
5.0 x 10-5
1.8 x 10-5
8.0 x 10-6
2.0 x 10-6 		-40.1
-43.0
-47.4
-51.0
-57.0 		-10.1
-13.0
-17.7
-21.0
-27.0 		62.9
60.0
55.6
52.0
46.0
7 x 10-3
5 x 10-3
3 x 10-3
2 x 10-3
1 x 10-3 		-43.1
-46.0
-50.5
-54.0
-60.0 		76.9
74.0
69.5
66.0
60.0 		9.8 x 10-7
5.0 x 10-7
1.8 x 10-7
8.0 x 10-8
2.0 x 10-8 		-60.1
-63.0
-67.4
-71.0
-77.0 		-30.1
-33.0
-37.4
-41.0
-47.0 		42.9
40.0
35.6
32.0
26.0
7 x 10-4
5 x 10-4
3 x 10-4
2 x 10-4
1 x 10-4 		-64.1
-66.0
-70.5
-74.0
-80.0 		56.9
54.0
49.5
46.0
40.0 		9.8 x 10-9
5.0 x 10-9
1.8 x 10-9
8.0 x 10-10
2.0 x 10-10 		-80.1
-83.0
-87.4
-91.0
-97.0 		-50.1
-53.0
-57.4
-61.0
-67.0 		22.9
20.0
15.6
12.0
6.0
7 x 10-5
5 x 10-5
3 x 10-5
2 x 10-5
1 x 10-5 		-84.1
-86.0
-90.5
-94.0
-100.0 		36.9
34.0
29.5
26.0
20.0 		9.8 x 10-11
5.0 x 10-11
1.8 x 10-11
8.0 x 10-12
2.0 x 10-12 		-100.1
-103.0
-107.4
-111.0
-117.0 		-70.1
-73.0
-77.4
-81.0
-87.0 		2.9
0
-4.4
-8.0
-14.0
7 x 10-6
5 x 10-6
3 x 10-6
2 x 10-6
1 x 10-6 		-104.1
-106.0
-110.5
-114.0
-120.0 		16.9
14.0
9.5
6.0
0 		9.8 x 10-13
5.0 x 10-13
1.8 x 10-13
8.0 x 10-14
2.0 x 10-14 		-120.1
-123.0
-127.4
-131.0
-137.0 		-90.1
-93.0
-97.4
-101.0
-107.0 		-17.1
-20.0
-24.4
-28.0
-34.0
7 x 10-7
5 x 10-7
3 x 10-7
2 x 10-7
1 x 10-7 		-124.1
-126.0
-130.5
-134.0
-140.0 		-3.1
-6.0
-10.5
-14.0
-20.0 		9.8 x 10-15
5.0 x 10-15
1.8 x 10-15
8.0 x 10-16
2.0 x 10-16 		-140.1
-143.0
-147.4
-151.0
-157.0 		-110.1
-113.0
-117.4
-121.0
-127.0 		-37.1
-40.0
-44.4
-48.0
-54.0

Conversion Between Field Intensity (Table 1) and Power Received (Table 2).

Power received (watts or milliwatts) can be expressed in terms of field intensity (volts/meter or µv/meter) using equation [3]:

or in log form:

[4] 10 log Pr = 20 log E + 10 log G - 20 log f + 10 log (c2/480 2)

Then

[5] 10 log Pr = 20 log E1 + 10 log G - 20 log f1 + K4

Values of K4 (dB)
Pr E1 f1(Hz) f1(MHz) f1(GHz)
Watts
(dBW) 		volts/meter 132.8 12.8 -47.2
µv/meter 12.8 -107.2 -167.2
mW
(dBm) 		volts/meter 162.8 42.8 -17.2
µv/meter 42.8 -77.2 -137.7

The derivation of equation [3] follows:

EquationReferenceTerms
PD= E2/120 Eq [1], pg 4-1.1(v2/ohm)
Ae = 2G/4 Eq [8], pg 3-1.7 (m2)
Pr = PDAe Eq [2], pg 4-3.3 (W/m2)(m2)
therefore:
Pr = ( E2/120 )(2G/4) 		(v2/m2ohm)(m2)
= c /f pg 2-3.1 (m/sec)(sec)
therefore:
Pr = ( E2/480 2 )( c 2 G/f 2)		Equation [3]

 (v2/m2ohm)(m2/sec2)(sec2) or v2/ohms = watts

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