FIELD  AND  POWER  DENSITY

ATMOSPHERIC REFRACTION

Due to variations in the density of the atmosphere, electromagnetic radiation is refracted according to the varying refractive index with height. Variations decreases linearly with height under standard conditions (no localized disturbances). As a result, the wave gradually curves until, if the wavelength and angle are correct, the signal bends back toward the earth (see graphic below).

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 (P_D) terminology is more often used than field strength.

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

where P_D 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 P_D are in mW/cm², then P_D (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/Z_o. 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.

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, A_e relates free space power density P_D with received power, P_r , i.e. P_r = P_D A_e. A_e 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.

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 P_r = 20 log E + 10 log G - 20 log f + 10 log (c²/480 ²)

Then

[5] 10 log P_r = 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:

Equation	Reference	Terms
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