Using a waveguide-to-coax adapter as an EMC antenna

Didier Juges, July 28, 1999

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It is current practice to use a waveguide to coax adapter feeding a spectrum analyzer to check microwave equipment for RF leakage. While this is useful to pinpoint the origin of a leak if one is suspected, it is more difficult to use the readings to ascertain compliance to regulatory requirements such as MIL-STD-461 or Part 15 of FCC Rules.
To convert a power reading in dBm into a field strength in dBµV/m, the antenna factor of the device used must be computed.
The antenna factor is the term used in EMC testing to convert a voltage or power level fed by an antenna to an EMI analyzer into the field strength units (usually dBµV/m) of the electromagnetic field producing that voltage or power.
In a 50 ohm system, the antenna factor (expressed in dB(m^-1)) of an antenna of absolute gain G (expressed in dB) at a wavelength L (expressed in meters) is:
    AF[dB(m^-1)] = 19.8 - 20*log(L[m]) - 20*log(G[dB])
To convert a voltage reading V (expressed in dB(µV)) into an electric field E (expressed in dB(µV/m),) use the following formula:
    E[dB(µV/m)] = V[dB(µV)] + AF[dB(m^-1)]
To convert a power reading P across a 50 ohm impedance (expressed in dBm) into an electric field E (expressed in dBµV/M,) use the following formula:
    E[dBµV/M] = P[dBm] + AF[dB(m^-1)] + 107

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Example:

Conventional wisdom has it that within its operating bandwidth, an open WR-90 waveguide adapter has a gain of approximately 7dB. Its antenna factor at 7.5 GHz (4.0 cm wavelength) is:     AF(dB(m^-1)) = 19.8 - 20*log(0.04) - 20*log(7) = 30.9 dB
If we measure -70 dBm on the receiver/spectrum analyzer, the E field is:
    E = -70 + 30.9 + 107 = +67.9 dBµV/m

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Reference:

John Osburn, EMC Antenna Parameters and Their Relationships, ITEM 1996 
source: http://www.ko4bb.com/Test_Equipment/EMI_Measurements.php

What is a Decibel ?
Decibel is the unit used to express relative differences in signal strength.
Decibel is expressed as the base 10 logarithm of the ratio of the power of two signals : dB = 10 x Log₁₀ (P1/P2)
Where Log₁₀ is the base 10 logarithm and P1 and P2 are the powers to compare
(Log₁₀ is different from Ln or LN = Neparian Logarithm, base e logarithm)
Signal amplitude can also be expressed in dB. Since power is proportional to the square of a signal's amplitude, dB is expressed as follows : dB = 20 x Log₁₀ (V1/V2)
Where V1 and V2 are the amplitudes to compare
1 Bell (not really used in current) = Log₁₀(P1/P2)
1 decibel (dB) = 1 Bell / 10 = 10 * Log₁₀(P1/P2)
dBr = dB (relative) = dB = 10 * Log₁₀(P1/P2)

base 10 Logarithm rules
Log10 (AxB) = Log10 (A) + Log10 (B)
Log10 (A/B) = Log10 (A) - Log10 (B)
Log10 (1/A) = - Log10 (A)
Log10 (0,01) = - Log10 (100) = -2
Log10 (0,1) = - Log10(10) = - 1
Log10 (1) = 0
Log10 (2) = 0,3
Log10 (4) = 0,6
Log10 (10) = 1
Log10 (20) = 1,3 Log10(2 x 10) = Log10(2) + Log10(10) = 1 + 0,3
Log10 (100) = 2
Log10 (1 000) = 3
Log10 (10 000) = 4

Logarithm and dB (decibel)
Power Ratio	dB = 10 x log10 (Power Ratio)
AxB	x dB = 10 x Log10(A) + 10 x Log10(B)
A/B	x dB = 10 x Log10(A) - 10 x Log10(B)
1/A	x dB = + 10 x Log10(1/A) = - 10 x Log10(A)
0,01	- 20 dB = - 10 x Log10(100)
0,1	- 10 dB = 10 x Log10(1)
1	0 dB = 10 x Log10(1)
2	3 dB = 10 x Log10(2)
4	6 dB = 10 x Log10(4)
10	10 dB = 10 x Log10(10)
20	13 dB = 10 x (Log10(10) + Log10(2))
100	20 dB = 10 x Log10(100)
1 000	30 dB = 10 x Log10(1 000)
10 000	40 dB = 10 x Log10(10 000)

dBm = dB milliwatt = 10 x Log10 (Power in mW / 1 mW)
Power	Ratio	dBm = 10 x Log10 (Power in mW / 1 mW)
1 mW	1 mW / 1 mW = 1	0 dBm = 10 x Log10(1)
2 mW	2 mW / 1 mW = 2	3 dBm = 10 x Log10(2)
4 mW	4 mW/1mW=4	6 dBm = 10 x Log10(4)
10 mW	10 mW/1mW=10	10 dBm = 10 x Log10(10)
0,1 W	100 mW/1mW=100	20 dBm = 10 x Log10(100)
1 W	1000 mW/1mW=1000	30 dBm = 10 x Log10(1 000)
10 W	10 000mW/1mW=10 000	40 dBm = 10 x Log10(10 000)

dBW = dB Watt = 10 x Log10 (Power in W / 1 W)
Power	Ratio	dBW = 10 x Log10 (Power in W / 1 W)
1 W	1 W / 1 W = 1	0 dBW = 10 x Log10(1)
2 W	2 W / 1 W = 2	3 dBW = 10 x Log10(2)
4 W	4 W / 1 W = 4	6 dBW = 10 x Log10(4)
10 W	10 W / 1 W = 10	10 dBW = 10 x Log10(10)
100 mW	0,1 W / 1 W = 0,1	-10 dBW = -10 x Log10(10)
10 mW	0,01 W / 1 W = 1/100	-20 dBW = -10 x Log10(100)
1 mW	0,001W/1W=1/1000	-30 dBW = -10 x Log10(1000)

Power/Voltage Gain
dB	Power Ratio	Voltage Ratio		dB	Power Ratio	Voltage Ratio
0	1,00	1,00		10	10,00	3,16
1	1,26	1,12		11	12,59	3,55
2	1,58	1,26		12	15,85	3,98
3	2,00	1,41		13	19,95	4,47
4	2,51	1,58		14	25,12	5,01
5	3,16	1,78		15	31,62	5,62
6	3,98	2,00		16	39,81	6,31
7	5,01	2,24		17	50,12	7,08
8	6,31	2,51		18	63,10	7,94
9	7,94	2,82		19	79,43	8,91
10	10,00	3,16		20	100,00	10,00

Attenuation (dB) = 10xLog₁₀(Pin/Pout) = 20xLog₁₀(Vin/Vout)
Gain (dB) = 10xLog₁₀(Pout/Pin) = 20xLog₁₀(Vout/Vin)
Gain (dB) = - Attenuation (dB)
Power (W) = voltage (V) x Voltage (V) x Z (Ohm) = V x V / Z = V x I
So Gain(dB) = 10 x Log₁₀(Pout/Pin) = 10 x Log₁₀((Vout x Vout / Zout) / (Vin * Vin / Zin))
if Zin = Zout, Gain(dB) = 10 x Log₁₀(Vout x Vout / Vin /Vin) =
10 x (Log₁₀(Vout / Vin) + Log₁₀(Vout / Vin)) = 10 x 2 x Log₁₀(Vout / Vin) = 20 x Log₁₀(Vout / Vin)
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