Calculating microwave opacity from a weakly absorbing gas mixture using a resonator requires measuring the quality factor of that resonator which necessitates accurately determining the center frequency (f(sub O)) and the half power bandwidth (Delta-f) of a noisy resonant line. The center frequency can be determined very accurately and varies very little over many measurements (a few kHz at GHz frequencies or a few hundredths of a percent). The greater source of error in estimating the Q of a resonator come from the bandwidth measurements. The half power bandwidth is determined essentially by eye-fitting a curve over a noisy resonant line and measuring with a spectrum analyzer

Deboer, David

English

NASA Technical Reports Server

NASA-CR-193389
APPENDIX B
H2S Data Reduction
David DeBoer
July 7, 1993
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1 Calculating Absorption of a Lossy Gas Using Q
Calculating microwave opacity from a weakly absorbing gas mixture using a resonator requires
measuring the quality factor of that resonator which necessitates accurately determining the center
frequency (.fo) and the half power bandwidth (a/) of a noisy resonant fine. The center frequency
can be determined very accurately and varies very little over many measurements (a few kHz at
GHs frequencies or a few hundredths of a percent}. The greater source of error in estimating the Q
of a resonator comes from the bandwidth measurements. The half power bandwidth is determined
essentially by eye-fitting a curve over a noisy resonant line and measuring with a spectrum analyser.
Assuming uncorrelated measurement error (an unbiased observer} the best estimation of the
bandwidth is obviously the mean of many measurements. Therefore,
N1
at. at,
i=l
where a]_ are the individual bandwidth measurements. To determine the accuracy of the measure-
ments we calculate the sample variance,
N
1=
i=l
and finally the variance of our estimation from the *true" bandwidth due to electrical noise,
(2)
where tu is the =student-t" for the la confidence level given a Ganuian distribution. Recall that
the student-t is a distribution to characterize the confidence level of a finite sample set where the
degrees of freedom are the number of samples. For ten measurements, t¢ -- 1.1 [5, pp. 255, 260].
One can also calculate some figures of merit to determine the stationarity of the data [5, pp.
255-6]. Divide the ten measurements into two groups of five and calculate the sample variance
(s_t, s_) and mean (mr, re.j) of each group where sx > s2 and ml > m2. We can define a figure of
merit for stationarity of the variance as Fo = 0.156s_/s]. If Pg._._hen the variance iJ likely to
be stationary. For the mean, we define to = 0.974(ml - rr_)/_/sz = + s_ as the figure of merit and if
to < 1 then the mean is likely to be stationary. If either one of these conditions does not hold the
measurement is not stationary and should be considered suspect.
There are also instrument errors which must be taken into consideration as welL These stem
from the limited accuracy of the spectrum analyser and are calculated as follows [2]:
ao _ lO-Sfo + 0.15RBW + 0.05SPAN + Lo + 250 Hs (4)
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on _ 10-_BW + 200N + 4Lsw Hi (5)
where ao is the standard deviation of the center frequency measurement due to the spectrum ana-
lyser accuracy, aA the standard deviation of the bandwidth measurement, RBW is the resolution
bandwidth, L the least significant digit, BW the bandwidth and N the harmonic number (see tables
1 aad 2).
Freq. band S X K
RBW[H,] 3OO 1000 3OOOO
SPAN {kHs] 50 100 2000
N 1 2 4
Lo 10 100 1000
Lsw 10 100 1000
Table 1: Spectrum analyser parameters
ao [kHs] aA [Hz]
S 25 240
X 90 800
K 320 4815
Table 2: Instrument uncertaintyfor the frequency bands.
Itisthen possibletoestimate the measured bandwidth and characterizeitsuncertainty.What we
measure from a conventionM spectrum analyser,however, isnot the true spectrum of the resonator
but rather the convolutionofthe resonatorspectrum and the sweep filterspectrum. IfY(f) isthe
output, X(/) the input and H(f) the filter,then
Y(f) = H(f) * X(f). (6)
In the time domain, we have y(t) = h(t)z(t) where z/(t) is the inverse Fourier transform of Y(f) and
so on. Therefore, the signs] spectrum is,
X(f) = Y'ly(t)lh(t)] {7)
where Y'[.] denotes a Fourier "/Yansform. If we assume a Gaussian filter and a Gaussian input then
the actual signs] bandwidth is simply
AfQ = _/Ap - RBW _ (8)
where AfQ isthe "true"halfpower bandwidth and AI isthe measured mean bandwidth. The Q
of the resonatoristhen simply
fo (9)Q;
We will discuss the error in the next section.
To cs]culate the absorption recall that
a ,/,_1 + (e,,/(,)2_ 1 (10)
= V's/1 + (e"/e')2 + 1
B3
0.2
0.15
_0.I
0.05
0 o
Figure 1: ,'-/_ vs.
I ) I e"
-- Full ._
! l |
0.1 0.2 0.3 0.4
_"/_' for full expression (eq. i0) and approximation (eq. 11).
where _ = 2_r/A (with the gas present), _ is the absorption and _ = d-je" the complex permittivity.
Since _,t _ e, for gases in the microwave region this reduces to
- _ --. (11)
2c'
Note that this is field attenuation--to get power attenuation one must multiply this by 2.
The Q of a dielectric, Qd, is given by
1 _(r _t
= = -- _ - (12)Qd tan6 we" + o _"
where the approximation holds for a gas at microwave frequencies. Therefore, we see that
,_ 1 (13)
_ 2Q---_= _ Q-_
To determine Qd we must examine the Q of a lossy, coupled resonator.
The measured quality factor of an evacuated resonator can be given by [1, p. 404]
1 I 1 I
Qr--_ = Qc---_+ Q,_(1) + Q,_(2) (14)
and in the presence of a lossy gas
1 1 1 1 1
QT--7= Qo--7+ _ + _ + Q..(,)
where
Qr,.g = the total (measured) Q without and with the gas
Qce,g = cavity Q without and with the gas
Qe_.o(z,21 = external Q on both sides without and with the gas
Qd = Q of the gas.
(15)
Since a symmetrical cavity is used, assume that Qeg._(z) = Qeg,_(2) - Qeg.v. Additionally 11, p.
404]
-- = -- (16)
2QTo,_ Qe¢._
B4
where t 0 and t_ are the transmissivities of the resonator at resonance with and without the gas
respectively (assuming a lossleu gas). Substituting into equations 14 and 15, subtracting and solving
for 1/Qd yields
1 1-V_ _ 1-V_ 1 1
+ (17)Qd Qr¢ Qr, Qc_ Qco"
Note that Qc_.o takes conductivity losses and diffraction losses into account. The Q for finite
conductivity, Qa, of a circular cavity varies as l/v/_ [3, p. 352] we Bee therefore that
= i_ _ 1. 08)Q_,_____.
QCr,O
For diffraction, likewise, the ratios of the Q's will be on the order of the ratios of the wavelengths
and also equal to about 1, therefore Qc_ e# Qco. Substituting these approximations into 17 yields
1 _ 1 -..._ 1 - '_v (19)
Qd Qrg Qr_
Recall that tg assumes a lossless gas. Actually, the measured transmissivity, to,.,ea° , equals toe-an.Is
or
to = to,m.eaoe ad'1! (20)
where dell = QA/2w isthe effectivepath lengthof the resonator.We can then solveiterativelyfor
the actual a.
We thereforesee that
lr (1-_g 1 -- V_'_v_ Nepers/m (21}
a = _ \ Qro QTv ]
where A is in meters. In dB, we can express a as
/
=s.6s6i dB/m. (22)\ Qro Qr_ ]
The above analysis neglects the effects of dielectric loading [4]. Using an amount of lossless
gas with the same refractivity value as the lossy gas in lieu of merely an evacuated resonator will
remove the effects of the dielectric loading, provided you are far enough from the line center of the
absorptionlineso that anomalous dispersioni_not present.
2 Calculating the Error
Define a quantity
= g(/o_, Af#, fou, A f,) = _¢Afg "i, A/, (23)
/. /o.
where A f0 and Afu are the loaded and unloaded half power bandwidths and fog and _o_ are the
loaded and unloaded center frequencies. We define this quantity since we wish to ignore the I/A
dependence in front of a in the interest of symmetry, Then to first order
Og 6/o_+ 6_Io+ 61o_+
_ .,,.1 ,:4,
= r_ - ro.
B5
Note that this ignores variation in _/= 1 - V_.
Assume that < 5_ >= 0 that is, we axe dealing with sero-mean processes; therefore
o_ffi<6_2>=< r_ > + < r_ > -2 < ror. > (25)
The bandwidth variation consists of instrument accuracy and noise; i.e. 6Af_ = 6Afs A + 5Afjv_.
Throughout, we will assume that electrical noise and the spectrum analyser accuracy are uncorre-
late& Since we have assumed sero-mean processes and uncorrelated accuracy/noise then
2 (29)< 6_ > = o"o
< 6AI,_ > = O_ +,_,. (30)
The uncertainty of the measured gas absorption, % is then
8.686_"
+ _cr¢ dB/m (31)
where we have neglected the uncertainty in the measurement of A. There are three cases that we
will consider: the uncorrelated case, the "worst" correlation case and the =best D correlation case.
2.1 Uncorrelated Case
In the uncorre]ated case, where naturally all variations are uncorrelated with one another, we find
that
_.'_ LQ_[°°2 o_,,] (32)< r,_>= +,4 +
< Far, >= o. (33)
2.2 Worst Case
In the worst case scenario (the greatest error), 6)¢,6Afg and 6roSAry terms axe completely correlated
while the other terms are completely anti-correlated and, of course, the electrical noise terms are
uncorrelated with the instrument accuracy terms. This yields
"/_ ro.o2 20c_'A 1 (3,)
<r_> = _ lq_+4+=_,+ Q, j
0"o0"_ 0"o0 A ]
_,_o [ _2o +o_+ + . (35)
< r, ro > = fo.lo_[QgQ,, _ Q,, J
This is the case typically used for the error, since a few small error terms have been neglected in
this derivation.
where
r_ > = I_ Q_ < > Q,
_,_. [<61o_61o_> <6Io_Af,> <6Io,6A1,>.12, )< r, rg > = fojo_ qgq,, + < 5Af°6Af" > qg - q,_
Q' = z_--7,," (28)
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2..$ Best Case
The best case scenario is the reverse of the worst case; that is 6f_6Afg and 6fgSAf.terms are
completely anti-correlated while the other terms are completely correlated. This yields
"_[ °_°-. a_v, 2aoa_ ]Q, (36)<r_> = _ _+_+
"_._, [ _o' _o_ _o_l (37)
References
[1] Ginzton, E.L., Microwave Measurements, 1957, McGraw-HiU Book Co., Inc.
[2] HP Manual for HP8562B
[3] Posar, D. M., Microwave Engineering, 1990, Addison-Wesley Publishing Company, Inc.
[4] Spilker, T.R., New L_boratory Measurements on Ammonia's Inversion Spectrum, With Impli-
cations for Planetary Atmosphere's, J. Geophys. Res., 1993, 98:5539
[5] Warner, F.L., Microwave Attenuation Measurements, 1977, Peter Peregrinus Ltd.


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