IEEE Instrumentation & Measurement - September 2023 - 29

can be ignored, then, according to [3], the resonant frequency
( f1
), allows the direct computation of the capacitance of C with:
ClZ
= −1 10 1
ωβ
/ cot()
(1)
where Z0 and l are respectively the characteristic impedance
and the physical length of the resonant coaxial-line; and β1l is
the electrical-line length given by:
βπ
112l = lf c/
where c is the speed of light, since the line's dielectric is air.
The Q factor of the system (Qm
(2)
) is affected by two additional
losses sources: the finite Q factor of the line and the test
fixture resistance (rf
). Hence, for de-embedding the capacitor's
ESR, two additional equations are needed. These equations are
based on the properties of, respectively, an open-circuited line
whose Q factor: ′Q1 at f1
and the short-circuited line whose Q factor: Q1 at f1
by rf. If we assume that Qm Q ′1, Q1, and f0
,
capacitor's ESR and the test fixture resistance (rf
is not affected by neither the ESR nor rf
is affected
;
are known, then the
) can be computed
as follows. Using the normal definition of the Q-factor
for a passive system [4], [5] demonstrated:
ESR Z= ()()(
1/Qm
× ()−
0 10
π ββ/ / /
 ()/ 1′  −Qrf


with:
r Z ff Q Q/ 1
f =()() ()−
π 0 10 1
//4 11/
 ()′ 
(4)
The determinations of the parameters in (3) and (4) are detailed
in the calibration section that follows.
Calibration
Measurement of the Q Factors
The open-circuited Q-factors are first measured by removing
the plunger and the capacitor C shown in Fig. 1. According to
[3], the resonance frequencies are:
f kf ()k integer
20k 2=
(5)
at which can be measured the corresponding open-circuited
Q-factors: Q2k
. The short-circuited Q-factors are then measured
by pushing the plunger shown in Fig. 1 in contact with the center
conductor. According to [3], the resonance frequencies are:
f12k = +() ()k integer
+
12kf0
(6)
at which can be measured the corresponding short-circuited
Q-factors: Q k12+ .
Expressions (5) and (6) show that the Q-factors can only be
measured at resonance frequencies that differ from the system's
resonant frequency ( f1
September 2023
). As the Q factors are a function
f1
with:
r rf f (1 x′)
f ASTM
() = () −
01 0
/
, f0
(12)
Separately, these equations are correct, but their association
hides an error that can be revealed by substituting in (12):
(11) and (7) applied to Q′0
. After rearrangement, we obtain:
r ()f ASTM
=()()
× ()() −()()
π 0 10
0 1
4


IEEE Instrumentation & Measurement Magazine
Z ff
Q ff Q f f
11
//
// //
−x′′−
2 1 2
x


(13)
29
4 111
11
1/ cot
2
f f sin l)− ( )( )
2
l


(3)
Frequency Extrapolation of the Measured Q Factors: The
solution specified by the ASTM [2] consists in replacing the
exponent 0.5 of the classical square root function [3], [4] by
experimental exponents whose symbols are: x and x′, for
the short-circuited and the open-circuited line, respectively.
Moreover, for minimizing the extrapolation errors, the line's
resonance frequencies must be closest to the system's resonant
frequency ( f1
). According to (5) and (6), an application of these
constraints yields, for the open-circuited line Q-factor ( ′Q1):
′
QQ =()x′
1 2 12f f
//
and for the short-circuited line Q-factor (Q1):
QQ f f
1 0 1
// 0
=()x
(7)
of frequency due to the skin effect, it is mandatory to perform
some frequency extrapolations that are presented in the next
paragraphs.
(8)
The exponents x and x′ are determined by remarking that on
a logarithmic plot, the exponential functions (7) and (8) are
linear functions whose slopes are x and x′, respectively. For determining
each slope, a second resonance closest to f1
is chosen.
Applying these constraints, yields:
x ′ = () ()log Q Q log f f
= () ()3 0 30
4 2 42
x log Q Q log f f
/
/
/ /
/ /
that is presented in the next section.
(9)
(10)
The procedure specified by the ASTM [2] follows the aforementioned
one with the exception of the frequency extrapolation of
rf
Frequency Extrapolation of the Test Fixture's Resistance Specified
by the ASTM: The ASTM [2] specifies the following two
step frequency extrapolation of rf
is added to the symbol of rf
. Note: The subscript (ASTM)
derived with the ASTM method,
whereas no subscript is added to the proposed corrected expression.
First
step: according to [5], the test fixture resistance (r0
computed at f0
with:
r
00 Q Q0
= 41 1
Z
)( ) −
(π // / 0
 ()′ 
) is
(11)
Second step: according to [2], extrapolation of r0 from f0
to
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