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

IEEE Instrumentation & Measurement - September 2023 - Cover1

IEEE Instrumentation & Measurement - September 2023 - Cover2

IEEE Instrumentation & Measurement - September 2023 - Contents

IEEE Instrumentation & Measurement - September 2023 - 2

IEEE Instrumentation & Measurement - September 2023 - 3

IEEE Instrumentation & Measurement - September 2023 - 4

IEEE Instrumentation & Measurement - September 2023 - 5

IEEE Instrumentation & Measurement - September 2023 - 6

IEEE Instrumentation & Measurement - September 2023 - 7

IEEE Instrumentation & Measurement - September 2023 - 8

IEEE Instrumentation & Measurement - September 2023 - 9

IEEE Instrumentation & Measurement - September 2023 - 10

IEEE Instrumentation & Measurement - September 2023 - 11

IEEE Instrumentation & Measurement - September 2023 - 12

IEEE Instrumentation & Measurement - September 2023 - 13

IEEE Instrumentation & Measurement - September 2023 - 14

IEEE Instrumentation & Measurement - September 2023 - 15

IEEE Instrumentation & Measurement - September 2023 - 16

IEEE Instrumentation & Measurement - September 2023 - 17

IEEE Instrumentation & Measurement - September 2023 - 18

IEEE Instrumentation & Measurement - September 2023 - 19

IEEE Instrumentation & Measurement - September 2023 - 20

IEEE Instrumentation & Measurement - September 2023 - 21

IEEE Instrumentation & Measurement - September 2023 - 22

IEEE Instrumentation & Measurement - September 2023 - 23

IEEE Instrumentation & Measurement - September 2023 - 24

IEEE Instrumentation & Measurement - September 2023 - 25

IEEE Instrumentation & Measurement - September 2023 - 26

IEEE Instrumentation & Measurement - September 2023 - 27

IEEE Instrumentation & Measurement - September 2023 - 28

IEEE Instrumentation & Measurement - September 2023 - 29

IEEE Instrumentation & Measurement - September 2023 - 30

IEEE Instrumentation & Measurement - September 2023 - 31

IEEE Instrumentation & Measurement - September 2023 - 32

IEEE Instrumentation & Measurement - September 2023 - 33

IEEE Instrumentation & Measurement - September 2023 - 34

IEEE Instrumentation & Measurement - September 2023 - 35

IEEE Instrumentation & Measurement - September 2023 - 36

IEEE Instrumentation & Measurement - September 2023 - 37

IEEE Instrumentation & Measurement - September 2023 - 38

IEEE Instrumentation & Measurement - September 2023 - 39

IEEE Instrumentation & Measurement - September 2023 - 40

IEEE Instrumentation & Measurement - September 2023 - 41

IEEE Instrumentation & Measurement - September 2023 - 42

IEEE Instrumentation & Measurement - September 2023 - 43

IEEE Instrumentation & Measurement - September 2023 - 44

IEEE Instrumentation & Measurement - September 2023 - 45

IEEE Instrumentation & Measurement - September 2023 - 46

IEEE Instrumentation & Measurement - September 2023 - 47

IEEE Instrumentation & Measurement - September 2023 - 48

IEEE Instrumentation & Measurement - September 2023 - 49

IEEE Instrumentation & Measurement - September 2023 - 50

IEEE Instrumentation & Measurement - September 2023 - 51

IEEE Instrumentation & Measurement - September 2023 - 52

IEEE Instrumentation & Measurement - September 2023 - 53

IEEE Instrumentation & Measurement - September 2023 - 54

IEEE Instrumentation & Measurement - September 2023 - 55

IEEE Instrumentation & Measurement - September 2023 - 56

IEEE Instrumentation & Measurement - September 2023 - 57

https://www.nxtbook.com/allen/iamm/26-6

https://www.nxtbook.com/allen/iamm/26-5

https://www.nxtbook.com/allen/iamm/26-4

https://www.nxtbook.com/allen/iamm/26-3

https://www.nxtbook.com/allen/iamm/26-2

https://www.nxtbook.com/allen/iamm/26-1

https://www.nxtbook.com/allen/iamm/25-9

https://www.nxtbook.com/allen/iamm/25-8

https://www.nxtbook.com/allen/iamm/25-7

https://www.nxtbook.com/allen/iamm/25-6

https://www.nxtbook.com/allen/iamm/25-5

https://www.nxtbook.com/allen/iamm/25-4

https://www.nxtbook.com/allen/iamm/25-3

https://www.nxtbook.com/allen/iamm/instrumentation-measurement-magazine-25-2

https://www.nxtbook.com/allen/iamm/25-1

https://www.nxtbook.com/allen/iamm/24-9

https://www.nxtbook.com/allen/iamm/24-7

https://www.nxtbook.com/allen/iamm/24-8

https://www.nxtbook.com/allen/iamm/24-6

https://www.nxtbook.com/allen/iamm/24-5

https://www.nxtbook.com/allen/iamm/24-4

https://www.nxtbook.com/allen/iamm/24-3

https://www.nxtbook.com/allen/iamm/24-2

https://www.nxtbook.com/allen/iamm/24-1

https://www.nxtbook.com/allen/iamm/23-9

https://www.nxtbook.com/allen/iamm/23-8

https://www.nxtbook.com/allen/iamm/23-6

https://www.nxtbook.com/allen/iamm/23-5

https://www.nxtbook.com/allen/iamm/23-2

https://www.nxtbook.com/allen/iamm/23-3

https://www.nxtbook.com/allen/iamm/23-4

https://www.nxtbookmedia.com