Table 12. Expanded and Assigned Total Uncertainties of Using the Type-2 Bridge to Measure
Two-Terminal Coaxial Mica Capacitors and Open-Circuit Terminations at 1 kHz

(All values are in ppm, except nominal values of capacitors)

Coaxial Mica Capacitors Open-Circuit Terminations
Nominal Value of Capacitors (pF) 1000 5000 to 100 000 0.172 2.67
Components of
Standard Uncertainties
Temperature 5 15 150 150
Relative Humidity 16 16 400 100
Hysteresis 20 20 20 20
Voltage 2 2 5 5
Type B Standard Uncertainty (u,) 26.17 29.75 427.70 181.45
Assigned uy, 27 30 450 200
Type A Standard Uncertainty (u,) 10 10 200 60
Expanded Uncertainty (U) 57.58 63.25 984.89 417.61
Assigned
Total Uncertainty (U,) 60 65 1000 450

51



Table 11 contains the calibration uncertainties for two types of HF coaxial air capacitors. The lower-
valued type, with nominal values that range from 1 pF to 20 pF, has a relatively flat frequency
response to a few hundred megahertz (see Fig. 3 in Ref. 12). These capacitors can be employed as
low-capacitance terminations for any two-port device. The mid-valued type, with nominal values
that range from 50 pF to 1000 pF, has negligible changes in effective capacitance with frequency up
to the megahertz range (see Fig. 2 in Ref. 11).’ These capacitors are used mainly to calibrate
capacitance bridges and other instruments at high frequencies from their reference values at 1 kHz.

The higher-valued coaxial capacitance standards use mica as a dielectric. The nominal values of
these capacitors range from 0.001 pF to 0.1 uF. Like the mid-valued type, these capacitors also have
very low changes in effective capacitance below 100 kHz (see Fig. 1 in Ref. 12). The main
application for this type of capacitor is to calibrate instruments at higher frequencies from their

reference values at 1 kHz. Table 12 gives the various calibration uncertainties for this type of
capacitor.

The open-circuit terminations are useful as capacitance standards at low frequency for calibration
bridges, and as shield caps for open-circuit lines in establishing initial conditions of line length and
signal phase [10]. There are two types of open-circuit terminations, depending on the effective
position of termination (plane position). The plane positions of these terminations are 0.26 cm and
4 cm, which have capacitance values of 0.172 pF and 2.670 pF, respectively, at low frequencies.
NIST provides calibration services for these terminations at a frequency of 1 kHz with the calibration
uncertainties given in Table 12.

4, UNCERTAINTY ANALYSIS FOR THE TYPE-12 SYSTEM

The original unity-ratio admittance bridge, known as the No. 12 type, was developed by Bell
Laboratories in 1942 [13] for precision measurements of capacitance up to 1.11 F and conductance
up to 1000 pS. Later on, an improved Type-12 capacitance bridge was designed, also by Bell
Laboratories, with a capacitance divider to measure low capacitance values and multi-range, direct
reading conductance standards. A complete schematic diagram of the Type-12 bridge is shown in
Fig. 5a. Specifications of bridge components and internal standards, and operational procedures of
the Type-12 bridge are described in detailed by Wilhelm [14].

There are two mica capacitance dials, three air capacitance dials, and two conductance (resistance)
dials, all with ten steps in each dial, in the Type-12 bridge. The mica dials range from 0.1 pF to 1 pF
and from 0.01 uF to 0.1 uF, and the air dials range from 0.001 uF to 0.01 uF, 0.0001 uF to
0.001 uF, and 0.00001 pF to 0.0001 pF. A 0.005 pF mica capacitor, measured using the Type-2
bridge, is used as a transfer standard to calibrate the 0.01 uF steps of both the mica capacitance and
the air capacitance dials of the Type-12 bridge. All other steps of mica and air capacitance dials are
calibrated from the 0.01 uF step, used as a reference. The conductance dials range from 10 pS to
100 pS and from 100 pS to 1000 4S, and are calibrated against resistance standards. Both
capacitance and conductance dials are adjusted to obtain a final balance at the time of mcasurcment

17


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Table 17a. Summary of Expanded and Assigned Total Uncertainties in Capacitance Measurements Using
the Type-12 Bridge to Measure Two-Terminal Mica Capacitors at 100 Hz, 1kHz, and 10 kHz

(All values are in ppm - except nominal values of the unknown capacitance standards)

Unknown Standard C, (uF) 0.001 | 0.002 | 0.005} 0.01 | 0.02 | 0.05 | 0.1 0.2 | 05 1.0

( Frequency = 100 Hz)

Type-A Standard Uncertainty (u,) | 46 | 25 | 25 | 38 | 25 | 25 | 25 | 25 | 25 | 25

Type-B Standard Uncertainty (u,) 65 40 25 40 40 40 40 40 40 40

Expanded Uncertainty (U) 159.26] 94.34 | 70.71 | 110.34] 94.34 | 94.34 | 94.34 | 94.34 | 94.34 | 94.34

Assigned Total Uncertainty (U,) 160 | 120 | 100 | 120 | 120 | 120 | 120 | 120 | 120 | 120
oo : —

( Frequency = 1 kHz )

Type-A Standard Uncertainty (w,) 40 35 30 30 30 30 25 25 25 25

Type-B Standard Uncertainty (w,) | 65 | 40 | 25 | 40 | 40 | 40 | 40 | 40 | 40 | 40

Expanded Uncertainty (U) 152.64] 106.30] 78.10 | 100.00] 100.00} 100.00) 94.34 | 94.34 | 94.34) 94.34

Assigned Total Uncertainty (U,) 160 | 120 | 100 | 120 | 120 | 120 | 120 120 | 120 120

rs ee ee ee

( Frequency = 10 kHz )

Type-A Standard Uncertainty (w,) 35 25 25 25 25 25 28 25 25 28

Type-B Standard Uncertainty (u,) 65 40 25 40 40 45 65 105 | 240 | 475

Expanded Uncertainty (U) 147.65] 94.34 | 70.71 | 94.34 | 94.34 | 102.96] 141.55]215.87] 482.60] 951.65

Assigned Total Uncertainty (U,) 160 | 120 | 100 | 120 } 120 | 120 | 150 | 250 | 500 | 1000

60


AV Varo

Figure 5b. Simplified Circuit Diagram of the Type-12 Capacitance Bridge.

71


10:1 (and 1:10) ratio, respectively.

3.2 Dial Corrections

The internal capacitors of the Type-2 bridge are a set of eight three-terminal air capacitors, C, to Cg,
with adjustable dials, as shown in Fig. 3. One side of each capacitor is connected to the detector,
and the other side is connected to the transformer via dial switches, which have the range from -0.1
to +1.0 of full voltage. As shown in Fig. 2a, when a dial is set at any value from +0.1 up to +1.0,
the capacitor is connected to the "S" side of the bridge. When a dial is set to -0.1, it is connected
to the "X" side of the bridge. The -0.1 positions of the dials are useful in ratio measurements, or
measuring capacitors of lower than nominal values. Among these internal capacitors, which range
from 100 pF to 10° pF, only C;, the 1 pF capacitor is purposely temperature compensated. Since
air capacitors vary with the room temperature and humidity, these are calibrated prior to each
measurement. All dial capacitors are calibrated at a frequency of 1 kHz, and dial corrections are
applied to the measurement results.

3.2.1 Calibration of the Dial Capacitors

The nominal values of the internal capacitors are shown in Fig. 3 and given as:

C, = 100 pF = 10G,, (12)
C, = 10 pF = 10G,,

C,= 1 pF = 10G,

Cy = 10° pF = 10C,, and

Cy = 10° pF.

i

The first step in the calibration of capacitors C, to Cg is to calibrate C, against a 10 pF reference
standard using a 1:10 ratio by setting C, to the +1.0 switch position and connecting the reference
capacitor to the - 0.1 tap on the "X" side of the transformer. Afterward, internal calibrations of dial
corrections are performed by using C; (at the +1.0 switch position) as reference to calibrate C, (at
the - 0.1 switch position) and using C, (at the - 0.1 switch position) to calibrate C, (at the +1.0
switch position). Other internal capacitors can be calibrated similarly by running up and down the
range. The balance equation comparing C; with Cony can be written as:

Cj + j= m (Cera + Any * Lr KK)» (13)

where C; and C(,1) are nominal values of C; and C41) , respectively,


Primary
Capacitance
Laboratory

Impedance
Calibration
Laboratory

Reference Bank
4x 10 pF Oil-Bath Type
Fused-Silica Capacitance Standards

Transfer Standard

---| 10 pF Air-Bath Type

Fused-Silica Capacitance Standard

Reference Bank
4x10 pF Oil-Bath Type

Customer's

Fused-Silica

Fused-Silica Capacitance Standards

Capacitance Standards|

Secondary References

2x 100 pF & 2 x 1000 pF Air-Bath Type

Fused-Silica Capacitance Standards

Check Standards Customer's
1 pF, 10 pF, Nitrogen Dielectric,
2x 100 pF and 2 x 1000 pF Air Dielectric, and
Nitrogen Dielectric HF Coaxial
Capacitance Standards Capacitance Standards

Figure 1. Block Diagram of the Farad Transfer and Calibration Process Using

the Type-2 Bridge.

66


The ratio resistors are always set to be equal, (R, = R, ) so that Eq. (52) becomes:
Cy = (Cu - Co), (53)

which is the equation used to calculate the value of the unknown capacitor.

4.3.1.2 Analysis
The relationship among the variances of the terms of Eq. (52) can be estimated by:
= rye 2. 2a pp? 2 2 2
Var(Cy) = (A) (Cy - Co)” Sta + (AY (TL + (Ry Ry) V (Spe1° + Spe0° (54)

where 5,4, S;,1, and 5.9 are the standard uncertainties in the ratio ( contributed by the resistors) and
the air capacitor dials of both measurements, respectively.

By substituting Eq. (53) and ( R, = R, ) into Eq. (54), it becomes:
Var(C,) = 1 (4) Cx Sra]? +5075 (55)
2 2 2
where S10" = Spey +Sy¢9

From Eq. (39), the uncertainty contributed by the ratio resistors for two-terminal low-capacitance
measurements, [(2) C, 5,,], is 9 ppm (see the first three columns of Table 15a). The capacitance
uncertainties associated with each dial reading, s, and s,,,, can be estimated from the

0
corresponding terms of Eq. (43), as: re

2 2 2 2, (A)
Sro0 = (S650 * Sto F S50 * Se)» and

(A)

2 2 2 2
Spot = (Sari + Stpr” + Sasi + Seg.) (56)

The combined uncertainty of the internal air dial capacitors, s,., is estimated to be between 18 ppm
and 62 ppm. (See the first three columns of Tables 13a and 15a).

At frequencies of 100 Hz and 1 kHz, the uncertainties due to lead impedance are negligible for
capacitor values up to 0.005 pF. The uncertainty of evaluating plate P2, 0.012 pF, is found to be
between 3 ppm and 12 ppm.

At a frequency of 10 kHz, the lead impedance including series inductance is calculated from
Eq. (47), and the combined uncertainties in the bridge components range from 4 ppm to 12 ppm.

The Type-B standard uncertainties for two-terminal low capacitance value measurements, as well
as their individual uncertainty components, are included in the first three columns of Table 15a.

24


Table C-3. Data and Analysis of Conductance Corrections of ( 0.1 yF/step ) Mica Capacitance Dial
of the Type-12 Bridge at 10 kHz

(All values are in 0.001 pS, except the dial which is 0.1 »F/step)

DIAL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.x

6/67 492.308 | 1332.120]3119.400] 4189.440] 4649.620] 5955.480| 8594.100} 10269.700| 9946.360 | 13599.400
11/67 494.150] 1350.415|3135.955] 4212.320] 4746.648] 6035.588] 8716.003] 10398.200} 10250.800) 13701.600
5/68 494.148] 1332.136]3127.273] 4148.161] 4627.199} 5926.96 1! 8596.424| 10266.500} 10104.200} 13631.000}
9/69 486.015] 1327.123] 3093.330] 4176.263| 4637.970] 5938.353] 8662.91 0] 10264.940) 10127.130] 13492. 130)
12/70 503.190] 1387.243| 3 148.095] 4202.098| 4643.225} 5896.328] 8723.505| 10416.780} 10026.540} 13419.540}
9/72 488.310] 1347.368] 3120.208] 4302.298] 4807.788| 6 188.928} 9006.043) 10757.383} 10395.990} 13851.110}
6/74 441.985] 1194.242]2898.130]3874.418] 4197.330| 5365.218] 7934.030] 9787.693 | 9423.110 | 13694.270)
11/75 509.012} 1361.352]3197.980| 4337.532} 4874.010] 6246.2 12|9003.290} 10564.867| 10717.750| 15029.550
77 498.798 | 1333.823] 3044.683| 4012.843] 4292.203] 5370.488| 7791 .873] 9196.882 | 8557.492 | 11750.903
4/78 487.290 1322.732] 3090.928] 4127.648] 4585.618] 5815.363] 8360.133] 9562.302 | 9014.248 | 11810.493
11/80 518.512] 1381.355]3205.027| 431 1.050] 4810.923] 5885.795| 8535.668] 9385.540 | 9035.413 | 12330.785
8/82 509.590} 1353.365|3176.718]4185.795} 4634. 123] 5873.650|8586.728} 9899.805 | 9608.808 | 13451.135
9/83 498.020 1373.128]3057.235] 4127.068] 4719.675] 601 1.483) 8954.090} 1 1046.698] 11089.305] 14631.913
1/85 488.340] 1319.033]3133.200)4134.693) 4883.835| 6264.903| 9145.970) 10427.038] 10508.105] 14019.673
3/85 523.880} 1418.883] 3 164.990) 4217.098] 4887. 130} 6133.763] 8723.120] 10486.753} 10250.585| 13853.518
2/91 514.700 | 1352.630) 3 168.623] 4187.365] 4862.783] 6208.600) 894 1.918} 10604.585| 10574.028] 14193.370

Mean (nS) | 496.766] 1342.93 | 3117.61 | 4171.63 | 4678.76 | 5944.82 | 8642.24 | 10208.48 | 9976.867 | 13528.77
SD (nS) 18.750 | 47.883 | 74.158 | 112.207 | 199.359 | 266.430 | 369.292 | 509.454 | 686.186 | 894.493
Slope(nS/day) | 3.0E-03 | 4.4E-03 | 5.7E-03 | 2.7E-05 | 2.6E-02 | 2.5E-02 | 3.2E-02 | 1.1E£-02 | 3.3E-02 | 3.6E-02
Slope(nS/year)} 1.077 | 1.602 2.082 0.010 | 9.398 9.024 } 11.757 4.157 11.984 13.074
SDfit 17.639 | 48.080 | 75.149 | 116.146 | 193.896 | 267.313 | 371.903 | 526.409 | 704.537 | 920.661

SD(2/01) (nS)| 23.6814] 64.5503] 100.892] 155.934] 260.319] 358.886] 499.305] 706.7409} 945.8904] 1236.052

C3


measure the unknown capacitor of value equal to 1 uF. The combined uncertainty in the mica
capacitor dial readings, [('4) s,,], is estimated to be less than 16 ppm. The uncertainty in the air
capacitance dial readings, [(12) s,,], is estimated to be 5 ppm. (See the last column of Table 15b).

For C, = 1 pF, the uncertainty due to the external bridge components is estimated to be 5 ppm at
frequencies of 100 Hz and 1 kHz.

The Type-B standard uncertainties for measurements of three-terminal mica capacitors of value
equal to 1 wF, as well as their individual uncertainty components, at both 100 Hz and 1 kHz, are also
included in Table 15b.

4.4 Type A Standard Uncertainty of Capacitance Measurements

The evaluation of the Type A standard uncertainty for the Type-12 system in making capacitance
measurements is based on the repeatability and the effect of temperature variations on unknown
capacitors during measurements. Also involved are the stabilities of the NIST standards that are
used with an impedance meter as a comparator. An uncertainty analysis of each component that
contributes to the Type A standard uncertainty for both two- and three-terminal measurements is
given in the following sections.

4.4.1 Repeatability of Measurements

The procedure for using the Type-12 bridge to calibrate mica capacitors at NIST is to perform two
complete measurements on the unknown capacitor and to report the average of these two results as
the calibrated value. The range of these two measured values is compared with a given “limit” to
ensure that it is within this limit. In the case where the range is over the limit, additional
measurements are required to observe the variability of the unknown and to obtain an estimate of
the component of uncertainty due to repeatability. From NIST historical data of a large population
of standards, the range is 60 ppm, except for capacitors of 0.001 yF, for which the range is 100 ppm.
These limits correspond to the maximum bounds of error in repeatability of +30 ppm for all
capacitors except the 0.001 uF capacitors where the limits are +50 ppm. According to Eq. (4),

Sem = 17.3 ppm < 20 ppm, (for C, > 0.001 pF),

Sem = 28.9 ppm < 30 ppm, (for C, = 0.001 pF), (86)
where s,,, is the standard uncertainty due to repeatability of results of measurements of capacitance.
The values of s,,,, are included in Tables 16a and 16b.

4.4.2 Effect of Temperature Variations on Unknown Capacitors

As stated previously, the temperature coefficients of air and mica dielectric capacitors are
approximately 10 ppm/°C and 45 ppm/°C, respectively, and the temperature of the ICL is always

31


Table 15a. Summary of Components of Type B Standard Uncertainties in Capacitance Measurements Using
the Type-12 Bridge to Measure Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz

(All values are in ppm - except nominal values of the unknown capacitance standards)

Unknown Standard C, (uF) 0.001 | 0.002 | 0.005} 0.01 | 0.02 | 0.05 | 0.1 0.2 0.5 1.0
7
Components of Standard Uncertainties .
(At all Frequencies)
Ratio Resistor Arms (5,4) 9 9 9 18 18 18 18 18 18 18
Air Capacitor Dials (s,.) 62 35 18 9 9 9 9 9 9 9
( Frequency = 100 Hz)
Mica Capacitor Dials (sn) - - - 32 30 29 29 30 31 30
External Bridge Component (s,,) 12 6 3 2 1 1 1 1 3 5
Type-B Standard Uncertainty (U,) 63.79 | 36.63 | 20.35} 37.85 | 36.14 | 35.31 | 35.31 | 36.14 | 37.08 | 36.47
Assigned U, 65 40 25 40 40 40 40 40 40 40
ssigned U,
( Frequency = 1 kHz )
Mica Capacitor Dials (5) - - - 28 28 28 28 28 29 28
External Bridge Component (s,,) 12 6 3 2 1 1 1 1 3 5
Type-B Standard Uncertainty (U,) | 63.79 | 36.63 ] 20.35 | 34.54 | 34.50 | 34.50 | 34.50 | 34.50 | 35.43 | 34.84
Assigned U, 65 40 25 40 40 40 40 40 40 40
( Frequency = 10 kHz )
Mica Capacitor Dials (s,,) - - - 28 28 28 28 28 28 35
External Bridge Component (sy) 12 7 4 5 10 | 25 | 48 | 95 | 235 | 470
Type-B Standard Uncertainty (U,) 63.79 | 36.81 | 20.52 | 34.84 | 35.90 | 42.59 | 59.10 | 101.06) 237.52}471.73
Assigned U, 65 40 25 40 40 45 65 105 | 240 | 475

56


where C,, =2 C,, and Cy, = 2 Cy are dial readings of internal air capacitors at balance with C,
connected to arm AD and to arm CD, respectively.

4.3.3.2 Analysis
The relationship among the variances of the terms of Eq. (66) can be estimated by:
Var(Cy) = (AY Spex + (AY Stoo" > (67)

where 5,.1 and S;o9 are the standard uncertainties in the air capacitor dials of both measurements,
respectively.

Eq. (67) can also be expressed as:
Var(C,) = (A) 54617, (68)

2... 2 2
where Spo = Sto1 + Ste9 -
According to Eq. (66), C,, is independent of Rj and R,, so uncertainties contributed by the ratio
resistors will not affect this type of measurement. The uncertainties in the air dial readings are

estimated to be between 8 ppm and 31 ppm. (See the first three columns of Table 15b).

Uncertainties due to external bridge components are similar to uncertainties of two-terminal
measurements of the same nominal values, except that the capacitance of the plate, P3, is 0.006 pF,
and these are found to be less than 6 ppm.

The Type-B standard uncertainties of three-terminal low-capacitance value measurements are
included in Table 15b, as well as their individual uncertainty components for each capacitance value
at 100 Hz, 1 kHz, and 10 kHz.

4.3.4 Three-Terminal Measurements of High-Capacitance Values up to 0.5 nF
(0.005 pF <C, < 0.5 uF)

4.3.4.1 Basic Equations of Balance

Measurement procedures for capacitors of this type are similar to those of three-terminal low-
capacitance value measurements, except that internal mica capacitors are needed for the second
balance, and a dummy capacitor (a mica decade capacitance box) is connected into the arm CD for
both balances, as shown in Fig. 9. The equations of balance, as in Eqs. (64) and (65), become:

Ri (Car +O) = Ro (Cor + G ) (69)

and
Ri (Cy + Cn) = Ro (Coty +G, ), (70)

27


Table 16b. Summary of Components of Type A Standard Uncertainties in Capacitance Measurements Using
the Type-12 Bridge to Measure Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz

(All values are in ppm - except nominal values of unknown capacitance standards)

Unknown Standard C, (uF) 0.001 | 0.002 | 0.005 | 0.01 | 0.02 | 0.05 | 0.1 0.2 | 0.5 1.0

tb ae eee
—_———

Components of Standard Uncertainties,

( Frequency = 100 Hz )

Repeatability (5m) 30 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 20
Temperature Variation (5 tm) 10 10 10 15 15 15 15 15 15 20
Predicted Values (spa) 35 - - 15 - - 15 - - 10

Lype-A Standard Uncertainty (u,) | 47.17 | 22.36 | 22.36 | 29.15 | 25.00 | 25.00 | 29.15 | 25.00 | 25.00] 30.00

Assigned u, 48 25 25 30 25 25 30 25 | 25 30
=— ——

( Frequency = 1 kHz )

Repeatability (51m) 30 20 20 20 20 20 20 20 20 20
Temperature Variation (5 tm) 10 10 10 15 15 15 15 15 15 20
Predicted Values (s pa) 25 20 25 15 15 15 10 10 10 10

Type-A Standard Uncertainty (w,) | 40.31 | 30.00 | 33.54 | 29.15 | 29.15 | 29.15 | 26.93 | 26.93 | 26.93 | 30.00

Assigned u, 45 30 35 30 30 30 30 30 30 30

(Frequency = 10 kHz )

Repeatability (5m) 30 20 20 20 20 20 20 20 20 NA
Temperature Variation (5m) 10 10 10 15 15 15 15 15 15 NA
Predicted Values (spa) 35 - - 5 - - 35 - -

Type-A Standard Uncertainty (u,) | 47.17 | 22.36 | 22.36 | 25.50 | 25.00 | 25.00 | 43.01 | 25.00] 25.00} NA

Assigned u, 48 25 25 30 25 25 45 25 25 NA

59


Table C-6.

Data and Analysis of Conductance Corrections of ( 0.01 uF/step ) Mica Capacitance Dial
of the Type-12 Bridge at 10 kHz

(All values are in 0.001 1S, except the dial which is 0.01 F/step)

DIAL 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0x"
6/67 36.761 | 73.168 | 128.885 | 183.211 | 227.610 | 275.876 | 352.813 | 414.849 | 453.899 | 552.168
11/67 | 36.918 | 74.121 | 130.590] 183.374 | 227.313 | 276.509 | 355.158 | 417.483 | 457.254 | 556.228
5/68 37.018 | 73.790 | 129.590 | 183.465 | 227.440 | 275.523 | 353.373 | 414.815 | 455.213 | 551.212
9/69 37.225 | 68.573 | 124.640 | 179.008 } 222.140 | 272.695 | 350.403 | 412.085 | 445.650 | 544.008
12/70 | 37.393 | 75.473 | 132.605 | 186.113 | 230.045 } 280.003 | 357.448 | 419.518 | 460.445 | 559.753
9/72 35.573 | 72.020 | 125.523 | 178.633 | 220.370 | 267.068 | 341.600 | 401.893 } 442.503 | 538.615
6/74 35.543 | 70.820 | 125.855 | 175.913 | 215.863 | 263.580 | 334.605 | 392.242 | 428.757 | 496.662
11/75 | 37.330 | 74.835 | 131.513 } 187.008 } 230.460 | 280.780 | 358.585 | 424.585 | 464.905 | 563.802
v7 35.370 | 71.388 | 125.915 | 179.930 | 221.373 | 269.292 | 341.165 | 404.885 | 444.090 | 548.910
4/78 35.638 | 71.273 | 125.816 | 179.858 | 222.605 | 269.015 | 342.115 | 400.530 | 441.767 | 542.245
11/80 | 37.193 | 77.985 } 132.345 | 186.200 | 230.512 | 282.745 | 358.795 | 417.550 | 468.235 | 574.872
8/82 37.413 | 75.218 | 133.318 | 187.315 | 231.468 | 280.923 | 361.620 | 423.888 | 466.238 | 563.078
9/83 35.608 | 72.873 | 129.023 | 177.423 | 217.978 | 270.463 | 349.908 | 406.953 | 440.248 | 542.608
1/85 39.015 | 76.540 | 133.590 | 184.778 | 231.168 | 283.430 | 360.645 | 415.523 | 463.850 | 556.068
3/85 35.303 | 73.620 | 129.383 | 181.688 | 223.143 | 273.448 | 351.205 } 413.033 | 455.255 | 563.633
2/91 36.652 | 73.875 | 128.665 | 181.350 | 223.655 | 273.513 | 348.088 | 410.108 | 450.418 | 551.593
Mean (nS) | 36.6221 | 73.4733 | 129.204 | 182.204 | 225.196 | 274.679 | 351.095 | 411.871 | 452.42 | 550.341
SD(n$) | 1.0318 | 2.3405 | 2.9816 | 3.5414 | 4.9637 | 5.8793 | 7.8936 | 8.7564 } 11.173 | 17.2627
Slope(n$/day)| -9.2E-06] 2.8E-04 } 2.8E-04 | 2.7E-05 | -5.8E-05] 2.6E-04 | 2.5E-04 | -6.5E-05| 4.8E-04 | 1.2E-03
Slope(nS/year)| -0.0034 } 0.1029 | 0.1015 | 0.0100 | -0.0213 | 0.0953 | 0.0912 | -0.0237 | 0.1739 | 0.4556
SDfit 1.0678 | 2.2960 | 2.9904 | 3.6649 | 5.1355 | 6.0434 | 8.1419 | 9.0620 | 11.4910 17.538¢
SD(2/01) (nS)] 1.43354} 3.08257] 4.01479] 4.92038] 6.89472] 8.11363] 10.931} 12.1664] 15.4275] 23.546

USS, GOVEANMENT PRINTING OFF

0-78



4.7.1 Repeatability of Measurements

As pointed out in section 4.4.1 for repeatability of capacitance measurements, the range of a given
“limit” for conductance measurements is also contained in NIST internal documents for each
nominal value of capacitor at each frequency. Accordingly, the maximum bounds of error of
individual capacitor are obtained, and the standard uncertainty in conductance is estimated as:
Sgrm = (maximum bounds of error) /¥3 US, (95)

where s, is the standard uncertainty due to repeatability of results of measurements of
conductance and is given in Table 19a and 19b.

4.7.2 Stabilities of NIST Standards

Using the same method as in section 4.4.4, conductance values of unknown capacitors measured at
100 Hz, 1 kHz, and 10 kHz can be calculated by utilizing the “substitution method” [15] from data
obtained by employing an impedance meter as a comparator. The uncertainties of the predicted
values of conductance in unknown capacitors at a frequency of 100 Hz, 1 kHz, and 10 kHz, s,,4 are
given in Tables 19a and 19b as well. The “substitution method” is not used on some capacitors at
frequencies other than 1 kHz; therefore, these uncertainties are not applied for those frequencies.

The Type A standard uncertainty of conductance measurements is defined as the RSS of s, and
Or ‘grm
Sgpd and these are all summarized in Table 19a and 19b.

4.8 Expanded Uncertainty for Conductance Measurement Using the Type-12 System

The expanded uncertainties of conductance measurements (for each nominal value of mica capacitor)
are calculated from both Type A and Type B standard uncertainties with a coverage factor of k= 2
[2], and the results are provided in Table 19a and 19b for the two-terminal and three-terminal
configurations, respectively.

5. CONCLUSION

The revised uncertainties of NIST capacitance calibrations have been established, according to
NIST's policy on expression of uncertainty [2] and on the modification of measurement procedures
in Impedance Calibration Laboratory. Numerical results are also included in the NIST Calibration

Services Users Guide (SP 250, 1998 edition) [17].

37


10°. Therefore, by eliminating these terms, the approximate equation for the ratio corrections is:
a, + jb, =LL e+ Len) tie) ps M(B; +E 8p) 1/ (Cy + Cy), (A6)

where a, and 5, are in-phase and quadrature corrections applied to the 1:1 ratio.

The design of performing the above two measurements to obtain ratio corrections is an attempt to

eliminate errors in non-unity ratios. However, the error sources which occurred during both

measurements are similar and affect the extent to which the offset cannot be eliminated. Therefore,

the exact values of ratio corrections in Eq. (A5) can be expressed as:

(at+jb)=a, +e, + jb, +jey, (A7)

where e, and e, are the errors in the in-phase and quadrature corrections, respectively, of the 1:1
ratio, and these can be derived by combining Eqs. (A5), (A6), and (A7) as follows.

By including the in-phase and quadrature components, the first term in the numerator of the right
hand side of the Eq. (A5) can be written as:

(Lryit Ln) = Lr Gee) 1-idG) + Dry Goey) A- id )
=O LG HAL nen Ga) tL GAL nen AB)
The second term in the numerator of the right hand side of the Eq. (A5) can be written as:
PsM(¥83+L8,)= PsMI Lg (+i) + Le, tix) ]
= psMU(Lg; +L 8) ti gx; +L sey) )- (A9)

The numerator of the right hand side of the Eq. (A5) is the summation of Eqs. (A8) and (A9), which

1S: (

(Ly + Ln) + PsM(L 8+ L Bn) =
OL 4 K+ L nen Gn) t+ Cs M/o) (Ls; tL gy]
+jo [CLG +L ty en) + Os M/ ) © 87; +L 8ytp) ]- (A10)

The denominator of the right hand side of the Eq. (A5) can be written as:
(¥, + ¥,) =jwCs (1- jDF, ) + joCx (1- jDFx) = w [(Cs DF, + Cx DF,) + j(Cs + Cx)}.
Thus,

[1/(%e+ Yl = (1/0) {1/[(C, DF, + Cy DF,) + i(C, + GD}. (All)

A-2


850 10 Spoq

TMm
Ste

Stm

S,

uy and up

uy, and Ugh
and U,

U, and U,

Values of ratio resistors R, and R,

Contact resistance of the two balances.

Standard uncertainty due to lead impedance.

Standard uncertainties due to corrections of plates P2 and P3.

Standard uncertainty of external bridge components.

Standard uncertainty in the air capacitor dials due to dial corrections.

Standard uncertainty due to dial switching.

Standard uncertainty due to residual conductance.

Standard uncertainty in conductance corrections of the mica capacitor dials.
Standard uncertainty in conductance potentiometer.

Standard uncertainty of the predicted conductance value of the NIST standard.
Standard uncertainty due to centact resistance.

Standard uncertainty due to repeatability of measurements in conductance.
Standard uncertainty due to conductance standard.

Combined Standard uncertainty of the internal mica capacitor dials.

Standard uncertainties in the mica capacitor dials corresponding the readings of C,,
and C3.

Standard uncertainty in the mica capacitor dials due to the uncertainty of transfer
standard.

Standard uncertainty in the mica capacitor dials due to temperature variations.
Standard uncertainty in the mica capacitor dials due to dial corrections.
Standard uncertainty of the predicted capacitance value of the NIST standard.
Standard uncertainty contributed by the ratio resistors.

Combined Standard uncertainty of the internal air capacitor dials.

Standard uncertainties in the air capacitor dials corresponding the readings of C,, to
Cry:
Standard uncertainty due to repeatability of measurements in capacitance.
Standard uncertainty in the air capacitor dials due to the uncertainty of transfer
standard.

Standard uncertainty on unknown capacitor due to temperature variations.
Standard uncertainty in the air capacitor dials due to temperature variations.

Type A and Type B standard uncertainties of capacitance measurements.

Type A and Type B standard uncertainties of conductance measurements.
Expanded and total (assigned) uncertainties of capacitance measurements.
Expanded and total (assigned) uncertainties of conductance measurements.


TABLE OF CONTENTS

page
List of Tables .
List of Figures ..
List of Symbols
ABSTRACT 22... eee een ene e eee n eee eens 1
1. INTRODUCTION ........ 60 cece cece center ener e eee te eee 1
2. DESCRIPTION OF SYSTEMS AND MEASUREMENT PROCEDURES ...........- 2

2.1 Type-2 Capacitance Bridge and Standards

2.2. Type-12 Capacitance Bridge
UNCERTAINTY ANALYSIS FOR THE TYPE-2 SYSTEM ......... 0000s eee eeeee 4
3.1 Transformer Ratios ......... 6... e cece eee eee eee 5
3.1.1 Corrections and Uncertainties of the 1:1 Ratio . 75
3.1.2 Calibration and Uncertainties of the 10:1 ( and 1: .6
3.2 Dial Corrections 7
3.2.1 Calibration of Dial Capacitors ©... 2.00... cece cece cee eee eee 7
3.2.2 Uncertainties in Dial Corrections ........... 6.00 cece cece eee e cece eee 8
3.2.3. Analysis of Uncertainties in Dial Corrections 9
3.3. Uncertainties in External Bridge Components ... -10
3.3.1 Voltage Dependence of Standards . . -U
3.3.2 Frequency Dependence of Standards .................. 11
3.3.3 Hysteresis Effects in Temperature Corrections of Standards . 12
3.3.4 Lead Impedance ........ 0. ccc cece eee teen nen eee 12
3.3.5 Uncertainty in the Reference Standard ........... 00. ese e eee e eee eee 4
3.3.6 Combined Uncertainty of the External Bridge Components ... 14
3.4 Uncertainties of Fused-Silica and Nitrogen Dielectric Capacitors at 1 kHz . 14
3.5 Uncertainties of Fused-Silica and Nitrogen Dielectric Capacitors at 400 Hz ......15
3.6 Uncertainties of Fused-Silica and Nitrogen Dielectric Capacitors at 100 Hz 15
3.7 Uncertainties of Air and Mica Dielectric Capacitors ................5 16
3.7.1 Three-Terminal Air Capacitance Standards .. . . 16
3.7.2. Two-Terminal HF Coaxial Capacitance Standards and Terminations ..... 16
UNCERTAINTY ANALYSIS FOR THE TYPE-12 SYSTEM ..............000 000 7
4.1 Basic Equations of Capacitance Measurements 18
4.2. Type B Standard Uncertainty of Capacitance Measurements .
4.2.1 Ratio Resistors ........ 0... c eee e cece eee
4.2.2 Internal Capacitors .
4.2.3 External Bridge Components .
4.3. Type B Standard Uncertainty for Various Types of Capacitance Measurement ... . 23

4.3.1 Two-Terminal Measurements of Low-Capacitance Values
4.3.1.1 Basic Equations of Balance ........... 0.00. e cece eee eee 23

iti


where G,, is the conductance of the internal mica capacitors with C, connected to arm CD,
G, is the conductance dial readings without connecting C,,
Gy, is the conductance dial readings with C, connected to arm CD, and
Ky is the divider setting.

For the cases where mica capacitance dials are not used to perform conductance measurements, the
corresponding term of G,, is equal to zero.

In general, the Type-12 bridge is used for high-Q capacitance measurements, and conductance is the
minor component. Therefore, the uncertainty analysis of conductance measurement for each
nominal value of mica capacitor is based on a general measurement procedure for both two- and
three-terminal measurements at frequencies of 1 kHz, 100 Hz, and 10 kHz. There are five
components included in the Type B evaluation of standard uncertainty of conductance
measurements. These are discussed in the following sections.

4.6.1 Conductance Standard

According to [13], the conductance standard, which was devised by Young [16], has a deviation of
+0.2 % from its nominal value. Since there are two balances required for most measurements except
for an unknown C, = 1 pF, which required four balances, the maximum bounds of error due to the
limit of accuracy of the conductance standard are estimated to be:

teg,= +0.002 Qf G, wS (for, <1 pF), and
eg, = 40.0024)" G, US (for Cy = 1 HF), (87)
where +e,, are the maximum bounds of error in conductance measurements due to the limit of

accuracy in the conductance standard, G, (in pS). Accordingly, the uncertainties in conductance
standards are expressed as:

Ss 0.0016 G,  uS (for C, <1 pF ), and

gs

i]

Ss

gs = 0.0023 G, pS  (forC, =1 pF), (88)

where s,. are the standard uncertainties due to the limit of accuracy in the conductance standard,

Ss
(in us).
According to the conductance values of the check standards, the values of s,. for each nominal value

of capacitor at frequencies of 100 Hz, 1 kHz, and 10 kHz in the two-terminal and three-terminal
configurations are estimated and included in Tables 18a and 18b, respectively.

4.6.2 Contact Resistance

The contact resistance in the capacitance standard decade switches was estimated to be 0.005 Q [14].

34


Table 19b. Summary of Components of Type A, Expanded, and Assigned Uncertainties in Conductance
Measurements Using the Type-12 Bridge to Measure Three-Terminal Mica Capacitors at
100 Hz, 1 kHz, and 10 kHz

(All values are in pS - except nominal values of the unknown capacitance standards)

Unknown Standard C, (uF) 0.002 | 0.005 | 0.01 | 0.02 | 0.05 | 01 | 02 0.5 1.0
(Frequency = 100 Hz)
Repeatability (5 gm) 2.3E-05| 2.3E-05 | 2.3E-05| 2.9E-05| 4.0E-05| 7.2E-05] 1.2E-04| 1.8E-04} 4.6E-04| 1.2E-03
Predicted Values (5 goa) 1,5E-05 - - 7.4E-05 - - 1.9E-04 - - 4.6E-03
Type-A Std. Uncertainty (w,,)] 2.7E-05 | 2.3E-05 | 2.3E-05] 7.9E-05| 4.0E-05| 7.2E-05 | 2.2E-04} 1.8E-04) 4.6E-04| 1.2E-03
Type-B Std. Uncertainty (# g)| 1.0E-04] 1.0E-04] 1.0E-04} 1.7E-04| 1.8E-04) 4.1E-04] 7.7E-04 | 2.6E-03 | 4.7E-03 | 6.7E-03
Expanded Uncertainty (U,) | 0.00021] 0.00021} 0.00021 | 0.00037} 0.00037] 0.00083] 0.00161) 0.00528) 0.00950) 0.01359
Assigned
Total Uncertainty (Ug) 0.0002 } 0.0002 | 0.0002 | 0.0004 | 0.0004 | 0.0009 | 0.002 | 0.006 | 0.010 | 0.02
|
(Frequency = 1 kHz)
Repeatability (5 gm) 4.0E-05| 4.6E-05] 7.5E-05} 1.2E-04} 2.3E-04| 5.2E-04} 1.2E-03/ 2.3E-03 | 7.5E-03 | 2.3E-02
Predicted Values (5 gpa) 1.0E-04 | 2.0E-04 | 2.0E-04 | 3.0E-04| 1.0E-03 | 2.0E-03 | 6.0E-03} 1.5E-02} 5.5E-02] 9.5E-02
Type-A Std. Uncertainty (w,.)| 1.1E-04) 2.1E-04] 2.1E-04| 3.2E-04| 1.0E-03) 2.1E-03] 6.1E-03) 1.5E-02] 5.6E-02| 9.8E-02
Type-B Std. Uncertainty (w ,)| 1.0E-04] 1.0E-04] 1.0E-04} 2.0E-04| 2.7E-04| 8.2E-04 | 2.9E-03| 8.6E-03} 1.8E-02} 5.1E-02
Expanded Uncertainty (U,) | 0.00030} 0.00046} 0.00047) 0.00076] 0.00212} 0.00444} 0.01354] 0.03487} 0.11700] 0.22018
Assigned
Total Uncertainty (Uy) | 0.0003 | 0.0005 | 0.0005 | 0.001 | 0.003 | 0.005} 0.02 | 0.04 | 0.12 0.3
—_——————
(Frequency — 10 kHz)
Repeatability (5 gm) 1.8E-04| 2.3E-04| 5.8E-04} 1.2E-03) 2.3E-03 | 8.1E-03] 2.3E-02| 6.4E-02|3.5E-01] NA
Predicted Values (5 gp) 3.5E-05 - - 2.3E-03 - - 3.2E-02 - - NA
Type-A Std. Uncertainty (1 g.)| 1.8E-04|2.3E-04| 5.8E-04 | 2.6E-03 | 2.3E-03 | 8.1E-03 | 3.9E-02] 6.4E-02} 3.5E-01] NA
Type-B Std. Uncertainty (wg) | 1.5E-04] 1.5E-04} 1.9E-04| 3.1E-03] 5.3E-03 2.6E-02] 7.9E-02] 2.4E-01]1.7E+00| NA
Expanded Uncertainty (U,) | 0.0005 | 0.0006 | 0.0012 | 0.0081 | 0.0115 | 0.0548 | 0.1773 | 0.5001 | 3.4532] NA
Assigned
Total Uncertainty (U,.) 0.001 | 0.001 | 0.005 | 0.01 0.02 0.06 0.2 0.5 4.0 NA

65


Figure 8. Components of the Type-12 Bridge for Three-Terminal Low-Capacitance
Values Measurements.

74


NIST Measurement Services:

Error Analysis and Calibration NIST

Uncertainty of Capacitance Pablication

Standards at NIST 250-52

seer tee

|
|
|

8/19/1993 8/14/1994 8/9/1995 431996 7/20/1997 7 24/1998 TH9/1999 7/13/2000

Control Chart for Reference Standards



National Institute of Standards and Technology Special Publication 250-52
Natl. Inst. Stand. Technol. Spec. Publ. 250-52, 109 pages (Jan. 2000)
CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE
WASHINGTON: 2000

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325


See (Sar? Sip? + S45? tg?) , and (43)

2 2 2 2\(%
+ 5mp? +5457 + Sm¢?) (44)

where s,, and s,,, are the combined uncertainties in the air and mica capacitor dials, respectively,
which are also included in Tables 13a to 14b.

1)

2)

4.2.3 External Bridge Components
Capacitance Between Connectors

Two special plates, P2 and P3, have been constructed to connect to the Type-12 bridge so
that it can perform direct measurements on two-terminal and three-terminal mica capacitors,
respectively, without using cables. Each plate has been evaluated to obtain its series
inductance and resistance, and parallel capacitance. After these corrections are applied, the
maximum bounds of error introduced by connections are found to be + 0.02 pF and
+ 0.01 pF for plates P2 and P3, respectively. These values imply that, according to Faq. (4),
the uncertainty when using plate P2 is in the range from 0.012 ppm (in measuring 1 pF
capacitors) to 12 ppm (in measuring 0.001 pF capacitors), and the uncertainty when using
plate P3 is from 0.006 ppm to 6 ppm. Hence,

0.012 < Spp2 $ 12 ppm and 0.006 < Spp3 < 6 ppm, (45)

where Spp2 and Spp3 are the standard uncertainties due to the applied corrections of plates P2
and P3, respectively.

Series Lead Impedance

The effective capacitance of an unknown capacitor, C,, including series lead inductance is
taken to be (see derivation of Eq. (30a) in sect. 3.3.4):

Cese = Cy 1+ o Ly Cy), (46)

where Corr is the effective capacitance, L, is the series lead inductance, and w= 2nf in
which f is the applied frequency. The lead inductance of the Type-12 bridge is estimated

to be less than 0.2 nH, and the maximum error caused by lead impedance, @,7 is less than
( 2nf') (0.2 )10° (C, ), which is 8f7(C,) ppm. Accordingly,

Spyz = 4.6f7(C,) ppm, (47)

where sy is the standard uncertainty due to lead impedance, f is the applied frequency in
kilohertz (kHz), and C, is nominal value of the unknown capacitor in microfarad (\1F).

The combined uncertainty of the external bridge components, s},,is defined as the RSS of the above

22


In order to estimate the standard uncertainties in the ratio corrections, a few assumptions and
approximations are needed to apply to Eqs. (A15) and (A16). In general, all values of DF,, DF,,
of, Of,» x;, and x, are less than 10°, and ratios of Ly | (DF, + DF,)] and [x, / (DF, + DF,)] are less
than 1. Maximum bounds of errors + eg and + e,, in ratio corrections are estimated from the
following sources and related assumptions: and the standard uncertainties, s, and s,, are obtained
in accordance to Eq. (4) of the text.

9)

2)

3)

U i. . & i

According to the data from measurements performed on capacitors of various nominal
values, it was found that:

I Yg;+ Ley Ws 10% S and |( Lre; + Ln? Tren $ 10% pF (for C, = C, = 10 pF),
\(Xg;+ Le, < 107 S and |(Yri¢; + Lrntn)l < 107 pF (for C= =C.= 100 pF),

\(Yg; een Wes 10° S and (L764 Lrpepl < 10° pF (for C= C= 1000 pF), and
Ms 10

By appropriate multiplication in Eqs. (A15) and (A16), the maximum bounds of errors are
estimated, and the standard uncertainties are found to be:

Sq, < 0.001 ppm and s,, << 0.001 ppm,

where s,, and s,, are the standard uncertainties from measured data in the in-phase and
quadrature corrections of the 1:1 ratio, respectively.

U inties d hei ies of 1 : dard
Assume the external capacitors, C, and C,, are inaccurate by + 0.01%, the maximum bounds
of errors are estimated by substituting C=C, (1 = 107) and GHGs 10* ) into Eqs.
(A15) and (A16). Accordingly,

Sao << 0.001 ppm and s,. << 0.001 ppm.

where sy and s,. are the standard uncertainties of the external capacitance standards, in the
in-phase and quadrature corrections of the 1:1 ratio, respectively.

U inties d . ti
Assume the capacitance and conductance dials are inaccurate by + 1%, the maximum bounds
of errors are estimated by substituting c;= c; (1 + 0.01), c, =¢, (1 + 0.01), g =g (1+ 0.01),
and g,, = 8, (1 £0.01) into Eqs. (A15) and (A16). Accordingly,

5,3 << 0.001 ppm and s,3 << 0.001 ppm,

where s,; and sp; are the standard uncertainties of the dial readings, in the in-phase and
quadrature corrections of the 1:1 ratio, respectively.

A4


Figure 10. Components of the Type-12 Bridge for Three-Terminal Measurements of
Capacitors Larger Than 0.5 pF.

76


mainly used to calibrate mica-dielectric capacitors of exposed binding-post connectors in the range
from 0.001 pF to 1 wF. In general, calibration of customers’ capacitors is performed using two
measurements (at a 1:1 ratio of the resistance arms) to eliminate the effects of lead impedance. In
the two-terminal configuration, the bridge is balanced with and without connecting the unknown
capacitor to the capacitance arm. In the three-terminal configuration, the bridge is balanced by
connecting the unknown capacitor to each of the capacitance arms. For each configuration, final
results are calculated using dial readings from both measurements.

3. UNCERTAINTY ANALYSIS FOR THE TYPE-2 SYSTEM

A complete schematic diagram of the Type-2 bridge is shown in Fig. 2a. Bridge components and
operational procedures of the Type-2 system were described in detail by Cutkosky [1]. Figure 2b
is a simplified circuit diagram of the Type-2 bridge, where C, is the external reference capacitor, C,
is the unknown capacitor to be measured, Cg represents the internal dial capacitors, G, represents
the conductance of the internal dial resistors, and M is the multiplier of the conductance settings.
At balance, the detector current is equal to zero, and the detector is at virtual ground. The balance
is achieved by applying voltages from the appropriate taps of the transformer to the internal
capacitors and to the conductance control network. The general balance equation is:

It yt lg = hy (6)

where J, is the current through the reference capacitor, C, ,
i is the current provided by the capacitance network, Cq as described below,
is the current supplied by the conductance balance divider and network, and
B is the current through the unknown capacitor, Cy

The dial capacitance, Cy, is provided by a set of eight air capacitors whose values range from 100 pF
to 10° pF having a total capacitance up to 111 pF (see Fig. 3). G, is a set of four dials providing
conductance from 0.001 1S to 1 yS with a multiplier switch to extend the total range from 10° nS
to 1 uS. The dials are switches of a four-decade transformer divider that applies selected voltages
to a tuned phase shift network providing conductance balance through a number of ranges (see
Fig. 2a). Another switch is included for reversing the sign of the conductance component - for
allowing the unknown capacitor to have greater or lesser conductance than the internal capacitors
of the Type-2 bridge. It can be seen from Fig. 2a that each capacitor-switch combination provides
the equivalent of a decade of capacitance by selection of the voltage applied to the capacitor, and that
if an unknown capacitor is connected to the bridge as shown in Fig. 2b and the bridge balanced, the

capacitance of C, is equal to the value of C, plus the sum of the dial settings. The balance equation
can be written as:

E,Y,+ E,(L7;9;) + Ps Ey (MY g;) =- Ex No (6)

where E, and E, are voltages produced by the transformer windings on the reference, C, and the
unknown capacitor, C, sides of the bridge, respectively,

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46


Table B-6. Data and Analysis of the (0.01 F/step ) Mica Capacitor Dial Corrections of the
Type-12 Bridge at 10 kHz

[All values are in pF except the dials which are in (uF/step)]

Date Dial | 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0X

6/67 0.269 | 1.491 | 1.829 | 2.800 | 4.312 | 6.523 | 5.180 | 8.255 | 11.291 | 11.987
11/67 0.354 | 1.670 | 2.239 | 2.971 | 4.562 | 6.850 | 5.692 | 8.821 | 11.925 | 12.710
5/68 0.321 | 1.610 | 2.148 | 2.905 | 4.511 | 6.784 | 5.583 | 8.742 | 11.855 | 12.634
9169 0.325 1.568 1.946 | 2.729 | 4.289 | 6.519 | 5.175 8.282 | 11.338 | 12.113
12/90 0.353 | 1.701 | 2.211 | 2.942 | 4.530 | 6.833 | 5.616 | 8.754 | 11.908 | 12.699
9/72 0.339 | 1.623 | 1.994 | 2.631 | 4.182 | 6.451 | 5.130 | 8.254 | 11.316 | 12.092
6/74 0.360 | 1.613 | 1.998 | 2.716 | 4.319 | 6.509 | 5.092 | 8.164 | 11.097 | 11.857
11/75 0.352 | 1.692 | 2.082 | 2.686 | 4.229 | 6.510 | 5.135 | 8.271 | 11.389} 12.153
77 0.306 | 1.608 | 1.916 | 2.585 | 4.164 | 6.428 | 5.024 | 8.081 | 11.224 | 12.018
4/78 0.223 | 1.421 | 1.559 | 2.215 | 3.693 | 5.876 | 4.327 | 7.313 | 10.383 | 11.121
11/80 0.167 | 1.386 | 1.420 | 1.749 | 3.128 | 5.337 | 3.741 | 6.542 | 9.719 | 10.403
8/82 0.211 | 1.432 | 1.530 | 2.054 | 3.519 | 5.703 | 4.101 | 7.140 | 10.170 | 10.871
9/83 0.240 | 1.425 | 1.485 | 1.860 | 3.362 | 5.592 | 3.935 | 6.997 | 9.975 | 10.657
1/85 0.220 | 1.480 | 1.475 | 1.902 | 3.382 | 5.627 | 3.907 | 7.016 | 10.054 | 10.676
3/85 0.158 | 1.452 | 1.302 | 1.713 | 3.242 | 5.427 | 3.568 | 6.527 | 9.552 | 10.156
2/91 0.179 | 1.380 | 1.089 | 1.412 | 2.876 | 5.095 | 3.141 | 6.192 ] 9.239 | 9.917

N. Val. (pF) | 1.0F+05}2.0F+05|3.0F+05] 4.0F-+05] 5.0F+05] 6.0F+05] 7.0F+05] 8.0F+05| 9.0F+05] 1.0E+06
Mean (pF) 0.2736 | 1.5345 | 1.7639 | 2.3669 | 3.8938 | 6.1290 | 4.6467 | 7.7094 | 10.7772] 11.5040
SD (pF) 0.0734 | 0.1126 | 0.3547 | 0.5138 | 0.5648 | 0.5893 | 0.8161 | 0.8759 | 0.8937 | 0.9458

Slope (pF/day) |-1.4E-05]-1.6E-05|-7.6E-05]-1.2E-04|-1.3E-04|-1.3E-04]-1.9E-04]-2.0E-04|-2.0E-04)-2.1E-04|
( pF/year) -0.005 | -0.006 | -0.028 | -0.045 | -0.048 | -0.049 | -0.070 | -0.073 | -0.073 | -0.077
SDfit (pF) 0.047 | 0.081 | 0.162 | 0.166 | 0.205 | 0.224 | 0.274 | 0.335 | 0.341 | 0.370
SD(2/01) (pF) | 0.062 | 0.109 | 0.217 | 0.223 | 0.275 | 0.301 | 0.368 | 0.449 | 0.458 | 0.496

B-6



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67


where C,, and C,, are values of the internal air capacitors at balance with the unknown capacitor,
C, connected to arm AD, C,y and C,) are values of the internal air capacitors at balance with Cy
connected to arm CD, C,,, are values of the internal mica capacitors at the second balance, and C;,
is the value of the dummy capacitance box.
From Eqs. (69) and (70), the value of the unknown capacitor can be expressed as:

Cy = TRy/ (Rit RNC + CAVE R/ (CR + RCC - Cy ds (71)
The ratio resistors are always set to be equal (R, = R,), so Eq. (71) becomes:

Cy. = (AVC + (A) (Cya- Ca), (72)

which is the equation used to calculate the unknown capacitor.

4.3.4.2 Analysis
Rewrite Eq. (71) as:
Cy = ((Ry/Ry) / 1+ (Ry! Ry YI} Cay + (A) (Ry /Rq) / 1+ (Ry Ry H(Cyp = Ce) 73)
The relationship among the variances of the terms of Eq. (71) can be estimated by:
Var(Cy) = [1+ (RVR (Coy + (A) (Coy-Coodl Spa? + (CR) Ry) / U1 (Ry/ Ry YIP? Sy
+ (YY {(R,/Ry) / 1+ (Ry/ Ry I}? (Spot Seo” > (74)

whee 5,4 Sy» Spo1» aNd S,¢9 are the standard uncertainties in the ratio ( contributed by the resistors),
the mica capacitor dial, and the air capacitor dials of both measurements, respectively.

By substituting Eq. (72) and ( R; = R, ) into Eq. (74), it becomes:
Var(Cy) = [(%) Cy Spq P+ UA) Sp)? + (0%) 501” (75)
2

2. 2
where Spo = Spo + Sp¢9

As in the two-terminal high-capacitance value measurements, the uncertainty contributed by the
ratio resistors in the three-terminal case, [ (’2) C, 5,, ], is also 9 ppm. (See Table 15b).

Using an analysis similar to that in the case of two-terminal high-capacitance measurements, one
finds that the uncertainties in the internal mica capacitors for the three-terminal measurements,
[(4)s,q], are estimated to be less than 23 ppm. The uncertainties in the air-dial capacitors, [(14) 5,.],
are estimated to be less than 3 ppm. These are also given in Table 15b.

Uncertainties due to external bridge components are similar to those in making two-terminal

28


Table 1.

Dial Corrections of the Type-2 Bridge

[ All values are in ( 10° pF ), except nominal values of dials ]

Dial No. 1 2 3 4 5 6 7 8
Nominal
Value (pF) 100 10 1 0.1 0.01 0.001 0.0001 +~—-0.00001
4/25/94 5160.1 -890.59 3.741 -11.426 1.657 0.566 0.357 0.336
5/27/94 5169.1 -870.49 7.951 -11.405 1,560 0.556 0.356 0.336
6/1/94 5903.3 -756.47 = 13.353 - 10.365 1.664 0.566 0.357 0.336
8/4/94 7587.3 -478.27 = 34.973 -7.803 1.900 0.592 0.359 0.436
8/19/94 8078.1 -372.19 36.181 6.982 2.002 0.600 0.360 0.436
9/16/94 6801.0 -634.24 39.876 -8.712 1.829 0.583 0.358 0.336
10/31/94 7270.2 -517.18 36.682 8.132 1.887 0.589 0.459 0.446
11/28/94 6932.2 -549.18 22.982 -8.702 1.830 0.583 0.358 0.336
12/30/94 7051.8 -620.02 23.898 -8.910 1,929 0.591 0.459 0.446
1/17/95 6209.7 -733.73 11.127 -10.087 1.691 0.569 0.457 0.446
2/27/95 6769.5 -692.15 14.485 -9.752 1.725 0.572 0.457 0.446
3/17/95 6174.9 — -738.51 7149 -9.985 1.801 0.680 0.468 0.447
4/28/95, 6117.6 -742.64 7.235 -10.276 1.672 0.567 0.357 0.436
5/19/95 6442.8  -680.12 7.688 -9.731 1.727 0.573 0.357 0.436
Mean Value | 6547.7 ~662.56 19.094 -9.448 1.777 0.585 0.394 0.404
Standard
Deviation 839.0 146.07 13.052 1.302 0.126 0.030 0.051 0.053

40



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45


Table 2.

[ All values are in 10% pF ), except nominal values of dials ]

Standard Uncertainties in Dial Corrections of the Type-2 Bridge

Dial No. 3 4 5 6 7 8 2 1
Nominal
Value C; (pF) 1 0.1 0.01 | 0.001 | 1E-04] 1E-05 10 100
Mean Value
of Dial
Correction | 19.094 -9.448] 1.777 | 0.585 | 0.394 | 0.404 -662.56 | 6547.69
Standard Dev. | 13.052 1.302 | 0.126 | 0.030 | 0.051 | 0.053 146.07 | 839.0
No. of
Dials Used 3 3 3 1 1 1 4 5
Components of
Standard
Uncertainty
(O.1)spq_—- [0.0043] (0.1)sgq_—_-| 4E-04] 4E-05| 4E-06| 4E-07] 4E-08] (10) spa 43 43
ergs | 0.08 | erxs,7)1"| 0.08 | 0.08 | 0.05 | 0.05 | 0.05 fopse,?s,7°7) 148 | 13.1
S| 0.2 S| 0.2 0.2 0.2 0.2 0.2 S| 0.2 0.2
(0.1)s, 0.75 | (O.1)sq-1) | 0.100 | 0.025 | 0.023 | 0.022] 0.022] — (10)sq1) 10 130
ssQ 0.609 0.056 | 0.047 | 0.043 | 0.043 | 0.043 120.72 | 18920.7
RSS 0.780 0.237 | 0.217 | 0.207 | 0.207 | 0.207 10.99 137.6
Assigned
Uncertainty (s;)} 1 0.25 | 0.23 | 0.22 | 0.22 | 0.22 13 150

Remarks :
Spat
Bern's, I :
Spt

Sst

Standard uncertainty of the 1:10 ratio

Standard Deviation of the dial corrections for the k"" dial of reading r,

Standard uncertainty due to the interpolation in last dial

Uncertainty of reference standard

41



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70


Multiply both the numerator and the denominator of the right hand side of Eq. (All) by
((C, Df, + G, DF,) - i(C, + C,)] and eliminate the second order terms of DF, and Df,. Eq. (A11)
becomes:

(1/(¥, + Xl] =(1/ @) { [ (C, DF, + Cy DF) / (Cy + Cy) 71 - i[1(C, + C)I}-

In measurements using the 1:1 ratio, C, = C,, and

[(C, DF, + C, DF,)/(C, + C,)] = [((DF,+ DF,)/2]. Then,

[1/(% + ¥%)] = (1 /@) (DF, + DF) /[2 (Co + Cy} - j{L/L@(Cg+C)}}.  (A12)
The expression of Eq. (A5) can be obtained by the multiplication of Eq. (A10) and (A12). After
the separation of the real and imaginary parts, and the elimination of the second order terms of DF,,

DF,, df;, dy, ¥;, and x, the in-phase correction, a, and the quadrature correction, b of the 1:1 ratio
can be expressed as:

a ={(5M/o)[(4) (0 8;+ ¥ 8,) (DF, + DF) + (XL xt L Sp%q)l

+(L¢; +¥ ray )} /(Co+ Cy)
and

b= (MOye +L Tn On) (DF, + DF) - (47,6 H+ Ln on Gy DI
-(PsM/ oY 8+ ¥ B)}/(Co+ Cy. (Al3)
According to Eq. (A7),
€,=a-ay
and
ep7- a1. (A14)

By substituting a, and b, from Eq. (A6) and a and b from Eq. (A13) into Eq. (A14), the errors in
the in-phase and quadrature corrections of the 1:1 ratio become:

eg = (Ps M/w){(%) (Lg; + Lg_) (DF,+ DF) + (L 8x + L Sin) 1/(Co+ Cy), (AIS)
p= (A) (Le; +L pty) (DF e+ DF) - © rieifz +L rena ) 1! (Cy + Cy). (A16)

and

In general, the capacitance and conductance dial readings of the two measurements when C, and C,
are interchanged are in opposite signs, i.e., one is positive and the other is negative. Therefore, the
magnitude of summations of the dial readings of the two measurements (i and n) in Eqs. (A15) and

(A16) is usually less than each individual reading, and these can be estimated from the measured
data.

A-3


This resistance is considered as being in series with the internal capacitance of the decade switches.
Since there are two or more balances required for each measurement, and results are calculated from
the difference of these readings, the maximum bounds of error for each nominal value of C, are
estimated from the difference of the contact resistance of each balance as:

Seg, = £0? Cy (Ry Rea) (89)

where +e,,, are the maximum bounds of error due to contact resistance, and R,, and R,, are contact
resistance in the capacitance decade switches at the first and second balances, respectively.

Assuming the differences between R,, and R,, are less than 0.002 Q, Eq. (89) can be expressed as:

= 2 2
egy = £0.08 f° Cy’. (90)
Accordingly, the standard uncertainty in contact resistance, Sop is estimated as:
Sop = 0.046 f? C’. (91)

The values of s,.. for each nominal value of capacitor at frequencies of 100 Hz, 1 kHz, and 10 kHz
in the two-terminal and three-terminal configurations are included in Tables 18a and 18b,
respectively.

4.6.3 Conductance Potentiometer

According to [14], the limited resolution of the conductance potentiometer has an error in reading
the continuous adjusted dial of +0.1 uS, plus +0.1 pS in calibration, all divided by the conductance
divider K,. Accordingly, the standard uncertainty in conductance measurements as well as in
conductance calibrations due to the limited resolution of the conductance potentiometer is estimated
to be 0.058 1S. When two (or four) balances are needed for routine measurements the combined
uncertainty, defined as the RSS of the above three (or five) terms, (two (or four) in measurements
and one in calibration, all are 0.058 uS), is expressed as:

Sop 0.10/ Ky uS,  (forC,<1 uF), and

Ss

‘gp 0.13 / Ky ps, (for C, = 1 pF ), (92)

where s,,, is the standard uncertainty in conductance potentiometer, due to its limited resolution, and
kK, is the setting of the conductance divider.

According to the conductance divider, K,, being used in measurements, the values of s,, for each

nominal value of capacitors at frequencies of 100 Hz, 1 kHz, and 10 kHz in the two-terminal and
three-terminal configurations are estimated and included in Tables 18a and 18b, respectively.

35


NIST Special Publication 250-52

NIST MEASUREMENT SERVICES:

Error Analysis and Calibration Uncertainty
of Capacitance Standards at NIST

Y. May Chang

Electricity Division

Electronics and Electrical Engineering Laboratory
National Institute of Standards and Technology
Gaithersburg, MD 20899-8110

January 2000

Ent OF Co,
ae te
cS

0 * dg,
yn oe,
°
Me,
od

STares oF ©

U.S. Department of Commerce
William M. Daley, Secretary

Technology Administration
Dr. Cheryl L. Shavers, Under Secretary of Commerce for Technology

National Institute of Standards and Technology
Raymond G. Kammer, Director


THE SP 250 SERIES ON NIST MEASUREMENT SERVICES* — Continued

SP 250-45 Radiation Processing Dosimetry Calibration
Services: Manual of Calibration Procedures
SN003-003-03514-9 $4.00

SP 250-46 NIST Multifunction Calibration System
SNO003-003-03515-7 $2.75

SP 250-47 NIST Calibration Service for Capacitance
Standards at Low Frequencies
SN003-003-03549-1 $7.00

SP 250-48 Spectral Reflectance
SN003-003-03545-9 $14.00

SP 250-49 NIST Calibration Services for Gas Flow
Meters: Piston Prover and Bell Prover
Gas Flow Facilities
SN003-003-03560-2 $6.00

SP 260-51 Calibration Service of Optoelectronic
Frequency Response at 1319 nm for
Combined Photodiode/rt Power Sensor
Transfer Standards
SN003-003-03623-4

SP 250-52 Error Analysis and Calibration Uncertainty of
Capacitance Standards at NIST

SP 250-53 Calibration Service for Spectral Responsivity of
Laser and Optical-Fiber Power Meters at
Wavelengths Between 0.4 wm and 1.8 um
‘SN003-003-03624-2

* Entries containing a stock number (SNO03-003-) and price can be purchased from the Superintendent of Documents, U.S. Government
Printing Office, Washington, DC 20402-9325. GPO will accept checks, money orders, VISA, and MasterCard. For more infomation. or to place
an order, call (202) 512-1800. Be sure to cite the stock number cn all orders.

Entries containing PB numbers can be purchased from the National Technical Information Service, Soringfield, VA 22161. NTIS will accept
‘American Express in addition to the payment methods listed for GPO. For more in‘ormation call (703)487-4650; to place an order call
(800) 553-6487. Fax: (703) 321-8547. Be sure to cite the PB number on all orders

Entries without stock or PB numbers are in preparat’on.


Table 16a. Summary of Components of Type A Standard Uncertainties in Capacitance Measurements Using
the Type-12 Bridge to Measure Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz

(All values are in ppm - except nominal values of unknown capacitance standards)

( Frequency = | kHz )

Unknown Standard C, (uF) 0.001 | 0.002 | 0.005} 0.01 | 0.02 | 0.05 | 0.1 0.2 | 0.5 1.0
a
Components of Standard Uncertainties
( Frequency = 100 Hz )

Repeatability (5.m) 30 20 20 20 20 20 20 20 20 20

Temperature Variation (5 tm) 4 4 4 4 4 4 4 4 4 4

Predicted Values (4) 34 - - 32 - - 10 - - 10
Type-A Standard Uncertainty (u,) | 45.52 | 20.40 | 20.40 | 37.95 | 20.40 | 20.40 | 22.72 | 20.40 | 20.40 | 22.72

Assigned u, 46 25 25 38 25 25 25 25 25 25

eT

58

Repeatability (s,m) 30 20 20 20 20 20 20 20 20 20
Temperature Variation (5 tm) 4 4 4 4 4 4 4 4 4 4
Predicted Values (s pq) 25 25 20 15 20 15 10 10 10 10
Type-A Standard Uncertainty (w,) | 39.26 | 32.26 | 28.57 | 25.32 | 28.57 | 25.32 | 22.72 | 22.72 | 22.72 | 22.72
Assigned u, 40 35 30 30 30 30 25 25 25 25
oo
( Frequency = 10 kHz )
Repeatability (Sm) 30 20 20 20 20 20 20 20 20 20
Temperature Variation (5 tm) 4 4 4 4 4 4 4 4 4 4
Predicted Values (s 54) 15 - - 10 - - 15 - - 15
Type-A Standard Uncertainty (u,) | 33.78 | 20.40 | 20.40 | 22.72 | 20.40 | 20.40 | 25.32 | 20.40 | 20.40 | 25.32
Assigned u, 35 25 25 25 25 25



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44


APPENDIX C.

CONDUCTANCE DIAL CORRECTIONS OF THE TYPE-12 BRIDGE

Data and analyses of conductance corrections for each mica capacitance dial of the Type-12 bridge are
included in the following tables:

Table C-1. Data and Analysis of Conductance Corrections of ( 0.1 #F/step ) Mica Capacitance Dial
of the Type-12 Bridge at 100 Hz
(All values are in 0.001 1S, except the dial which is 0.1 4F/step)
DIAL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.X
6/67 15.422 | 29.480 | 50.507 | 51.425 | 66.792 | 80.853 | 101.950] 117.432] 131.360] 147.055
11/67 15.573 | 30.235 | 50.848 | 52.920 | 68.485 | 83.173 | 103.873] 119.520] 134.050] 149.875
5/68 15.743 | 30.173 | 51.198 | 52.458 | 68.240 | 82.723 | 103.808] 119.660] 134.058 | 150.110
9/69 15.693 | 30.395 | 51.225 | 54.055 | 69.765 | 84.570 | 105.435 | 121.320] 136.035 | 152.025
12/70 15.715 | 30.563 | 51.328 | 53.610 | 69.340 | 84.240 | 105.033 | 120.908 | 135.635 | 151.625
9/72 15.915 | 30.885 | 51.800 | 55.570 | 71.485 | 86.500 | 107.485 | 123.513 | 138.455 | 154.538
6/74 15.965 | 30.625 | 51.873 | 54.035 | 70.050 | 84.778 | 106.178] 122.265 | 136.845 | 153.253
11/75 15.710 | 30.388 | 51.223 | 53.695 | 69.333 | 84.185 | 105.025 | 120.893 | 135.780} 151.678
V77 15,858 | 30.230 | 51.670 | 52.705 | 68.508 | 82.953 | 104.418] 120.410 | 134.738} 151.003
4/78 15.490 | 29.505 | 50.995 | 51.713 | 67.243 | 81.280 | 102.760] 118.395} 132.310] 148.010
11/80 15.250 | 29.470 | 50.200 | 51.913 | 67.203 | 81.570 | 102.523 | 117.965 | 132.483 | 148.273
8/82 15.398 | 29.625 | 50.568 | 48.255 | 63.630 | 77.965 | 99.140 | 114.585] 128.813] 144.710
9/83 15.485 | 30.228 | 50.773 | 54.513 | 70.013 | 84.783 | 105.533 | 121.128] 135.940 | 151.943
1/85 15.400 | 30.315 | 50.933 | 55.215 | 70.668 | 85.605 | 106.255 | 121.870] 136.780} 152.570
3/85 15.518 | 29.993 | 51.063 | 53.140 | 68.680 | 83.713 | 104.778 | 120.455 | 134.840] 150.700
2/91* 14.625 | 28.197 | 48.720 | 49.175 | 63.505 | 76.885 | 97.022 | 111.475] 124.565 | 139.132
Mean (nS) | 15.5475 | 30.0192 | 50.9328 | 52.7748 | 68.3088 | 82.8610} 103.826} 119.487} 133.918] 149.781
SD (nS) 0.3191 | 0.6465 | 0.7473 | 1.9816 | 2.2459 | 2.6261 | 2.697 | 3.027 | 3.451 3.773
Slope(nS/day) | -7.8E-05|-1.3E-04| -1.5E-04] -2.2E-04] -3.1E-04] -3.5E-04| -4.0E-04|-4.9E-04| -5.4E-04] -6.2E-04
Slope(nS/year) | -0.0285 | -0.0469 } -0.0563 | -0.0798 } -0.1126 | -0.1295 | -0.1449 | -0.1772 | -0.1989 | -0.2267
SDfit 0.2512 | 0.5690 | 0.6478 | 1.9616 | 2.1653 | 2.5382 | 2.5701 | 2.8362 | 3.2443 | 3.5147
SD(2/01) (nS) | 0.3373 | 0.7639 | 0.8697 | 2.6335 | 2.9070 | 3.4077 | 3.4506 | 3.8078 | 4.3556 | 4.7188
* See Section 4.6.5

C-1



to provide capacitance and conductance values of the mica capacitors being calibrated by the Type-
12 bridge.
4.1 Basic Equations of Capacitance Measurements

Figure 5b is a simplified schematic of the Type-12 bridge showing the components for capacitance
measurements, where R, and R, are resistors with a nominal ratio of 1:1. Using C, and C,, as the
total capacitance of each of the other two arms, the basic equation of balance for making capacitance
measurements (when the voltage across detector D is zero) is:

R,C,=R, C,, (33)
where C, and C, are the total capacitances of arm AD and arm CD, respectively.
As shown in Fig. 5b, C, and C, can be expressed as:

Ci= C+ Cy t+ Cy (when C, is connected to arm AD) (34)
and

Qa, + Cy (when C, is connected to arm CD) , (35)
where C, is the unknown capacitor to be measured, C,, is the reading of internal mica dial
capacitors, and C, and C, are effective capacitances of arms AD and CD, respectively, with internal
air capacitors coupled so that:

C,+C,=C  (aconstant). (36)
Each dial of the air capacitors of the Type-12 bridge is arranged such that the dial reading indicates
the difference of the effective capacitance between arm AD and arm CD, plus a constant capacitance
C, as:

C= (Ca- CHC, G7)
where C,, is the capacitance value corresponding to the dial reading, r.
Substituting Eq. (36) into Eq. (37) gives:

C,=2C,. (38)
The internal air capacitors of the Type-12 bridge simulate three decade capacitors consisting of ten
0.001 wF (1000 pF) , ten 0.0001 pF (100 pF), and ten 0.00001 pF (10 pF) capacitors, and a
continuously variable capacitor providing a capacitance range of (11 + 0.5) pF.
Using the 10 pF/step dial as an example, where C = 50 pF, a set of four capacitors of 5 pF, 10 pF,

15 pF, and 20 pF, are switched back and forth individually from arm CD to arm AD when the dial
readings are increased from 0 to 1, 1 to 2, ........sse , and 9 to 10. The capacitance values corresponding

18


AG = 394x 10S < 4x 107 nS.

Table 4 gives a summary of the uncertainties due to lead impedance for measurements of various
nominal capacitance values at frequencies of 100 Hz, 400 Hz, and 1 kHz.

3.3.5 Uncertainty in the Reference Standard

The relative stability of eleven fused-silica dielectric capacitors was measured over a period of five
months in 1964 [6]. With the exception of one capacitor, which exhibited a fairly steady increase
in capacitance over time and has a large voltage dependence of capacitance, data taken on the others

indicated the maximum bounds of error are + 9.076 ppm. Therefore, the uncertainty of these
standards is taken to be:

5, = 0.044 ppm < 0.05 ppm,

where s, is the standard uncertainty of using the 10 pF fused-silica dielectric capacitors from the
PCL as reference.

3.3.6 Combined Uncertainty of the External Bridge Components

The combined uncertainty of the external bridge components is estimated as the RSS of the above
five components in sections 3.3.1 to 3.3.5 as:

Sop = (Syq’ +56" +54, +(AC/CY +52)! — ppm, (32)

where 5, is the combined standard uncertainty in the external bridge components using the Type-2
bridge to measure fused-silica and nitrogen dielectric capacitors. Table 5 contains a summary of sc,
for various nominal values of capacitors measured at 1 kHz.

After the combined Type B standard uncertainty is estimated by using a capacitance standard from
the PCL as reference to measure another capacitor, C,, this value is utilized as the uncertainty (s,)

of C, when it ( previously C,, ) is used as a reference to calibrate other capacitors later on, also
shown in Table 5.

3.4 Uncertainties of Fused-Silica and Nitrogen Dielectric Capacitors at 1 kHz

Table 5 is a summary of the calculations for the combined Type B standard uncertainties for
measurements of fused-silica and nitrogen dielectric capacitors of different nominal values at a
frequency of 1 kHz in the ICL, using the Type-2 bridge. Again, since only capacitance values are
reported, only the components in capacitance measurements are used for uncertainty calculations.
By taking the RSS value of all the above components, the uncertainty in measurements, using a
10 pF fused-silica capacitance standard from PCL as reference, is estimated to be 0.711 ppm.
Therefore, the combined Type B standard uncertainty of measuring an oil-bath type 10 pF fused-

14


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THE SP 250 SERIES ON NIST MEASUREMENT SERVICES*

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Far Ultraviolet Detector Standards
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By substituting Eq. (62) and ( R, = R, ) into Eq. (61), it becomes:

Var(Cy) < (Cy Spq)°+ 5p? + Spe" 5 (63)

22 2 2
where Sip = Stor + Syc0

From Eq. (39), the uncertainty contributed by the ratio resistors,

for the two-terminal high-
capacitance measurements is 18 ppm. (See Table'15a).

Star

The uncertainty of the internal mica capacitors is estimated from the dials being used. According
to Eq. (60), the value of the mica-dials correspond to the nominal value of the unknown capacitor.
Therefore, the combined uncertainty of the internal mica capacitors, s,,, is estimated to range
between 28 ppm and 35 ppm, from Eq. (44). (See Table 14a). The combined uncertainty of the
internal air-dial capacitors for the two-terminal high capacitance measurements, Sy is estimated to
be less than 9 ppm, from Eq. (43). (See Tables 13a, columns 4 to 10 and 15a).

Uncertainties due to the capacitance of plate P2 are + 0.02 pF and uncertainties in lead impedance
are functions of both frequency and capacitance values, as shown in Eq. (47). At frequencies of
100 Hz and 1 kHz, the combined uncertainties are less than 5 ppm. However, at a frequency of
10 kHz, these uncertainties increase substantially for high valued capacitors, and are calculated to
be as large as 475 ppm.

The Type-B standard uncertainties for two-terminal high-capacitance value measurements, as well

as their individual uncertainty components, are included in Table 15a.

4.3.3 Three-Terminal Measurements of Low-Capacitance Values
(0.001 uF < Cy < 0.005 uF)
4.3.3.1 Basic Equations of Balance
This type of measurement is similar to those of two-terminal low-capacitance measurements, except
that the unknown capacitor, C,, is connected to arm AD in the first balance and to arm CD in the
second balance, as illustrated in Fig. 8. In this case, the equations of balance are:
Ry Cay + Cy) = Ry Cy (64)

and

RX Co =R, (Cop * Cy), (65)

where C,, and C,, are the values of the selected internal capacitors at balance with C, connected
to arm AD, and C,, and C,, are the values at balance with C, connected to arm CD.

From Eqs. (64) and (65), the value of the unknown capacitor can be expressed as:
Cy =(A) (Cy - Cs (66)

26


Table 11. Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure
Two-Terminal Coaxial Air Capacitors at 1 kHz

(All values are in ppm, except nominal values of capacitors)

(Lower-Valued ) (Mid-Valued )
Nominal Value of
Capacitors (pF) 1 2 5 10 20 50 to 1000
Components of
Standard Uncertainties
Temperature 100 40 20 40 20 10
Relative Humidity 400 200 80 0 0 16
Hysterises 20 20 20 20 20 20
Voltage 5 5 5 5 5 5
Type B Standard Uncertainty (u,) | 412.83 205 85 45 28.72 27.95
Assigned uy, 415 205 85 45 30 28
Type A Standard Uncertainty (w,) 35 35 35 20 20 10
Expanded Uncertainty (U) 832.95 | 415.93 | 183.85 | 98.49 72.11 59.46
Assigned
Total Uncertainty (U,) 840 420 200 100 75 60

50



ay!

7

QO
n

\ |
7I
QD
*

Ex

Figure 2b. Simplified Circuit Diagram of the NIST Type-2 Capacitance Bridge.


4.2.1 Ratio Resistors

According to the specifications for the Type-12 bridge [13], the maximum bounds of error in the 1:1
ratio are + 30 ppm. Thus, according to Eq. (4), the uncertainty in the ratio of 1:1 is estimated as:

Spq = 17.3 ppm < 18 ppm, (39)
Ta

where s,, is the standard uncertainty contributed by the ratio resistors.

4.2.2 Internal Capacitors

The uncertainty components of both air and mica dial capacitors depend on their characteristics and
errors due to their calibrations and environmental variations, which are discussed as follows:

1) Dial Corrections

Data and analyses of capacitance dial corrections for the Type-12 bridge at various
frequencies (100 Hz, 1 kHz, and 10 kHz) are given in Appendix B. The capacitance
correction of each dial setting has been evaluated by fitting a regression line to the
corresponding data of 25 years to determine the predicted value. These corrections are
updated once a year and are used to measure two sets of check standards of the Type-12
bridge. The data histories of these check standards are analyzed to ensure that the linear
models of the dial corrections are valid. Thus, the standard deviations of the predicted values
of the dial corrections for a particular date can be obtained. The standard uncertainty of the
dial corrections is the RSS of the standard deviations of the predicted values for the dials
which depend on the nominal value of the capacitor being measured.

For the air capacitor dials, the standard deviations of the predicted values of the dial
corrections of those dials being used during measurements for the two-terminal and three-
terminal configurations range between 0.018 pF and 0.076 pF, as given in the first rows of
Tables 13a and 13b*. According to the nominal values of the unknown capacitors, the
respective standard uncertainties (in ppm) of the dial corrections of the air capacitors, sq, for
both the two-terminal and three-terminal measurements are estimated and also given in
Tables 13a and 13b (second rows).

For the mica capacitor dials, the standard uncertainties are estimated in the same manner as
in the air capacitor dials for the two-terminal and three-terminal measurements, respectively.
According to the dials being used during measurements and the nominal values of the
unknown capacitors, the standard uncertainty of the corrections of the mica capacitor dials,
Simp are included in Tables 14a and 14b, for the two-terminal and three-terminal

2 Starting from Table 13, the Tables are arranged that Table xxa and Table xxb are using
the "a" and the "b" to refer to the components of standard uncertainties for the two-terminal and
three-terminal measurements, respectively.

20


Table B-4. Data and Analysis of the ( 0.01 ,F/step ) Mica Capacitor Dial Corrections of the
Type-12 Bridge at 100 Hz

[All values are in pF except the dials which are in (jF/step)]

Date Dial 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0X

6/67 2.558 | 4.926 | 7.174 | 9.840 | 12.623 | 14.896 | 14.297 | 17.852 | 20.346 | 20.623
11/67 2.625 | 5.143 | 7.624 | 10.009 | 12.915 | 15.365 | 14.988 | 18.629 | 21.272 | 21.610
5/68 2.869 | 5.312 | 7.746 | 10.342 | 13.396 | 15.802 | 15.491 | 19.284 | 21.900 | 22.447
91/69 2.748 | 5.161 7.616 | 10.066 | 12.994 | 15.354 | 14.925 | 18.615 | 21.250 | 21.668
12/70 2.613 | 5.186 | 7.619 | 9.907 | 12.835 | 15.398 | 15.018 | 18.656 | 21.449 | 21.824
9/72 2.531 | 4.978 | 7.249 | 9.696 | 12.602 | 15.066 | 14.710 | 18.554 | 21.358 | 21.820
6/74 2.700 | 5.340 | 7.391 | 9.638 | 12.584 | 15.011 | 14.454 | 18.104 }| 20.857 | 21.272
11/75 2.733 | 5.162 | 7.615 | 9.838 | 12.801 | 15.117 | 14.655 | 18.411 } 21.117 | 21.615
V/77 2.694 | 5.178 | 7.534 | 9.940 | 12.989 | 15.458 | 14.922 | 18.643 | 21.339 | 21.753
4/78 2.373 | 4.774 | 6.707 | 8.948 | 11.758 | 14.134 | 13.794 | 16.750 | 19.338 | 19.506
11/80 2.462 | 4.778 | 6.880 | 8.789 | 11.673 | 14.044 | 12.596 | 16.207 | 18.829 | 19.183
8/82 2.573 | 5.109 | 7.083 | 9.094 | 12.029 | 14.328 | 13.611 | 17.282 | 19.888 | 20.209
9/83 2.560 | 5.155 | 7.132 | 9.092 | 11.990 | 14.445 | 13.767 | 17.450 | 20.127 | 20.537
1/85 2.667 | 5.210 | 7.462 | 9.317 | 12.252 | 14.854 | 14.049 | 17.711 | 20.489 | 20.781
3/85 2.671 | 5.264 | 7.200 | 9.133 | 11.992 } 14.577 | 13.778 | 17.519 | 20.262 | 20.551
2/91 2.589 | 5.152 | 6.584 | 8.570 | 11.486 | 14.342 | 13.384 | 16.967 | 19.256 | 18.972

N. Val. (pF) | 1.0E+05]2.0E+05]3.0E+05|4.0E+05] 5.0E+05| 6.0E+05|7.0F+05 8.0E+05|9.0F+05] 1.0R+06

Mean (pF) 2.6229 | 5.1143 | 7.2885 | 9.5137 | 12.4324] 14.8869| 14.2774 | 17.9146 | 20.5673 | 20.8982
SD (pF) 0.1190 | 0.1678 | 0.3499 | 0.5232 | 0.5589 | 0.5261 | 0.7566 | 0.8420 | 0.9011 | 1.0326

Slope (pF/day) |-1.3E-05) 1.6E-07 }-8.5E-05|-1.7E-04]-1.7E-04]-1.4E-04|-2.1E-04|-2.2E-04]-2.3E-04)-2.8E-04
( pF/year) -0.005 | 0.000 | -0.031 | -0.063 | -0.063 | -0.052 | -0.078 | -0.081 | -0.086 | -0.101
SDfit (pF) 0.118 | 0.174 | 0.277 | 0.269 | 0.329 | 0.380 | 0.516 | 0.622 | 0.676 | 0.752
SD(2/01) (pF) | 0.159 | 0.233 | 0.371 | 0.362 | 0.441 | 0.510 | 0.693 | 0.836 | 0.907 } 1.010

B-4


ERROR ANALYSIS AND CALIBRATION UNCERTAINTY
OF CAPACITANCE STANDARDS AT.NIST

YY. May Chang
Electricity Division
National Institute of Standards and Technology

ABSTRACT

This document presents the analysis of error sources that contribute to the total uncertainty of
capacitance calibrations at the National Institute of Standards and Technology (NIST). Based on
considerations of the measuring systems and calibration procedures, and data taken on NIST
working and check standards and customer’s standards, uncertainties in the calibrations for each
model and nominal value of capacitor are estimated. The results of the analysis are expressed as
expanded uncertainties using the coverage factor k = 2, in accordance with NIST Technical Note
1297. Also included are a detailed description and analysis for each component of error in the
evaluation of Type A and Type B standard uncertainties.

1. INTRODUCTION

Since 1960, a transformer-ratio capacitance-measuring system [1], known as the " Type-2 " bridge,
has been used in the National Bureau of Standards / National Institute of Standards and Technology
(NBS/NIST) impedance calibration laboratory (ICL) to transfer the farad from the primary
capacitance laboratory (PCL), via transfer standards, to NIST working standards and, ultimately, to
customer’s standards. The values of these standards are expressed in terms of measurements of the
NIST calculable capacitor, used to realize the farad in SI units. During the past ten years, high
quality commercial instruments, such as detectors and function generators, have become available
for use in the Type-2 system to improve its resolution. The establishment of a bank of oil-bath-type
fused-silica capacitors with predicable temperature corrections for use as reference standards in the
ICL has increased the stability of Type-2 system measurements. In 1993, NIST issued a new policy
on the expression of uncertainty associated with measurement results, as described in NIST
Technical Note (NIST TN) 1297 using the coverage factor k= 2 [2], consistent with the international
practice [2, 3]. Therefore, based on the above modifications, uncertainties in NIST capacitance
calibration have been reevaluated and reestablished for each model and type of capacitance standard
measured.

In brief, the combined standard uncertainty of a measured value is the combination of estimates of
two types of uncertainties. The first is the Type A standard uncertainty, defined as that which can
be evaluated by statistical methods. The other is the Type B standard uncertainty, which has no root
in formal statistics, but rather evaluated many times based solely on the experience of the

1


cf cf cf ca cf ccf Cy cd

Ne

100 10 1 10% 102 493 104 105

(pF )

Figure 3. Internal Capacitors of the Type-2 Bridge.

|

Figure 4. Lead Impedance of a Capacitor Connected to the Bridge.

69


Figure 9. Components of the Type-12 Bridge for Three-Terminal Measurements of
Capacitors up to 0.5 pF.

75


Figure 1.

Figure 2a.
Figure 2b.
Figure 3.
Figure 4.
Figure 5a.
Figure 5b.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

List of Figures

page
Block Diagram of the Farad Transfer and Calibration Process by Using
the Type-2 Bridge 0.0.6... cece eee cence nent e ne ennee 66
Schematic Diagram of the NIST Type-2 Capacitance Bridge ...............++- 67
Simplified Circuit Diagram of the NIST Type-2 Capacitance Bridge............ 68
Internal Capacitors of the Type-2 Bridge ........... 0. ce cece cece eee eee eee 69
Lead Impedance of a capacitor connected to the Bridge .. 6.6.6... 0c eee eee eee 69
Schematic Diagram of the Type-12 Capacitance Bridge ................0e00 ee 70
Simplified Circuit Diagram of the Type-12 Capacitance Bridge ............... 71
Components of the Type-12 Bridge for Two-Terminal
Low-Capacitance Values Measurements .........6. 00sec cece ese e eee eeeees 72
Components of the Type-12 Bridge for Two-Terminal
High-Capacitance Values Measurements .......... 0.0 e eee e eee e eee e eee nee 73
Components of the Type-12 Bridge for Three-Terminal
Low-Capacitance Values Measurements ............ 0 0c c eee ee cece eee eee 714
Components of the Type-12 Bridge for Three-Terminal Measurements
of Capacitors up to 0.5 WF oo... 6. cece ccc eee eee een e eee 75
Components of the Type-12 Bridge for Three-Terminal Measurements
of Capacitors Larger than 0.5 PF... 2.1... e cece cece cece eee nee eee 76
Simplified Circuit Diagram of the Type-12 Capacitance Bridge
Including the Conductance Components ...........0.e cece cece eee ence ee eee 77

vii


way Cc @
Ce

Ca
X®,
D

Figure 7. Components of the Type-12 Bridge for Two-Terminal High-Capacitance
Values Measurements.

73


Table B-2. Data and Analysis of the ( 0.1 F/step ) Mica Capacitor Dial Corrections of the
Type-12 Bridge at 1 kHz
[All values are in pF except the dials which are in (uF/step)]

Date Dial 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.x
6/67 9.522 | 20.953 | 34.653 } 43.932 } 53.654 | 64.848 | 76.408 | 87.139 | 98.631 | 106.029
11/67 9.705 | 22.182 | 34.121 | 46.703 | 56.383 | 68.800 | 78.367 | 89.047 | 101.917 109.401
5/68 10.244 | 22.408 | 35.507 | 46.347 | 56.786 | 68.780 | 79.704 | 91.092 | 103.511] 111.678
10/69 9.533 | 22.286 | 34.772 | 49.342 | 59.030 | 71.561 | 81.847 | 92.502 | 105.488] 112.861
12/70 9.542 | 22.709 | 35.576 | 48.770 | 58.502 | 71.422 | 82.079 | 92.883 | 106.127] 113.514
9/72 9.897 | 23.147 | 31.106 | 53.148 | 63.368 | 76.185 | 82.202 | 92.999 | 106.621} 114.550
6/74 10.752 | 23.997 | 33.094 | 52.792 | 63.621 | 76.659 | 83.574 | 95.366 | 108.613] 116.993
11/75 10.588 | 24.618 | 39.073 | 55.295 | 66.103 | 79.755 | 92.148 | 103.905 | 118.263 | 126.865
77 10.213 | 22.744 | 37.782 | 50.548 | 60.833 | 73.164 | 86.462 | 92.568 | 110.491] 118.474
4/78 9.394 | 21.063 | 36.221 | 48.260 | 57.992 | 69.640 | 83.019 | 93.405 | 105.327] 112.628
11/80 9.435 | 22.353 | 36.393 | 52.559 | 62.166 | 75.404 | 88.194 | 98.502 | 111.187] 118.503
8/82 9.446 | 22.688 | 37.819 | 56.923 | 66.559 | 79.701 | 93.087 | 103.668] 117.187] 124.176
9/83 9.592 | 23.459 | 33.441 | 58.878 | 68.678 | 82.418 | 90.272 | 101.104] 115.226} 122.771
1/85 9.724 | 24.240 | 38.454 | 61.502 | 71.393 | 85.677 | 97.786 | 108.552] 123.278] 130.692
3/85 9.171 | 21.622 | 34.525) 52.493 | 61.847 | 74.082 | 87.226 | 97.457 | 110.102] 116.931
2/91 9.857 | 22.480 | 33.344 | 56.339 | 66.521 | 79.029 | 88.098 | 98.671 | 111.643] 118.968

N. Val. (pF) | 1.0E+05}2.0E+05) 3.0E+05| 4.0E+05| 5.0E+05] 6.0E+05| 7.0E+05| 8.0E+05|9.0E+05] 1.0E+06

Mean (pF) 9.7884 | 22.6843 | 35.3676 | 52.1144 | 62.0898 | 74.8203 | 85.6546 | 96.1788] 109.601] 117.190
SD (pF) 0.4470 | 1.0358 | 2.1787 | 4.8477 | 4.9138 | 5.6067 | 5.8077 | 5.8919 | 6.494 | 6.523

Slope (pF/day) |-3.8E-05] 7.5E-05 | 1.2E-04| 1.4E-03] 1.4E-03] 1.5E-03 | 1.7E-03 | 1.6E-03 | 1.8E-03 | 1.7E-03
( pF/year) -0.014 | 0.027 | 0.045 0.526 | 0.517 | 0.564 | 0.624 0.601 0.643 0.617
SDfit (pF) 0.451 | 1.053 | 2.230 | 3.094 | 3.286 | 3.970 | 3.767 | 4.100 | 4.674 | 4.908
SD(2/01) (pF) 0.606 1.413 2.994 | 4.154 | 4.412 5.330 5.058 5.505 6.275 | 6.589

B-2



Table 13a. Summary of Components of Standard Uncertainties in Air Capacitor Dial Corrections
for Two-Terminal Measurements

Unknown Standard C,(uF) | 0.001 | 0.002 | 0.005 | 0.01 | 0.02 | 0.05 0.1 0.2 0.5 1.0
LL 1
Std Dev.of Pred. Values of .
Dial Correction (pF) 0.0601 | 0.0672 | 0.0759 | 0.0208 | 0.0208 | 0.0208 | 0.0208 | 0.0208 | 0.0276} 0.0213
Std Dev.of Pred. Values of +
Dial Correction (ppm) 60.136] 33.584] 15.184] 2.085 | 1.042 | 0.417 | 0.208 | 0.104 | 0.055 } 0.021
Components of
Standard Uncertainties (ppm)
Pred. Values of Dial Corr't (s 4,) | 60.136] 33.584] 15.184] 2.085 | 1.042 | 0.417 | 0.208 | 0.104 | 0.055 | 0.021
Temperature Variation (s,,) 6 6 6 6 6 6 6 6 6 6
Dial Switching (s 4.) 6 3 15 0.6 0.3 0.15 0.1 0.05 } 0.02 | 0.01
Transfer Standard (s,,) 6 6 6 6 6 6 6 6 6 6
RSS 61.027] 34.769] 17.459] 8.758 | 8.554 | 8.497 | 8.488 | 8.486 | 8.485 | 8.485
Assigned Standard Uncertainties
in Air Cap. Dial Corrections (s,.)| 62 35 18

52


Table 9. Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure
Three-Terminal Air Capacitors at Frequencies of 400 Hz and 1 kHz

(All values are in ppm, except nominal values of capacitors)

Nominal Value of

Capacitors (pF) 0.001 0.01 0.1 1 to 1000 10 000
Components of
Standard Uncertainties
Temperature 20 20 15 10 10
Relative Humidity 16 16 16 16 16
Hysterises 20 20 20 20 20
Voltage 5 5 5 5 5

Interpolation of Last Dial

(2x 10” pF) 200 20 2 0.2 to 0.0002 | 0.00002
Type B Standard Uncertainty (u,) | 202.68 38.48 30.17 27.95 27.95
Assigned uy 210 40 32 30 30
Type A Standard Uncertainty (w,) 950 90 30 30 60
Expanded Uncertainty (U) 1945.87 196.98 87.73 84.85 134.16
Assigned

Total Uncertainty (U,) 2000 200 100 100 150

48



Sq < 7.2 x 10° pF.

3) Table 1 gives a set of typical data of the average values and standard deviations of dial
corrections of the Type-2 bridge from 15 measurements taken over a period of one year.
Measurement records show that only the three lowest dials are needed during dial correction
measurements of C3. Looking at r; = 1.0 (which is a maximum) in Table 1, the standard
deviation of the dial corrections used during the calibration of C; is estimated from the
standard deviations of the last three dial corrections, (dials number 6, 7, and 8), also given
in Table 1. Thus,

(Dre 52) < 0.08 x 10° pF.

4) The maximum bounds of error due to the interpolation of the last capacitance dial is equal
to+(3x 107 ) pF. Therefore, the standard uncertainty of the interpolation of the last dial
becomes:

5, < 0.2 x 10% pF.

According to Eq. (20), the combined standard uncertainty for the third dial can be estimated by using
the RSS of the above components, which are also shown in Table 2, as:

53 = 0.75 x 10° pF <10° pF.

After s3 is determined, other s; can be estimated similarly from Eqs. (21) and (22). Table 2 is the
summary of standard uncertainties for each dial correction, s;, of the Type-2 bridge.

The combined standard uncertainty in the dial corrections during measurements depends on the
number of dials being used and is estimated as the RSS of standard uncertainties from those dials.
In general, the order of the dial number being used starts from dial number 8 and goes incrementally
to dial number 1. Table 3 gives the estimated combined standard uncertainties of dial corrections,
S¢q, for capacitors of various nominal values, according to the number of dials being used during
measurements.

3.3 Uncertainties in External Bridge Components

As mentioned in sections 1 and 2, the transfer of the farad to the impedance calibration laboratory
(ICL) is from a group of NIST-made 10 pF fused-silica capacitance standards in the primary
capacitance laboratory (PCL). The primary reference in the ICL is a group of the same type of
capacitors. The construction and stability of these standards, including results from measurements
of their performance, were discussed in detail by Cutkosky and Lee [6]. The analysis of
measurement uncertainties when using fused-silica capacitance standards as the reference is mainly
based upon the test results from [6]. During routine calibrations when nitrogen dielectric capacitors
are used as the reference standards, the uncertainty analysis is based on the performance of such
capacitors as published by the manufacturer [7].

10


11]

[12]

[13]

[14]

[15]

[16]

(17]

“Capacitance Standards with Precision Connectors," The General Radio Experimenter, vol.
41, no. 9, September, 1967.

"More Coaxial Capacitance Standards," The General Radio Experimenter, vol. 42, no. 5,
May, 1968.

W. D. Voelker, "An Improved Capacitance Bridge for Precision Measurements," Bell Labs
Record, vol. 20, pp. 133-137, Jan., 1942.

H. T. Wilhelm, "The Type-12 Capacitance Bridge," Bell Labs Memorandum, MM 70-4735-
9, Case 70052-213, File 37954-15, May, 1970.

Y. M. Chang and S. B. Tillett, "Calibration Procedures for Inductance Standards using a
Commercial Impedance Meter as a Comparator," Natl. Inst. Stand. Internal Report (NISTIR)
4466, Nov., 1990.

C. H. Young, Conductance Standard, U. S. Patent 2-326-274.

"NIST Calibration Services Users Guide," Natl. Inst. Stand. Technol. Special Publication
250, 1998 edition, U. S. Government Printing Office, Washington, DC 20402.

39


Table 4. Uncertainties in Lead Impedance Using the Type-2 Bridge to Measure
Capacitors at Frequencies of 100 Hz, 400 Hz, and 1 kHz

Frequency 1 kHz 400 Hz 100 Hz
Nominal Value AC/C AC/C AC/C
of Capacitor (pF) (ppm) (ppm) (ppm)
1 << 0.0002 << 0.0002 << 0.0002
10 0.0002 << 0.0002 << 0.0002
100 0.00197 0.00032 << 0.0002
1000 0.0197 0.0032 0.0002
10 000 0.197 0.032 0.00197

43


@)
o
©)

Ca Ce

D

Figure 6. Components of the Type-12 Bridge for Two-Terminal Low-Capacitance
Values Measurements.


Y, and Y, are admittances of Cc, and C,, respectively,

r; is the dial reading of i dial applying voltage to capacitor ¢;, where 0 < r; < 1, in
increments of 0.1,

y; is the admittance of ¢;,

g; is the effective admittance associated with the settings of the i conductance dial coupled
with the frequency compensation network,

Psis a constant whose value can be taken as either plus or minus 0.5, and

M is the multiplier of the conductance dial settings, where 10° < M< 1.

On the secondary side of the transformer, besides the +1 taps, a pair of + 0.1 taps are also externally
available for the reference and unknown capacitors to be connected to obtain 1:1, 1:10, and 10:1
ratios.

At present, this bridge is not used to measure the conductance or loss in standard capacitors.

The uncertainty associated with capacitance measurements using the Type-2 bridge is comprised of

two types, as given in Eq. (3). The evaluation of the components of the Type B standard uncertainty
takes into account errors in

- the transformer ratios,

- the main dial corrections (i.e., values of the Cg capacitor set),
- frequency dependence of the reference and bridge,

- similar voltage dependence,

- temperature corrections,

~- lead impedance, and

- the uncertainty of the reference standards.

The evaluation of the Type A standard uncertainty is based on the variability of the measurement
data. A detailed analysis for each type of uncertainty is given in the following sections.
3.1 Transformer Ratios
3.1.1 Corrections and Uncertainties of the 1:1 Ratio

Corrections for errors of the 1:1 ratio of the Type-2 bridge can be determined with two
measurements by using two capacitors of nominally equal value. One measurement is taken with
C, and C, connected as shown in Fig. 2b. The other measurement is taken with C, and Cy,
interchanged. The balance equation of the second measurement can be written as:

ESV + Es(L ry ¥n) + Ps Es (MY 8) =- Ey Ys, (7)

where parameters with subscript n have the same meanings as those in Eq. (6) with subscript i, and
M' is the multiplier of the second balance. M’ is normally equal to M.

By combining Eqs. (6) and (7), (with M = M") the result becomes:


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47


43.1.2 Analysis... 0... eee cece eee e eet e nee
4.3.2 Two-Terminal Measurements of High-Capacitance Values
4.3.2.1 Basic Equations of Balance
43.2.2 Analysis......... 0.0 cece cece eee eee eee :
4.3.3. Three-Terminal Measurements of Low-Capacitance Values ............ 26
4.3.3.1 Basic Equations of Balance eee
43.3.2 Analysis..................... wees
4.3.4 Three-Terminal Measurements of High-Capacitance Values
uptoO.5 pF ......... cee eee
4.3.4.1 Basic Equations of Balance . :
4.3.4.2 Amalysis......... 0.0. c eee eee eee . 28
4.3.5 Three-Terminal Measurements of Capacitors Larger Than 0.5 pF .
4.3.5.1 Basic Equations of Balance ..............
4.3.5.2 Analysis 00.0.0... cece eee ee eee
4.4 Type A Standard Uncertainty of Capacitance Measurements ...............005
4.4.1 Repeatability of Measurements ............ 0... cece cence eee eens
4.4.2 Effect of Temperature Variations on Unknown Capacitors .
44.3 Stabilities of NIST Standards ....... 0... o cece eee eee eee
4.5 Expanded Uncertainty for Capacitance Measurement
Using the Type-12 System... 2.2... 0. cece cece eee tence e nee ees
4.6 Type B Standard Uncertainty of Conductance Measurements .
4.6.1 Conductance Standard
4.6.2 Contact Resistance ....... 6.06. cece cece cece enter teen eees
4.6.3 Conductance Potentiometer
4.6.4 Residual Conductance ..
4.6.5 Dial Corrections . .
4.7 Type A Standard Uncertainty of Conductance Measurements .
4.7.1 Repeatability of Measurements
4.7.2 Stabilities of NIST Standards
4.8 Expanded Uncertainty for Conductance Measurement

Using the Type-12 System ..... 0.6... eee e cece cece eee eee eee eee 37

5. CONCLUSION

6. ACKNOWLEDGMENTS ..... 0.6.6 c cece cence etn n tener ene ete n tees 38

APPENDIXES 1.2.2.6 occ ence nner ee ene e ten en eens A-1

APPENDIX A. Uncertainty Analysis of the Corrections to the 1:1 Ratio ............. A-1
APPENDIX B. Capacitance Dial Corrections of the Type-12 Bridge ...
APPENDIX C. Conductance Dial Corrections of the Type-12 Bridge



Table 19a. Summary of Components of Type A, Expanded, and Assigned Uncertainties in Conductance
Measurements Using the Type-12 Bridge to Measure Two-Terminal Mica Capacitors at
100 Hz, 1 kHz, and 10 kHz

(All values are in 1S - except nominal values of the unknown capacitance standards)

Unknown Standard C, (uF) | 0.001 | 0.002 | 0.005 | 0.01 0.02 0.05 0.1 0.2 10
Components
(Frequency = 100 Hz)
Repeatability (Sym) 2.3E-05| 2.3E-05| 2.3E-05] 2.9E-05| 4.0E-05 | 7.2E-05] 1.2E-04] 1.8E-04/ 4.6B-04| 1.2E-03
Predicted Values (5 gpa) 1.0E-04 - - 1.2E-04 - - 2.1E-04 - - 5.0E-03
Type-A Std. Uncertainty (w.,) | 1.0E-04] 2.3E-05} 2.3E-05] 1.2E-04] 4.0E-05] 7.2E-05| 2.4E-04] 1.8E-04] 4.6E-04| 5.1E-03
Type-B Std. Uncertainty (14,) | 1.0E-04| 1.0E-04] 1.0E-04| 1.6E-04/ 1.7E-04| 1.8E-04} 4.1E-04} 7.7E-04} 2.9B-03 | 4.8E-03
Expanded Uncertainty (U,) }0.00029| 0.00021 | 0.00021 | 0.00040] 0.00035] 0.00039) 0.00094} 0.00158} 0.00590) 0.01399)
Assigned
Total Uncertainty (U,,) 0.0003 | 0.0002 | 0.0002 | 0.0004 | 0.0004 | 0.0004 | 0.001 | 0.002 | 0.006 | 0.02
—t
(Frequency = 1 kHz)
Repeatability (5 gan) 4.0E-05| 4.6E-05| 7.5E-05] 1.2E-04| 2.3E-04] 5.2E-04| 1.2E-03] 2.3E-03] 7.5E-03| 2.3E-02
Predicted Values (ga) 1.0E-04} 2.0E-04 | 2.0E-04] 3.0E-04| 1.0E-03 | 2.0E-03 | 6.0E-03 | 1.5E-02] 5.5E-02| 9.5E-02
Type-A Std. Uncertainty (w.,) | 1.1E-04] 2.1E-04|2.1E-04] 3.2E-04] 1.0E-03} 2.1E-03] 6.1E-03] 1.5E-02] 5.6E-02| 9.8E-02
Type-B Std. Uncertainty (u 4) | 1.0E-04] 1.0E-04) 1.0E-04} 1.4E-04] 2.0E-04| 3.0E-04] 9.3E-04| 3.4E-03| 1.4E-02| 4.8E-02
Expanded Uncertainty (U,) | 0.00030} 0.00046] 0.00047] 0.00070} 0.00209} 0.00418} 0.01236] 0.03112) 0.11473] 0.21824
Assigned
Total Uncertainty (U,) 0.0003 | 0.0005 | 0.0005 | 0.001 0.003 | 0.005 0.02 0.04 0.12 0.3
eee aes ees ee
(Frequency = 10 kHz)
Repeatability (5 gn) 1.8E-04| 2.3E-04| 5.8E-04] 1.2E-03 | 2.3E-03| 8.1E-03 | 2.3E-02] 6.4E-02| 3.5E-01 | 1.2E+00|
Predicted Values (5 goa) 1.0E-04 - - 3.3E-03 - - 3.5E-02 - - 1.,0E+00}
Type-A Std. Uncertainty (w,,) | 2.0E-04| 2.3E-04| 5.8E-04| 3.5E-03 | 2.3E-03) 8.1E-03 | 4.2E-02| 6.4E-02| 3.5E-01]1.5E+00|
Type-B Std. Uncertainty (ug) | 1.5E-04] 1.6E-04] 1.9E-04] 1.5E-03 | 3.6E-03) 1.3E-02] 5.3E-02| 2.0E-01|1.2E+00)4.8E+00|
Expanded Uncertainty (U,) | 0.0005 | 0.0006 | 0.0012 | 0.0076 | 0.0085 | 0.0314 | 0.1356 | 0.4116 | 2.4667 | 10.0424
Assigned
Total Uncertainty (U,) 0.001 | 0.001 | 0.005 | 0.01 0.02 0.05 0.2 0.5 3.0 10.0
————— 1 =<

64



to the dial readings are obtained in accordance with Eq. (36) and with the following procedure for
switching the capacitors:

At dial reading r = 0, all four capacitors are connected to arm CD, then
Cy = 50 pF and C, = 0; therefore, C,= 0.

At dial reading r = 1, the 5 pF capacitor is connected to arm AD, the other three are connected to arm
CD, then

C, = 45 pF and C, = 5 pF; therefore, C, = 10 pF.

At dial reading r = 2, the 10 pF capacitor is connected to arm AD, the other three are connected to
arm CD, then

C, = 40 pF and C, = 10 pF; therefore, C, = 20 pF.

At dial reading r = 10, all four capacitors are connected to arm AD, and then
C, = 0 and C, = 50 pF; therefore, C, = 100 pF.

Measurements of two- and three-terminal mica capacitors with nominal values up to 0.005 pF are
made using internal air capacitors only.

The Type-12 bridge also has a set of internal decade mica capacitors of ten 0.01 yF and ten 0.1 pF
which are employed in the arm AD, together with the air capacitors, to measure mica capacitors with
nominal values larger than 0.005 pF. As mentioned previously, a 0.005 uF mica capacitor is used
as a transfer standard to obtain the dial corrections for the 0.01 pF step of both air and mica dial

capacitors from Type-2 bridge measurements. All other steps of air and mica dial capacitors are
calibrated against the 0.01 uF step.

Again, the total unccrtainty associated with Type-12 bridge capacitance measurements contains both
Type A and Type B standard uncertainties. An analysis of each type of uncertainty is contained in
the following sections.

4.2 Type B Standard Uncertainty of Capacitance Measurements

The evaluation of the components of the Type B standard uncertainty for the Type-12 bridge is based
on uncertainties in the ratio of resistors, R, and R,, in the internal air and mica capacitors, and in the
external bridge components. An analysis of each component at various frequencies is discussed in
general to cover the entire range of capacitance calibrations. The Type B standard uncertainty for

each type of capacitance measurement depends on the measurement procedure and internal standards
used for the calibrations.

19


Table 18a. Summary of Components of Type B Standard Uncertainties in Conductance Measurements Using
the Type-12 Bridge to Measure Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz

(All values are in 1S - except nominal values of the unknown capacitance standards)

Unknown Standard C, (uF) | 9.001 0.002 | 0.005 | 0.01 | 0.02 | 0.05 0.1 0.2 0.5 1.0
Components of

Standard Uncertainties
(Frequency = 100 Hz)

Accuracy of G, (ss) 4.8E-07 | 6.4E-07 | 1.3E-06 | 1.6E-06} 3.2E-06| 3.2E-06| 2.4E-05] 3.2E-05 | 1.6E-04 | 2.8E-04
Contact Resistance (5 g.) 4.6E-10 | 1.8E-09| 1.2E-08 | 4.6E-08} 1.8E-07] 1.2E-06] 4.6E-06| 1.8E-05 | 1.2E-04 | 4.6E-04
Conductance Pot. (sg) 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.0E-04 1.0E-04] 1.0E-04] 1.0E-04] 1.0E-04 | 1.0E-04 | 1.3E-04
Residu. Conduct. (s gc) 1.2E-06 | 1.2E-06 | 1.2E-06 | 1.2E-06] 1.2E-06] 1.2E-06] 1.2E-06] 1.2E-06 | 1.2E-06 | 1.2E-04

Mica Capacitor Dials (gm) - - - 1.2E-04} 1.4E-04] 1.5E-04| 3.9E-04} 7.6E-04 | 2.9E-03 | 4.7E-03
Type-B Std. Uncertainty (x4) | 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.6E-04} 1.7E-04] 1.8E-04} 4.1E-04| 7.7E-04 | 2.9E-03 | 4.8E-03

(Frequency = 1 kHz)

Accuracy of G, (54s) 2.4E-06 | 3.2E-06 | 9.6E-06 | 1.6E-05] 3.2E-05] 3.2E-05| 9.6E-05] 1.6E-04 | 5.6E-04 | 2.3E-03
Contact Resistance (s ,,) 4.6E-08 | 1.8E-07 | 1.2E-06 | 4.6E-06} 1.8E-05] 1.2E-04) 4.6E-04] 1.8E-03 | 1.2E-02 | 4.6E-02
Conductance Pot. (5 .)) 1.0E-04 | 1.0E-04 | 1,0E-04 | 1.0E-04] 1.0E-04} 1.0E-04| 1.0E-04| 1.0E-04 | 1.0E-04 | 1.3E-04
Residu. Conduct. (s gc) 1.2E-05 | 1.2E-05 | 1.2E-05 | 1.2E-05] 1.2E-05] 1.2E-05} 1.2E-05] 1.2E-05 | 1.2E-05 | 1.2E-0!

Mica Capacitor Dials (5m) - - - 9.0E-05 | 1.7E-04] 2.6E-04] 8.0E-04] 2.9E-03 | 8.7E-03 | 1.5E-02
Type-B Std. Uncertainty (u .,) | 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.4E-04] 2.0E-04} 3.0E-04] 9.3E-04| 3.4E-03 | 1.4E-02 | 4.8E-0!

(Frequency = 10 kHz)

Accuracy of G, (s.;) 1.6E-05 | 1.9E-05 | 3.2E-05 | 4.8E-05] 9.6E-05| 9.6E-05] 1.6E-03) 3.2E-03 | 1.6E-02 | 1.4E-0
Contact Resistance (sg) 4.6E-06 | 1.8E-05 | 1.2E-04 | 4.6E-04| 1.8E-03] 1.2E-02|4.6E-02] 1.8E-01 | 1.2E+00 | 4.6E+0
Conductance Pot. (sg) 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.0E-04} 1.0E-04] 1.0E-04] 1.0E-04] 1.0E-03 | 1.0E-03 | 1.3E-0
Residu. Conduct. (s gc) 1.2E-04 | 1.2E-04 | 1.2E-04 | 1.2E-04| 1.2E-04] 1.2E-04} 1.2E-03} 1.2E-03 ] 1.2E-03 | 1.2E-0:

Mica Capacitor Dials (5 gm) - - - 1.4E-03} 3.1E-03 | 6.9E-03 | 2.7E-02| 6.5E-02 | 2.6E-01 | 1.2E+0
Type-B Std. Uncertainty (x 9,) | 1.5E-04 | 1.6E-04] 1.9E-04 } 1.5E-03 | 3.6E-03] 1.3E-02] 5.3E-02] 2.0E-01 } 1.2E+00 | 4.8E+0

62


Table 14b. Summary of Components of Standard Uncertainties in Mica Capacitor Dial Corrections

for Three-Terminal Measurements

Unknown Standard C, (uF) 0.01 0.02 0.05 0.1 0.2 0.5 1.0
Components of
Standard Uncertainties (ppm)
(At all Frequencies)
Temperature Variation (5 mp) 26 26 26 26 26 26 26
Dial Switching (s 4.) 0.6 0.3 0.15 0.1 0.05 0.02 0.01
Transfer Standard (s m¢) 6 6 6 6 6 6 6
(At Frequency = 100 Hz)
Std Dev.of Pred. Values of
Dial Correction (pF) 0.2331 | 0.3617 | 1.0096 | 2.5727 | 7.1997 | 11.8816 | 16.8030
Std Dev.of Pred. Values of
Dial Correction (ppm) (5 mr) 23.310 | 18.085 | 20.192 | 25.727 | 35.999 | 23.763 | 16.803
RSS. 35.436 | 32.236 | 33.463 | 37.066 | 44.810 | 35.731 31.533
Assigned Standard Uncertainties
in Mica Cap. Dial Corrections (s ,) 36 33 34 38 45 36 32
(At Frequency = 1 kHz)
Std Dev.of Pred. Values of
Dial Correction (pF) 0.1208 | 0.2414 | 0.6075 | 1.4130 | 4.1541 | 6.5888 | 9.3179
Std Dev.of Pred. Values of
Dial Correction (ppm) (8 ;n:) 12.080 | 12.070 } 12.150 } 14.130 } 20.771 } 13.178 | 9.318
RSS 29.297 | 29.288 | 29.320 | 30.194 | 33.814 | 29.760 | 28.263
Assigned Standard Uncertainties
in Mica Cap. Dial Corrections (8) 30 30 30 31 34 30 29
oo
(At Frequency = 10 kHz)
Std Dev.of Pred. Values of
Dial Correction (pF) 0.1087 | 0.2229 } 0.4965 | 1.2880 ] 3.2637 | 21.8635} NA
Std Dev.of Pred. Values of
Dial Correction (ppm) (5 mr) 10.870 | 11.145 9.930 12.880 | 16.319 | 43.727 NA
RSS 28.819 | 28.919 | 28.472 | 29.629 | 31.278 | 51.225 NA
Assigned Standard Uncertainties
in Mica Cap. Dial Corrections (s,) 29 29 29 30 32 52 NA

55


Therefore, no additional errors due to frequency dependence are incurred except for calibrations
performed at frequencies other than 1 kHz.

3.3.3 Hysteresis Effects in Temperature Corrections of Standards

Each NIST-made 10 pF fused-silica dielectric capacitor has a known temperature coefficient, which
is used to obtain a temperature correction during measurements. Errors in the temperature
measurements and their effects are negligible (<10~ ). However, hysteresis effects of temperature
on the capacitance of fused-silica dielectric capacitors are known to exist and will introduce errors
in the corrections. As described in [6], changes of capacitance of a group of nine such capacitors
were observed when measured at 25 °C before and after being subjected to 50 ‘C. These results are
employed to estimate the uncertainty due to temperature changes in the environment. Utilizing the
average change and the standard deviation of the mean, it is estimated to be:

Ste < 0.084 ppm < 0.1 ppm,

where s;,_is the standard uncertainty in temperature correction of using fused-silica capacitors as
reference.

In routine calibrations of nitrogen and air dielectric capacitors, no temperature corrections are
applied: Standard uncertainties associated with the nitrogen dielectric working standards (used as
reference standards ) due to variations of room temperature during measurements are estimated to
be less than 1 ppm, assuming the temperature variation is within + 0.5 °C. Thus,

Ste. $ 1 ppm,
where s;,, is the standard uncertainty from temperature changes of nitrogen dielectric capacitors
used as references.

3.3.4 Lead Impedance

In general, the effects of leads that are used to connect a capacitor, C, to a bridge can be represented
by the capacitor in series with a resistor and an inductor, as shown in Fig. 4. The expression for the
effective impedance in Fig. 4 can be shown to be:

Z={Rt+(G (Got eC) }+ jofL)-[C/Ge+0'C)]}, (23)
where Z is the effective impedance of the capacitor C,

R, is the series resistance of leads,

L, is the series inductance of leads, and

G, is the parallel conductance of the capacitor C.

The dissipation factors, DF,,, of the fused-silica, nitrogen, and air capacitors are all less than 10°;
therefore, the magnitude of (G,’+w°C”) is approximately equal to w’C’, since G.? < (10")w"C?.

12


measurements of the same nominal values, except that the capacitance of the plate, P3, is 0.006 pF,
and these are estimated to be less than 3 ppm at frequencies of 100 Hz and 1 kHz. However, at a

frequency of 10 kHz, these uncertainties increase substantially for high-valued capacitors, and are
calculated to be as large as 235 ppm.

The Type-B standard uncertainties of three-terminal high-capacitance value measurements are
included in Table 15b, as well as their individual uncertainty components for each capacitance value
at 100 Hz, 1 kHz, and 10 kHz.

4.3.5 Three-Terminal Measurements of Capacitors Larger Than 0.5 uF

4.3.5.1 Basic Equations of Balance

Due to range limitations of the internal mica capacitors of the Type-12 bridge, four balances are
needed to measure unknown capacitors larger than 0.5 1F, and measurements are performed at
frequencies only up to 1 kHz. The dummy capacitor box, C},, is connected across the arm CD and
its value is set to either 0 or the nominal value of the unknown capacitor, C,, as shown in Fig. 10.

Using C,,= 1 uF as an example, the measurement sequence and equations of balance in accordance
to Fig.10 are described as follows:

1) Connect C, across arm AD, set the mica dials to zero, and the value of C, to 1 uF. Use the
air dials to obtain a balance, and record the reading as C,,. The equation of balance is:

Ry Cart Cy) = Ry (Coy + Cp). (76)

2) Disconnect C,, and leave Cy at 1 uF. Use both the air and mica dials to obtain a balance,
and record the readings as C,) and C,,), respectively. The cquation of balance is:

Ry (Cap + Cy) = Ro (Cog + Cy). (77)

3) Connect C, across arm CD, and set C, to zero. Use both the air and mica dials to obtain a
balance, and record the readings as C,, and C,,,3, respectively. The equation of balance is:

Ry (Cy3 + Cay) = Ry (Cog + Cx). (78)

4) Disconnect C,, set both mica dials and C, to zero. Use the air dials to obtain a balance, and
record the reading as C,4. The equation of balance is:

Ry Cag = Ry Cog: (79)

The numerical subscripts of Eqs. (76) to (79) indicate the values of mica and air capacitors in the
corresponding measurements.

The value of the unknown capacitor, Cy can be expressed as:

29


Table 10. Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure
Three-Terminal Air Capacitors at 100 Hz

(All values are in ppm, except nominal values of capacitors)

Nominal Value of
Capacitors (pF) 0.01 0.1 1 10 to 1000
Components of
Standard Uncertainties
‘Temperature 20 15 10 10
Relative Humidity 16 16 16 16
Hysterises 20 20 20 20
Voltage 5 5 5 5
Frequency 1 1 1 1
Interpolation of Last Dial
(2x 10° pF) 200 20 2 0.2 to 0.002
Type B Standard Uncertainty (#,) 202.69 36.15 28.04 27.97
Assigned uy 220 40 30 30
Type A Standard Uncertainty (w,) 600 100 30 30
Expanded Uncertainty (U) 1278.12 215.41 84.85 84.85
Assigned
Total Uncertainty (U,) 1300 230 100 100

49


silica capacitor in ICL is assigned to be 0.72 ppm, including the components of transformer ratio,
dial corrections, and external bridge components. Other calculations using the oil-bath type
capacitor as the reference standard utilize s,, = 0.72 ppm as the uncertainty.

The Type A standard uncertainty is estimated from the measurement data on the repeatability of
measurements. A calibration of customer’s fused-silica dielectric capacitor standard consists of at
least five sets of measurements performed over a two-week period, and the pooled standard deviation
of within and between days measurements is employed as the Type A standard uncertainty. The
expanded uncertainty of such capacitors is calculated from both Type A and Type B standard
uncertainties with a coverage factor of k= 2 [3], as given in Eq. (3). For nitrogen dielectric capacitor
standards, the Type A standard uncertainty is taken from results of calibrations of a large population
of standards. In an individual calibration, the capacitor is measured a few times to ensure that the
use of the population statistics is valid. In the case where the Type A standard uncertainty is over
the limit, the total uncertainty is increased according to the data. Table 6 gives the expanded and
assigned uncertainties for calibrations of fused-silica and nitrogen dielectric capacitance standards
at 1 kHz, including both Type A and Type B standard uncertainties.

3.5 Uncertainties of Fused-Silica and Nitrogen Dielectric Capacitors at 400 Hz

The error analysis for the calibration of fused-silica and nitrogen dielectric capacitors at a frequency
of 400 Hz is similar to that at 1 kHz. Since the assigned value of the oil-bath-type fused-silica
reference capacitor in the ICL is measured at 1 kHz, there is an uncertainty of 0.9 ppm in frequency
dependent error, when using it as a reference to calibrate other capacitors at 400 Hz.

Table 7 is a summary of the expanded and assigned total uncertainties for calibrations of fused-silica
and nitrogen dielectric capacitors at a frequency of 400 Hz.

3.6 Uncertainties of Fused-Silica and Nitrogen Dielectric Capacitors at 100 Hz

The error analysis for the calibration of fused-silica and nitrogen dielectric capacitors at a frequency
of 100 Hz is similar to that at 1 kHz with two exceptions:

1) At 100 Hz, due to lower applied voltage, the resolution of the bridge is lower than at 1 kHz,
such that an uncertainty exists in the 10°° pF dial, instead of the 10° pF dial. The maximum
bounds of error is estimated to be + (3 x10°5) pF. According to the nominal values of the
unknown standards, the standard uncertainty to be included in the combined uncertainties
at measurement, sy, is :

Sop <2 ppm (for 1 pF capacitors),
Sg.) < 0.2 ppm (for 10 pF capacitors),
Sj < 0.02 ppm (for 100 pF capacitors), and
5s, < 0.002 ppm (for 1000 pF capacitors),

where s,; is the standard uncertainty in the interpolation of the last dial.

15


4) Uncertainties estimated from the interpolation of last dials

The sensitivity of the conductance dials is such that changes of the + 1 step on the fourth dial
do not affect the balance. This introduces a standard uncertainty to the in-phase correction
of:

Sag S 0.002 ppm,

where s,, is the standard uncertainty in the in-phase correction of the 1:1 ratio from the
interpolation of the last dial.

The sensitivity of the capacitance dials is such that the difference between two steps of the
last dial can be estimated to gain another digit of accuracy. Such interpolation introduces the
maximum bounds of error of + (3 x 10”) pF to the dial readings, which affects the
quadrature correction according to the nominal values of test capacitors:

Spq < 0.02 ppm (forC,=C,= 10pF),
Sp4 < 0.002 ppm (for C,=C,= 100 pF), and
Spq < 0.0002 ppm (for C, = C, = 1000 pF ),

where s,,, is the standard uncertainty in the quadrature correction of the 1:1 ratio from the
interpolation of the last dial.

3) U inties due to load admi :

An accurate method to analyze and measure the components of loading errors and lead
impedance was discussed and calculated in [5]. The maximum bounds of errors in ratio
corrections owing to loading effect and lead impedance are estimated. Accordingly,

Sqs < 0.004 ppm and sys < 0.019 ppm,

where s, and s,, are the standard uncertainties due to load admittance and lead impedance,
in the in-phase and quadrature corrections of the 1:1 ratio, respectively.

In routine calibration of capacitors using the Type-2 bridge, only capacitance values are reported,
even though both capacitance and conductance dials are used to obtain a balance. Therefore, only
the in-phase standard uncertainties of the ratio corrections are to be included in the Type B standard
uncertainty. The combined standard uncertainties that contribute to 1:1 ratio corrections, Sy and Sp»

are defined as the square root of the sum-of-squares (RSS) of each component, which are estimated
to be:

Sq= (Sgt? +5 gy? + 5y37 4 Sqq7 4 S57)" < 0.005 ppm, (Al?)

and
Sp=(sb,? + 5b)? + sb” , sby 4 sds?) < 0.03 ppm. (A18)

A-5


4.6.4 Residual Conductance

According to [14], the maximum bounds of error due to the limit of resolution in residu
conductance in arms AD and CD are functions of frequency and given as:

lp, = +(0.02)f/K, pS, (93)
where #eg¢ are the maximum bounds of error due to the residual conductance. Accordingly,
Sgq = (0.0115) f/Ky uS, (94)

where s,, is the standard uncertainty due to the residual conductance,
f is the applied frequency in kHz, and
Ky is the setting of the conductance divider.

The values of s,, for each nominal value of capacitor at frequencies of 100 Hz, 1 kHz, and 10 kHz

in the two-terminal and three-terminal configurations are included in Tables 18a and 18b,
respectively.

4.6.5 Dial Corrections

Data and analyses of conductance corrections of internal mica dial capacitors for the Type-12 bridge
at frequencies of 100 Hz, 1 kHz, and 10 kHz are given in Appendix C. As discussed in section 4.2.2,
the standard uncertainty of the dial corrections is estimated from the predicted values of the dials
being used during measurements, depending on the nominal value of the capacitor being measured.
The standard uncertainty in conductance corrections of the internal mica capacitors, s,_,, for the two-
terminal and three-terminal measurements of capacitors C, > 0.01 uF at frequencies of 100 Hz,
1 kHz, and 10 kHz are estimated and given in Table 18a and 18b, respectively.

The Type B standard uncertainties for the Type-12 bridge measurement of conductance at
frequencies of 1 kHz, 100 Hz, and 10 kHz, defined as the RSS of the above uncertainties, are also
given in Tables 18a and 18b.

4.7 Type A Standard Uncertainty of Conductance Measurements

The evaluation of the Type A standard uncertainty for the Type-12 system in making conductance
measurements is similar to that in making capacitance measurements. It is based on the repeatability
in measurement results, the variation of conductance corrections of the capacitance dials during
measurements, and the stability of the NIST conductance standards used with an impedance meter
as a comparator. An analysis of each term is given in the following sections.

36


Figure 11. Simplified Circuit Diagram of the Type-12 Capacitance Bridge Including
the Conductance Components.

77


Table 14a. Summary of Components of Standard Uncertainties in Mica Capacitor Dial Corrections
for Two-Terminal Measurements

Unknown Standard C, (uF) ] 0.01 0.02 0.05 0.1 0.2 0. 0

Components of
Standard Uncertainties (ppm)

(At all Frequencies)

Temperature Variation (5 mp) 26 26 26 26 26 26 26
Dial Switching (5 4.) 0.6 0.3 0.15 0.1 0.05 0.02 0.01
Transfer Standard (5 m¢) 6 6 6 6 6 6 6

(At Frequency = 100 Hz)

Std Dev.of Pred. Values of

Dial Correction (pF) 1585 2331 4412 | 1.0096 | 2.5727 | 7.7378 | 11.8816
Std Dev.of Pred. Values of

Dial Correction (ppm) (Sm) 15.850 | 11.655 | 8.824 } 10.096 | 12.864 | 15.476 | 11.882

RSS 31.042 | 29.119 | 28.105 | 28.530 | 29.622 | 30.846 | 29.209
Assigned Standard Uncertainties
in Mica Cap. Dial Corrections (s ») 32 30 29 29 30 31 30
(At Frequency = 1 kHz)
Std Dev.of Pred. Valucs of
Dial Correction (pF) 0.0702 | 0.1208 | 0.3019 | 0.6075 | 1.413 | 4.4166 | 6.5888
Std Dev.of Pred. Values of

Dial Correction (ppm) (5 mz) 7.020 | 6.040 | 6.038 | 6.075 | 7.065 | 8.833 | 6.589

RSS 27.598 | 27.360 | 27.358 | 27.366 | 27.603 | 28.107 | 27.485
Assigned Standard Uncertainties
in Mica Cap. Dial Corrections (s ,) 28 28 28 28 28 29 28
(At Frequency = 10 kHz)
Std Dev.of Pred.Values of
Dial Correction (pF) 0.0627 | 0.1087 | 0.275 } 0.4965 | 1.288 3.991 | 21.8635

Std Dev.of Pred. Values of
Dial Correction (ppm) (5 mr) 6.270 5.435 5.500 4.965 6.440 7.982 | 21.864

RSS. 27.417 | 27.233 | 27.245 | 27.142 | 27.450 | 27.852 | 34.497

Assigned Standard Uncertainties
in Mica Cap. Dial Corrections (s ») 28 28 28 28 28 28 35

54


APPENDIX A. UNCERTAINTY ANALYSIS OF THE CORRECTIONS TO THE 1:1 RATIO

Rewrite the balance equation and the corrections to the 1:1 ratio of the Type-2 bridge as shown in
Eqs. (6) to (9) of the text as Eqs. (A1) to (A4) as:

E,Y,+ E, (2 1;9j) + Ps Ey (MY 8;)=- Ex Vy, (Al)
where E, and E, are voltages produced by the transformer windings on the reference, C, and the

unknown capacitor, C, sides of the bridge, respectively,

Y, and Y, are admittances of C, and Cy , respectively,

‘ nis the dial Teading of i” dial applying voltage to capacitor ¢;

yj is the admittance of ¢;,
g;is the effective admittance associated with the settings of the i i conductance dial coupled
with the frequency compensation network,
P;is a constant whose value can be taken as either plus or minus 0.5, and
Mis the multiplier of the conductance dial settings, where 10° S<M<l.
Corrections for errors of 1:1 ratio of the Type-2 bridge can be determined in two measurements by
using two capacitors of nominally equal value. One measurement is taken with C, and C, connected

as shown in Fig. 2b. The other measurement is taken with C, and C, interchanged. The balance
equation of the second measurement can be written as:

Ey Yq Ey (Dn) + Ps By (MY, 8) =~ By Yo (a2)

where parameters with subscript n have the same meanings as those in Eq. (A1) with subscript i,
and M’ is the multiplier of the second balance. M’ is normally equal to M.

By combining Eqs. (A1) and (A2), (with M=M’ ) the result becomes:

(Ey + EY et Yt Ey (L794 L ty + Yq) + Ps EM (LD 8;+L 8,)=0. (A3)
Assuming E, = 1 V as the reference, and by the definition of 1:1 ratio, (E,./ E,) = - 1, then:

Ey =-(1+a+jb), : (A4)
where a and b are the in-phasc and quadrature corrections to the ratio, respectively.
Substitution of Eq. (A4) into Eq. (A3) gives:

(a+ jb)= (CL +L yn) + PsM(L 8+ L By ))/ (gt Yq) }- (A5)
Let ¥, = joC, (1 - jDFy), yp = joc, (1 - jdfy ), and gy = gy (1 + jx, ), where DF, and df, are the

dissipation factors of the respective capacitors Cy, and ¢, , and x, is the phase correction of the
resistor network used in the conductance dials. The magnitudes of DF, , df, , and x, are all less than

A-l


two components, which are estimated from each type of measurement, as:

(”)

She = (Shp? +5527) (48)

where s,,, is equal to either Spp2 OF Spp3» depending on which plate (either P2 or P3) is being used.
These are given in Tables 15a and 1

The Type B component of standard uncertainty for capacitance measurements made with the
Type-12 bridge, U,, is defined as the RSS of the four components, 5,., 5;¢5 S,,, and sy, in Eqs. (39),
(43), (44), and (48). This uncertainty is analyzed in detail for each type of measurement in the
following section and the numerical results are given in Tables 15a and 15b for the two-terminal and
three-terminal measurements, respectively.

4.3 Type B Standard Uncertainty for Various Types of Capacitance Measurement

4.3.1 Two-Terminal Measurements of Low-Capacitance Values
(0.001 pF < C, < 0.005 pF)
4.3.1.1 Basic Equations of Balance

This type of measurement covers the range of capacitors with nominal values from 0.001 pF to
0.005 jF, where only internal air-dial capacitors are used to balance the bridge. Using internal air
capacitors, the bridge is balanced twice, with and without the unknown capacitor connected to the
arm CD, as shown in Fig. 6. According to Eq. (33), the equations of balance are given as:

Ry Cay = Ry Coo (49)
and

Ry Cy = Ry (Co + Cy), (50)
where C,o and Coo are capacitance values of internal air capacitors, C, and C,, at balance without

connecting the unknown capacitor C,, and C,, and C,, are the corresponding capacitance values at
balance with C, connected to arm CD.

From Eggs. (49) and (50), the value of the unknown capacitor can be written as:

Cy = [E+ (R,/ Ry) 1 (Car - Cao) (51)
By substituting Eq. (38) into Eq. (51), C, can be expressed in terms of dial readings as:

Cy = CAYL T+ (Ry/ Ry) 1 (Ca - Co), (52)

where C,, =2 C,, and Cy =2 Go are dial readings of the internal air capacitors at balance, with and
without connecting C, to arm CD, respectively.

23


Uy, Ug, & Uy, Combined, Type A and Type B standard uncertainties.

Uand U, Expanded and total (assigned) uncertainties.

Y,, Y,,&y; Admittance of C,, C, and c;

Y and Z Effective admittance and impedance of the capacitor, where Y = (1/Z).

2. The Type-12 System

C,andC, Total capacitance of arm AD and arm CD.

C,andC, — Effective internal capacitors on arm AD and arm CD.

Cy, and C, Capacitance of mica capacitor dial readings and of air capacitor dial reading, r.
C,andC, — Unknown capacitor and its value.

Cy Nominal value of capacitor C,.

For two-terminal measurements:
Cao and C,9 Internal capacitors on arm AD and arm CD without connecting C, to arm CD.
C,, and Cy Internal capacitors on arm AD and arm CD with C, connecting to: arm CD.
C,o and C,,; Dial readings of air capacitors with and without connecting C, to arm CD.

For three-terminal measurements:

C,, and C,; Internal capacitors on arm AD and arm CD with C, connecting to arm AD.
om and ro Internal capacitors on arm AD and arm CD with CG connecting to arm CD.
C,, and Cj. _ Dial readings of air capacitors with C, connecting to arm AD and arm CD.
(or Cy > 0.5 pF)

Cur to C,4 Intemal capacitors on arm AD for the four respective measurements.

Cy to C,4 Internal capacitors on arm CD for the four respective measurements.

rom to C,4 Dial readings of air capacitors for the four respective measurements.

Cl & Cy)3 Dial readings of mica capacitors for the second and third measurements.

C, and C, Capacitance decade box and its value.

Ch Nominal value of capacitance decade box C,,.

Cote Effective capacitance of a capacitor C,.

Coz Maximum error caused by lead impedance.

eds Maximum error due to dial switching.

&ge Maximum error due to residual conductance.
Maximum error due to contact resistance.

og Maximum error due to conductance standard.

fe Applied frequency.

G, and G, Effective internal conductance on arm AD and arm CD.

Gy, and G, conductance of mica capacitor dial readings and conductance dial reading, r.
G, and G, Conductance standard and its value.

G, and G, Conductance of unknown capacitor and its value.

kK, and Ky Divider resistor and its value.

Ly Series inductance of leads.

R,andR, _ Ratio resistors on arm AB and arm CB.

1K


6. ACKNOWLEDGMENTS

The author would like to acknowledge the members of the MCOM Subcommittee for Capacitance
Calibration Services; Carroll Croarkin (Chairman), Barry Bell, Norman Belecki, Andrew Koffman,
and John Mayo-Wells for their valuable suggestions and contributions of the analysis. Special
thanks to Robert Palm for his support in the graphs of the manuscript.

(1)

[2]

Bl

[4]

[5]

{6]

[7]

[8]

[9]

{10]

7, REFERENCES

R. D. Cutkosky, "Capacitance Bridge -- NBS Type 2," National Bureau of Standards Report
7103, March, 1963.

B.N. Taylor and C. E. Kuyatt, "Guidelines for Evaluating and Expressing the Uncertainty

of NIST Measurement Results," Natl. Inst. Stand. Technol. Technical Note 1297, September,
1994.

International Bureau of Weights and Measures (BIPM) Working Group on the Statement of
Uncertainties, Metrologia, vol. 17, pp. 73-74, 1981.

Y. M. Chang and S. B. Tillett, "NIST Calibration Service for Capacitance Standards at Low

Frequencies," Natl. Inst. Stand. Technol. Special Publication 250 Series (NIST SP250-47),
April, 1998.

R. D. Cutkosky and J. Q. Shields, "The Precision Measurement of Transformer Ratios," IRE
Trans. Instrum., vol. I-9, pp. 243-250, September, 1960.

R. D. Cutkosky.and L. H. Lee, "Improved Ten-Picofarad Fused Silica Dielectric Capacitor,"
J. Res. Nat. Bur. Stand., vol. 69C (Eng. and Instr.) no. 3, pp. 173-179, July-September, 1965.

"A Highly Stable Reference Standard Capacitor, "The General Radio Experimenter, vol. 37,
no. 8, August, 1963.

J. Q. Shields, "Voltage Dependence of Precision Air Capacitors," J. Res. Nat. Bur. Stand.,
vol. 69C (Eng. and Instr.), no. 4, pp. 265-274, Oct.-Dec. 1965.

Y. M. Chang and S. B. Tillett, "Recent Developments in the Capacitance Calibration
Services at the National Institute of Standards and Technology," Proceedings of NCSL, July,
1998.

"General Radio Catalog 73" General Radio Company, 1973.

38


List of Symbols

1. Type-2 System

aandb In-phase and quadrature corrections of the 1:1 ratio.
C, to Cy Dial capacitors of the Type-2 bridge.
C, to Cg Nominal values of capacitors C, to Cy.

C, to Cg Real values of capacitors C, to Cy.

C,, Cy, & Cg Reference capacitor, unknown capacitor, and total dial capacitors.
Cy, Cy, & Cq Nominal values of capacitors C,, C,, and Cy

Cy, Cy, & Cy Real values of capacitors C,, C,, and Cy.

(AC/C) Uncertainty in the capacitance due to the lead impedance.
d, to dg Dial corrections of C, to Cg.

d, Deviation of the reference capacitor from its nominal value.
DF, Dissipation factor of the capacitor.

emb Maximum bound of error.

E,and Ey Applied voltages to C, and Cy.

G,and AG Parallel conductance of C, and the uncertainty of Go. th

G, and g; Conductance of the internal dial resistors and admittance of thei dial.
Ty Tay Ty & I, Current through C,, Cg, C,, and G,.

k Coverage factor for confidence level, normally equal to 2.

L, Series inductance of leads.

Mand M Multiplier of the conductance and its setting.

m Ratio of measurement, either 1 or 10.

Ps A constant, either plus or minus 0.5.

" Dial reading of ¢;.

Lr eK Sum of dial readings giving the capacitance value of ¢, , fork>i+1.

R Series resistance of leads.

Sq and s}, Standard uncertainties of the in-phase and quadrature corrections of 1:1 ratio.

Seb Combined standard uncertainty of the external bridge components.

Sod Combined standard uncertainty of dial corrections.

Se Standard uncertainty of the capacitor due to frequency dependence.

s;and 5j Standard deviations of the components of the Type A and Type B standard
uncertainties. th

Sk Standard uncertainty of the dial correction of the k dial.

syand sy, Standard uncertainties of interpolations of the last dial at 1 kHz and 100 kHz.

SRq and spp, Standard uncertainties of the in-phase and quadrature corrections of 1:10 ratio.

Sy Standard uncertainty of the reference standard, due to its stability.

Ss Combined standard uncertainty of the reference standard in measurements.

Seq Equivalent standard deviation (refer as the standard uncertainty) of a component.

St¢ and s4,, Standard uncertainties in temperature corrections of fused-silica and nitrogen

dielectric capacitors.

Sy, ands,,, | In-phase and quadrature standard uncertainties of the capacitor due to voltage

dependence.

viii


maintained at 23 °C + 1 °C. Uncertainties in unknown capacitors due to temperature changes are
dependent on the length of time needed to perform measurements and to connect additional
components. If an external decade capacitance box is needed (such as in three-terminal calibrations
of capacitors with values from 0.01 pF and up), the temperature coefficient of the box will also
contribute additional uncertainties in measurements. The maximum bounds of error due to
temperature changes are equal to +TC*(AT). According to Eq. (4),

Stm = TC*(AT)/V¥3_ ppm, (87)

where 5;,,, is the standard uncertainty component owing to the effect of temperature changes in the
ICL on the values of the unknown capacitor, TC is the temperature coefficient (in ppm/°C), and AT
is the maximum change in temperature (in °C) during the measurement.

In two-terminal measurements, it takes less than 10 min to complete the measurement on one
unknown capacitor. The maximum change in temperature during measurement is 0.15 °C, which
corresponds to an uncertainty, s;,,, of 4 ppm in capacitance, as shown in Table 16a.

The time required to complete three-terminal measurements on one unknown capacitor is less than
20 min, which corresponds to a maximum change in temperature of less than 0.3 °C, and introduces
an uncertainty, s,,,, of 10 ppm in the low-value unknown capacitors (C, < 0.005 pF). For
0.005 pF < C, < 0.5 pF, the values ofa capacitance box are needed for balances. Although the value
of the capacitance box does not enter into the calculation of unknown capacitors, the variation in its
capacitance values during measurements is reflected in the balance readings and the final value of
the unknown capacitor, C,, as shown in Eqs. (68) and (69). The combined uncertainty due to
temperature changes that occur during measurement, s,,,, is estimated to be 15 ppm, which is the
RSS of uncertainties in the unknown capacitor and in the capacitance box. For C, > 0.5 uF, it may
take as long as 30 min to complete one measurement, which corresponds to a maximum change in
temperature up to 0.4°C. The combined uncertainty due to temperature changes that occur during
measurement, 5;,,, is estimated to be 20 ppm, which is the RSS of uncertainties for the unknown
capacitor and for the capacitance box. The values of s;,, are included in Table 16b.

4.4.3 Stabilities of NIST Standards

Since 1993, the calibration procedures for measuring mica capacitors at 1 kHz at NIST have been
modified by using commercial impedance meters to compare a customer's standard with a NIST
standard of the same type over a short time. Then, the value for a customer's standard is calculated
from the predicted value of the NIST standard by using the “substitution method”. This method is
similar to the procedures used for inductance standards calibration, and is described in detail in [15].
The Type-12 bridge is used to measure the NIST standards (for updating data bases from which
predicted values of NIST standards are derived), and to calibrate customers’ standards at frequencies
other than 1 kHz. From three to five years of data on NIST standards at 1 kHz, the standard
uncertainties of the predicted values of both two- and three-terminal measurements of NIST

standards, Spd > are estimated to be between 10 ppm and 25 ppm, which are included in the Tables
16a and 16b.

32


3.3.1 Voltage Dependence of Standards

The voltage dependence of a group of ten 10 pF capacitors was observed in [6] by measuring their
capacitance and dissipation factors from 100 V to 200 V. Using the average values of the
measurements, as well as the systematic and standard errors, the results are:

Syq = 3.54 x 10° < 0.004 ppm
and
Syp = 2.73 x 10° < 0.003 ppm,

where s,,, and s,,, are the in-phase and quadrature standard uncertainties, respectively, of the fused-
silica capacitance standards duc to voltage dependence. This dependency is believed not to change
with time.

The uncertainty of measuring the voltage dependence of 100 pF and 1000 pF air capacitors is
reported to be in the order of 0.001 ppm [8].

3.3.2 Frequency Dependence of Standards

The results of comparing a group of ten 10 pF fused-silica capacitors with two 10 pF air capacitors
at three frequencies (159 Hz, 1592 Hz, and 15 920 Hz) were presented in [6]. Since the transfer of
the farad from the PCL to the ICL involves measurements of a transportable 10 pF fused-silica
dielectric capacitor at 1592 Hz in the PCL and measurements of that same standard in the ICL at
1 kHz, the uncertainty due to frequency dependence is first estimated between frequencies of
1592 Hz and 1 kHz. The uncertainties due to frequency dependence for frequencies of 400 Hz and
100 Hz are estimated between frequencies of 1 kHz and 400 Hz, and 1 kHz and 100 Hz, respectively.
Recently, additional data were obtained from the PCL for three 10 pF fused-silica capacitors (#113,
#112, and #125) at frequencies of 1592 Hz, 1 kHz, 400 Hz, and 100 Hz. Using a natural logarithmic
fit to all the data of capacitor #113, which has the largest frequency dependence, the maximum
values of errors due to frequency dependence at frequencies of 1 kHz, 400 Hz, and 100 Hz are
estimated to be 1.17 ppm, 1.50 ppm, and 3.0 ppm, respectively [9].

According to Eq. (4),

Spe < 0.68 ppm < 0.7 ppm (between frequencies of 1592 Hz and 1 kHz)
Sp < 0.87 ppm < 0.9 ppm (between frequencies of 1 kHz and 400 Hz)
Sp < 1.73 ppm <1.8 ppm (between frequencies of 1 kHz and 100 Hz)

where sp is the standard uncertainty of the 10 pF fused-silica capacitors due to frequency
dependence from the PCL as reference.

The reference standards in the ICL are maintained at a frequency of 1 kHz, and the majority of the
calibration measurements, including the determination of dial corrections, are performed at 1 kHz.

11


(Ey +E + K+ Ey (Lit Ldn) + Ps Ee M(L 8+ L 8) =0. (8)
Assuming E, = 1 V as the reference, and by the definition of 1:1 ratio, (E, / E,) = - 1, then:
E,=-(1+a+jb), Q)

where a is the in-phase correction to the ratio, and
bis the quadrature correction to the ratio.

The errors in the in-phase and quadrature corrections to the 1:1 ratio that are incurred with the
measured data, external capacitance standards, dial readings, interpolation of last dials, load
admittance, and lead impedance are shown in Appendix A. The combined standard uncertainties
that contribute to the 1:1 ratio correction are estimated, also given in Appendix A, as:

Sq < 0.005 ppm!
and

Sp < 0.03 ppm, (10)

where s, and s, are the standard uncertainties in the in-phase and quadrature corrections of the 1:1
ratio, respectively.

In routine calibrations of capacitors using the Type-2 bridge, only capacitance values are reported
even though both capacitance and conductance dials are used to obtain a balance. Therefore, only
the in-phase standard uncertainties of the ratio corrections are included in the Type B standard
uncertainty.

3.1.2 Calibration and Uncertainties of the 10:1 ( and 1:10) Ratio

The calibration method and error analysis related to the 10:1 (and 1:10) ratio transformer are
discussed in detail by Cutkosky and Shields [5]. Based on the numerical results contained in [5],
standard uncertainties in the corrections at the 1:10 ratio are incurred with voltage variations,
temperature variations, and the repeatability of measurements. By combining these components with
those at 1:1 ratios, given in Eq. (10), total standard uncertainties that contribute to 10:1 (and 1:10)
ratio corrections are estimated as:

5pq < 42.25 x 10° ppm <0.043 ppm
and
Spp < 32.49 x 10° ppm <0.033 ppm, (ly

where sp, and Spy, are the standard uncertainties in the in-phase and quadrature corrections of the

' The term uncertainty as used in this document refers to the relative standard uncertainty
when the unit is expressed in ppm [2].


Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

Table 11.

Table 12.

Table 13a.

Table 13b.

List of Tables

Page
Dial Corrections of the Type-2 Bridge 1.0.0... 0... e cece cece e tence eee 40
Standard Uncertainties in Dial Corrections of the Type-2 Bridge ............... 41
Combined Uncertainties in Dial Corrections Using the Type-2 Bridge to
Measure Capacitors of Nominal Values of 1 pF, 10 pF, 100 pF, and 1000 pF ..... 42
Uncertainties Due to Lead Impedance Using the Type-2 Bridge to
Measure Capacitors at Frequencies of 100 Hz, 400 Hz, and 1 kHz............-- 43
Summary of Components of Type B Standard Uncertainties Using the Type-2
Bridge to Measure Fused-Silica and Nitrogen Dielectric Capacitors at 1 kHz ..... 44
Expanded and Assigned Uncertainties Using the Type-2 Bridge to Measure
Fused-Silica and Nitrogen Dielectric Capacitors at 1 KHZ ............ eee eee 45
Expanded and Assigned Uncertainties Using the Type-2 Bridge to Measure
Fused-Silica and Nitrogen Dielectric Capacitors at 400 Hz .............e0 eee 46
Expanded and Assigned Uncertainties Using the Type-2 Bridge to Measure
Fused-Silica and Nitrogen Dielectric Capacitors at 10OHZ ..............0000 47
Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure
Three-Terminal Air Capacitors at Frequencies of 400 Hz and 1 kHz ............ 48
Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure
Three-Terminal Air Capacitors at 100 HZ «1... eee eee eee eee eee eens 49
Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure
Two-Terminal Coaxial Air Capacitors at 1 KHZ ....... 0... eee cece eee eee ee 50
Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure
Two-Terminal Coaxial Mica Capacitors and
Open-Circuit Terminations at 1 KHZ... 0.0... eee ee cece teen eee eee 51
Summary of Components of Standard Uncertainties in Air Capacitor
Dial Corrections for Two-Terminal Measurements ............00. 00 eee seen 52
Summary of Components of Standard Uncertainties in Air Capacitor
Dial Corrections for Three-Terminal Measurements .............-0eseseeeee 53


Table 17b. Summary of Expanded and Assigned Total Uncertainties in Capacitance Measurements Using
the Type-12 Bridge to Measure Three-Terminal Mica Capacitors at 100 Hz, 1kHz, and 10 kHz

(All values are in ppm - except nominal values of the unknown capacitance standards)

Unknown Standard C, (uF) 0.001 | 0.002] 0.005} 0.01 | 0.02 | 0.05} 0.1 } 0.2 } 0.5 1.0
( Frequency = 100 Hz)
Type-A Standard Uncertainty (u,) | 48 25 25 30 25 25 30 25 25 30
Type-B Standard Uncertainty (uy) 35 25 15 25 25 25 25 25 25 25
Expanded Uncertainty (U) 118.81} 70.71 | 58.31 | 78.10} 70.71 | 70.71 | 78.10 | 70.71 | 70.71] 78.10
Assigned Total Uncertainty (U,) 120 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 100
( Frequency = 1 kHz )
Type-A Standard Uncertainty (w,) | 45 30 35 30 30 30 30 30 30 30
Type-B Standard Uncertainty (u,) | 35 25 15 20 20 20 20 20 20 20
Expanded Uncertainty (U) 114.02} 78.10 | 76.16 | 72.11 | 72.11) 72.11 { 72.01, 72.11) 72.11) 72.11
Assigned Total Uncertainty (U,) 120 100 100 100 100 100 100 100 100 100
( Frequency = 10 kHz )
Type-A Standard Uncertainty (u,) 48 25 25 30 25 25 45 25 25 NA
Type-B Standard Uncertainty (uy) 35 25 15 20 25 35 55 100 | 240 NA
Expanded Uncertainty (U) 118.81} 70.71 | 58.31 | 72.11 | 70.71 | 86.02 | 142.13]206.16} 482.60} NA
Assigned Total Uncertainty (U,) 120 | 100 | 100 | 100 | 100 | 100 150 | 250 | 500 NA

61


2) Since the assigned value of the oil-bath type fused-silica reference capacitor in the ICL is
measured at 1 kHz, there is an uncertainty component of 1.6 ppm from its frequency
dependency, when using it as a reference to calibrate other capacitors at 100 Hz.

With the above assumptions, the expanded and assigned iotal uncertainties for capacitance
calibrations of fused-silica and nitrogen dielectric types of capacitors at 100 Hz is given in Table 8.

3.7 Uncertainties of Air and Mica Dielectric Capacitors

NIST/Gaithersburg also provides calibration services at a frequency of 1 kHz for air and mica
dielectric capacitance standards in three-terminal or two-terminal high-frequency (HF) coaxial
arrangements for plugging into a capacitance bridge or an impedance instrument directly. These
capacitors are used mainly to calibrate the bridge or instrument at high frequencies, with its reference
at 1 kHz. Stabilities of these capacitors are not as high as those with fused-silica and nitrogen
dielectrics and they are affected more readily by the environment. Evaluation of the Type B standard
uncertainty for these capacitors is based on the manufacturer's published information on sources of
instability for these types of capacitors [7, 10] and on the specifications for each nominal value and
type of capacitor. Evaluation of the Type A standard uncertainty is estimated from the measurement
data; the capacitors are measured a few times to ensure that the standard deviation is within a given
limit commensurate with the assigned total uncertainty based on population statistics. Assuming the
laboratory conditions during measurements are such that the temperature change is less than +0.5 °C
and the relative humidity change is less than + 8 %, the combined Type B standard uncertainty and
the expanded uncertainties (coverage factor of k = 2 ) for each type of capacitor are estimated. The
results are shown in tables 9, 10, 11, and 12.

3.7.1 Three-Terminal Air Capacitance Standards

The terminals of these air dielectric capacitors are arranged to connect directly to the terminals of
certain commercial capacitance bridges. Adapters also can be utilized to connect them to other
impedance (LRC) meters and instruments. Table 9 gives the expanded and assigned total
uncertainties (and their components) of these capacitance standards at 400 Hz and 1 kHz, with
nominal values that range from 0.001 pF to 10 000 pF. These capacitors can be used at 100 Hz, as
well as at frequencies higher than 1 kHz. The expanded and assigned total uncertainties of three-
terminal air capacitors at 100 Hz is shown in table 10.

3.7.2 Two-Terminal HF Coaxial Capacitance Standards and Terminations

The terminals of these capacitors, with the combination of certain commercial precision coaxial
connectors, can be connected directly to instruments to obtain low lead inductance and high stability
at high frequency (HF) [11, 12]. Therefore, these capacitors are mainly used for measurements in
the radio frequency range, with their reference values determined at 1 kHz. There are three types
of HF coaxial capacitance standards available, as characterized by their nominal values and dielectric
materials.


metrologist.

The combined standard uncertainty, u,, is defined as the "RSS" (root-sum-of-squares) of both types,
as:

y
tg = [ ug? + uy? 17 (1)

where uy is the combined standard uncertainty, and
u,, and 1, are Type A and Type B standard uncertainties, respectively.

According to the guidelines recommended by the International Bureau of Weights and Measures [3],
the overall, or expanded uncertainty, U, is expressed as:

U=kugy (2a)
or

U=k [LJ + Ds 1, (2b)

where U is the expanded uncertainty, k is the coverage factor to be chosen on the basis of the
approximate confidence level desired, and s; and s; are the standard deviations of the components
of Type A and equivalent standard deviations of the components of Type B standard uncertainties,
respectively. The coverage factor used at NIST to calculate U is generally k = 2, indicating a level
of confidence of approximately 95 percent.

Therefore, the total uncertainty, U, assigned to capacitance calibration at NIST is calculated
according to the following equation:

U, = 2[ D6 + D5)" 7}. (3)
In this document, unless otherwise stated, the uniform distribution is used to obtain the equivalent

standard deviation of the components of the Type B standard uncertainties from the maximum
bounds of error for each source of errors. Thus,

Sod = Cmp/ V3, (4)
where s,, is the equivalent standard deviation, as one of the s; in Eqs. (2b) and (3), (hereafter referred

as the standard uncertainty) of a typical component of the Type B standard uncertainty, and + ep
are its maximum bounds of error.

2. DESCRIPTION OF SYSTEMS AND MEASUREMENT PROCEDURES

There are two major capacitance measuring systems in the ICL at NIST. One is the Type-2 bridge,
which is mainly used to calibrate capacitors of nominal values up to 10 000 pF at frequencies of


2)

3)

4)

measurements, respectively.
Temperature variations

The temperature coefficients of air and mica capacitors are 10 ppm/°C and 45 ppm/°C,
respectively. The temperature in the ICL is always maintained at (23 + 1) °C. Therefore,
the maximum bounds of error owing to temperature effects for the readings of the air and
mica capacitance dials are +10 ppm and + 45 ppm, respectively. According to Eq. (4),
Stp = 5.77 ppm < 6ppm, and Simp = 25.98 ppm < 26 ppm, (40)

where s,, and s,,, refer to the standard uncertainties in the air and mica dial capacitors,
respectively, due to temperature variations, which are included in Tables 13a to 14b.

Dial switching

The maximum bounds of error due to the switching of dials, + eg, are estimated to be
+ 0.01 pF, which will affect only the measurements of low capacitance. Accordingly,

Sqg = 5.77 ppm < 6 ppm (for C= 0.001 pF),

Sqy = 0.58 ppm < 0.6 ppm (forC=0.01 wF),
Sds 0.06 ppms< 0.1 ppm (forC=0.1 wF), and
Sqy = 0.006 ppm< 0.01 ppm (forC=1 pF) . (41)

where sq, is the standard uncertainty due to the switching of dials, also included in
Tables 13a to 14b.

Uncertainty of transfer standard

From numerous measurements, the calibrated value of the transfer standard, a 0.005 pF mica
capacitor has an uncertainty of less than 0.03 pF, which will cause an uncertainty of less than
6 ppm in the dial corrections of both air and mica dial capacitors. Thus,

Ste < 6 ppm, and sip < 6 ppm, (42)

where s;pand s,,¢ are the standard uncertainties in the air and mica capacitor dial corrections,

respectively, due to the uncertainty of the transfer standard, which are given in Tables 13a
to 14b.

For measurements of capacitors with nominal values up to 0.005 F, (only air-dial capacitors
are needed), stp is used as the standard uncertainty. However, for capacitors of nominal
values larger than 0.005 pF, the RSS of sy and spp is used to estimate the standard
uncertainty from the dial corrections in measurements.

The combined uncertainty of the internal capacitors is defined as the RSS of the above components,
which are estimated for each type of measurement as:

21


Table B-8. Data and Analysis of the (100 pF/step) Air Capacitor Dial Corrections of the
Type-12 Bridge

(All values are in pF )

Date’ Dial 100 200 300 400 500 600 700 800 900 1000

6/67 -0.003 | 0.125 | 0.092 | 0.075 | 0.067 | 0.200 | 0.167 | 0.165 | 0.302 | 0.300
11/67 -0.001 | 0.130 | 0.096 | 0.077 | 0.076 | 0.211 | 0.176 | 0.125 | 0.307 | 0.309
5/68 -0.002 | 0.127 | 0.096 | 0.083 0.071 0.204 0.166 | 0.165 0.298 | 0.297
9/69 0.000 | 0.132 | 0.098 | 0.082 | 0.076 | 0.209 | 0.173 | 0.175 | 0.309 | 0.308
12/70 -0.004 | 0.129 | 0.095 | 0.077 | 0.075 | 0.204 | 0.174 | 0.174 | 0.310 | 0.311
9/72 -0.004 | 0.131 | 0.097 | 0.083 | 0.078 | 0.211 | 0.180 | 0.172 | 0.312 | 0.314
6/74 0.000 | 0.133 | 0.096 | 0.089 | 0.086 | 0.215 | 0.179 | 0.179 | 0.308 | 0.305
11/75 -0.002 | 0.130 | 0.097 | 0.083 | 0.084 | 0.217 | 0.185 | 0.184 | 0.316 | 0.315
V/77 -0.002 | 0.130 | 0.093 | 0.085 | 0.079 | 0.211 | 0.184 | 0.176 | 0.313 | 0.308
4/78 -0.003 | 0.126 | 0.096 | 0.088 | 0.087 | 0.213 | 0.180 | 0.180 | 0.311 | 0.309
11/80 0.000 | 0.136 | 0.103 | 0.092 | 0.091 | 0.228 | 0.199 | 0.206 | 0.321 | 0.320
8/82 0.014 | 0.143 | 0.108 | 0.094 | 0.093 | 0.224 | 0.189 | 0.188 | 0.326 | 0.322
9/83 0.001 | 0.134 | 0.103 | 0.093 | 0.091 | 0.223 | 0.192 | 0.190 } 0.329 | 0.327
1/85 0.001 | 0.131 | 0.107 | 0.102 | 0.101 | 0.221 | 0.189 | 0.188 | 0.322 | 0.325
3/85 0.001 } 0.129 | 0.101 | 0.092 | 0.093 | 0.226 | 0.192 | 0.193 | 0.318 | 0.321
2/91 0.013 | 0.138 | 0.097 | 0.093 | 0.081 } 0.223 | 0.210 | 0.208 | 0.318 | 0.339

N. Val. (pF) 100 200 300 400 500 600 700 800 900 1000
Mean (pF) 0.0004 | 0.1315 | 0.0984 | 0.0868 | 0.0831 | 0.2150 | 0.1834 | 0.1793 | 0.3138 | 0.3144
SD (pF) 0.0053 | 0.0046 | 0.0047 | 0.0074 | 0.0093 | 0.0085 | 0.0116 | 0.0191 | 0.0084 | 0.0108
Slope (pF/day) | 1.3E-06 | 9.9E-07| 1.1E-06 | 2.4E-06 | 2.7E-06 | 2.8E-06 | 4.0E-06 | 5.8E-06 | 2.6E-06 | 3.7E-06
( pF/year) 0,000 | 0.000 | 0.000 | 0.001 | 0.001 | 0.001 | 0.001 | 0.002 | 0.001 | 0.001
SDfit (pF) 0.004 | 0.004 | 0.004 | 0.004 | 0.006 | 0.004 | 0.005 | 0.012 | 0.005 | 0.005
SD(2/01) (pF) | 0.006 | 0.005 | 0.005 | 0.005 | 0.008 | 0.006 | 0.006 | 0.016 | 0.007 | 0.007

B-8



APPENDIX B. CAPACITANCE DIAL CORRECTIONS OF THE TYPE-12 BRIDGE

Data and analyses of capacitance corrections for each capacitor dial of the Type-12 bridge are included

in the following tables:

Table B-1. Data and Analysis of the ( 0.1 F/step ) Mica Capacitor Dial Corrections of the
Type-12 Bridge at 100 Hz

[All values are in pF except the dials which are in (uF/step)]

Daié~Dial | 0.1 02 03 04 0.5 06 | 07 08 | 09 | 0x

6/67 19.701 | 41.053 | 97.318 | 88.878 | 108.694] 129.700} 183.656] 204.267| 225.604] 242.514
11/67 20.437 | 44.477 | 98.434 | 95.788 | 116.348] 140.055] 191.927| 213.334 | 237.464] 255.166
5/68 21.847 | 45.248 | 101.395} 96.267 | 118.168] 141.237] 195.189] 217.845 | 241.382] 260.284
9/69 20.758 | 44.866 | 99.509 | 100.527] 121.418] 145.294] 197.802| 219.432 | 243.710] 261.716
12/70 20.664 | 45.464 | 100.178] 98.560 | 119.320] 143.764] 196.294 | 217.848| 242.717] 260.552
972 20.817 | 45.897 | 96.451 | 106.001 127.050] 151.815] 199.822| 221.694] 246.911 | 265.385
6/74 21.012 | 44.577 | 96.139 | 99.552 | 120.756] 143.991] 193.474 | 215.618 | 239.273 | 257.560
11/75 21.435 | 46.830 | 103.188] 105.313] 126.560] 151.373] 205.533 227.700] 253.190| 271.885
77 21.693 | 45.409 | 104.405] 100.598] 122.436] 145.847 | 202.762 | 225.370] 249.734| 268.604
4/78 19.716 | 41.565 | 99.931 | 93.988 | 113.837] 135.478] 191.719] 212.355] 234.427| 251.531
11/80 19.338 | 42.613 | 99.161 | 98.911 | 118.419] 141.564] 195.577| 215.905] 239.642| 255.905
8/82 20.726 | 45.563 | 104.109] 109.993] 130.962] 155.511 | 211.717| 233.171 | 257.810] 275.475
9/83 20.642 | 46.292 | 98.191 | 111.796] 132.763] 158.175] 208.107] 229.569] 255.559] 273.476
1/85 20.874 | 48.402 | 104.174] 117.362] 138.158] 164.957] 218.316] 240.047] 267.163 | 284.982
3/85 21.416 | 46.567 | 103.820] 107.933] 129.529] 154.597] 209.513 | 231.897| 257.120] 275.731
2/91* 20.367 | 43.857 | 99.149 | 106.297] 126.824] 150.029] 201.141] 222.516] 246.128] 263.833
N. Val. (pF) | 1.0E+05 |2.0E+05]3.0E+05| 4.0E+05] 5.0E+05] 6.0E+05| 7.0E+05| 8.0E+05]9,0E+05| 1.0E+06
Mean (pF) | 20.7152 | 44.9175] 100.347] 102.360] 123.203 | 147.087] 200.159] 221.786] 246.115 | 264.037
SD(pF) | 0.7072 | 1.9211 | 2.836 | 7.365 | 7.628 | 8.928 | 8.843 | 9.131 | 10.375 | 10.701
Slope (pF/day) | -1.5E-05] 1.9E-04] 3.9E-04 | 2.0E-03 | 2.0E-03 | 2.2E-03 | 2.3E-03 | 2.3E-03 | 2.5E-03] 2.4E-03
(pF/year) | -0.006 | 0.071 | 0.143 | 0.721 | 0.718 | 0.796 | 0.829 | 0.823 | 0.897 | 0.886
SDfit (pF) | 0.731 | 1.916 | 2.732 | 5.363 | 5.763 | 7.045 | 6.706 | 7.147 | 8.361 | 8.850

SD(2/01) (pF) | 0.981 | 2.573 | 3.668 | 7.200 | 7.738 | 9.459 | 9.004 | 9.595 | 11.225 | 11.882

* See Section 4.2.2
B-1



100 Hz, 400 Hz, and 1 kHz. The other is the Type-12 bridge, which is a resistance ratio capacitance
measuring system for calibration of capacitors of nominal values from 0.001 pF to 1 pF at
frequencies of 66 Hz, 100 Hz, 400 Hz, 1 kHz, and 10 kHz. Since 1992, commercial impedance
meters have also been used as capacitance comparators, after they have been characterized using
NIST standards, in an effort to automate some of the calibration services. A detailed description of
the capacitance calibration service at NIST is given in [4].

2.1 Type-2 Capacitance Bridge and Standards

The NIST Type-2 bridge is a transformer-ratio bridge used for comparing unknown capacitors of
coaxial connectors with external reference standards whose values are well defined. A set of eight
internal air capacitors with values of 100 pF, 10 pF, .......... , 10° pF are used to balance the bridge
during measurements. Values of these internal capacitors are selected by means of dial settings to
balance the bridge at 1:1, 1:10, or 10:1 ratios. During the transfer of the farad, an NIST-made 10 pF
air-bath-type fused-silica capacitor from the PCL is used as the reference to measure a group of 10
pF oil-bath-type fused-silica capacitors in the ICL. These capacitors, which are also NIST-made and
have known temperature coefficients, serve as the primary reference in the ICL. Changes in their
temperatures are precisely monitored via internal resistive sensors of copper, and are used to adjust
their assigned values of capacitance at the time of measurement. The primary reference is employed
to measure two secondary reference sets, each having one 10 pF and one 100 pF commercial air-
bath-type fused-silica capacitors in a temperature-controlled oven. The secondary references are
used for routine calibrations. Also, the primary reference is directly employed to calibrate
customer’s fused-silica capacitors.

The ICL also has a number of capacitance check standards. Among these are six commercial dry-
nitrogen-dielectric, parallel-plate capacitors housed in the Styrofoam containers originally supplied
as protectors during shipment. Their nominal values are 1 pF, 10 pF, 100 pF (2), and 1000 pF (2).
There is also an NIST-made, 10 000 pF capacitor. These check standards are used to ensure that the
measuring system remains under control and are used also as a part of the process of calibrating
capacitors having dielectrics other than fused-silica, as shown below.

Calibration procedures for customer’s capacitors having nitrogen and air dielectrics have two steps.
The first step is to use the secondary reference to measure the check standards. Secondly, the check
standards are used as working standards to measure the customers’ capacitors. Values of all
reference and check standards are kept in a database. They are used in control charts to ensure that
the Type-2 bridge is in good operational condition. Figure 1 is a block diagram illustrating the
transfer of the farad and the calibration measurement process using the Type-2 system. Prior to
performing any measurements, the internal capacitors of the Type-2 bridge are calibrated against the
reference standard to obtain dial corrections used in the calculation of that day’s measurement
results.

2.2 Type-12 Capacitance Bridge

The Type-12 bridge is a resistance ratio bridge having internal capacitors as its reference and is

3


Table B-7. Data and Analysis of the (1000 pF/step) Air Capacitor Dial Corrections of the
Type-12 Bridge

(All values are in pF )

Date Dial | 1000 | 2000 | 3000 | 4000 | 5000 | 6000 | 7000 | 8000 | 9000 | 10000

6/67 0.060 | 0.010 | 0.189 | 0.109 | 0.189 | 0.154 | 0.334 | 0.403 | 0.353 | 0.403
11/67 0.075 | 0.017 } 0.253 | 0.118 | 0.204 | 0.156 | 0.388 |} 0.463 | 0.392 | 0.460
5/68 0.055 | 0.009 | 0.184 | 0.113 | 0.185 | 0.145 | 0.317 | 0.379 | 0.336 | 0.391
9/69 0.069 | 0.021 | 0.232 | 0.136 } 0.214 | 0.172 | 0.381 | 0.454 | 0.400 | 0.463
12/70 0.074 | 0.025 | 0.244 | 0.150 | 0.241 | 0.196 | 0.406 | 0.487 | 0.431 | 0.503
9/72 0.072 | 0.040 | 0.208 | 0.146 | 0.231 | 0.211 | 0.394 | 0.473 | 0.434 | 0.499
6/74 0.069 | 0.040 | 0.198 | 0.183 | 0.259 | 0.239 | 0.396 | 0.465 | 0.428 | 0.485
11/75 0.081 | 0.050 | 0.244 | 0.213 | 0.305 | 0.279 | 0.461 | 0.545 | 0.509 | 0.588
1/77 0.071 | 0.049 | 0.184 | 0.219 | 0.298 | 0.286 | 0.425 | 0.491 | 0.466 | 0.531
4/78 0.070 | 0.064 | 0.192 | 0.235 | 0.313 | 0.307 | 0.432 | 0.502 | 0.481 0.543
11/80 0.071 | 0.062 | 0.242 | 0.233 | 0.330 | 0.318 | 0.509 | 0.591 | 0.563 | 0.642
8/82 0.086 | 0.071 | 0.241 | 0.248 | 0.336 | 0.338 | 0.510 | 0.593 | 0.580 | 0.653
9/83 0.090 | 0.070 | 0.259 | 0.256 | 0.357 | 0.347 | 0.535 | 0.629 | 0.607 | 0.695
1/85 0.087 | 0.071 | 0.244 | 0.261 | 0.366 | 0.361 | 0.529 | 0.624 | 0.605 | 0.690
3/85 0.094 | 0.080 | 0.231 0.277 | 0.380 | 0.375 | 0.523 0.613 | 0.587 | 0.671
2/91 0.109 | 0.098 | 0.226 | 0.255 | 0.352 | 0.366 | 0.523 | 0.629 | 0.631 | 0.724

N. Val. (pF) 1000 2000 3000 4000 5000 | 6000 | 7000 8000 | 9000 | 10000
Mean (pF) 0.0771 | 0.0486 | 0.2232 | 0.1970 | 0.2850 | 0.2656 | 0.4414 | 0.5213 | 0.4877 | 0.5588
SD (pF) 0.0135 | 0.0269 | 0.0263 | 0.0595 | 0.0671 | 0.0834 | 0.0725 | 0.0828 | 0.0970 | 0.1086
Slope (pF/day) | 4.4E-06] 1.0E-05 | 3.3E-06 | 2.1E-05 | 2.4E-05 | 3.0E-05 | 2.5E-05 | 2.9E-05 | 3.5E-05 | 3.9E-05
( pF/year) 0.002 | 0.004 | 0.001 | 0.008 | 0.009 | 0.011 | 0.009 | 0.011 | 0.013 | 0.014
SDfit (pF) 0.007 0.005 0.026 0.020 0.022 0.022 0.028 0.032 0.026 0.032
SD(2/01) (pF) | 0.010 | 0.007 | 0.034 | 0.027 | 0.029 | 0.029 | 0.037 | 0.042 ]) 0.035 | 0.042

B-7


Table C-2. Data and Analysis of Conductance Corrections of ( 0.1 F/step ) Mica Capacitance Dial
of the Type-12 Bridge at 1 kHz
(All values are in 0.001 pS, except the dial which is 0.1 1 F/step)
DIAL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.X
6/67 47.305 | 98.877 | 264.330 | 224.372 | 271.098 | 326.986 | 500.838 | 558.528 | 600.240 | 676.897
11/67 46.940 | 100.430 | 263.095 | 229.102 | 276.858 | 335.962 | 506.872 | 563.892 | 607.910 | 685.677
5/68 47.805 | 99.947 | 266.455 | 226.535 | 274.364 | 330.679 | 505.564 | 563.470 | 606.765 | 684.400
9/69 47.680 | 101.745 | 265.495 | 235.212 | 282.337 | 341.020 | 513.247 | 571.053 | 615.033 | 693.146
12/70 48.053 | 103.612 | 266.732 | 233.383 | 280.935 | 340.675 | 513.425 | 573.657 | 619.455 | 696.497
9/72 48.202 | 102.982 | 266.970 | 241.348 | 288.850 | 347.110 | 521.205 | 580.365 } 622.810 } 702.902
6/74 48.520 | 101.800 | 268.048 | 233.778 | 282.025 | 339.182 } 514.950 | 575.485 } 616.887 | 695.442
11/75 48.898 | 103.762 | 270.010 | 237.880 | 286.902 | 345.262 | 520.682 | 581.692 | 635.092 } 705.292
1/77 49.190 | 102.397 | 270.695 | 231.745 } 281.047 | 338.747 | 516.430 | 577.637 | 622.715 | 702.235
4/78 47.653 | 98.143 | 266.622 | 225.987 | 274.072 | 328.845 | 504.327 | 562.692 | 602.840 | 680.885
11/80 49.485 | 104.973 | 272.487 | 241.390 | 290.853 | 349.962 | 525.185 | 583.917 | 631.317 | 711.745
8/82 49.868 | 105.278 | 272.753 | 243.995 | 294.873 | 352.798 | 530.568 | 588.847 | 637.482 | 715.372
9/83 50.190 | 107.250 | 272.320 | 249.523 | 299.703 | 360.835 | 535.808 | 593.048 | 645.178 | 727.558
1/85 49.015 | 108.588 | 271.920 | 251.525 | 302.480 | 363.743 | 536.508 | 598.165 | 647.708 | 728.600
3/85 49.383 | 104.635 | 271.035 | 237.648 | 288.115 | 348.248 522.268 582.600 | 629.233 | 708.830
2/91 49.877 | 104.365 | 272.027 | 238.170 | 288.995 | 347.640 | 521.587 | 581.903 | 624.710] 705.013
Mean (nS) | 48.629 | 103.049 | 268.812} 236.35 | 285.219 | 343.606 | 518.092 | 577.309 | 622.836 | 701.281
SD (nS) 1.0010 | 2.8719 | 3.2009 } 7.9901 | 9.0811 | 10.5141} 10.7484 | 11.3459} 14.3714] 15.2700
Slope(nS/day)) 3.3E-04 | 7.5E-04 | 1.1E-03 | 2.0E-03 | 2.5E-03 } 2.8E-03 | 3.0E-03 | 3.2E-03 | 3.8E-03 ) 4.2E-03
Slope(nS/year)) 0.1187 | 0.2750 | 0.3878 | 0.7221 | 0.9075 } 1.0133 | 1.0896 }| 1.1755 } 1.3767 | 1.5399
SDfit 0.5267 | 2.1373 | 1.5763 | 6.2415 | 6.4693 | 7.7757 | 7.5324 | 7.7394 | 10.6895 } 10.7667
SD(2/01) (AS)} 0.70714} 2.86942] 2.11632] 8.37964] 8.68548] 10.4394] 10.1127] 10.3907 14.3514] 14.4551

C-2



Table 15b. Summary of Components of Type B Standard Uncertainties in Capacitance Measurements Using
the Type-12 Bridge to Measure Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz

(All values are in ppm - except nominal values of the unknown capacitance standards)

57

Unknown Standard C, (uF) 0.001 | 0.002 | 0.005} 0.01 | 0.02 | 0.05) 0.1 0.2 | 0.5 [10 |
Components of Standard Uncertainties 7
(At all Frequencies)
Ratio Resistor Arms (5 ,.) - - - 9 9 9 9 9 9 9
Air Capacitor Dials (5 ,<) 31 20 8 3 3 3 3 3 3 5
( Frequency = 100 Hz)
Mica Capacitor Dials (5m) - - - 18 17 17 19 23 18 16
External Bridge Component (s},) 6 3 2 1 1 1 1 1 3 5
Type-B Standard Uncertainty (U,) _ | 31.58 | 20.22} 8.25 | 20.37] 19.49 19.49 | 21.26 | 24.90 | 20.57 | 19.67
Assigned U, 35 25 15 25 25 25 25 25 25 25
( Frequency = 1 kHz )
Mica Capacitor Dials (sq) - - - 15 15 15 16 17 15 15
External Bridge Component (s4;) 6 3 2 1 1 1 1 1 3 5
Type-B Standard Uncertainty (U,) 31.58 | 20.22 | 8.25 | 17.78] 17.78 | 17.78 | 18.63 | 19.49} 18.00 | 18.87
Assigned U, 35 25 15 20 20 20 20 20 20 20
—
( Frequency = 10 kHz )
Mica Capacitor Dials (5m) - - - 15 15 15 15 16 26 NA
External Bridge Component (s4,) 6 4 3 5 10 25 48 95 235 | NA
Type-B Standard Uncertainty (U,) | 31.58 | 20.40] 8.54 | 18.44] 20.37 30.66 | 51.18 | 96.80 |236.62} NA
L Assigned U, 35 25 15 20 25 35 [5 100 | 240 | NA


Table 18b. Summary of Components of Type B Standard Uncertainties in Conductance Measurements Using
the Type-12 Bridge to Measure Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz

(All values are in pS - except nominal values of the unknown capacitance standards)

(Frequency = 10 kHz)

ee

Unknown Standard C,(uF) | 0.001 | 0.002 | 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0
Components of
Standard Uncertainties
(Frequency = 100 Hz)
Accuracy of G, (5s) 3.2E-07 | 3.2E-07 | 8.0E-07] 1.3E-06 | 3.2E-06 | 3.2E-06 | 3.2E-05 | 8.0E-05 | 1.9E-04 | 3.5E-04
Contact Resistance (s,,) | 4.6E-10| 1.8E-09] 1E-08 | 4.6E-08 | 1.8E-07 | 1.2E-06 | 4.6E-06 | 1.8E-05 | 1.2E-04 | 4.6E-04
Conductance Pot. (5 ,,) 1.0E-04 | 1.0E-04 | 1.0E-04) 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.0B-04 | 1.0E-04| 1.3E-04
Residu. Conduct. (s,.) 1.2E-06 | 1.2E-06] 1E-06 | 1.2E-06| 1.2E-06 | 1.2E-06 | 1.2E-06 | 1.2E-06 | 1.2E-06] 1.2E-04
Mica Capacitor Dials (5 gm) - - - 1.4E-04 | 1.5E-04 | 3.9E-04 | 7.6E-04 | 2.6E-03 | 4.7E-03 | 6.7E-03
Type-B Std. Uncertainty (ug)| 1.0E-04 | 1.0E-04 | 1.0E-04} 1.7E-04 | 1.8E-04 | 4.1E-04 | 7.7E-04 | 2.6E-03 | 4.7E-03 | 6.7E-03
—— EE SSS
(Frequency = 1 kHz)
Accuracy of G, (55) 1.6E-06 | 1.6E-06 | 8.0E-06] 1.3E-05 | 3.2E-05 | 3.2E-05 | 1.1E-04| 1.6E-04 | 5.6E-04 | 3.5E-03
Contact Resistance (Sr) | 4.6E-08 | 1.8E-07} 1E-06 | 4.6E-06 | 1.8E-05 | 1.2E-04 | 4.6E-04 | 1.8E-03 | 1.2E-02 | 4.6E-02
Conductance Pot. (Sp) 1.0E-04 | 1.0E-04 | 1.0E-04] 1.0E-04 | 1.0E-04 | 1.0E-04} 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.3E-04
Residu. Conduct. (5 g.) 1.2E-05 | 1.2E-05| 1E-05 | 1.2E-05] 1.2E-05 | 1.2E-05 | 1.2E-05 | 1.2E-05 | 1.2E-05 | 1.2E-05
Mica Capacitor Dials (sm) - - - 1.7E-04 | 2.4E-04 | 8.0E-04 | 2.9E-03 | 8.4E-03 | 1.4E-02 | 2.0E-02
Type-B Std. Uncertainty (ug,)| 1.0E-04 | 1.0E-04 | 1.0E-04} 2.0E-04 | 2.7E-04 | 8.2E-04 | 2.9E-03 | 8.6E-03 | 1.8E-02 | 5.1E-02

Type-B Std. Uncertainty (u,,) | 1.5E-04

Accuracy of G, (5s) 4.8E-06 | 8.0E-06 | 1.6E-05| 3.2E-05 | 8.0E-05 | 8.0E-05 | 1.6E-03 | 3.2E-03 | 1.6E-02] NA
Contact Resistance (sg) | 4.6E-06 | 1.8E-05 | 1.2E-04] 4.6E-04 | 1.8E-03 | 1.2E-02 | 4.6E-02 | 1.8E-01] 1.2E+00] NA
Conductance Pot. (sg) 1.0E-04 | 1.0E-04 | 1.0E-04) 1.0E-04 | 1.0E-04 | 1.0E-04 | 1.0E-04] 1.0E-03]| 1.0E-03} NA
Residu. Conduct. (s gc) 1.2E-04 | 1.2E-04 | 1.2E-04] 1.2E-04] 1.2E-04 | 1.2E-04 | 1.2E-03 | 1.2E-03 | 1.2E-03] NA
Mica Capacitor Dials (5 gm) - - - 3.1E-03 | 4.9E-03 | 2.4E-02 | 6.5E-02 | 1.6E-01}1.2E+00] NA
1.5E-04 | 1.9E-04] 3.1E-03 | 5.3E-03 | 2.6B-02 | 7.9E-02 | 2.4E-01|1.7E+00| NA

63


ai and dj;,1) are dial corrections of C; and C¢41) » respectively,
= 10 is the ratio, and
Ty cz is the sum of dial readings giving the capacitance value at balance, for k> i+1.

The balance equation for calibrating C, (where C, = 1 pF) with the 10 pF reference standard (C,),
according to Eqs. (12) and (13), is:

Co + dy = 10(C,+d; +Yrp cy), (14)
where C, (= 10 pF) is the nominal value of the reference standard,

d, is the deviation of the reference from its nominal value, and

&7;, cy is the sum of the dial readings giving the capacitance value at balance, for k > 3.
The dial corrections are related similarly by:

d;= 10 (dey) + Le CK) (k>it+b, (15)
and, hence (from Eq. (14)),

dy = Ol) dy - Dry cg. (k>3). (16)
Therefore, dial corrections for each of the internal capacitors can be determined from a known
reference capacitor and dial readings. After the preliminary results of dial corrections are obtained,

from Eqs. (15) and (16), the final values of the dial corrections can be determined by including the
corrections of those dials used in dial correction measurements.

3.2.2 Uncertainties in Dial Corrections
Rewrite Eqs. (15) and (16) by including the dial corrections of the internal capacitors, d, as:
4; = 10 (dey + ED rp eg tL rp ay), Ck i+1), and (17)

d,=(0.1) d.-(Lrpey td rpdg), (k>3), (18)

where c, is the nominal value of the a capacitor, c,, and
d,, is the correction to the K capacitor, ¢,.

The standard uncertainties of each internal capacitor can be estimated from Eqs. (17) and (18).
As mentioned in the previous section, (3.2.1), the first step is to calibrate the third dial, C3 by using
a 10 pF reference standard. Therefore, the standard uncertainty of the correction of the third dial,

d, is determined first. The relationship of the variance of each term in Eq. (18) can be written as:

Var (d3)=(O.1) (pq +55") +E ry Var (dy) +57, (19)



Table B-9. Data and Analysis of the (10 pF/step) Air Capacitor Dial Corrections of the
Type-12 Bridge

(All values are in pF )

Date Dial 10 20 30 40 50 60 70 80 90 100

6/67 0.007 | -0.057 | 0.005 | 0.047 | 0.059 | -0.015 | 0.047 | 0.059 | -0.004 | 0.007
11/67 0.007 | -0.060 | 0.006 | 0.051 | 0.059 | -0.006 | 0.060 | 0.068 | 0.003 | 0.012
5/68 0.006 | -0.056 | 0.009 0.055 0.065 | -0.001 0.060 0.067 0.000 0.009
9/69 0.006 | -0.061 | -0.001 | 0.054 | 0.061 | -0.003 | 0.062 | 0.069 | 0.003 | 0.014
12/70 0.006 | -0.059 | 0.004 } 0.050 | 0.062 | -0.005 | 0.059 | 0.072 | 0.005 | 0.011
9/72 0.004 | -0.062 | 0.001 | 0.047 | 0.060 | -0.005 | 0.056 | 0.069 | 0.004 | 0.013
6/74 0.006 | -0.058 | 0.007 | 0.056 | 0.067 | 0.003 | 0.069 | 0.072 | 0.005 | 0.011
1/75 0.005 | -0.059 | 0.006 | 0.053 | 0.058 | -0.006 | 0.059 | 0.063 | -0.001 | 0.011
V/77 0.006 | -0.057 | 0.009 | 0.048 | 0.053 | -0.008 | 0.054 | 0.071 | 0.007 | 0.013
A/T8 0.009 | -0.053 | 0.004 | 0.048 | 0.054 | -0.008 | 0.056 | 0.072 | 0.010 | 0.017
11/80 0.006 | -0.061 | -0.001 | 0.051 | 0.057 | -0.008 } 0.055 | 0.066 | 0.014 | 0.018
8/82 0.011 | -0.053 | 0.010 | 0.061 | 0.061 | -0.005 | 0.070 | 0.081 | 0.016 } 0.019
9/83 0.009 | -0.052 | 0.007 | 0.056 | 0.064 | 0.006 | 0.073 | °0.079 | 0.010 | 0.018
1/85 0.010 | -0.053 | 0.004 | 0.046 | 0.057 | 0.000 | 0.064 | 0.070 | 0.008 | 0.019
3/85 0.015 | -0.052 | 0.008 | 0.050 | 0.069 | 0.005 | 0.064 | 0.068 | 0.009 | 0.016
2/91 0.013 | -0.037 } 0.041 | 0.079 | 0.092 | 0.041 | 0.106 | 0.101 } 0.031 } 0.015

N. Val. (pF) 10 20 30 40 50 60 70 80 90 100
Mean (pF) 0.0079 | -0.0556 | 0.0074 | 0.0533 | 0.0624 | -0.0009| 0.0634 | 0.0717 | 0.0075 | 0.0139
SD (pF) 0.0031 | 0.0060 | 0.0095 | 0.0080 } 0.0090 | 0.0124 | 0.0131 | 0.0094 | 0.0082 | 0.0037
Slope (pF/day) | 8.6E-07] 1.7E-06] 1.9E-06] 1.5E-06 | 1.6E-06 | 3.0E-06 } 3.3E-06 | 2.4E-06 } 2.6E-06} 1.1E-06
( pF/year) 0.000 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.000
SDfit (pF) 0.002 | 0.004 | 0.008 | 0.007 | 0.008 | 0.010 | 0.010 | 0.007 | 0.005 | 0.002
SD(2/01) (pF) | 0.003 | 0.006 | 0.011 | 0.010 | 0.011 | 0.013 | 0.014 | 0.010 | 0.006 | 0.003

B-9


Therefore, Eq. (23) can be written as:

Z=[R+(G,/w'C J] +jo[L)-(C/wC (24)
Thus, the equivalent equation of admittance, Y = (1/Z) is expressed as:

Y=(@'C’)/[(Rw’'C + G,)+j oC(w'L,C-1)), (25)
where Y is the effective admittance of the capacitor.
The value of Z, is less than 10°H and G, is less than 10° S. By multiplying the numerator and
the denominator of Eq. (25) by [( RX, © C+ G,)-joc( wo L, C- 1 )] and eliminating all second
order terms of L; and G, , Eq. (25) becomes:

Y=[(G,+ RoC )+j oC(1-w L,C)]/ [1+ 2, °C +2R G-207° LC], (26)

For a one meter long coaxial cable, R, and L, were measured to be approximately 0.1 Q and 0.5 »H,
respectively. For measurements of capacitors of nominal values up to 10‘ pF, at frequencies up to
1592 Hz, Eq. (26) can be reduced to:

Y=[(G,+ RoC) +j oC (1 -w" L,C)]/ [1-207 L,C]. (27)

By multiplying the numerator and the denominator of Eq. (27) by (1+2 wo 1, C), and eliminating
the second order terms of L, and G,, Eq. (27) becomes:

Y=(G.+R, WO’ C+2R'L,C)+j oC +o LC). (28)

Furthermore, the third term in Eq. (28), (2 Rj @* L; C’) is less than 10"'° S, which is negligible as
compared with G, and (Ry oC ). Therefore, Eq. (28) becomes:

Y=(G,+Rw C)+joC(1+0' LC). (29)
The effective values of capacitance and conductance of the standard with lead impedance are:
Cope = C(1 40° LC) = C[1+ (4C/C)], and (30)
Gepp = (Go+Rjw’C?) = (G+ AG), G1)

where (AC/ C) and AG are uncertainties in the capacitance and conductance values of a capacitor
due to the lead impedance.

For measurements using the fused-silica dielectric transfer standard from PCL, C = 10 pF, estimated
uncertainties in the lead impedance are calculated to be:

(AC/C) =197x 10" —< 0.0002 ppm, and

13


Table 3. Combined Uncertainties in Dial Corrections Using the Type-2 Bridge to Measure
Capacitors of Nominal Values of 1 pF, 10 pF, 100 pF, and 1000 pF

Nominal Value
of Capacitor (pF) 1 10 100 1000
No. of Dials Used
Routinely 4 5 5 6
Assigned Combined
Uncertainty
Seq (10° pF) * 0.45 0.51 0.51 112
Sea (ppm) 0.45000 0.05100 0.00510 0.00112
Remark :

* From number of dials used to perform measurements, as given in the following table:

No.of dials used 1 2 3 4 5 6 7 8
Dial No. 8 7&8 6to8 Sto8 4to8 3to8 2to8 1to8
Assigned Combined
Uncertainty
Sea ( 10° pF) 0.22 0.31 0.38 0.45 0.51 1.12 13.05 150.57

42



Table B-5. Data and Analysis of the ( 0.01 F/step ) Mica Capacitor Dial Corrections of the
Type-12 Bridge at 1 kHz
[All values are in pF except the dials which are in (uF/step)]

Date Dial 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0X
6/67 1.165 | 2.741 | 3.516 | 4.629 | 6.035 | 7.545 | 5.495 | 7.686 | 9.454 | 8.387
11/67 1.185 } 2.851 | 3.739 | 4.507 | 5.937 | 7.535 | 5.583 | 7.754 | 9.534 | 8.500
5/68 1.244 | 2.875 | 3.726 | 4.625 | 6.098 | 7.662 | 5.693 | 7.922 | 9.720 | 8.719
10/69 1.218 | 2.808 | 3.583 | 4.466 | 5.914 | 7.429 | 5.315 | 7.550 | 9.295 | 8.261
12/70 1.178 | 2.888 | 3.664 | 4.449 | 5.870 | 7.553 | 5.513 | 7.704 | 9.584 | 8.522
9/72. 1.234 | 2.895 | 3.524 | 4.283 | 5.730 | 7.354 | 5.238 | 7.454 | 9.338 | 8.300
6/74 1.278 | 2.905 | 3.696 | 4.466 | 5.981 | 7.546 | 5.482 | 7.754 | 9.557 | 8.587
11/75 1.251 | 2.957 | 3.760 | 4.493 | 5.999 | 7.652 | 5.648 | 7.921 | 9.774 | 8.805
V/77 1.184 | 2.818 | 3.416 | 4.205 | 5.654 | 7.228 | 5.032 | 7.256 | 9.097 | 8.093
4/78 1.093 | 2.646 | 3.084 | 3.855 | 5.193 | 6.704 | 4.337 | 6.450 | 8.216 | 7.066
11/80 1,099 | 2.706 | 3.205 | 3.801 | 5.138 | 6.687 | 4.348 | 6.460 | 8.181 | 7.128
8/82 1.086 | 2.674 | 3.068 | 3.659 | 4.999 | 6.570 | 4.159 | 6.302 | 8.055 | 6.906
9/83 1.125 | 2.850 | 3.245 | 3.872 | 5.237 | 6.877 | 4.507 | 6.657 | 8.492 | 7.375
1/85 1.137 | 2.835 | 3.332 | 3.682 | 5.134 | 6.792 | 4.494 | 6.666 | 8.501 | 7.429
3/85 1.028 | 2.652 | 2.860 | 3.233 | 4.534 | 6.110 | 3.503 | 5.562 | 7.375 | 6.188
2/91 1.054 | 2.715 | 2.881 | 3.465 | 4.801 | 6.407 | 3.621 ] 5.685 | 7.494 | 6.362

N. Val. (pF) | 1.0E+05}2.0E+05] 3.0E+05] 4.0E+05] 5.0E+05] 6.0E+05| 7.0E+05] 8.0E+05] 9.0E+05] 1.0E+06,
Mean (pF) 1.1599 | 2.8010 | 3.3937 | 4.1056 | 5.5159 | 7.1032 | 4.8730 | 7.0489 | 8.8542 | 7.7893
SD (pF) 0.0747 | 0.0986 | 0.3064 | 0.4498 | 0.5031 | 0.5043 | 0.7345 | 0.7935 | 0.8052 | 0.8622

Slope (pF/day) |-2.1E-05}-1.8E-05|-9.7E-05|-1.6E-04|-1.7E-04]-1.7E-04|-2.5E-04|-2.7E-04|-2.7E-04|-2.8E-04|
( pF/year) ~0.008 | -0.006 | -0.035 | -0.057 | -0.063 | -0.061 | -0.091 | -0.097 | -0.097 | -0.102
SDfit (pF) 0.052 | 0.090 | 0.172 | 0.180 } 0.225 | 0.253 | 0.335 | 0.379 | 0.407 | 0.453
SD(2/01) (pF) | 0.070 | 0.121 | 0.231 | 0.241 | 0.302 | 0.339 | 0.450 | 0.509 | 0.547 | 0.608

B-5



4.3.2 Two-Terminal Measurements of High-Capacitance Values
(0.005 pF < Cy < 1 pF)
4.3.2.1 Basic Equations of Balance

Measurement procedures of this type are the same as those for low-capacitance values, except that
the mica dial capacitors are needed when the unknown is connected to arm CD, as shown in Fig. 7.
The equations of balance, corresponding to Eqs. (49) and (50) become:

Ri Cap = Ry Cy (57)
and

Ry (Car + Co.) = Ro (Cor + Cx), (58)
where C9 and Coo are the values of the air capacitors at balance without connecting the unknown
capacitor C,; C,, and C,, are the values of the air capacitors, and C,, is the value of the mica
capacitors at balance with C, connected to arm CD.
The value of the unknown capacitor can be expressed as:

Cy = CR Ry Cyt (4) LE + CRY RQ) ICC - Go) (59)

where C,,, is the mica dial reading, and C,,= 2 C,, and C,) = 2C,o are air-dial readings of internal
capacitors at balance, with and without connecting C, to arm CD, respectively.

The ratio resistors are always set to be equal, (R, =, ) so that Eq. (59) becomes:
Cy = Gat (Cn - Go )s (60)

which is the equation used to calculate the value of the unknown capacitor.

4.3.2.2 Analysis
The relationship among the variances of the terms of Eq. (59) can be estimated by:
Var(C,) = [Con + CAM Coy - Cro IP Sq” + (Ry Ry)” Spm?
+ AP EL+ (RI RAI (Sper. * S00") (61)

where 5,4, Spy» Spe 1» ANd S,o9 are the standard uncertainties in the ratio ( contributed by the resistors),
the mica capacitor dial, and the air capacitor dials of both measurements, respectively.

According to Eq. (60), the first term of Eq. (61) can be expressed as:
1 2. 2 2. 2
[Cu + CAM Cy - Cro Spa < (CQ Sa (62)

25


where Var ( d; ) is the variance of the dial correction of the third dial, d,,
Ska is the in-phase standard uncertainty in the 1:10 ratio, as given in Eq. (11),
s, is the standard uncertainty of the reference standard,
% is the dial reading for the ge dial, where k >3, th
Var (dj, ) is the variance of the dial correction of the k dial, where k >3,
s, is the standard uncertainty of interpolation of the least significant capacitance dial incurred
during measurement, and
¢;, does not have a variance.

Then the standard uncertainty for the correction of the third dial, s, can be expressed as:
55 =[ (01) spy? + (01562 +E C252) +571, (20)
where s;, is the standard deviation of the dial correction of the gh dial, where k >3.

After s, is determined, the standard uncertainties of other dials, s;, can be estimated from s, by using
the appropriate expression similar to Eq. (20).

For capacitors C, to Cg, the standard uncertainties for the dial corrections can be expressed,
according to Eq. (17), as:

2, (4)

5;=[(0.1) Spy + 01) 5G ry FLO 52) +1, (kD i+ Land 4<i< 8). 21)

Similarly, standard uncertainties in dial corrections of C, and C, are given as:

5,= (0)? spq? + (10) 5674.1) HOY LOY 545711 (&>i+1 and i=1 or 2). (22)

3.2.3 Analysis of Uncertainties in Dial Corrections

As mentioned previously, only capacitance values are reported. The component of errors from the
cross product of quadrature errors in the 10:1 (or 1:10 ) ratio and the internal capacitors is negligible.
Therefore, only the components of the in-phase standard uncertainties of the dial corrections are
included in the Type B standard uncertainty. Based upon Eqs. (20), (21), and (22), the standard
uncertainty, s3, for the dial correction d; of GC is determined first, and the standard uncertainties of
other dials, s; , are obtained from s;. Estimated values of each term of Eq. (20) are given as follows:

1) According to Eq. (11) in section 3.1.2, the in-phase standard uncertainty for the 10:1 (and
1:10) ratio corrections, sp,, is estimated to be less than 0.043 ppm. For C; = 1 pF, this
becomes:

Spq < 0.043 x 10° pF.

2) The combined standard uncertainty of the reference standard, s,., is less than 0.72 ppm,
which will be discussed in section 3.4, (i.., s,, < 0.72 ppm). For C, = 10 pF, then we have:


Table 13b. Summary of Components of Standard Uncertainties in Air Capacitor Dial Corrections
for Three-Terminal Measurements

Unknown Standard C, (uF) | 0.001 | 0.002 | 0.005} 0.01 | 0.02 | 0.05 0.1 0.2 0.5 1.0

Std Dev.of Pred. Values of
Dial Correction (pF) 0.0607 | 0.0749 | 0.0636 | 0.0224 | 0.0241 | 0.0183 | 0.0266} 0.0191 | 0.0264 0.0324
Std Dev.of Pred. Values of
Dial Correction (ppm) 60.744 | 37.448} 12.723) 2.238 | 1.205 | 0.365 | 0.266 | 0.095 | 0.053 | 0.032

Components of
Standard Uncertainties (ppm)

Pred. Values of Dial Corr't (sg) | 60.744 | 37.448} 12.723] 2.238 | 1.205 | 0.365 | 0.266 | 0.095 | 0.053 | 0.032
Temperature Variation (sp) 6 6 6 6 6 6 6 6 6 6

Dial Switching (s 4.) 6 3 15 0.6 0.3 0.15 0.1 0.05 | 0.02 | 0.01
Transfer Standard (sj) 6 6 6 6 6 6 6 6 6 6
RSS 61.627 | 38.515] 15.366] 8.796 | 8.576 | 8.494 | 8.490 | 8.486 | 8.485 | 8.485

Assigned Standard Uncertainties
in Air Cap. Dial Corrections (s,.)] 62 39 16 9 9 9 9 9 9 9

53


Table 14a. Summary of Components of Standard Uncertainties in Mica Capacitor

Dial Corrections for Two-Terminal Measurements .............0.0 cece ee eeee
Table 14b. Summary of Components of Standard Uncertainties in Mica Capacitor

Dial Corrections for Three-Terminal Measurements .............0-000ee ee eee
Table 15a. Summary of Components of Type B Standard Uncertainties in

Capacitance Measurements Using the Type-12 Bridge to Measure

Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10kHz ................
Table 15b. Summary of Components of Type B Standard Uncertainties in

Capacitance Measurements Using the Type-12 Bridge to Measure

Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz ...............
Table 16a. Summary of Components of Type A Standard Uncertainties in

Capacitance Measurements Using the Type-12 Bridge to Measure

Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz ................
Table 16b. Summary of Components of Type A Standard Uncertainties in

Capacitance Measurements Using the Type-12 Bridge to Measure

Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz ...............

Table 17a. Summary of Expanded and Assigned Total Uncertainties in
Capacitance Measurements Using the Type-12 Bridge to Measure
Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz ..............-5

Table 17b. Summary of Expanded and Assigned Total Uncertainties in
Capacitance Measurements Using the Type-12 Bridge to Measure
Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10kHz ...............

Table 18a. Summary of Components of Type B Standard Uncertainties in
Conductance Measurements Using the Type-12 Bridge to Measure
Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz ................

Table 18b. Summary of Components of Type B Standard Uncertainties in
Conductance Measurements Using the Type-12 Bridge to Measure
Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10kHz .............

Table 19a. Summary of Components of Type A, Expanded, and Assigned Uncertainties
in Conductance Measurements Using the Type-12 Bridge to Measure
Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz ..............-.

Table 19b. Summary of Components of Type A, Expanded, and Assigned Uncertainties
in Conductance Measurements Using the Type-12 Bridge to Measure
Three-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz

vi


Table B-3.

Data and Analysis of the (0.1 F/step ) Mica Capacitor Dial Corrections of the

Type-12 Bridge at 10 kHz

[All values are in pF except the dials which are in (uF/step)}

Date’ Dial 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 0.x
6/67 16.633 | 40.841 | 49.829 | 109.127] 133.238] 173.159] 197.804 | 234.249 | 267.953 | 334.641
11/67 17.537 | 43.049 | 51.619 | 113.617] 138.727] 180.194] 193.636 | 232.127 | 265.649 | 333.446
5/68 17.580 } 42.757 } 51.198 | 112.780] 137.699) 178.588} 192.175 | 229.817} 263.211 | 331.357
9/69 17.023 | 42.341 }| 51.152 | 113.782] 137.885] 179.716] 195.162} 230.105 | 263.758 | 330.093
12/70 17.577 | 43.579 | 53.528 | 116.540} 141.480} 183.752] 198.649 | 236.748 | 271.662] 338.677
172 17.200 | 43.264 | 47.719 | 116.766] 140.603] 183.300) 192.219} 233.629] 268.911 | 338.315
6/74 17.242 } 42.382 | 46.622 | 113.742] 137.876] 179.084] 190.984 | 214.931 | 247.191 | 316.558
11/75 17.373 | 43.618 | 53.638 | 117.938] 143.700] 185.088 | 200.033 | 234.903 | 264.293 | 333.768
1/77 17.433 | 43.129 | 53.667 | 117.288} 141.523] 182.779} 197.197 | 234.093 | 268.804 | 336.119
4/78 16.649 | 41.610 | 52.254 | 114.918] 138.599) 180.768 | 194.839 | 224.878 | 252.769 | 321.523
11/80 15.938 } 42.038 | 52.608 | 116.776] 140.616] 180.812] 213.659 | 251.150] 290.010} 352.910
8/82 16.626 | 42.233 | 52.724 | 120.318] 145.889] 187.436 | 200.815 | 239.526 | 264.850} 328.029
9/83 16.262 | 42.149 | 47.889 | 118.916] 143.913] 186.143 | 202.890] 238.802 | 281.866 | 349.791
1/85 16.406 | 39.757 | 49.519 } 114.702] 139.098] 178.572 195.466 | 225.324 | 251.888 | 306.074
3/85 16.218 | 41.479 | 52.377 | 115.708] 138.774} 178.980] 195.691 | 231.559] 265.150} 330.821
2/91 16.440 | 41.815 | 50.474 | 114.584] 137.736] 178.309] 197.036 | 232.733 | 265.200} 381.138

N. Val. (pF) | 1.0E+05]2.0E+05)3.0E+05| 4.0E+05| 5.0E+05} 6.0E+05] 7.0E+05 | 8.0E+05|9.0E+05] 1.0E+06)
Mean (pF) | 16.8836] 42.2526] 51.0511 | 115.469} 139.835] 181.043] 197.391 | 232.786 | 265.823 | 335.204
SD (pF) 0.5486 } 1.0266 | 2.2112] 2.681 | 3.050 | 3.570 | 5.433 | 7.740 | 10.413 | 16.652

Slope (pF/day) |-1.5E-04)-1.7E-04|-3.4E-06] 4.9E-04 | 3.9E-04 | 2.8E-04 | 7.0E-04 | 4.2E-04 | 2.5E-04 | 2.1E-03
( pF/year) -0.054 | -0.061 | -0.001 | 0.177 | 0.141 0.103 | 0.255 | 0.153 | 0.091 0.749
SDfit (pF) 0.397 | 0.959 | 2.289 | 2.431 | 2.973 | 3.612 | 5.288 | 7.928 | 10.756 | 16.285

SD(2/01) (pF) | 0.532 1.288 | 3.073 | 3.264 | 3.991 4850 | 7.100 | 10.644 | 14.441 | 21.863



Cy = ER / (Ry + Ry) CC + Cs) + C2) (0 Cpa = Ga + Cys - Cea (80)

where C,, to C,, and C,,) to G,,3 are air and mica dial readings, respectively, corresponding to each
measurement.

The ratio resistors are always set to be equal (R, = R,), so Eq. (80) becomes:
Cy = (A) (Cyr + Cons) + (2) (Cyn = Cr) + Cys - Cry) (81)

which is the equation used to calculate the value of the unknown capacitor.

4.3.5.2 Analysis
Rewrite Eq. (80) as:
Cy = (RYURg) / T+ (RyRy MH Cina + Cons + (A) WC Con = Cor + (Cy = Cog Je (82)
The relationship among the variances of the terms of Eq. (82) can be estimated by:
Var(C,) = [1 +(R,/ Ry N* (Cn + Cons) Spa”

ECR Ry)/ [1+ (RI Ry DY? (Sa + Spa)

+ (AY ( Sse + Sten. + 553° + Sted ) (83)
where s,., Sm Sms Stow Sez? Ste3 and. Spo4 are the standard uncertainties in the ratio ( contributed
by the resistors), the mica capacitor dials, and the air capacitor dials of the four measurements,
respectively.

According to Eq. (81), the first term of Eq. (83) can be expressed as:

(Cina Cons) Sta” (2g Sys (84)
By substituting Eq. (84) and ( R, = R, ) into Eq. (83), it becomes:

Var(C,) < (CA) Cy Seq +1CA) Sal? +A) 5401". (85)
where Sm = Sm + Sma and Sto. = Sto” + Sten + Ste” + Stoq 2

In Eq. (85), the first term represents uncertainties in the ratio; the second and third terms represent
uncertainties in the mica and air dial readings, respectively.

As in the three-terminal high-capacitance value measurements, the uncertainty contributed by the

ratio resistors, [(12) C, s,q], is also 9 ppm. (See Table 15b). According to Eq. (81), two balances
using the internal mica capacitors and four balances using the internal air capacitors are required to

30




Similar measurements have also been performed on four NIST standards at frequencies of 100 Hz
and 10 kHz, and standard uncertainties of their predicted values are also given in the Tables 16a and
16b. The Type-12 bridge is employed to perform measurements on mica capacitors with nominal
values other than these four values at frequencies other than 1 kHz.

Tables 16a and 16b provide summaries of the uncertainty components of Type A evaluation of
standard uncertainties for the Type-12 bridge in measuring two-terminal and three-terminal mica
dielectric capacitors, respectively, at frequencies of 100 Hz, 1 kHz, and 10 kHz. The Type A
standard uncertainties for capacitance measurements using the Type-12 system are defined as the
RSS of these components (s,.5, Str» andsyq), and also are included in Tables 16a and 16b.

4.5 Expanded Uncertainty for Capacitance Measurement Using the Type-12 System

The expanded uncertainty for each nominal value of mica capacitor is calculated from both Type A
and Type B components of uncertainty with a coverage factor of k = 2 [2]. Tables 17a and 17b give
the expanded and assigned total uncertainties for two-terminal and three-terminal capacitance
measurements, respectively, using the Type-12 system.

4.6 Type B Standard Uncertainty of Conductance Measurements

The Type-12 bridge, which has a conductance standard as well as mica and air capacitance standards,
was designed to be direct-reading for capacitance and conductance. The analysis of the uncertainties
in conductance measurements is based on considerations taken into account by the designers of the
Type-12 bridge [14].

During measurements, both capacitance and conductance dials are adjusted to obtain a balance from
which to determine the capacitance and conductance values of the unknown capacitor. As shown
in Fig. 5a, the conductance standard, G,, of the Type-12 bridge consists of two decades with ten
100 pS steps, and ten 10 pS steps, plus a continuously variable standard with a range of -1 pS to
11 pS. Similar to the internal air capacitors, G, is a differential-type standard, independently
connected to arms AD and CD such that the sum of the conductance in both arms is a constant.
There is a divider resistor, K,, which serves as a range shifter, so that the reading of the conductance
standard can be divided by 1; 10, 100, 1000, or 10 000. Specifications and operational procedures
for the conductance standard of the Type-12 bridge are described in detail by Wilhelm [14]. Figure
11 is a simplified diagram of the Type-12 bridge including the conductance components in each arm,
where (G,/K,) and (G,/K,) are effective conductance adjustments to arm AD and arm CD,
respectively, and the quantity of (G, + G, ) is equal to a constant. G, provides readings in both
decades and the variable dial to cover the required range and keeps the sum of G, and G, a constant.
Equations used to calculate the conductance of unknown capacitors are similar to those used to
calculate capacitance with C,,, C,,, and C,, being replaced by G,,, (G,/K,), and (G,./K,),
respectively. For example, in the two- terminal high-capacitance measurements, fiom Eq. (60), the
conductance value, G,, of the unknown capacitor Cy, is calculated from the two balances as:

Gg = Gm (Gy = Ga VKys (86)

33