NIST

Special

Publication

250-52 . .

NAT'L INST. OF STAND & TECH m

Error Analysis and Calibration

Uncertainty of Capacitance

Standards at NIST

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Control Chart for Reference Standards

Y May Chang

U.S. Department of Commerce

Technology Administration

National Institute of Standards and Technology
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'At Boulder, CO 80303. ^Some elements at Boulder, CO.
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

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
National Institute of Standards and Technology Special Publication 250-52

Natl. Inst. Stand. Technol. Spec. Publ. 250-62, 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
TABLE OF CONTENTS

page

List of Tables v

List of Figures vii List of Symbols viii ABSTRACT 1

L INTRODUCTION 1

2. DESCRIPTION OF SYSTEMS AND MEASUREMENT PROCEDURES 2

2.1 Type-2 Capacitance Bridge and Standards 3

2.2 Type- 12 Capacitance Bridge 3

3. UNCERTAINTY ANALYSIS FOR THE TYPE-2 SYSTEM 4

3.1 Transformer Ratios 5 3.1.1 Corrections and Uncertainties of the 1:1 Ratio 5

3.1.2 Calibration and Uncertainties of the 10:1 ( and 1:10) Ratio 6

3.2 Dial Corrections 7

3.2.1 Calibration of Dial Capacitors 7

3.2.2 Uncertainties in Dial Corrections 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 11

3.3.2 Frequency Dependence of Standards 11

3.3.3 Hysteresis Effects in Temperature Corrections of Standards 12

3.3.4 Lead Impedance 12

3.3.5 Uncertainty in the Reference Standard 14

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-Sihca 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 16

3.7.1 Three-Terminal Air Capacitance Standards 16

3.7.2 Two-Terminal HF Coaxial Capacitance Standards and Terminations 16

4. UNCERTAINTY ANALYSIS FOR THE TYPE- 1 2 SYSTEM 17

4.1 Basic Equations of Capacitance Measurements 18

4.2 Type B Standard Uncertainty of Capacitance Measurements 19

4.2.1 Ratio Resistors 20

4.2.2 Internal Capacitors 20

4.2.3 External Bridge Components 22

4.3 Type B Standard Uncertainty for Various Types of Capacitance Measurement .... 23

4.3.1 Two-Terminal Measurements of Low-Capacitance Values 23

4.3.1.1 Basic Equations of Balance 23

iii
4.3.1.2 Analysis 24

4.3.2 Two-Terminal Measurements of High-Capacitance Values 25

4.3.2.1 Basic Equations of Balance 25

4.3.2.2 Analysis 25

4.3.3 Three-Terminal Measurements of Low-Capacitance Values 26

4.3.3.1 Basic Equations of Balance 26

4.3.3.2 Analysis 27

4.3.4 Three-Terminal Measurements of High-Capacitance Values

up to 0.5 ^iF 27

4.3.4.1 Basic Equations of Balance 27

4.3.4.2 Analysis 28

4.3.5 Three-Terminal Measurements of Capacitors Larger Than 0.5 |xF 29

4.3.5.1 Basic Equations of Balance 29

4.3.5.2 Analysis 30

4.4 Type A Standard Uncertainty of Capacitance Measurements 31

4.4.1 Repeatability of Measurements 31

4.4.2 Effect of Temperature Variations on Unknown Capacitors 31

4.4.3 Stabilities of NIST Standards 32

4.5 Expanded Uncertainty for Capacitance Measurement

Using the Type- 12 System 33

4.6 Type B Standard Uncertainty of Conductance Measurements 33

4.6.1 Conductance Standard 34

4.6.2 Contact Resistance 34

4.6.3 Conductance Potentiometer 35

4.6.4 Residual Conductance 36

4.6.5 Dial Corrections 36

4.7 Type A Standard Uncertainty of Conductance Measurements 36

4.7.1 Repeatability of Measurements 37

4.7.2 Stabilities of NIST Standards 37

4.8 Expanded Uncertainty for Conductance Measurement

Using the Type- 12 System 37

5. CONCLUSION 37

6. ACKNOWLEDGMENTS 38

7. REFERENCES 38

TABLES 40

FIGURES 66

APPENDIXES A-1

APPENDIX A. Uncertainty Analysis of the Corrections to the 1 :1 Ratio A-1

APPENDIX B. Capacitance Dial Corrections ofthe Type- 12 Bridge B-1

APPENDIX C. Conductance Dial Corrections ofthe Type- 12 Bridge C-1

iv
List of Tables

Page

Table 1 . Dial Corrections of the Type-2 Bridge 40

Table 2. Standard Uncertainties in Dial Corrections of the Type-2 Bridge 41

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 42

Table 4. Uncertainties Due to Lead Impedance Using the Type-2 Bridge to Measure Capacitors at Frequencies of 1 00 Hz, 400 Hz, and 1 kHz 43

Table 5. 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

Table 6. Expanded and Assigned Uncertainties Using the Type-2 Bridge to Measure

Fused-Silica and Nitrogen Dielectric Capacitors at 1 kHz 45

Table 7. Expanded and Assigned Uncertainties Using the Type-2 Bridge to Measure

Fused-Silica and Nitrogen Dielectric Capacitors at 400 Hz 46

Table 8. Expanded and Assigned Uncertainties Using the Type-2 Bridge to Measure

Fused-Silica and Nitrogen Dielectric Capacitors at 100 Hz 47

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 48

Table 10. Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure

Three-Terminal Air Capacitors at 100 Hz 49

Table 1 1 . Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure

Two-Terminal Coaxial Air Capacitors at 1 kHz 50

Table 12. Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure

Two-Terminal Coaxial Mica Capacitors and

Open-Circuit Terminations at 1 kHz 51

Table 13a. Summary of Components of Standard Uncertainties in Air Capacitor

Dial Corrections for Two-Terminal Measurements 52

Table 13b. Summary of Components of Standard Uncertainties in Air Capacitor

Dial Corrections for Three-Terminal Measurements 53

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

Dial Corrections for Two-Terminal Measurements 54

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

Dial Corrections for Three-Terminal Measurements 55

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 56

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 57

Table 1 6a. 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 58

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 59

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 60

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 10 kHz 61

Table 18a. Summary of Components of Type B Standard Uncertainties in

Conductance Measurements Using the Type- 12 Bridge to Measure

V - Two-Terminal Mica Capacitors at 100 Hz, 1 kHz, and 10 kHz 62

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 63

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 64

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 65

vi
List of Figures

page

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

the Type-2 Bridge 66

Figure 2a. Schematic Diagram of the NIST Type-2 Capacitance Bridge 67

Figure 2b. Simplified Circuit Diagram of the NIST Type-2 Capacitance Bridge 68

Figure 3. Internal Capacitors of the Type-2 Bridge 69

Figure 4. Lead Impedance of a capacitor connected to the Bridge 69

Figure 5 a. Schematic Diagram of the Type- 12 Capacitance Bridge 70

Figure 5b. Simplified Circuit Diagram of the Type-12 Capacitance Bridge 71

Figure 6. Components of the Type-12 Bridge for Two-Terminal

Low-Capacitance Values Measurements 72

Figure 7. Components of the Type-12 Bridge for Two-Terminal

High-Capacitance Values Measurements 73

Figure 8. Components of the Type-12 Bridge for Three-Terminal

Low-Capacitance Values Measurements 74

Figure 9. Components of the Type-12 Bridge for Three-Terminal Measurements

of Capacitors up to 0.5 ^iF 75

Figure 10. Components of the Type-12 Bridge for Three-Terminal Measurements

of Capacitors Larger than 0.5 i^F 76

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

Including the Conductance Components 77

vii
List of Symbols

1. Type-2 System

a and b In-phase and quadrature corrections of the 1:1 ratio.

Cj to Cg Dial capacitors of the Type-2 bridge.

Cj to Cg Nominal values of capacitors Cj to Cg.

Cj to Cg Real values of capacitors Cj to Cg.

Cg, Cj, & Cjj Reference capacitor, unknown capacitor, and total dial capacitors.

Cg, Cj, & Cji Nominal values of capacitors C^, C^, and C^.

Cg, C^, & C^ Real values of capacitors Cg, C^, and Cj.

(AC / C) Uncertainty in the capacitance due to the lead impedance.

i/j to Dial corrections of Cj to Cg.

Jg Deviation of the reference capacitor from its nominal value.

DF^ Dissipation factor of the capacitor,

^mb Maximum bound of error. and E^^ Applied voltages to Cg and C^.

and AG Parallel conductance of C, and the uncertainty of G^.

Gj. and g- Conductance of the internal dial resistors and admittance of the / dial,

/g, /j, & /g Current through Cg, C^j, C^, and G^.. k Coverage factor for confidence level, normally equal to 2.

Z,j Series inductance of leads.

M and M Multiplier of the conductance and its setting,

m Ratio of measurement, either 1 or 10.

/>5 A constant, either plus or minus 0.5.

r- Dial reading of c^-.

Y/]^ cj^ Sum of dial readings giving the capacitance value of , for A: > / + 1

.

i?j Series resistance of leads.

and Standard uncertainties of the in-phase and quadrature corrections of 1 : 1 ratio.

s^^ Combined standard uncertainty of the external bridge components.

s^^ Combined standard uncertainty of dial corrections.

5f Standard uncertainty of the capacitor due to frequency dependence.

Si and Sj Standard deviations of the components of the Type A and Type B standard

uncertainties.

sj^ Standard uncertainty of the dial correction of the dial.

5j and Standard uncertainties of interpolations of the last dial at 1 kHz and 100 kHz.

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

5g Standard uncertainty of the reference standard, due to its stability.

Combined standard uncertainty of the reference standard in measurements.

5g^ Equivalent standard deviation (refer as the standard uncertainty) of a component.

s^^ and Standard uncertainties in temperature corrections of fused-silica and nitrogen

dielectric capacitors.

s^.^ and In-phase and quadrature standard uncertainties of the capacitor due to voltage

dependence.

viii
u^, u^, & Combined, Type A and Type B standard uncertainties.

U and Expanded and total (assigned) uncertainties.

Fg, 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

Cj and C2 Total capacitance of arm AD and arm CD.

and Q Effective internal capacitors on arm AD and arm CD.

Cj^ and Capacitance of mica capacitor dial readings and of air capacitor dial reading, r. and Unknown capacitor and its value.

Cjj. Nominal value of capacitor C^.

For two-terminal measurements:

C^Q and Qo Internal capacitors on arm AD and arm CD without connecting to arm CD.

C^j and Internal capacitors on arm AD and arm CD with connecting to arm CD.

Cj.Q and Cj., Dial readings of air capacitors with and without connecting to arm CD.

For three-terminal measurements:

C^j and Internal capacitors on arm AD and arm CD with connecting to arm AD.

and C^2 Internal capacitors on arm AD and arm CD with connecting to arm CD.

Cj.} and Cj.2 Dial readings of air capacitors with connecting to arm AD and arm CD.

(For > 0.5 i^F) to Internal capacitors on arm AD for the four respective measurements.

Qi Q4 Internal capacitors on arm CD for the four respective measurements.

Cj.j to Cj.4 Dial readings of air capacitors for the four respective measurements.

Cj^2 ^ ^m3 ^^^^ readings of mica capacitors for the second and third measurements.

Cjj and Capacitance decade box and its value.

C|lj Nominal value of capacitance decade box C^.

Cg£f Effective capacitance of a capacitor C^,.

^jj2, Maximum error caused by lead impedance.

e^^ Maximum error due to dial switching.

^gj. Maximum error due to residual conductance.

^gj. Maximum error due to contact resistance.

Maximum error due to conductance standard.

Applied frequency.

and Effective internal conductance on arm AD and arm CD.

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

and Conductance of unknown capacitor and its value.

Kg and Divider resistor and its value.

Z,j Series inductance of leads.

Rj and R2 Ratio resistors on arm AB and arm CB.

ix
i?l and i?2 Values of ratio resistors Rj and

R^^ and Rj.2 Contact resistance of the two balances.

Standard uncertainty due to lead impedance.

Sjjp2 and S|jp3 Standard uncertainties due to corrections of plates P2 and P3.

s^^ Standard uncertainty of external bridge components.

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

s^^ 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.

5gpj^ Standard uncertainty of the predicted conductance value of the NIST standard.

5gj. ^ ; Standard uncertainty due to contact 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.

s^2 Standard uncertainties in the mica capacitor dials corresponding the readings of

andCj^3.

' Standard uncertainty in the mica capacitor dials due to the uncertainty of transfer

standard.

5j^p Standard uncertainty in the mica capacitor dials due to temperature variations.

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

5pj^ • Standard uncertainty of the predicted capacitance value of the NIST standard.

Standard uncertainty contributed by the ratio resistors.

5j.^ Combined Standard uncertainty of the internal air capacitor dials.

Sj.j,Q to Sj.^4 Standard uncertainties in the air capacitor dials corresponding the readings of C^q to

Cj.4.

Standard uncertainty due to repeatability of measurements in capacitance.

s^^ Standard uncertainty in the air capacitor dials due to the uncertainty of transfer

standard.

Standard uncertainty on unknown capacitor due to temperature variations.

5^p Standard uncertainty in the air capacitor dials due to temperature variations,

and Type A and Type B standard uncertainties of capacitance measurements,

and Wgi^ Type A and Type B standard uncertainties of conductance measurements.

U and Expanded and total (assigned) uncertainties of capacitance measurements.

C/g and L/g^ Expanded and total (assigned) uncertainties of conductance measurements.

X
ERROR ANALYSIS AND CALIBRATION UNCERTAINTY

OF CAPACITANCE STANDARDS AT NIST

Y. 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,m 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

quahty 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 pohcy

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
metrologist.

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

as:

r 2 , 2-,'/2

where is the combined standard uncertainty, and

and Wj^ are Type A and Type B standard uncertainties, respectively.

According to the guidehnes recommended by the Litemational Bureau of Weights and Measures [3],

the overall, or expanded uncertainty, U, is expressed as:

U=ku^, (2a)

or U=k [T{s^)'+nsjf]"\ (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 and Sj 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 = l, indicating a level

of confidence of approximately 95 percent.

Therefore, the total uncertainty, assigned to capacitance calibration at NIST is calculated

according to the following equation:

^ = 2[E(5^.)'+%)']'''- (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,

^sd ^mb //3, (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 ± e^y^

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
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 )iF to 1 ^iF 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 1 00 pF, 1 0 pF, , 1 0"^ 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 : 1 0, or 1 0: 1 ratios. During the transfer of the farad, an NIST-made 1 0 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 fiised-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 fiised-silica capacitors.

The ICL also has a number of capacitance check standards. Among these are six commercial drynitrogen-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 fiised-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
mainly used to calibrate mica-dielectric capacitors of exposed binding-post connectors in the range

from 0.001 |iF to 1 |iF. 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 is the extemal reference capacitor,

is the unknown capacitor to be measured, represents the internal dial capacitors, Gj. 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:

where /g is the current through the reference capacitor, , is the current provided by the capacitance network, C^ as described below,

' I is the current supplied by the conductance balance divider and network, and

7^ is the current through the unknown capacitor, .

The dial capacitance, C^j, is provided by a set of eight air capacitors whose values range from 100 pF

to 10"^ pF having a total capacitance up to 1 1 1 pF (see Fig. 3). G^. is a set of four dials providing

conductance from 0.001 |xS to 1 [iS with a multiplier switch to extend the total range from 10"^ \iS to 1 |j,S. 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 is equal to the value of plus the sum of the dial settings. The balance equation can be written as:

E,Y^ + E^(lr,y,)+p,E^{Mlg,) =-E^Y^, (6)

where and E^ are voltages produced by the transformer windings on the reference, and the

unknown capacitor, sides of the bridge, respectively.

4
and are admittances of and C^^ , respectively,

r,- is the dial reading of i dial applying voltage to capacitor c,-, where 0 < r,- < 1, in

increments of 0.1,

V; is the admittance of c,-,

• th g- is the effective admittance associated with the settings of the i conductance dial coupled

with the frequency compensation network,

/»5 is a constant whose value can be taken as either plus or minus 0.5, and

Mis the muhipHer 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 C^j 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

and connected as shown in Fig. 2b. The other measurement is taken with C ^ and

interchanged. The balance equation of the second measurement can be written as:

+ E 'nyn )+PsEs{M'lgJ = - F3, (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:

5
(8)

Assuming E^ = lY as the reference, and by the definition of 1:1 ratio, (E^ I E^) - - 1, then:

E^ = -{l+a+ib), (9)

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

b is 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:

< 0.005 ppm'

and

5^,<0.03 ppm, (10)

where and 5^ 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 ofmeasurements. 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:

< 42.25 X 10'^ ppm < 0.043 ppm

and

< 32.49 X

10"^ ppm < 0.033 ppm, (11)

where and ^j^^^ 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].

6
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, Cj 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 C3, 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:

100 pF = 10 C2, (12)

10 pF

C3 - 1 pF = 10 C4,

C7 = 10"^ pF = 10 Cg, and

Cg = 10"^ pF.

The first step in the calibration of capacitors Cj to Cg is to calibrate C3 against a 10 pF reference

standard using a 1:10 ratio by setting C3 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 C3 (at the +1.0 switch position) as reference to calibrate Cj (at

the - 0.1 switch position) and using C3 (at the - 0.1 switch position) to calibrate C4 (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 C^^+j^ can be written as: Q + di=m (C(,.+i) + + cj^ ), (13)

where and C/^•^^^ are nominal values of C^- and ^n+\\ , respectively.

7
di and c/^^-^j^ are dial corrections of C^- and C^^-+j^ , respectively,

/« = 1 0 is the ratio, and

cj^ is the sum of dial readings giving the capacitance value at balance, for k> i+l.

The balance equation for calibrating C3 (where C3 = 1 pF) with the 10 pF reference standard (C^),

according to Eqs. (12) and (13), is:

C^ + d^ - 10(C3-f J3 +Er^c^), (14)

where ( = 10 pF) is the nominal value of the reference standard,

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

is the sum of the dial readings giving the capacitance value at balance, for A: > 3.

The dial corrections are related similarly by:

^/=10(^(,-+i) + Zr^c^), {k>i+l), (15)

and, hence (from Eq. (14)),

d, = {OA)d^ - Irj^cj^ (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, dj^ as:

^/ - 10 ( + lri^Ck + lrj^dj^),(k> i+\ ), and (17)

d,- iO.l) d^- il rj^c^ + l rj^dj^), ik>3), (18)

til where Cy^ is the nominal value of the k capacitor, c^, and

dj^ is the correction to the capacitor, c^.

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 1 0 pF reference standard. Therefore, the standard uncertainty of the correction of the third dial,

J3 is determined first. The relationship of the variance of each term in Eq. (18) can be written as:

Var ( ^3 ) = ( 0.1 f (.r/ + + l Var ( ^/^ ) + s^, (19)

8
where Var ( t/j ) is the variance of the dial correction of the third dial, d^^, s^^ is the in-phase standard uncertainty in the 1:10 ratio, as given in Eq. (1 1),

5„ is the standard uncertainty of the reference standard,

th r^^ is the dial reading for the ^ dial, where A: >3,

Var ( ) is the variance of the dial correction of the k dial, where k >3,

5| is the standard uncertainty of interpolation ofthe least significant capacitance dial incurred

during measurement, and

Cj^ does not have a variance.

Then the standard uncertainty for the correction of the third dial, s-^ can be expressed as:

^3 = [ ( 0.1 ^Ra' + ( 0.1 )^s' + E ( ) + ^i' ]

^^^'^

, (20)

th where sj^ is the standard deviation of the dial correction of the k dial, where k >3.

After 53 is determined, the standard uncertainties of other dials, 5^-, can be estimated from 53 by using

the appropriate expression similar to Eq. (20).

For capacitors C4 to Cg, the standard uncertainties for the dial corrections can be expressed,

according to Eq. (17), as:

^/ = [(0-l)'^Ra' + (0-l)'^(/-i)' + E('-/^^') + ^lV'^^\ (^>/ + land 4</<8). (21)

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

= [(10)' ^Ra' (10)' + I) +(10)'L(r/ sj^)+s:^] ^^^'\ {k>i+\ and ^ = 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 intemal 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, 53, for the dial correction d^^ of C3 is determined first, and the standard uncertainties of

other dials, , are obtained from 53. Estimated values of each term of Eq. (20) are given as follows:

1) According to Eq. (1 1) in section 3.1.2, the in-phase standard uncertainty for the 10:1 (and

1:10) ratio corrections, s^^, is estimated to be less than 0.043 ppm. For Cj^^ \ pF, this

becomes:

< 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.e., s^^ < 0.72 ppm). For = 10 pF, then we have:

9
5„„ < 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 rj^ = 1.0 (which is a maximum) in Table 1, the standard

deviation of the dial corrections used during the calibration of C3 is estimated from the

standard deviations of the last three dial corrections, (dials number 6, 7, and 8), also given

in Table 1 . Thus,

( ILrj^ 5/ f'^'^ < 0.08 X 10"^ pF.

4) The maximum bounds of error due to the interpolation of the last capacitance dial is equal

to ± (3 X 10"^

) pF. Therefore, the standard uncertainty of the interpolation of the last dial

becomes:

5i < 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 53 is determined, other 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^^, 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 1 0 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 nifrogen 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
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:

s^^ = 3.54 X 10"^ < 0.004 ppm

and

s^^ = 2.73 X 10-^ < 0.003 ppm,

where s^^ and s^^ are the in-phase and quadrature standard uncertainties, respectively, of the fusedsiHca capacitance standards due to voltage dependence. This dependency is believed not to change

with time.

The uncertainty of measuring the vohage 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 1 0 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 ftised-silica capacitors (#113,

#1 12, 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),

< 0.68 ppm < 0.7 ppm (between frequencies of 1592 Hz and 1 kHz)

Sf- < 0.87 ppm < 0.9 ppm (between frequencies of 1 kHz and 400 Hz)

5f < 1.73 ppm < 1.8 ppm (between frequencies of 1 kHz and 100 Hz)

where s^- is the standard uncertainty of the 10 pF fiised-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
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 NlST-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:

s^^ < 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,

^tci - 1 PP"^'

where s^^^ is the standard uncertainty fi"om 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 = { i?i + [ G, / (G/ + o) V)] } + j 0) { II

- [ C / (G/ + w'c" )] }, (23)

where Z is the effective impedance of the capacitor C,

i?l is the series resistance of leads,

Lj is the series inductance of leads, and

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^^^WC^) is approximately equal to w^C^ since G^^ < (10''°)o)^Cl

12
Therefore, Eq. (23) can be written as: Z= + ( G^. / (oV )] + j 0) [ Ij - ( C/ w^C^ )], (24)

Thus, the equivalent equation of admittance, 7= (1/Z) is expressed as:

Y = ( (o^C^ )/[(Ri o)^C^ + ) + j G)C ( G)^ Z-i C -

1 )], (25)

where Y is the effective admittance of the capacitor.

The value ofZj is less than 10'^ H and is less than 10''° S. By multiplying the numerator and

the denominator of Eq. (25) by [( i?j a)^C^ + G^)-} wC ( o)^ Lj C - 1 )] and eliminating all second

order terms of Zj and G^ , Eq. (25) becomes:

7=[(Gc + i?ia)V)+j a)C(l -a)^ZiC)]/[l+i?iO)^d + 2i?iG(,-2o)^ZlC], (26)

For a one meter long coaxial cable, and Zj were measured to be approximately 0. 1 Q and 0.5 |iH,

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^ + R^(ji)^d ) + i oiC {I - (J- L^C)] / [\ -2oy^L^C]. (27)

By multiplying the numerator and the denominator of Eq. (27) by ( 1 + 2 O) Z-j C ), and eliminating

the second order terms of Zj and G^, Eq. (27) becomes:

7= ( G^ + i?l (0^ + 2 i?i oj^ ) + j ooC (1 + G)^ C ). (28)

Furthermore, the third term in Eq. (28), (2 i?j o)"* Zj C^) is less than 10'^^ S, which is negligible as compared with G^ and (R^ o)^ ). Therefore, Eq. (28) becomes:

F=(Gj, + i^ia)^d)+ja)C(l +co^ZiC). (29)

The effective values of capacitance and conductance of the standard with lead impedance are:

Qff= C( 1 +a)'i:iC) = C[ 1 + (^C/C)],and (30)

Geff= (G, + i?iO)2c2) =(G,+ zlG), (31)

where (AC I 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 = 1 0 pF, estimated

uncertainties in the lead impedance are calculated to be:

{ACIC) = 1.97 x

10"'° < 0.0002 ppm, and

13
AG = 3.94 X lO-'^ S < 4 X 10"' nS

Table 4 gives a summary of the uncertainties due to lead impedance for measurements of various

nominal capacitance values at frequencies of 1 00 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 ± 0.076 ppm. Therefore, the uncertainty of these

standards is taken to be:

5g = 0.044 ppm < 0.05 ppm,

where 5g 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:

^cb = ( -^va' + ^f ' + ^tc

' + / C + s^^ ppm, (32)

where s^^ is the combined standard uncertainty in the external bridge components using the Type-2

bridge to measure frised-silica and nitrogen dielectric capacitors. Table 5 contains a summary ofs^^

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 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 fiised-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 fased14
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 fiised-silica dielectric capacitor standard consists of at least five sets ofmeasurements 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 ofA: = 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 fiised-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 fiised-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 fiised-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 ofthe expanded and assigned total uncertainties for calibrations of fiised-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 fiised-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 xlO"^) pF. According to the nominal values of the

unknown standards, the standard uncertainty to be included in the combined uncertainties

at measurement, ^^^i^ is :

< 2 ppm (for 1 pF capacitors),

< 0.2 ppm (for 10 pF capacitors),

< 0.02 ppm (for 100 pF capacitors), and

< 0.002 ppm (for 1000 pF capacitors),

where is the standard uncertainty in the interpolation of the last dial.

15
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 total 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 ofthe 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 A: = 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 threeterminal 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

ofHF coaxial capacitance standards available, as characterized by their nominal values and dielectric

materials.

16
Table 1 1 contains the calibration uncertainties for two types ofHF coaxial air capacitors. The lowervalued 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 iiF to 0.1 \xF. 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 1.1 1 |iF and conductance

up to 1000 |xS. 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 intemal 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 jiF to 1 ^F

and from 0.01 nF to 0.1 |iF, and the air dials range from 0.001 |iF to 0.01 ^F, 0.0001 ^F to 0.001 )xF, and 0.00001 )iF to 0.0001 |liF. A 0.005 |iF mica capacitor, measured using the Type-2

bridge, is used as a fransfer standard to calibrate the 0.01 )iF 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 \iF step, used as a reference. The conductance dials range from 10 jiS to 100 )iS and from 100 |liS to 1000 )iS, and are calibrated against resistance standards. Both

capacitance and conductance dials are adjusted to obtain a final balance at the time of measurement

1
to provide capacitance and conductance values of the mica capacitors being calibrated by the Type12 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 Rj and Rj are resistors with a nominal ratio of 1 : 1 . Using Cj and C2 as the

total capacitance of each of the other two arms, the basic equation ofbalance for making capacitance

measurements (when the voltage across detector D is zero) is:

where Cj and C2 are the total capacitances of arm AD and arm CD, respectively.

As shown in Fig. 5b, Cj and C2 can be expressed as:

Ci = C^ + C^+ Q (when is connected to arm AD) (34)

and

€2 = + Q (when is connected to arm CD) , (35)

where is the unknown capacitor to be measured, is the reading of internal mica dial

capacitors, and and Q are effective capacitances of arms AD and CD, respectively, with internal

air capacitors coupled so that:

C^ + Q = C (a constant). (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^-C^) + C, (37)

where is the capacitance value corresponding to the dial reading, r.

Substituting Eq. (36) into Eq. (37) gives:

C, = 2Q. (38)

The internal air capacitors of the Type-12 bridge simulate three decade capacitors consisting of ten

0.001 ^iF (1000 pF) , ten 0.0001 |^F (100 pF), and ten 0.00001 ^F (10 pF) capacitors, and a continuously variable capacitor providing a capacitance range of (1 1 ± 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,

1 5 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, , and 9 to 10. The capacitance values corresponding

18
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

Q = 50 pF and = 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

Q = 45 pF and Q = 5 pF; therefore, = 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

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

At dial reading r = 10, all four capacitors are connected to arm AD, and then

Q = 0 and = 50 pF; therefore, = 100 pF.

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

The Type-12 bridge also has a set of internal decade mica capacitors of ten 0.01 [iF and ten 0.1 \iF

which are employed in the arm AD, together with the air capacitors, to measure mica capacitors with

nominal values larger than 0.005 |iF. As mentioned previously, a 0.005 ^iF mica capacitor is used

as a transfer standard to obtain the dial corrections for the 0.01 )liF 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 \i¥ step.

Again, the total uncertainty 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 ofthe components ofthe Type B standard uncertainty for the Type-12 bridge is based on uncertainties in the ratio of resistors, Rj and Rj, 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
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:

= 17.3 ppm < 18 ppm, (39)

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 ofthe 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 threeterminal 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, s^^, 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, s , are included in Tables 14a and 14b, for the two-terminal and three-terminal

^ 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
measurements, respectively.

2) 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),

= 5.77 ppm < 6 ppm, and = 25.98 ppm < 26 ppm, (40)

where s^^^ and s refer to the standard uncertainties in the air and mica dial capacitors,

respectively, due fo temperature variations, which are included in Tables 13a to 14b.

3) Dial switching

The maximum bounds of error due to the switching of dials, ± e^^ are estimated to be

± 0.01 pF, which will affect only the measurements of low capacitance. Accordingly,

5jjg = 5.77 ppm < 6 ppm (for C = 0.001 |nF), = 0.58 ppm < 0.6 ppm (for C= 0.01 |iF), = 0.06 ppm < 0.1 ppm (for C= 0.1 |xF), and = 0.006 ppm < 0.01 ppm (forC=l |iF) . (41)

where s^^ is the standard uncertainty due to the switching of dials, also included in

Tables 13a to 14b.

4) Uncertainty of transfer standard

From numerous measurements, the calibrated value ofthe transfer standard, a 0.005 |iF 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,

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

where and 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 |xF, (only air-dial capacitors

are needed), s^^ used as the standard uncertainty. However, for capacitors of nominal

values larger than 0.005 i^F, the RSS of s^^ and s^f^ 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
^rc^C-^dr +^tp ^-^ds +-^tf ) '^nd (43)

= ( ^mr ^ + ^mp ^ + ^ds

^ + '^mf^

) ^ (44)

where s^^ and are the combined uncertainties in the air and mica capacitor dials, respectively,

which are also included in Tables 13a to 14b.

4.2.3 External Bridge Components

1) 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 Eq. (4),

the uncertainty when using plate P2 is in the range fi*om 0.012 ppm (in measuring 1 |xF

capacitors) to 12 ppm (in measuring 0.001 \xF capacitors), and the uncertainty when using

plate P3 is from 0.006 ppm to 6 ppm. Hence,

0.012 <

5|jp2 < 12 ppm and 0.006 < s^^^ - ^ ppm, (45)

where s^^j ^bps standard uncertainties due to the applied corrections of plates P2

and P3, respectively.

2) 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):

<^eff =Q(l + ^'^iQ)' (46)

where C^ff is the effective capacitance, is the series lead inductance, and co = lizf in

which / is the applied frequency. The lead inductance of the Type- 12 bridge is estimated

to be less than 0.2 |iH, and the maximum error caused by lead impedance, e^^j^ is less than

( iTzf) ^

( 0.2 )10'^ (C^ ), which is 8/^ (Q) ppm. Accordingly,

^bz = 4.6/'(q) ppm, (47)

where s^.^ is the standard uncertainty due to lead impedance, / is the applied frequency in

kilohertz (kHz), and is nominal value of the unknown capacitor in microfarad (|xF).

The combined uncertainty of the extemal bridge components,

^i^^., is defined as the RSS of the above

22
two components, which are estimated from each type of measurement, as:

, 2 , 2 X ('/2) "^br^^-^bp +'^bz ) ' (48)

where s^^^ is equal to either S)^^^ -^bpS' depending on which plate (either P2 or P3) is being

These are given in Tables 15a and 15d.

used.

The Type B component of standard uncertainty for capacitance measurements made with the

Type- 12 bridge, U|j, is defined as the RSS of the four components, s^^, s^^, s^, and St^^ 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 ^iF < C^^ < 0.005 i^F)

4.3.1.1 Basic Equations of Balance

This type of measurement covers the range of capacitors with nominal values from 0.001 |iF to 0.005 |xF, 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:

^iCao = ^2CcO (49)

and

^iCai=/?2(Qi + CJ, (50)

where C^q Qo ^'"^ capacitance values of internal air capacitors, and C^., at balance without

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

From Eqs. (49) and (50), the value of the unknown capacitor can be written as: q = [i + (/?i/i?2)](qi-qo), (51)

By substituting Eq. (38) into Eq. (51), can be expressed in terms of dial readings as: q = C/2)[l+(/?,//?2)](C„-C,o), (52)

where Cj.j = 2 and = 2 are dial readings of the internal air capacitors at balance, with and

without connecting to arm CD, respectively.

23
The ratio resistors are always set to be equal, {R^ = Rj) so that Eq. (52) becomes:

q = ( - C,o ), (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:

Var(q) = C/2)' ( - Qo)' ^ra' + ('/^)' ( [1 + ( / ^2 ) ]' ( ^rci' + ^rco' )' (54)

where s^^, s^^^, and s^^q 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^ = Rj) into Eq. (54), it becomes:

Var(Q = [C/2)q5,j' + 5rc'' (55)

where sj= s^^^^ + s^J .

From Eq. (39), the uncertainty contributed by the ratio resistors for two-terminal low-capacitance

measurements, [(V2) s^^], is 9 ppm (see the first three columns of Table 15a). The capacitance

uncertainties associated with each dial reading, s^^q and s^^^, can be estimated from the

corresponding terms of Eq. (43), as: - / 2 , 2 , 2 , 2 . ('/2) J •^rcO ~ y ^dro + -^tpO ^ '^dsO ^ -^tfo ) '

^rci=('^dri ^-^tpi ^-^dsi +^tfi ) • (56)

The combined uncertainty of the intemal air dial capacitors, s^^, is estimated to be between 1 8 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 |iF. 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
4.3.2 Two-Terminal Measurements of High-Capacitance Values

(0.005 ^iF < C^^ < 1 ^iF)

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:

and

^i(Qi + C^) = ^2(Qi + qX (58)

where C^q and are the values of the air capacitors at balance without cormecting the unknown

capacitor C^; and Q, are the values of the air capacitors, and is the value of the mica

capacitors at balance with connected to arm CD.

The value of the unknown capacitor can be expressed as:

(i?i/i?2)C^+ C/2) [l+(i?i/i?2)](CrrQoX (59)

where Cj^ is the mica dial reading, and = 2 and C^q = 2C^q are air-dial readings of internal

capacitors at balance, with and without connecting to arm CD, respectively.

The ratio resistors are always set to be equal, ( i?j = i?2 ) that Eq. (59) becomes:

Q= C^ + (C,,-qo), (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(q) = [C^ + C/2)( - )]' ^ra' + ( / ^2 )' ^m'

+ {y2f [\+{R,/R2)]\ 5,,,' + s^J ), (61)

where s^^, s^, s^.^^, and ^^,q 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:

[ C^ + {V2)i - Qo)]' ^ra' ^ (Q )' ^ra'' (62)

25
By substituting Eq. (62) and ( = i?2 ) into Eq. (61), it becomes:

Var(q) <(C^s^^f + sJ + , (63)

2 2 where s^^ = +^^^0 •

From Eq. (39), the uncertainty contributed by the ratio resistors, s^^, for the two-terminal highcapacitance measurements is 18 ppm. (See Table 15a).

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, s^^, 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

1 00 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.

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:

4.3.3 Three-Terminal Measurements of Low-Capacitance Values

( 0.001 \iF < < 0.005 ^iF)

4.3.3.1 Basic Equations of Balance

^1 ( + <^x) ~^2 Qi (64)

and

^1 ^a2 ~ -^^2 (Q2 + ^x)' (65)

where C^j and C^^ are the values of the selected internal capacitors at balance with connected

to arm AD, and and Q2 are the values at balance with connected to arm CD.

From Eqs. (64) and (65), the value of the unknown capacitor can be expressed as: q =C/2)(q2-C,i), (66)

26
where Cj.j = 2 C^, and = 2 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(q) = C/2)^,,i' + C/2)^,

2

'rcO ' (67)

where s^.^^ and s^^q are the standard uncertainties in the air capacitor dials of both measurements,

respectively.

Eq. (67) can also be expressed as:

where s^^ = s^^^ + s^^q .

According to Eq. (66), C^. is independent of and R2, 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 1 5b, as well as their individual uncertainty components for each capacitance value

at 100 Hz, 1 kHz, and 10 kHz .

Measurement procedures for capacitors of this type are similar to those of three-terminal lowcapacitance 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:

Var(q) = [(y2)^rc]'' (68)

4.3.4 Three-Terminal Measurements of High-Capacitance Values up to 0.5 |iF

(0.005 ^F < < 0.5 |iF)

4.3.4.1 Basic Equations of Balance

(69)

and

^i(Ca2 +C^)= ^2(Q2 + q I (70)

27
where C^j and are values of the internal air capacitors at balance with the unknown capacitor,

connected to arm AD, and C^2 values of the internal air capacitors at balance with C

connected to arm CD, are values of the internal mica capacitors at the second balance, and Cj,

is the value of the dummy capacitance box.

From Eqs. (69) and (70), the value of the unknown capacitor can be expressed as: Q = [i?i/(i?i + i?2)]Cm + (V^2)[i?i/(i?, + /?2)](Cr2-C,i ). (71)

The ratio resistors are always set to be equal {R^ = R2), so Eq. (71) becomes:

q= {V2)C^ + (%) (C,2-Cr, ), (72)

which is the equation used to calculate the unknown capacitor.

4.3.4.2 Analysis

Rewrite Eq. (71) as:

C^={(R,/R2)/[l+(R,/R2)]}C^ + {y2){{R,/R2)/[l+{R^/R2mC,2-C,,). (73)

The relationship among the variances of the terms of Eq. (71) can be estimated by:

Var(Q) = [1+ (i?i/i?2)]"' [C^ + C/2) (Cri-C,o)]' ^ra' + ((^1 ^^i) / [1 + (i^i / )]}' ^m'

+ {V2f {{R, IR2) / [1 + (i?i / i?2 )]} ' (^rci'+ ^rco' I (74)

where s^^, s^, s^^^, and 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 ( i?j = i?2 ) i^^o Eq. (74), it becomes:

Var(q) = {{V2) q s^^ f + ms^] ' +

[(^4) J ^ (75)

where 5^./ = s^J + s^J .

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

ratio resistors in the three-terminal case, [ (Vz) s^^ ], 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,

[(y2)5j^], are estimated to be less than 23 ppm. The uncertainties in the air-dial capacitors, [('74) s^^], 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
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 1 5b, 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

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 |liF, and measurements are performed at frequencies only up to 1 kHz. The dummy capacitor box, Cjj, 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^ = I jiF as an example, the measurement sequence and equations of balance in accordance

to Fig. 10 are described as follows:

1) Connect across arm AD, set the mica dials to zero, and the value of to 1 i^F. Use the

air dials to obtain a balance, and record the reading as Cj.]. The equation of balance is:

^i(Cai+ = ^2(Cci + Cb). (76)

2) Discormect C^, and leave Cy^ at 1 \iF. Use both the air and mica dials to obtain a balance,

and record the readings as Cj.2 and C^2-> respectively. The equation of balance is: MC^i + C^i) =^2(Cc2 + Cb). (77)

3) Connect across arm CD, and set Cj, to zero. Use both the air and mica dials to obtain a

balance, and record the readings as Cj.3 and C^2-> respectively. The equation of balance is:

^i(Q3 + C^3) = ^2(Cc3 + CJ. (78)

4) Disconnect C^, set both mica dials and C^j to zero. Use the air dials to obtain a balance, and

record the reading as Cj.4. The equation of balance is:

^1 Q4 = Ri Cc4- (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, can be expressed as:

29
q = [i?i/(i?,+/?2)](C^2+C^3) + C/^)[(Cr2-Cri) + (C,3-C,4)L (80)

where Cj.j to Cj.4 and to Qn3 mica dial readings, respectively, corresponding to each

measurement.

The ratio resistors are always set to be equal = R2), so Eq. (80) becomes:

Q = (V2) ( + ) + C/^) [( - ) + ( Q3 - ^4 )l (81)

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

4.3.5.2 Analysis

Rewrite Eq. (80) as: q = {(R./R^) I [1+ {R,IR2 )]}( C^2 + C^3 ) + ('/^ ) [( - q, ) + ( C,3 - C,4 )]. (82)

The relationship among the variances of the terms of Eq. (82) can be estimated by:

Var(q) = [1 + ( i?, / )]"' (C^2 + C^z f

^ +{(i?,/i?2)/[l+(^,/i?2)]}'(^m2' + ^m3')

+ iV2)\s^^,' + s^J + s^J + s^J), (83)

where s^^, s^2^ s^^^ s^^^, s^^2^ s^^^ ^rc4 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:

(Cm2 + C^3)'^ra' M2CJ^,,^ (84)

By substituting Eq. (84) and ( i?j = i?2 ) i^^^o Eq. (83), it becomes:

Var(q) < [(^/z) q f + m5J

' + m5,J ^ (85)

u 2 2, 2,2 2, 2, 2, 2 where = + s^^ ands^^ = s^^^ + s^^2 +

'rc3 +^rc4 •

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, [(Vi) s^J, 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
measure the unknown capacitor of value equal to 1 |iiF. The combined uncertainty in the mica

capacitor dial readings, [(Vi) s^], is estimated to be less than 16 ppm. The uncertainty in the air

capacitance dial readings, [(V2) s^.^], is estimated to be 5 ppm. (See the last column of Table 15b).

For = 1 |iF, 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 |liF, 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 |iF, 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 \iF capacitors where the limits are ±50 ppm. According to Eq. (4),

= 17.3 ppm < 20 ppm, (for > 0.001 ^F),

= 28.9 ppm < 30 ppm, (for Q = 0.001 ^iF), (86)

where is the standard uncertainty due to repeatability of results of measurements of capacitance.

The values of 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
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 \iF 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),

s^^ = TC*(AT)//3 ppm, (87)

where 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 1 6a.

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 |iF). For

0.005 [iF <C^< 0.5 )iF, the values of a 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 > 0.5 p.F, 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, s^^, is estimated to be 20 ppm, which is the RSS of uncertainties for the unknown

capacitor and for the capacitance box. The values of 5^^^ 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

fi"om 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 ofNIST standards are derived), and to calibrate customers' standards at fi-equencies

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, s^^ , are estimated to be between 10 ppm and 25 ppm, which are included in the Tables

16a and i6b.

32
Similar measurements have also been performed on four NIST standards at frequencies of 1 00 Hz

and 1 0 kHz, and standard uncertainties of their predicted values are also given in the Tables 1 6a 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^^, s^^, and^p^j), 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 ^ = 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 \iS steps, and ten 10 |iS steps, plus a continuously variable standard with a range of -1 |nS to 11 plS. Similar to the internal air capacitors, 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. Kg, 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

1 1 is a simpHfied diagram of the Type- 12 bridge including the conductance components in each arm,

where (GJK^ and (GJK^) are effective conductance adjustments to arm AD and arm CD,

respectively, and the quantity of (G^ + G^, ) is equal to a constant. Gg 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^, Cj.j, and Cj.2 being replaced by G^, {G^^IK^, and {G^2^K^,

respectively. For example, in the two- terminal high-capacitance measurements, from Eq. (60), the

conductance value, G^, of the unknown capacitor C^, is calculated from the two balances as:

G^ = G^+ {G^,-G^2)IK (86)

33
where is the conductance of the internal mica capacitors with C^ connected to arm CD,

Gj-i is the conductance dial readings without connecting C^,

Gj.2 is the conductance dial readings with C^^ connected to arm CD, and

Kg is the divider setting.

For the cases where mica capacitance dials are not used to perform conductance measurements, the

corresponding term of 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 = 1 I^F, which required four balances, the maximum bounds of error due to the

limit of accuracy of the conductance standard are estimated to be:

i^gj = ±0.002 (2/'^^^ Gg ^S (for < 1 ), and

±gg5 = ±0.002 (4/^^^^ Gg |iS (for q = 1 |iF ), (87)

where ±e^^ are the maximum bounds of error in conductance measurements due to the limit of

accuracy m the conductance standard, Gg ( in |iS). Accordingly, the uncertainties in conductance

standards are expressed as:

5gg = 0.0016 Gg )iS (for < 1 i^F ), and

5gg = 0.0023 Gg ^iS (forCj^l^iF), (88)

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

Gg ( in ^iS).

According to the conductance values of the check standards, the values of^gg for each nominal value

of capacitor at fi-equencies 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
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 are estimated from the difference of the contact resistance of each balance as:

±^gr - ±0/ ( - i?r2 ) » (89)

where ±^g^ are the maximum bounds of error due to contact resistance, and R^^ and R^j ^^e contact

resistance in the capacitance decade switches at the first and second balances, respectively.

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

i^gr = ±OMfC^\ (90)

Accordingly, the standard uncertainty in contact resistance, s^, is estimated as:

= 0.046fC^\ (91)

The values of^g^. 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 \iS 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 \xS. 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 ^S), is expressed as:

= 0.10 / |nS , (for < 1 ^iF ), and

5gp = 0.13 / ^iS , (for Q = 1 ^iF ), (92)

where is the standard uncertainty in conductance potentiometer, due to its limited resolution, and

ATg is the setting of the conductance divider.

According to the conductance divider, K^, being used in measurements, the values of 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
4.6.4 Residual Conductance

According to [14], the maximum bounds of error due to the limit of resolution in residual

conductance in arms AD and CD are functions of frequency and given as:

±^g,= ±(0.02)// ^S, (93)

where ±^__ are the maximum bounds of error due to the residual conductance. Accordingly,

= (0.01 15)// ^iS, (94)

where s is the standard uncertainty due to the residual conductance,

/is the applied frequency in kHz, and

K is the setting of the conductance divider.

The values of 5„„ for each nominal value of capacitor at frequencies of 100 Hz, 1 kHz, and 10 kHz

in the two-tenninal 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 ofthe internal mica capacitors, s^, for the twoterminal and three-terminal measurements of capacitors > 0.01 |nF 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
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:

^grm (maximum bounds of error) / /3 |iS, (95)

where 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 utiHzing 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, 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

Sgpjj, 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 ofconductance measurements (for each nominal value ofmica capacitor)

are calculated from both Type A and Type B standard uncertainties with a coverage factor ofk = 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
6. ACKNOWLEDGMENTS

The author would Hke 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.

7. REFERENCES

[1] R. D. Cutkosky, "Capacitance Bridge ~ NBS Type 2," National Bureau of Standards Report

7103, March, 1963.

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

ofNIST Measurement Results," Natl. Inst. Stand. Technol. Technical Note 1297, September,

1994.

[3] International Bureau of Weights and Measures (BIPM) Working Group on the Statement of

Uncertainties, Metrologia, vol. 17, pp. 73-74, 1981.

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

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

April, 1998.

[5] R. D. Cutkosky and J. Q. Shields, "The Precision Measurement of Transformer Ratios," IRE

Trans. Instrum., vol. 1-9, pp. 243-250, September, 1960.

[6] 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.

[7] "A Highly Stable Reference Standard Capacitor, "The General Radio Experimenter, vol. 37,

no. 8, August, 1963.

[8] 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.

[9] 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.

[10] "General Radio Catalog 73" General Radio Company, 1973.

38
"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. Wilhebn, "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 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) 1 AA 100 1 A

10 1 0.1 0.01 A AA 1 0.001 A AAA 1 0.0001 A AAAA

1

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 7.149 -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
Table 2. Standard Uncertainties in Dial Corrections of the Type-2 Bridge

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

Dial No. 3 4 5 6 7 8 2 1

Nominal

Value Ci (pF) 1 0.1 0.01 0.001 lE-04 lE-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

-3J J

-J 1111

1 A

Components of

Standard

Uncertainty

(0.1)5 Ra

2 2.-,(i/2)

0.0043

0.08

(0.1)5Ra

2 2., (1/2) 4E-04

0.08

4E-05

0.08

4E-06

0.05

4E-07

0.05

4E-08

0.05

(10)5Ra 4.3

1.48

43

13.1

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

(0.1)53 0.75 (0.1)5(,.l) 0.100 0.025 0.023 0.022 0.022 (10)5(,+,) 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

Standard uncertainty of the 1:10 ratio

th Standard Deviation of the dial corrections for the k dial of reading

Standard uncertainty due to the interpolation in last dial

Uncertainty of reference standard

Remarks

5,

41
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

5ed (lO-'pF) * 0.45 0.51 0.51 1.12

(ppm) 0.45000 0.05100 0.00510 A AA 1 1 O 0.001 12

Remark

* From number of dials used to perform measurements, as given in the followin i, laUie.

No.of dials used 1 2 3 4 5 6 7 8

Dial No. 8 7&8 6 to 8 5 to 8 4 to 8 3 to 8 2 to 8 1 to 8

Assigned Combined

Uncertainty

Sod (lO-'pF) 0.22 0.31 0.38 0.45 0.51 1.12 13.05 150.57

42
Table 4. Uncertainties in Lead Impedance Using the Type-2 Bridge to Measure

Capacitors at Frequencies of 1 00 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

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45
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OS cO

O 3 HU

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46
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47
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 lO'^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

210 40 32 30 30

Type A Standard Uncertainty (wj 950 90 30 30 60

Expanded Uncertainty (U) 1945.87 196.98 87.73 84.85 134.16

Assigned

Total Uncertainty 2000 200 100 100 150

48
Table 10. Expanded and Assigned Total Uncertainties Using the Type-2 Bridge to Measure

Three-Terminal Air Capacitors at 1 00 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"%F) 200 20 2 0.2 to 0.002

Type B Standard Uncertainty (Wb) 202.69 36.15 28.04 27.97

Assigned 220 40 30 30

Type A Standard Uncertainty (mJ 600 100 30 30

Expanded Uncertainty {U) 1278.12 215.41 84.85 84.85

Assigned

Total Uncertainty (f/,) 1300 230 100 100

49
Table 1 1 . 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 A AA 400 O AA 200 80 A

0

A0 16

Hysterises 20 20 20 20 20 20

Voltage 5 5 5 5 5 5

Type B Standard Uncertainty (Wb) 412.83 205 85 45 28.72 27.95

Assigned Wb 415 205 85 45 30 28

Type A Standard Uncertainty (wj 35 35 35 20 20 10

Expanded Uncertainty (U) 832.95 415.93 183.85 98.49 72.11 59.46

Assigned

Total Uncertainty (f/,) 840 420 200 100 75 60

50
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 z L

cJ

cJ

Type B Standard Uncertainty (wt,) 26.17 29.15 427.70 181.45

Assigned 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 (Ui) 60 65 1000 450

51
Table 13a. Summary of Components of Standard Uncertainties in Air Capacitor Dial Corrections

for Two-Terminal Measurements

Unknown Standard ()iF) 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0

OlU L/CV.OI rrcU.Vd.lUC3 Ol Dial Correction (pF)

StdDev.ofPred.Values of

Dial Correction (ppm)

0.0601

60.136

0.0672

33.584

0.0759

15.184

0.0208

2.085

0.0208

1.042

0.0208

0.417

0.0208

0.208

0.0208

0.104

0.0276

0.055

0.0213

0.021

Components of

Standard Uncertainties (ppm)

Pred.Values of Dial Corr't (s^^)

Temperature Variation (s ,p) Dial Switching (s^^) Transfer Standard (s tf)

60.136

6

6

6

33.584

6

36

15.184

6

1.5

6

2.085

6

0.6

6

1.042

6

0.36

0.417

6

0.15

6

0.208

6

0.1

6

0.104

6

0.05

6

0.055

6

0.02

6

0.021

6

0.01

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 ^c) 62 35 18 9 9 9 9 9 9 9

52
Table 13b. Summary of Components of Standard Uncertainties in Air Capacitor Dial Corrections

for Three-Terminal Measurements

Unknown Standard ()iF) 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0

Std Dev.ot rred. Values ot Dial Correction (pF)

Std Dev.ofPred.Values of

Dial Correction (ppm)

0.0607

60.744

0.0749

37.448

0.0636

12.723

0.0224

2.238

0.0241

1.205

0.0183

0.365

0.0266

0.266

0.0191

0.095

0.0264

0.053

0.0324

0.032

Components of

Standard Uncertainties (ppm)

Pred.Values of Dial Corr't {s^,)

Temperature Variation {s ,p) Dial Switching (s^^

Transfer Standard {s tf)

60.744

6

6

6

37.448

6

3

6

12.723

6

1.56

2.238

6

0.6

6

1.205

6

0.36

0.365

6

0.15

6

0.266

6

0.16

0.095

6

0.05

6

0.053

6

0.02

6

0.032

6

0.01

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 (5 ^c) 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

Unknown Standard (fiF) 0.01 0.02 0.05 0.1 0.2 0.5 1.0

Components of

Standard Uncertainties (ppm)

(At all Frequencies)

Temperature Variation {s ^p)

Dial Switching (s ^s) Transfer Standard (s ^f)

26

0.6

6

26

0.3

6

26

0.15

6

26

0.1

6

26

0.05

6

26

0.02

6

26

0.01

6

(At Frequency - 100 Hz)

Std Dev.ofPred.Values of

Dial Correction (pF)

Std Dev.of Pred.Values of

Dial Correction (ppm) (s ^r)

.1585

15.850

.2331

11.655

.4412

8.824

1.0096

10.096

2.5727

12.864

7.7378

15.476

11.8816

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 (5^) 32 30 29 29 30 31 30

(At Frequency = 1 kHz)

Std Dev.of Pred.Values of

Dial Correction (pF)

Std Dev.of Pred.Values of

Dial Correction (ppm) (s^^)

0.0702

7.020

0.1208

6.040

0.3019

6.038

27.358

0.6075

6.075

1.413

7.065

4.4166

8.833

6.5888

6.589

RSS 27.598 27.360 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)

Std Dev.of Pred.Values of

Dial Correction (ppm) (5^^)

0.0627

6.270

0.1087

5.435

0.275

5.500

0.4965

4.965

1.288

6.440

3.991

7.982

21.8635

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

for Three-Terminal Measurements

Unknown Standard (jiF) 0.01 0.02 0.05 0.1 0.2 0.5 1.0

Components of

Standard Uncertainties (ppm)

(At all Frequencies)

Temperature Variation (^^p)

Dial Switching (s^^)

Transfer Standard (s ^f)

26

0.6

6

26

0.36

26

0.15

6

26

0.16

26

0.05

6

26

0.02

6

26

0.01

6

(At Frequency = 1 00 Hz)

Std Dev.ofPred.Values of

Dial Correction (pF)

Std Dev.ofPred.Values of

Dial Correction (ppm) (s^^)

0.2331

23.310

0.3617

18.085

1.0096

20.192

2.5121

25.727

7.1997

35.999

11.8816

23.763

16.8030

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.ofPred.Values of

Dial Correction (pF)

Std Dev.ofPred.Values of

Dial Correction (ppm) (^^r)

0.1208

12.080

0.2414

12.070

0.6075

12.150

1.4130

14.130

4.1541

20.771

6.5888

13.178

9.3179

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 (s^) 30 30 30 31 34 30 29

(At Frequency = 1 0 kHz)

Std Dev.ofPred.Values of

Dial Correction (pF)

Std Dev.ofPred.Values of

Dial Correction (ppm) (5n,r)

0.1087

10.870

0.2229

11.145

0.4965

9.930

1.2880

12.880

3.2637

16.319

21.8635

43.727

NA

NA

RSS 28.819 28.919 28.472 29.629 31.278 51.225 NA

Assigned Standard Uncertainties

in Mica Cap. Dial Corrections (5^) 29 29 29 30 32 52 NA

55
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 (|iF) 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0

Components of Standard Uncertainties

(At all Frequencies)

Ratio Resistor Arms (s^^) 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 (5^) - - - 32 30 29 29 30 31 30

External Bridge Component (5br) 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 Ub 65 40 25 40 40 40 40 40 40 40

( Frequency = 1 kHz )

Mica Capacitor Dials (s^) - - - 28 28 28 28 28 29 28

External Bridge Component (s ^r) 12 6 3 2 1 1 1 1 3 5

Type-B Standard Uncertainty (Ub) 63.79 36.63 20.35 34.54 34.50 34.50 34.50 34.50 35.43 34.84

Assigned Ub 65 40 25 40 40 40 40 40 40 40

( Frequency = 1 0 kHz )

Mica Capacitor Dials (s 28 28 28 28 28 28 35

External Bridge Component (^br) 12 7 4 5 10 25 48 95 235 470

Type-B Standard Uncertainty (Ub) 63.79 36.81 20.52 34.84 35.90 42.59 59.10 101.06 237.52 471.73

Assigned Ub 65 40 25 40 40 45 65 105 240 475

56
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)

Unknown Standard (ju,F) 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0

Components of Standard Uncertainties

(At all Frequencies)

Ratio Resistor Arms {s^^ - - - 9 9 9 9 9 9 9

Air Capacitor Dials {s rc) 31 20 8 3 3 3 3 3 3 5

( Frequency = 1 00 Hz )

Mica Capacitor Dials {s ^ - - - 18 17 17 19 23 18 16

External Bridge Component {s ^r) 6 3 2 1 1 1 1 1 3 5

Type-B Standard Uncertainty (Ub) 31.58 20.22 8.25 20.37 19.49 19.49 21.26 24.90 20.57 19.67

Assigned Uj, 35 25 15 25 25 25 25 25 25 25

( Frequency = 1 kHz )

Mica Capacitor Dials {s „) - - - 15 15 15 16 17 15 15

External Bridge Component {s ^r) 6 3 2 1 1 1 1 1 3 5

Type-B Standard Uncertainty (Ub) 31.58 20.22 8.25 17.78 17.78 17.78 18.63 19.49 18.00 18.87

Assigned Ub 35 25 15 20 20 20 20 20 20 20

( Frequency = 1 0 kHz )

Mica Capacitor Dials (5^) 15 15 15 15 16 26 NA

External Bridge Component {s br) 6 4 3 5 10 25 48 95 235 NA

Type-B Standard Uncertainty (Ub) 31.58 20.40 8.54 18.44 20.37 30.66 51.18 96.80 236.62 NA

Assigned Ub 35 25 15 20 25 35 55 100 240 NA

57
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)

Unknown Standard ()iF) 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 = 1 00 Hz )

Repeatability (^m,) 30 20 20 20 20 20 20 20 20 20

Temperature Variation 4 4 4 4 4 4 4 4 4 4

32 10 10

Type-A Standard Uncertainty (w^) ZU.4U on /in ZU.4U 5 1 .yj on /in on /in ZU.4U OO TO zz. /z on /in ZU.4U on /in ZU.4U 00 "70 ZZ. /z

Assigned 46 25 25 38 25 25 25 25 25 25

( Frequency = 1 kHz )

Repeatability (s^) 30 20 20 20 20 20 20 20 20 20

Temperature Variation (s tm) 4 4 4 4 4 4 4 4 4 4

Predicted Values ^

v

25 25 20 15 20 15 10 10 10 10

Type-A Standard Uncertainty (u^) JZ.ZO 00zo.D / o*; "JO 0 0 ^"7 ZO.J / Oiv TO Z J.JZ 00 TO ZZ. /z OO "70 ZZ. /z OO 70 ZZ. /Z OO OO ZZ. /z

Assigned 40 35 30 30 30 30 25 25 25 25

( Frequency = 1 0 kHz )

Repeatability (s^) 30 20 20 20 20 20 20 20 20 20

Temperature Variation (s tm) 4 4 4 4 4 4 4 4 4 4

Predicted Values (s pj) 15 10 15 15

Type-A Standard Uncertainty (w^) 33.78 20.40 20.40 22.72 20.40 20.40 25.32 20.40 20.40 25.32

Assigned 35 25 25 25 25 25 28 25 25 28

58
Table 1 6b. 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 {\xF) 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 = 1 00 Hz )

Repeatability (5^,) 30 20 20 20 20 20 20 20 20 20

Temperature Variation (s tm) 10 10 10 15 15 15 15 15 15 20

PrpHirtpH Valiips 35 15 1

5

10

Type-A Standard Uncertainty (m^) 47.17 22.36 22.36 29.15 O C f\f\ 25.00 O C f\f\ 25.00 29.15 25.00 O C f\f\ 25.00 T A A A 30.00

Assigned 48 25 25 30 25 25 30 25 25 30

( Frequency = 1 kHz )

Repeatability (s^) 30 20 20 20 20 20 20 20 20 20

Temperature Variation (s tm) 10 10 10 15 15 15 15 15 15 20

Predicted Values {s pj) Zj zu

1 c

1

J

1

J

1 D

1 (\ lU 1 u1 (\ 1 nlU

Type-A Standard Uncertainty (wj 40.31 30.00 33.54 29.15 29.15 29.15 26.93 26.93 26.93 30.00

Assigned 45 30 35 30 30 30 30 30 30 30

( Frequency = 1 0 kHz )

Repeatability (s^) 30 20 20 20 20 20 20 20 20 NA

Temperature Variation (s tm) 10 10 10 15 15 15 15 15 15 NA

Predicted Values (s pj) 35 5 35

Type-A Standard Uncertainty (uj 47.17 22.36 22.36 25.50 25.00 25.00 43.01 25.00 25.00 NA

Assigned 48 25 25 30 25 25 45 25 25 NA

59
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, IkHz, and 10 kHz

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

Unknown Standard (\xF) 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 (w^) 46 25 25 38 25 25 25 25 25 25

Type-B Standard Uncertainty (w^) 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 (t/j) 160 120 100 120 120 120 120 120 120 120

( Frequency - 1 kHz )

Type-A Standard Uncertainty (u^) 40 35 30 30 30 30 25 25 25 25

Tvne-R StandflrH TJncertaintv ^Wl^ 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 Uncertaintv (11.^ 160 120 100 120 120 120 120 120 120 120

( Frequency = 10 kHz )

Type-A Standard Uncertainty (Wg) 35 25 25 25 25 25 28 25 25 28

Type-B Standard Uncertainty (Wb) 65 40 25 40 40 45 65 105 240 475

Expanded Uncertainty {IJ^ 147.65 94.34 70.71 94.34 94.34 102.96 141.55 215.87 482.60 951.65

Assigned Total Uncertainty {V^ 160 120 100 120 120 120 150 250 500 1000

60
Table 1 7b. Summary of Expanded and Assigned Total Uncertainties in Capacitance Measurements Using

the Type- 1 2 Bridge to Measure Three-Terminal Mica Capacitors at 1 00 Hz, 1 kHz, and 1 0 kHz

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

Unknown Standard C , (pr) 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 (Wb) 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 (Ui) 120 100 100 100 100 100 100 100 100 100

( Frequency = 1 kHz )

Type-A Standard Uncertainty (u^) 45 30 35 30 30 30 30 30 30 30

Tvne-R StandarH T IncertaintA/ 35 25 1 S 20 20 20 20 20 20 20

Expanded Uncertainty {U) 114.02 78.10 76.16 72.11 72.11 72.11 72.11 72.11 i2.n 72.11

Assigned Total Uncertainty ([/,) 120 100 100 100 100 100 100 100 100 100

( Frequency = 1 0 kHz )

Type-A Standard Uncertainty {u^ 48 25 25 30 25 25 45 25 25 NA

Type-B Standard Uncertainty {u^ 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 (f/,) 120 100 100 100 100 100 150 250 500 NA

61
Table 1 8a. 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 - except nominal values of the unknown capacitance standards)

Unknown Standard (|aF) 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 = 1 00 Hz)

Accuracy of Gs (^gs) 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 (Sg^) 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. (s gp) l.OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 1. OE-04 l.OE-04 1.3E-04

Residu. Conduct, (s^^) 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 (Wgi,) l.OE-04 l.OE-04 l.OE-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 Gs (s^^) 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. {s^p) l.OE-04 l.OE-04 1 .OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 1.3E-04

Residu. Conduct, (s^^) 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-05

Mica Capacitor Dials (s gn,)

- - - 9.0E-05 1.7E-04 2.6E-04 8.0E-04 2.9E-03 8.7E-03 1.5E-02

lype-B Std. Uncertainty (Wgb) 1 .Ob-04 l.OE-04 l.OE-04 1 AX^ f\ A 1.4E-04 2. OE-04 3. OE-04 A r* C\ A 9.3E-04

'> AT^ AO 3.4E-03 1 AT^ A'^

1 .4E-02 A or* A'l 4.8E-02

(Frequency = 10 kHz)

Accuracy of Gs (5gs) 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-01

Contact Resistance (5 gj 4.ob-06 1 or AC 1.8b-05 1.2L-04 A HT^ f\ A 4.6h-04 1.8h-03 1 .2b-02 A ^iV A'^ 4.ob-02 1 or? A t 1.8t-01 1 TC J AA

1 .2h+00 A /Z'C I AA 4.6h+00

Conductance Pot. (Sgp) l.OE-04 l.OE-04 l.OE-04 l.OE-04 1. OE-04 l.OE-04 l.OE-04 l.OE-03 l.OE-03 1.3E-01

Residu. Conduct, (^g,,) 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-02

Mica Capacitor Dials (^gm) 1.4E-03 3.1E-03 6.9E-03 2.7E-02 6.5E-02 2.6E-01 1.2E+00

Type-B Std. Uncertainty (Wgb) 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

62
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 jiS - except nominal values of the unknown capacitance standards)

Unknown Standard (|^F) 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0

Comnonents of

Standard Uncertainties

(Frequency = 100 Hz)

Accuracy of G; (s gs) 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 lE-08 4.6E-08 1.8E-07 1.2E-06 4.6E-06 1.8E-05 1.2E-04 4.6E-04

Conductance Pot. (Sgp) l.OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 l.OE-04 1. OE-04 1. OE-04 l.OE-04 1.3E-04

Residu. Conduct. (Sg^) 1.2E-06 1.2E-06 lE-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 (^gn,)

- - - 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 (Wgt) l.OE-04 l.OE-04 l.OE-04 1.7E-04 1.8E-04 4.1E-04 7.7E-04 2.6E-03 4.7E-03 6.7E-03

(Frequency = 1 kHz)

Accuracy of Gs (^gs) 1.6E-06 1.6E-06 8.0E-06 1.3E-05 3.2E-05 3.2E-05 l.lE-04 1.6E-04 5.6E-04 3.5E-03

Contact Resistance (^g^) 4.6E-08 1.8E-07 lE-06 4.6E-06 1.8E-05 1.2E-04 4.6E-04 1.8E-03 1.2E-02 4.6E-02

Conductance Pot. (5gp) l.OE-04 l.OE-04 l.OE-04 l.OE-04 1. OE-04 l.OE-04 l.OE-04 1. OE-04 1. OE-04 1.3E-04

Residu. Conduct. (Sg^) 1.2E-05 1.2E-05 lE-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 (s^^) - - - 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 (Mgb) 1 .OE-04 1 .OE-04 1 .OE-04 2.0E-04 2.7E-04 8.2E-04 2.9E-03 8.6E-03 1.8E-02 5.1E-02

(Frequency = 10 kHz)

Accuracy of G; (s^^) 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

V^UIllaLl IvcSlblallCC gj) 1 .oC-UJ

1 OP C\A A fiV C\A 1 8P C\'X

1 .oc-yjj 11 .oJ_/-U 8P 011

1 9p+nn NA

Conductance Pot. (^gp) l.OE-04 1. OE-04 l.OE-04 l.OE-04 l.OE-04 I. OE-04 1. OE-04 l.OE-03 l.OE-03 NA

Residu. Conduct, (s^^) 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 (s gn,) 3.1E-03 4.9E-03 2.4E-02 6.5E-02 1.6E-01 1.2E+00 NA

Type-B Std. Uncertainty (Wgb) 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

63
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 - except nominal values of the unknown capacitance standards)

Unknown Standard {[iF) 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0

Components

(Frequency = 1 00 Hz)

Repeatability (s^^) 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 (s gpj) l.OE-04 _ _ 1.2E-04 _ _ 2.1E-04 _ _ 5.0E-03

Type-A Std. Uncertainty (WgJ l.OE-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 (Wgt) l.OE-04 l.OE-04 l.OE-04 1.6E-04 1.7E-04 1.8E-04 4.1E-04 7.7E-04 2.9E-03 4.8E-03

Expanded Uncertainty (C/g) 0.00029 0.00021 0.00021 0.00040 0.00035 0.00039 0.00094 0.00158 0.00590 0.01399

Assigned

Total Uncertainty (t/g,) 0.0003 0.0002 0.0002 0.0004 0.0004 0.0004 0.001 0.002 0.006 0.02

(Frequency = 1 kHz)

Repeatability (Sgm,) 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 (s gpj) l.OE-04 2.0E-04 2.0E-04 3.0E-04 l.OE-03 2.0E-03 6.0E-03 1.5E-02 5.5E-02 9.5E-02

Type-A Std. Uncertainty (MgJ l.lE-04 2.1E-04 2.1E-04 3.2E-04 l.OE-03 2.1E-03 6.1E-03 1.5E-02 5.6E-02 9.8E-02

Type-B Std. Uncertainty (Wgt) l.OE-04 l.OE-04 l.OE-04 1.4E-04 2.0E-04 3.0E-04 9.3E-04 3.4E-03 1.4E-02 4.8E-02

Expanded Uncertainty (f/g) 0.00030 0.00046 0.00047 0.00070 0.00209 0.00418 0.01236 0.03112 0.11473 0.21824

Assigned

Total Uncertainty (f/gt) 0.0003 0.0005 0.0005 0.001 0.003 0.005 0.02 0.04 0.12 0.3

(Frequency = 10 kHz)

Repeatability (s^^) 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 (^gpd) l.OE-04 - - 3.3E-03 - - 3.5E-02 - - l.OE+00

Type-A Std. Uncertainty (WgJ 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 (Wgb) 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 (t/g) 0.0005 0.0006 0.0012 0.0076 0.0085 0.0314 0.1356 0.4116 2.4667 10.0424

Assigned

Total Uncertainty (f/g,) 0.001 0.001 0.005 0.01 0.02 0.05 0.2 0.5 3.0 10.0

64
Table 1 9b. 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 |uS - except nominal values of the unknown capacitance standards)

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

Components

(Frequency = 100 Hz)

Repeatability (Sgm,) 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 (5gpd) 1.5E-05 7.4E-05 1 .9E-04 4.6E-03

Type-A Std. Uncertainty (WgJ 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 (Wgb) l.OE-04 l.OE-04 l.OE-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 iU„.) 0.0002 0.0002 0.0002 0.0004 0.0004 0.0009 0.002 0.006 0.010 0.02

(Frequency = 1 kHz)

Reneatahilitv ^ 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 {s^^ l.OE-04 2.0E-04 2.0E-04 3.0E-04 l.OE-03 2.0E-03 6.0E-03 1.5E-02 5.5E-02 9.5E-02

Type-A Std. Uncertainty (m g

J

l.lE-04 2.1E-04 2.1E-04 3.2E-04 l.OE-03 2.1E-03 6.1E-03 1.5E-02 5.6E-02 9.8E-02

Type-B Std. Uncertainty (Wgb) l.OE-04 l.OE-04 l.OE-04 2.0E-04 2.7E-04 8.2E-04 2.9E-03 8.6E-03 1.8E-02 5.1E-02

Expanded Uncertainty (f/g) 0.00030 0.00046 0.00047 0.00076 0.00212 0.00444 0.01354 0.03487 0.11700 0.22018

Assigned

loiai uncenainiy yu U.UUUj U.UUUj U.UUUj U.UU 1 U.UUj u.uuo u.uz U-Ut U. 1 z U.J

(Frequency = 1 0 kHz)

Repeatability (^g^,) 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 (.^gpd) 3.5E-05 2.3E-03 3.2E-02 NA

Type-A Std. Uncertainty (WgJ 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 (Wgb) 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 (JJ

0.0005 0.0006 0.0012 0.0081 0.0115 0.0548 0.1773 0.5001 3.4532 NA

Assigned

Total Uncertainty (t/g,) 0.001 0.001 0.005 0.01 0.02 0.06 0.2 0.5 4.0 NA

65
Primary

Capacitance

Laboratory

Impedance

Calibration

Laboratory

Reference Bank

4x lOpF Oil-Bath Type

Fused-Silica Capacitance Standards

Transfer Standard

10 pF Air-Bath Type

Fused-Silica Capacitance Standard

Reference Bank

4xl0pF Oil-Bath Type

Fused-Silica Capacitance Standards

Secondary References

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

Fused-Silica Capacitance Standards

Customer's

Fused-Silica

Capacitance Standards

Check Standards Customer's

1 pF, 10 pF, Nitrogen Dielectric,

2xl00pFand 2xl000pF 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
67
-1

Ex

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

68
Ci|_

^^^^

C2L C2L C4L Cj_

^^^^

C6\_Cjj_ cj_

100 10 1 10"' 10"2 iQ-' 10-'' 10-5

(pF)

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

Gc

R L, —^^/w

c

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

69
70
Figure 5b. Simplified Circuit Diagram of the Type- 12 Capacitance Bridge.

71
Figure 6. Components of the Type- 12 Bridge for Two-Terminal Low-Capacitance

Values Measurements.

72
ure 7. Components of the Type- 12 Bridge for Two-Terminal High-Capacitance

Values Measurements.

73
Figure 8. Components of the Type- 12 Bridge for Three-Terminal Low-Capacitance

Values Measurements.

74
Figure 9. Components of the Type- 12 Bridge for Three-Terminal Measurements of

Capacitors up to 0.5 |iF.

75
B

D

Figure 10. Components of the Type- 12 Bridge for Three-Terminal Measurements of

Capacitors Larger Than 0.5 |j,F.

76
gure 1 1 . Simplified Circuit Diagram of the Type- 12 Capacitance Bridge Including

the Conductance Components.

77

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. (Al) to (A4) as:

E,Y^ + E^(lriy,)+p,E^(Mlg.) = -E^Y^, (Al)

where E^ and E^ are voltages produced by the transformer windings on the reference, Cj^ and the

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

y_ and 7-, are admittances of C„ and C„ , respectively,

o A. a A

ri is the dial reading of / dial applying voltage to capacitor c^-

,

is the admittance of c^-, g^is the effective admittance associated with the settings of the conductance dial coupled

with the frequency compensation network,

jf?5 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'^ < M < 1.

Corrections for errors of 1 : 1 ratio of the Type-2 bridge can be determined in two measurements by

using two capacitors ofnominally equal value. One measurement is taken with and connected

as shown in Fig. 2b. The other measurement is taken with and interchanged. The balance

equation of the second measurement can be written as:

EsYx^^s(T^Vn)'-P5EsiM'lg,) = -E^Y^ (A2)

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

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

(^s + ^x)(^s+ ^x) + ( I ^l^n^yn)+P5Es^ii:Si + lgn)-0. (A3)

Assuming = 1 V as the reference, and by the definition of 1 : 1 ratio, {E^ I E^) — - 1 , then:

E^=-il+a+ib), (A4)

where a and b are the in-phase and quadrature corrections to the ratio, respectively.

Substitution of Eq. (A4) into Eq. (A3) gives:

i^+}b)-{[{lnyi + lr^yn)+PsM{lgi + lg„)]/{Y^+Y^)}. (A5)

Let - jwCj^ (1 - jDF^ ), y^^ = jo)Cj^ (1 - jd/y. ), and = gj^ (1 + jxj^ ), where DF^^ and df^ are the

dissipation factors of the respective capacitors Cj^ and Cj^ , and Xj^ is the phase correction of the

resistor network used in the conductance dials. The magnitudes ofDFy^ , dfy^ , and jcj. are all less than

A-1
10'^. Therefore, by eliminating these terms, the approximate equation for the ratio corrections is:

«i + j^i - [ ( L c. + L r„ cJ + j(-l/0))/.5

^( L ^/ + Z ] / + Q, (A6)

where and are in-phase and quadrature corrections apphed 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:

{a^]h) = a^^e^^]h^\]e^^, (A7)

where 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:

- ( Z riyt + L^nyn) = ln 0""^/ ) ( ^-idfi ) + E (i^c^ )

( i-

)

-^[ilr.c.df. + lr^c^df„)+j{Y.r^c, + lr„c^)]. (A8)

The second term in the numerator of the right hand side of the Eq. (A5) can be written as:

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

is:

^[(l n dfi ^lr^c^df^) + (p,M/o,){l g- g„)]

+ c. + lr^c^) + (p,M/ o)) (l g^^ +1 g^„) ]. (AlO)

The denominator of the right hand side of the Eq. (A5) can be written as:

(Fg + Y^) = jo)Cs (1- ji)Fg ) + jwCx (1- jZ)Fx) = o) [(Cs DF^ + Cx DF^ + j(Cs + Cx)].

Thus,

[l/CFg + r^)] =(l/G)){l/[(CsDF3 + qZ)F^)+j(C3 + Q)]}. (All)

A-2
Multiply both the numerator and the denominator of the right hand side of Eq. (All) by

[(Cg Df^ + Q " J(Q Q)] eliminate the second order terms of DF^ and Df^ . Eq. (Al 1)

becomes:

[1 / (Fg + Y^)] = (1/ 0)) { [ (C3 DF^ + qZ)F^) / (Cg + Q'

] - j[l/(C3 + CJ]}.

In measurements using the 1:1 ratio, Cg - C^, and

[(Cg DFg + C^ DFj^) / (Cg + C^)] = [(DFg + DF^) 1 2]. Then,

[1 / (7g + Y^)] = (1 / o)){(Z)Fg + Z)F^) / [2 (Cg + C^)]} - j{l / [ o) (Cg + C^) ]}. (A12)

The expression of Eq. (A5) can be obtained by the muhiplication of Eq. (AlO) and (A12). After

the separation of the real and imaginary parts, and the elimination of the second order terms of DF^,

DF^, df-, df^, x^, and the in-phase correction, a, and the quadrature correction, ^ of the 1:1 ratio can be expressed as: a={{p,MI a))[(l4) (E + E g„) ( DF^ +DF^) + (l g^. + I g^^)]

+ ( I r^i + I rnCn )) I {C, + C^),

and

b = {[(^/2)(Zr.c^. + E r^c^)iDF^ + DF^)-{lr,c.df. + lr„c^df„)]

-iPsM/ioXlg. + l g^)} I (Cg + C^). (A13)

According to Eq. (A7),

and

e^ = h-h\. (A14)

By substituting and from Eq. (A6) and a and h from Eq. (A13) into Eq. (A14), the errors in

the in-phase and quadrature corrections of the 1:1 ratio become:

= (p, M/0))[C/2) (Lg^. + Ig^) (DF^ + DF^) + (1 g^- + ^ g^^) ] / (Cg + C^), (A15)

and

% - m(L + L r„c„ ) (DFg + DF^) - (£ r^^^f, + I r^c^df^ ) ] / (Cg + C^). (A16)

In general, the capacitance and conductance dial readings of the two measurements when and

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 (/ 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
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 Z)Fg, DF^^,

df-, df^, x^, and are less than 10'^, and ratios of [x^ I (DF^ + DF^)] and [x^ I (DF^ + DF^)] are less

than 1 . Maximum bounds of errors ± and ± e^^, in ratio corrections are estimated from the

following sources and related assumptions; and the standard uncertainties, and s^, are obtained

in accordance to Eq. (4) of the text.

1) Uncertainties estimated from the measured data

According to the data from measurements performed on capacitors of various nominal

values, it was found that:

l( Lgi + lgn)\^ 10"! S and |( lr.c. + lr^c„)\ < pF (for C^-C^- 10 pF),

l( Igi + I>„ )l ^

10'' S and |( lr.c. + lr^c^)\ < lO'' pF (for C^ = C^= 100 pF),

l( Igi + lgn )l ^

10-' S and |( Ir-c- + lr^cj\ <10"' pF (for C^-C^= 1000 pF), and

M<10"'^

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

5^1 < 0.001 ppm and s^^ « 0.001 ppm,

where and ^^^j are the standard uncertainties from measured data in the in-phase and

quadrature corrections of the 1:1 ratio, respectively.

2) Uncertainties due to the inaccuracies of the external capacitance standards

Assume the external capacitors, C„ and C^, are inaccurate by ± 0.01%, the maximum bounds

of errors are estimated by substituting Cy = C^(l±10 ) and Q = Q(1±10" ) into Eqs.

(A1 5 ) and (A1 6) . Accordingly,

« 0.001 ppm and s^2 0.001 ppm,

i where and 51^2 are the standard uncertainties of the external capacitance standards, in the

in-phase and quadrature corrections of the 1:1 ratio, respectively.

3) Uncertainties due to the dial readings

Assume the capacitance and conductance dials are inaccurate by ± 1%, the maximum bounds

of errors are estimated by substituting = (1 ± 0.01), c^=c^ (1 ± 0.01), =g^ (1 ± 0.01),

and g-^ (1 ± 0.01) into Eqs. (A15) and (A16). Accordingly,

« 0.001 ppm and 51,3

« 0.001 ppm,

where 5^3 and 5^3 are the standard uncertainties of the dial readings, in the in-phase and

quadrature corrections of the 1:1 ratio, respectively.

A-4
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:

5^4 < 0.002 ppm,

where 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 infroduces 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:

5,34 < 0.02 ppm (for C^ = C^= 10 pF),

s^^ < 0.002 ppm {for C^ = C^= 100 pF), and

5^4 < 0.0002 ppm (for = Q= 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) Uncertainties due to load admittance and lead impedance

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,

5^5 < 0.004 ppm and s^^ < 0.019 ppm,

where 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, and s^, are defined as the square root of the sum-of-squares (RSS) of each component, which are estimated

to be:

= ( -^ai^ + ^a2

^ + ^ + ^a4 % ^a5^

) ^^'^^ < ^.005 ppm, (A17)

and = ( 5bi^ +sb2^ + sb^ ^ +

5b4 % 5b5 ^

) ^^^^^ < 0.03 ppm. (A18)

A-5

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

Data and analyses of capacitance corrections for each capacitor dial of the Type- 1 2 bridge are included

in the following tables:

Table B-1. Data and Analysis of the ( 0.1 fiF/step ) Mica Capacitor Dial Corrections of the

Type- 12 Bridge at 100 Hz

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

Date-"\Dial 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 O.X

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

9/72 20.817 45.897 96.451 106.001 127.050 151.815 199.822 221.694 246.911 265.385

6/ /4 21.012 44. D / / 96. 139 nn ceo 120. / j6 1 /I '5 nn 1 143.991 1 no An A 193.4/4 lie c^ o ZlD.olo 239.2 15 OCT c £ir\

1 1/75 21.435 46.830 103.188 105.313 126 560 151.373 205.533 227 700 253.190 271.885

1/77 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 1 1 1.796 132.763 158.175 208.107 255.559 273.476

1/85 20.874 48.402 104.174 117.362 138.158 164.957 218.316 240.047 267.163 284.982

Jl oJ to. JO / 1 KtJ .OZU 1 07 OH 190 S90 900 S

1

911 807 9^7 1 90 97*^ 71

1

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) l.OE+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 l.OE+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
Table B-2. Data and Analysis of the ( 0.1 jiiF/step ) Mica Capacitor Dial Corrections of the

Type- 12 Bridge at 1 kHz

[All values are in pF except the dials which are in (fj.F/step)]

Date^^^ial 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 O.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

1/77 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

j/OJ 07.1/1 171 z 1 .ozz ^10 1 .64 847/ 870 / .zzo 07 A'\l 1 1 A 1 no 1 lU.lUZ 1 1 A OT 1 1 lO.yj 1 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) l.OE+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 l.OE+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 B-3. Data and Analysis of the ( 0.1 p-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 (fxF/step)]

Date"-"»..^ial 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 O.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

7/72 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

4/70 16.649 41.610 52.254 11/1 A 1 o1 14.918 1 '7 O C C\C\ 138.599 180.768 194.839 224.878 252.769 321.523

11/80 15.938 42.038 52.608 1 16.776 140 616 180 812 213.659 251.150 290 010 352.910

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 1 A A f\ 16.440 At O 1 £ 41.815 50.474 lid C O A1 14.584 137.736 178.309 197.036 232.733

f\ /' r ''\ f\r\ 265.200 T O 1 1 o 381.138

N. Val. (pF) l.OE+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 l.OE+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 4.850 7.100 10.644 14.441 21.863

B-3
Table B-4. Data and Analysis of the ( 0.01 )a.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 (fiF/step)]

Date"""^^Dial 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 O.OX

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

9/69 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 15.1 17 14.655 18.41

1

21.117 21.615

1/77 2.694 5.178 7.534 9.940 12.989 15.458 14.922 18.643 21.339 21.753

/I /TO 4/ /o Z.J / J 4. / /4 A 1C\1 C Q/l 8 1 1 7<Q i 1 . / JO 14. 1 j4 1 1 IQA 1 J . /y4 10. /jU 1 Q TJQ ly.J JO ly.jUo

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 1 1 .992 14.577 13.778 17.519 20.262 20.551

z/y 1 z.Jo9 J. 1 JZ o. jo4 O.J /U 11/1 0/;1 1 .4oo 1 /t T /I 0 11 IQA 1 j.jo4 lo.yo / ly.zjo 1 0 OTO lo.y /z

N. Val. (pF) l.OE+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 l.OE+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
Table B-5. Data and Analysis of the ( 0.01 |j.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 (fiF/step)]

Date"^"^ Dial 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 O.OX

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

1/77 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

1 1 /8n 1 l/oU 1 noQ L. /UO J .ZU J1 801 J .ou 1 J. 1 JO f\O.OoA87/ A 1/18 't. o c+ou 8o. 181 1 O 1 7 198

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 6.192 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) l.OE+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 l.OE+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
Table B-6. Data and Analysis of the ( 0.01 jiF/step ) Mica Capacitor Dial Corrections of the

Type-12 Bridge at 10 kHz

[All values are in pF except the dials which are in (|j,F/step)]

Date^^Dial 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 O.OX

6/67 0.269 1.491 1.829 2.800 4.312 6.523 5.180 8.255 11.291 11.987

1 1/67 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

9/69 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

1/77 0.306 1.608 1.916 2.585 4.164 6.428 5.024 8.081 11.224 12.018

4/ /O U.ZZj 1 AO 1 1 .DJy Z.Z 1 J j.VyJ j.O /O A 197 7/.JIJ 111 1 n 181 11 101 1 1 . IZ

1

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

") /Q 1 Z/y 1 1 .JoU 1 .voy 1 /I 1 0 1 .4 1 Z Z.O /O j.Vyj 1 1 /t 1 J. 141 A 1 Q7 o.iyz O OlO Q 0 1 7 y.yi 1

N. Val. (pF) l.OE+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 l.OE+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
Table B-7. Data and Analysis of the ( 1 000 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 10 000

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 1ll nil (\ r\i 1 U.U /

1

u.U4y U. 1 OH A O 1 o C\ TOO U.ZoO (\ /IK \)Ay 1 U.400 U.J j 1 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

jIoj n AQ/1 u.uy4 U.UoU U.ZJ 1 U.Z / / U.JoU U.i / J U.DZj n /CI "J U.Oli U.Do /

(\ ci^ U.O / 1 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 10 000

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 l.OE-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 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

;4. 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

n/75 -0.002 0.130 0.097 0.083 0.084 0.217 0.185 0.184 0.316 0.315

. ... 1/77 -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

Q/8'? y/oj U.UU 1 n 1 lA n 1 ni u.uy 1 U.ZZJ u. 1 yz n 1 on U. jZV f\U.jZ111/ 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 l.lE-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
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

1 1/67 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 0.01

1

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

11/75 0.005 -0.059 0.006 0.053 0.058 -0.006 0.059 0.063 -0.001 0.011

1/77 0.006 -0.057 0.009 0.048 0.053 -0.008 0.054 0.071 0.007 0.013

4/78 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

Q/CJ y/oj n nno u.uuy -U.Ujz U.UU / U.Ujo U.Uo4 U.UUo U.U /J u.u /y U.UlU U.U 1 o 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 l.lE-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

APPENDIX C. CONDUCTANCE DIAL CORRECTIONS OF THE TYPE- 1 2 BRIDGE

Data and analyses of conductance corrections for each mica capacitance dial of the Type- 1 2 bridge are included in the following tables:

Table C-1. Data and Analysis of Conductance Corrections of ( 0.1 |a.F/step ) Mica Capacitance Dial

of the Type- 12 Bridge at 100 Hz

(All values are in 0.001 jiS, except the dial which is 0.1 fiF/step)

DIAL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 O.X

yjl \j 1

1 5 499 ^99y .'~rO\j 480 J\J 50 .J\J 5071 51 495 66yjyj . 799 1 80 851 101 950 1 1 7 419 1 1 1 160 147 OSS

\mi 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 1 19.660 134.058 150.1 10

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

1 1/75 15.710 30.388 51.223 53.695 69.333 84.185 105.025 120.893 135.780 151.678

1/77 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 61.2A3 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
Table C-2. Data and Analysis of Conductance Corrections of ( 0. 1 ^iF/step ) Mica Capacitance Dial

of the Type- 1 2 Bridge at 1 kHz

(All values are in 0.001 |j.S, except the dial which is 0.1 |j,F/step)

DIAL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 O.X

0/0 /

/I "7 0 AC Vo.O / / Z04.J JU zz4.J /Z TZI r>QO z / 1 .UVo jzo.yoo DUU.ojo J JO. JZO 0UU.Z4U 0 /o.oy /

1 1 icn

1 I/O / A 0/1 C\ 1 f\(\ A1(\ zzV. iUz Z /O.OJO jUo.o I a joj.oyz ou /.y lu OOJ.O / / ^ //CO j/OO 4 /.oUj OQ O/l 7 yy.y4 / OA/; /I ^ < ZOO.4.) J zzO.jj J z /4.J04 J ju.o /y jU-).jo4 jOj.4 /U OUO. /OJ Oo4.4UU

9/69 47.680 101.745 265.495 235.212 282.337 341.020 513.247 571.053 615.033 693.146

1 9/70 4RHO.WOS"? J J 1 UJ .O 1 z 9^ifi zoo. 719 / .JZ 911 181 9R0 zoo."QISJ J 140 67S S 1 1 49 S JS71/ J .OfiS7 J / 61O 1 0".tj4SSJ 606 070.407

/

9112 48.202 102.982 266.970 241.348 288.850 347.110 521.205 580.365 622.810 702.902

Ho. JZW ini 800 968 ZOo.UHo 0/18 Zj911J. 778 / /

o

989 ZOZ.UZ09SJ HQ7. 11 89OZ '^ 1 4 Q'^0 'n7^ 48'^ J / J .HO J 616 O 1 o.oo887/ 60'> 07J .HHZ 449

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 1 07 250 979 120 249 521 9QQ 701 160 815 515 808 591 048 645 VJ" .I/O 178 727 558

1/85 49.015 108.588 251.525 302.480 363.743 536.508 598.165 647.708 728.600

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 iiimi 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 l.lE-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 l.llSl 7.5324 7.7394 10.6895 10.7667

SD(2/01)(nS) 0.70714 2.86942 2.11632 8.37964 8.68548 10.4394 10.1127 10.3907 14.3514 14.4551

C-2
Table C-3. Data and Analysis of Conductance Corrections of ( 0. 1 jj,F/step ) Mica Capacitance Dial

of the Type- 12 Bridge at 10 kHz

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

DTAT 0 1 0 2 0 1 0 4 0 5 0 6 0 7 0 8 0 9 0 X

O/O / 4yz.JUo 1 J JZ. 1 ZU T 1 1 Q Af\r\ /1 1 CO /I /in 4o4V.0zU jVj j.4oU C<0/1 \C\(\ 1 (\mQ ir\(\

I uzoV. /UU W40.jOU 1 J jyv.4uu

1 I/O / /10/I 1 '^n 4y4. 1 ju 1 1 en 4 1 'n 1 1 QCC J 1 J j.yj J 494Z 1 9 'Xl(\ 1 Z. JZU 474A 4 /40.04O ^48 f.cW's ^88 0/1 87 1 0.UUJ nni 11 n'?Q8 UJ70.900 zuu 11 09'^0 UZjU.oUU 800 1'?701 ^^00 1 J /U 1 .DUU D/Oo 404 1 48 1 J JZ. 1 JO 1 1 97 97'? J 1 Z / .Z / J 4148 4 1 A1 1 40. 1 0 1 4^97 40Z / . 100 1 yy 'n09/^ 0/i1 J 7ZO.7O 1 0

8'^0A jy0.4Z4 494 1 09A^ ^(\(\

1 UZOO. JUU 11 01U 1 04U4.ZUU 900 \'X(.'i

i joj \1 .UUU noo

9/69 486.015 1327.123 3093.330 4176.263 4637.970 5938.353 8662.910 10264.940 10127.130 13492.130

19/70 iz/ /w 1 J O / .ZHJ 1 1 48 00'^ 4909 008 4f>41 totJ .ZZ 99^J JO7O. JZO 879'? 0 / ZJ . JUSO'^J 11 04110./780OU 1 009^^ '^40 1 UUZD. JHU

9/72 488.310 1347.368 3120.208 4302.298 4807.788 6188.928 9006.043 10757.383 10395.990 13851.1 10

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 6246.212 9003.290 10564.867 10717.750 15029.550

1/77 498.798 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 4311.050 4810.923 5885.795 8535.668 9385.540 9035.413 12330.785

o/oZ 8/89 ^no 'nOO 1 J jj.jOj 1 1 7 1 8 J 1 10.

1

4oj. lyj 40j4. Af.'XA 11 Zj JO / J .OjU 0

oco/: JoO. 708 /Z6 Q8QO yow.ouj 80^; 0/^08 youo.ouo 808 \'i.A'^\ 1 'J^ 1J4J 1 . 1 J J 9/83 498 020 1373.128 3057 235 AMI 068 4719.675 601 \J\J 1 1 483 r V 8954 \j y ^ r . 090 \j y \j 1 1046 698 1 1089 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 3164.990 4217.098 4887.130 6133.763 8723.120 10486.753 10250.585 13853.518

2/91 514.700 1352.630 3168.623 4187.365 4862.783 6208.600 8941.918 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 l.lE-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

C-3
Table C-4. Data and Analysis of Conductance Corrections of ( 0.01 uF/step ) Mica Capacitance Dial

of the Type- 12 Bridge at 100 Hz

(All values are in 0.001 |liS, except the dial which is 0.01 |j,F/step)

DT AT 0 01 0 03 0 04 0 05 0 06 0 07 0w. 08wo 0 00 0 OY

o/o / i.5U / Z.643 C A 1 C 5.015 6.565 6.083 y.zi /

1 1 COA 1 1 .580 \ 1 { AT 1 J. lOz 14.25 / 15. /82

1 1 /^n

1 1/67 1.520 Z.680 C AAO 5.098 c c 6.655 O 1 c 8.165 A T O A 9.320 1 1 TO '? 1 1.723 13.273 14.450 1 ^ A'O C 16.035

5/68 1 C A1 1.543 O T 1 A 2.710 5.138 6.698 8.285 9.465 1 1 oc\o1 1.898 13.465 14.628 16.193

9/69 1.603 2.765 5.165 6.760 8.315 9.463 11.888 13.448 14.590 16.160

vZI /U 1 ^Tv 1 .J /

J

0 7/1 "J Z. / 4

J

1 Q1 J.iyj O. /ZU o.ZoU 1 1 00^ 1 1 .OOJ 1 J.4oU 14.0JO 1 o.ZUj

9/72 : 1.478 2.660 5.143 6.700 8.255 9.450 11.973 13.573 14.785 16.365

1 . JOO Z. / Z J f,O.o830jU 8 360 1 1 QQ3 1 1 '^8^ 1 J .Jo J 1It.4 7Q8 lyo 1 6 '^90

1 0. jZU

11/75 1.518 2.683 5.218 6.750 8.245 9.503 11.850 13.440 14.570 16.088

1/77 1.430 2.555 5.048 6.598 8.228 9.430 11.855 13.440 14.590 16.115

4/78 1.490 2.628 4.980 6.478 7.950 9.103 11.465 13.023 14.185 15.728

11/80 1.465 2.635 4.993 6.490 8.065 9.130 11.440 12.985 14.130 15.593

8/82 1.478 2.563 4.895 6.410 7.923 9.118 11.438 13.038 14.180 15.648

9/83 1 coo1 .583 o o c c z.o55 C IOC5. 125 C 1 0 6.5 13 O AiCA 8.060 A 1 no9. 1 /8 1 1 nf\c1 1 . /05 IT oo o 13.228 14.263 1 c nc\f\ 15.790

1/85 1.613 2.730 5.145 6.523 8.070 9.195 11.603 13.125 14.293 15.825

3/85 1.520 2.728 5.128 6.563 8.080 9.220 11.613 13.158 14.313 15.838

2/91 1.857 2.987 5.337 6.717 8.070 8.880 11.190 12.730 13.397 14.665

Mean (nS) 1.5480 2.7056 5.1168 6.6231 8.1521 9.2936 11.6937 13.2546 14.3804 15.8969

SD (nS) 0.0981 0.1062 0.1110 0.1217 0.1321 0.1907 0.2307 0.2452 0.3420 0.4072

Slope(nS/day) 1.5E-05 1.6E-05 4.0E-06 -1.9E-05 -3.0E-05 -5.1E-05 -5.8E-05 -6.1E-05 -9.0E-05 -l.lE-04

Slope(nS/year) 0.0056 0.0057 0.0015 -0.0071 -0.0108 -0.0186 -0.0213 -0.0222 -0.0330 -0.0402

SDfit 0.0923 0.1013 0.1144 0.1143 0.1099 0.1391 0.1775 0.1910 0.2525 0.2938

SD(2/01)(nS) 0.12393 0.13601 0.15356 0.15347 0.14756 0.18677 0.23836 0.25649 0.33897 0.39448

C-4
Table C-5. Data and Analysis of Conductance Corrections of ( 0.01 uF/step ) Mica Capacitance Dial

of the Type- 12 Bridge at 1 kHz

(All values are in 0.001 \xS, except the dial which is 0.01 |iF/step)

DIAL A A 1 0.01 A AO O.OZ A A^ 0.03 A Ayl 0.04 A AC A A/C 0.06 A f\n 0.0 / A AO 0.08 A An 0.09 A AV O.Oa

6/67 5.743 9.678 16.308 22.865 28.638 32.697 39.490 45.435 49.357 55.620

11/67 5.747 9.888 16.678 23.072 28.872 33.133 40.125 45.860 49.870 55.987

5/68 5.778 9.775 16.505 23.045 28.835 32.940 39.813 45.813 49.832 56.142

9/69 5.798 9.818 16.533 23.013 28.865 33.032 39.927 45.962 50.045 56.322

12/70 5.785 9.995 16.883 23.275 29.055 33.432 40.415 46.432 50.720 56.987

9/72 5.840 10.003 16.707 23.185 28.975 33.157 40.120 46.087 50.302 56.652

fillA J .ouo 0".O8n1 J 1 f,O.J"SOIJ 90 OSS 1'^ 1 70 Aft O'^O

11/75 5.658 9.728 16.508 22.91^ 28.873 33.115 40.135 46.252 50.412 56.935

1/77 5.700 9.758 16.370 22.943 28.693 33.050 39.928 46.060 50.310 56.882

4/78 5.638 9.615 16.178 22.620 28.350 32.490 39.077 44.880 48.905 55.043

11/80 5.765 10.050 16.935 23.273 29.130 33.418 40.260 46.403 50.460 56.905

8/82 5.760 9.863 16.560 22.878 28.853 33.140 40.085 46.375 50.520 57.1 10

9/83 5.758 10.023 16.825 23.138 28.860 33.345 40.420 46.643 51.475 57.865

1/85 5.578 9.870 16.720 22.760 28.635 33.143 40.200 46.363 50.548 56.990

3/85 5.743 9.990 16.765 22.905 28.693 33.098 40.068 46.043 50.440 56.763

2/91 5.780 9.920 16.473 22.868 28.738 33.160 39.875 45.707 49.972 56.815

Mean (nS) 5.7424 9.8617 16.5963 23.0008 28.8202 33.0956 39.9988 46.0216 50.2008 56.5930

SD (nS) 0.0680 0.1303 0.2094 0.1851 0.1949 0.2389 0.3358 0.4336 0.5799 0.6626

Slope(nS/day) -7.4E-06 1.7E-05 1.3E-05 -2.3E-05 -1.6E-05 2.2E-05 2.1E-05 3.8E-05 8.0E-05 1.3E-04

Slope(nS/year) -0.0027 0.0062 0.0049 -0.0083 -0.0058 0.0082 0.0077 0.0138 0.0293 0.0460

SDfit 0.0674 0.1267 0.2136 0.1812 0.1969 0.2396 0.3427 0.4367 0.5583 0.5923

SD(2/01)(nS) 0.09046 0.17008 0.28677 0.24331 0.26441 0.32166 0.46009 0.58627 0.74954 0.79517

C-5
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 |j,S, except the dial which is 0.01 ^F/step)

Dl AT 0 01 0 07 0 03 0 04 0 05 0 06 0 07 0yj ,yjo 08 0yj .yjy 09 0yj .yj/\- OX

o/o / JO. /o 1 /j. 1 OO 1 ZO.OOJ \ Q1 O 1 1 1 oi.z 1 1 ZZ /.O 1 U Z / J.O /O 1 CO O 1 1 jjZ.o 1

J

A \ A 0/10 ACI OOO c o 1 /;o DDz.lOo

11/0/ JO.y 1 6 1 A 1 O 1 /4. 1 z 1

1 1 A COA 1 Qi inA

1 o3.3 /4 ZZ /.J 1 J z /o. juy T r c ICO /I 1 T AQI 41 /.40J 4j /.zj4 CC/; OOO JJO.ZZO

D/OO 3 /.Ul ft 15. /yU 1 on CAA 129. j90 1 Q1 ACQ. lo3.4o-) OT? A Af\ ZZ /.44U Z /j.JZj 3 j3.3 /3 /I 1 /I O 1 c 414. ol J

/ICC Til 4-)-).zl3 C C 1 O 1 o jD 1 .z Iz 9/69 37.225 68.573 124.640 179.008 222.140 272.695 350.403 412.085 445.650 544.008

1 9/70 J111 .Jyj IQI 7"^ All 1 JZ.OUJ 11 86ou. Wl1 1

J

ZJ W.UHJ 9X0 00'? 1S7 44R 41 Q ^ 1 R t-OW.'t't J Jjy. IJJ 9112 35.573 72.020 125.523 178.633 220.370 267.068 341.600 401.893 442.503 538.615

6/14 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

1/77 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 ITT A 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 (nS) 1.0318 2.3405 2.9816 3.5414 4.9637 5.8793 7.8936 8.7564 11.173 17.2627

Slope(nS/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 0.4550

SDfit 1.0678 2.2960 2.9904 3.6649 5.1355 6.0434 8.1419 9.0620 11.4910 17.5384

SD(2/01)(nS) 1.43354 3.08257 4.01479 4.92038 6.89472 8.11363 10.931 12.1664 15.4275 23.5465

*U.S. GOVERNMENT PRINTING OFFICE: 2000-460-746-17124

C-6

THE SP 250 SERIES ON NIST MEASUREMENT SERVICES*

SP 250-1 Spectral Radiance Calibrations PB87 179883 SP 250-2 Far Ultraviolet Detector Standards PB87227609

SP 250-3 Radiometric Standards in the Vacuum Ultraviolet PB87227625

SP 250-4 Fricke Dosimetry in High-Energy Electron Beams

PB881 10374 SP 250-5 Alpha-Particle Calibrations PB881 68620

SP 250-6 Regular Spectral Transmittance PB881 08550

SP 250-7 Radiance Temperature Calibrations PB881 23674

SP 250-8 Spectral Reflectance PB881 09905

SP 250-9 Calibration of Beta-Particle-Emitting Ophthalmic

Applicators

PB881 08535

SP 250-1 0 Radioactivity Calibrations with the "4tt" Gamma

Ionization Chamber and Other Radioactivity Calibration Capabilities

PB881 23708

SP 250-1 1 Dosimetry for High Dose Applications

PB88201587

SP 250-12 Neutron Personnel Dosimetry

PB87227617

SP 250-13 Activation Foil Irradiation with Californium

Fission Sources PB882 17443 SP 250-14 Activation Foil Irradiation by Reactor Cavity

Fission Sources PB882 17435 SP 250-15 Photometric Calibrations PB881 53747

SP 250-16 Calibration of X-Ray and Gamma-Ray

Measuring Instruments PB88211826

SP 250-17 The NBS Photodetector Spectral Response

Calibration Transfer Program

PB88201595

SP 250-18 Neutron Source Strength Calibrations PB88211818

SP 250-19 Calibration of Gamma-Ray-Emitting

Brachytherapy Sources PB891 93858

SP 250-20 Spectral Irradiance Calibrations PB88123781

SP 250-21 Calibration of Beta-Particle Radiation

Instrumentation PB88201579

SP 250-22 Platinum Resistance Thermometer Calibrations PB881 38367

SP 250-23 Liquid-in-Glass Thermometer Calibration Service PB89 128888

SP 250-24 Standard Cell Calibrations PB881 23690

SP 250-25 Calibration Service for Inductive Voltage Dividers SP 250-26 NBS Phase Angle Calibration Services PB88225636

SP 250-27 AC-DC Difference Calibrations PB892222616

SP 250-28 Solid-state DC Voltage Standard Calibrations PB881 68703

SP 250-29 Traceable Frequency Calibrations PB881 68364

SP 250-30 GOES Satellite Time Code Dissemination:

Description and Operation

PB881 68760

SP 250-31 Mass Calibrations PB89 153894

SP 250-32 A Calibration Service for 30 MHz Attenuation and Phase Shift PB88238324

SP 250-33 A Calibration Service for Voltage Transformers and High-Voltage Capacitors

PB882252903

SP 250-34 High Vacuum Standard and Its Use

PB891 93841 SP 250-35 The Calibration of Thermocouples

and Thermocouple Materials PB89209340

SP 250-36 A Calibration Service for Current Transformers PB91216770

SP 250-37 Photometric Calibrations PB97 148472 SP 250-38 NIST Leak Calibration Service PB92 149772 SP 250-39 NIST Pressure Calibration Service PB941 64043

SP 250-40 Absorbed-Dose Calibration of Ionization Chambers

in a ^°Co Gamma-Ray Beam

SN003-003-03034-1 $2.00

SP 250-41 Spectroradiometric Detector Measurements:

Part I - Ultraviolet Detectors and

Part II - Visible to Near-Infrared Detectors SN003-003-03550-5 $9.50

SP 250-42 Sprectroradiometric Detector Measurements:

Part III —Infrared Detectors SN003-003-03582-3 $5.25

SP 250-43 Radiance Temperature Calibrations SN003-003-03511-4 $10.00

SP 250-44 Radiation Processing Dosimetry Calibration Services and Measurement Assurance Program

SN003-003-03513-1

* Entries containing a stock number (SN003-003-) 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 intomation, or to place an order, call (202) 512-1800. Be sure to cite the stock number on all orders.

Entries containing PB numbers can be purchased from the National Technical Information Service, Springfield, VA 22161. NTIS will accept

American Express in addition to the payment methods listed for GPO. For more information 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 f^B numbers are in preparation.
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

SN003-003-03515-7 $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 250-51 Calibration Service of Optoelectronic

Frequency Response at 1319 nm for Combined Pfiotodiode/rf 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 Wavelengtfis Between 0.4 [xm and 1 .8 jjim SN003-003-03624-2

* Entries containing a stock number (SN003-003-) and price can be purchased from ttie Superintendent of Documents, U.S. Government

Printing Office, Wasfiington, 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 on all orders.

Entries containing PB numbers can be purchased from the National Technical Information Service, Springfield, VA 22161. NTIS will accept

American Express in addition to the payment methods listed for GPO. For more information 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 preparation.
O CO S

0) CD 9

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. o -c «^ E 32 raO