NIST Technical Note 1486

NIST Special Test Service for Four-Terminal-Pair 

Capacitance Standards from 0.01 μF to 100 μF

Svetlana Avramov-Zamurovic‡

Andrew D. Koffman†

Bryan C. Waltrip†

‡United States Naval Academy

Annapolis, Maryland, 21402 

†Quantum Electrical Metrology Division

National Institute of Standards and Technology

Gaithersburg, Maryland, 20899-8172

October 2007

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

National Institute of Standards and Technology

James M. Turner, Acting Director
ii

Table of Contents

 Page

List of Figures .......................................................................................iii

List of Tables ................................................................................................iv

1. Introduction .........................................................................................1

2. Description of Capacitance Scaling Method .....................................................2

2.1. Calibration of the Scaling Ratio ...............................................................4

2.2. Calibration of 10 nF Capacitor ..............................................................6

2.3. Calibration of 100 nF Capacitor ..............................................................6

3. Description of Graphical Control Software .....................................................7

3.1. Offset Measurements .......................................................................8

3.2. Selecting Voltage Ranges for the Capacitance Scaling Calibration .................9

4. Uncertainty Analysis for the Capacitance Scaling Measurement Procedure ...............10

4.1. Impedance Parameter Definitions ...................................................10

4.2. Capacitance Error .....................................................................12

4.3. Dissipation Factor Error .....................................................................14

5. Instrumentation Error ..............................................................................15

5.1. LCR Meter Uncertainty .....................................................................15

5.2. Automatic Capacitance Bridge Uncertainty ..........................................19

6. Capacitance Uncertainty ..............................................................................19

6.1. 10 nF Capacitance .....................................................................19

6.1.1. Additional Tests on 10 nF ...................................................25

6.2. 100 nF Capacitance .....................................................................26

6.3. 1 μF Capacitance ..............................................................................27

6.4. 10 μF Capacitance .....................................................................29

6.5. 100 μF Capacitance .....................................................................31

7. Dissipation Factor Uncertainty .....................................................................33

7.1. 10 nF Dissipation Factor .....................................................................33

7.2. 100 nF Dissipation Factor ............................................................35

7.3. 1 μF Dissipation Factor .....................................................................36

7.4. 10 μF Dissipation Factor .....................................................................36

7.5. 100 μF Dissipation Factor ............................................................37

8. Summary ................................................................................................38

9. Conclusion ................................................................................................39

10. References .......................................................................................39

Appendix A ................................................................................................41
iii

List of Figures

 Page

Figure 1. Four-terminal-pair capacitor model. .....................................................1

Figure 2. Commercial set of four-terminal-pair capacitors.............................................1

Figure 3. Capacitance scaling measurement system. ............................................3

Figure 4. Photograph of the measurement system. .....................................................4

Figure 5. Example of a user-friendly interactive window. ............................................8

Figure 6. Measurement setup for 4TP 1 nF capacitor using 3T capacitance bridge. ......21
iv

List of Tables

 Page

Table 1. Offset switch operation for the standard capacitor (STD) and the device under test 

(DUT) for different capacitance values over the frequency range of interest. ........9

Table 2. Selected test voltages for the capacitance scaling system. ........................10

Table 3. Selected test voltages for the 10 nF capacitor calibration. ........................11

Table 4. Selected test voltages for the 100 nF capacitor calibration. ........................11

Table 5. Selected test voltages for the 1 μF capacitor calibration. .................................11

Table 6. Selected test voltages for the 10 μF capacitor calibration. ........................11

Table 7. Selected test voltages for the 100 μF capacitor calibration. ........................11

Table 8. Nominal measured impedances given in ohms. ..........................................15

Table 9. Basic accuracy, A, obtained from manufacturer specifications, given in %. ......16

Table 10. Impedance proportional factor, Ka, values. ..........................................17

Table 11. Impedance proportional factor, Kb, values. ..........................................17

Table 12. LCR meter relative accuracy, AE, obtained from manufacturer specifications, 

given in %. ....................................................................................18

Table 13. Calibration accuracy, ACAL, obtained from manufacturer specifications, given in %.

................................................................................................18

Table 14. Combined standard uncertainty of the capacitance bridge as calibrated at NIST. The 

values are given in parts in 106. ............................................................19

Table 15. Actual LCR meter deviations when measuring 100 pF and 1 nF capacitors for 10 nF 

scaling ratio calibration. .....................................................................23

Table 16. Combined standard uncertainty of scaling ratio for 10 nF capacitor, in parts in 106.

................................................................................................23

Table 17. Uncertainty contributions for 10 nF capacitor, in parts in 106. ........................24

Table 18. Combined standard uncertainty of 10 nF capacitance. .................................25

Table 19. Frequency dependence of the 1 nF standard capacitor (nitrogen dielectric) measured 

using the NIST 4TP Bridge, in parts in 106. ..........................................25

Table 20. 10 nF capacitor measurements at several frequencies, in parts in 106. ...............25

Table 21. LCR meter deviations when measuring the 10 nF and 100 nF capacitors for the 

100 nF calibration. .....................................................................26

Table 22. Uncertainty contributions for the 100 nF capacitor, in parts in 106. ...............27

Table 23. Combined standard uncertainty of 100 nF capacitance. .................................27

Table 24. LCR meter deviations when measuring the 100 nF and 1 μF capacitors for the 1 μF

calibration. ..............................................................................28

Table 25. Uncertainty contributions for the 1 μF capacitor, in parts in 106.........................28

Table 26. Capacitance uncertainty for the 100 nF capacitor, in parts in 106. ...............28

Table 27. Combined standard uncertainty of 1 μF capacitance. .................................29

Table 28. LCR meter deviations when measuring the 1 μF and 10 μF capacitors for the 10 μF

calibration. ..............................................................................30

Table 29. Uncertainty contributions for the 10 μF capacitor, in parts in 106. ...............30

Table 30. Uncertainty contributions for the 10 μF capacitor in, parts in 106. Note that there are 

100 nF and 1 μF calibration steps as part of the 10 μF calibration. ...............30
v

List of Tables, cont’d

 Page

Table 31. Combined standard uncertainty of 10 μF capacitance. .................................31

Table 32. LCR meter deviations when measuring the 100 μF and 10 μF capacitors for the 

100 μF calibration. .....................................................................32

Table 33. Uncertainty contributions for the 100 μF capacitor, in parts in 106. ...............32

Table 34. Uncertainty contributions for the 100 μF capacitor, in parts in 106, cont’d. ......32

Table 35. Combined standard uncertainty of 100 μF capacitance. .................................33

Table 36. Summary of expanded capacitance uncertainty (k = 2), in parts in 106. ...............33

Table 37. Uncertainty contributions for the scaling ratio phase, in μrad. ........................34

Table 38. Combined standard uncertainty of 10 nF capacitor dissipation factor, in μrad.

................................................................................................35

Table 39. Combined standard uncertainty of 100 nF capacitor dissipation factor, in μrad.

................................................................................................36

Table 40. Combined standard uncertainty of 1 μF capacitor dissipation factor, in μrad.

................................................................................................36

Table 41. Combined standard uncertainty of 10 μF capacitor dissipation factor, in μrad.

................................................................................................37

Table 42. Combined standard uncertainty of 100 μF capacitor dissipation factor, in μrad.

................................................................................................37

Table 43. Summary of expanded dissipation factor uncertainties (k = 2), in μrad. ...............37

Table 44. Expanded capacitance uncertainties (k = 2), when the standard deviations of the 

measured reference capacitors are used as the Type A component, in parts in 106.

................................................................................................38

Table 45. Standard deviations of the measured reference capacitors, in parts in 106. ......38

Table 46. Average measured capacitance values for the set of ceramic standards. ................41

Table 47. Average measured dissipation factor values for the set of ceramic standards, in μrad.

................................................................................................41
1

1. Introduction

Precision LCR (impedance) meters typically operate with a four-terminal-pair (4TP) 

configuration and are calibrated using 4TP capacitance standards. The measurement system 

described herein has been developed in response to a need for improved LCR meter calibrations. 

Figure 1. Four-terminal-pair capacitor model.

Figure 2. Commercial set of four-terminal-pair capacitors.

Figure 1 shows a simple circuit model for a 4TP capacitance standard. A capacitance scaling 

method is used to calibrate 4TP standard capacitors of values from 10 nF to 100 μF. Aoki and 

Yokoi introduced this technique in 1997 [1]. References [1] and [2] describe the general method 

and provide a detailed uncertainty analysis. Aoki and Yokoi developed a calibration procedure 

LCUR LPOT HPOT HCUR

CDUT
2

based on reference [1]. Figure 2 shows a set of commercial standards** to which this calibration 

system may be applied. Note that convention allows for the standards to be identified with units 

of nF or μF. Therefore, the 0.01 μF standard is also referred to as the 10 nF standard and the 

0.1 μF standard is referred to as the 100 nF standard. Note, also, that the typical commercial set 

of 4TP ceramic capacitance standards contains values from 10 nF to 10 μF. The 100 μF capacitor 

must be obtained separately. 

The capacitance scaling system uses a commercial automatic capacitance bridge to measure 

reference points, and it uses an LCR meter along with a single-decade inductive voltage divider 

(IVD) and interface circuitry (together called Scaling Fixture) as an impedance comparator. The 

capacitance scaling system measures 4TP capacitors in decade (10:1) steps from 10 nF to 

100 μF. The measurements are performed at frequencies of 100 Hz, 1 kHz, 10 kHz, and 

100 kHz. 

The initial procedure used at NIST with the capacitance scaling system was described in 

reference [3]. This Technical Note more fully describes the method, modified procedure, and 

error analysis.

2. Description of Capacitance Scaling Method

Modern instrumentation is designed for automated control and can be used to create custom 

calibration procedures. In the case when very precise and specialized tests are necessary for a 

metrology application, it is particularly challenging to establish computer control of an entire 

procedure. Accurate calibration of capacitors that range in value from 0.01 μF to 100 μF over the 

frequency range from 100 Hz to 100 kHz is desired. 

There are several instruments available commercially to measure the impedance of a capacitor. 

LCR meters are general impedance-measuring instruments that have limited uncertainty, while 

automatic capacitance bridges are available commercially with very low uncertainties but with 

more limited measurement ranges. 

An automatic capacitance bridge is very convenient for measuring three-terminal (3T) standard 

capacitors with precision, reliability, and uncertainty at metrological levels. Measurement errors 

are on the order of parts in 106. The measurement uncertainty depends directly on the standard 

capacitor that resides in one branch of the bridge as well as on the inductive voltage divider used 

to scale the measured capacitance to that standard. The limitations of the automatic bridge 

include a limited frequency band and limited capacitance range. A single-frequency commercial 

capacitance bridge has been available for several years. Recent versions allow measurement 

from near dc to 20 kHz. The capacitance measurement upper limit is typically around 1 F but 

measurement uncertainties grow significantly near the upper limit.

Since the capacitance bridge is a 3T instrument and the measured capacitors and the LCR meter 

are 4TP, the 3T-to-4TP conversion must be addressed. The details of the measurement setup and 

 

** Identification of commercial products does not imply endorsement by the U.S. Government, nor does it imply that such 

products are necessarily the best available.
3

their analysis are given in section 6 (see Figure 6). The study showed negligible influence of the 

lead impedance on the capacitance measurements. 

LCR meters that operate in a four-terminal-pair configuration have been widely available for 

many years. They operate on the principle of sourcing a known current through unknown 

impedance and measuring the voltage drop. The impedance is determined based on the 

magnitude and phase of the known current and the measured voltage. Instruments typically 

operate over a broad frequency range (up to MHz) and can measure inductors, capacitors, and 

resistors with a wide range of values and losses. Uncertainties are typically on the order of 

fractions of a percent for the most accurate meters. 

Both the automatic capacitance bridge and the LCR meter play major roles in evaluating

capacitors. The goal is to calibrate capacitors up to 100 F over the frequency range from near 

dc to 100 kHz. The reference measurements are made using an automatic capacitance bridge. All 

of the remaining measurements are performed using the LCR meter and are traceable to the 

reference measurements.

Figure 3. Capacitance scaling measurement system.

The goal of the newly developed measurement procedures is to use commercially available 

instrumentation as much as possible and design specialized equipment so it is easily integrated 

into an automatic measuring system.

The capacitance scaling system consists of an LCR meter, an automatic capacitance bridge, and a 

custom-made interface (Scaling Fixture), as shown schematically in Fig 3. Figure 4 shows a

photograph of the measurement system.

Capacitors calibrated using this procedure have values of 10 nF, 100 nF, 1 μF, 10 μF and 100 μF. 

The scaling fixture uses a fixed 10:1 ratio IVD. Reference 100 pF and 1 nF standard capacitors 

are used to calibrate the scaling fixture and the 1 nF reference is also used to begin the 

Scaling Fixture

LCR Meter Interface

4TP

Capacitor

DUT

IVD

Capacitance 

Bridge Control 

Software

PC
4

Power Supply for Scaling Fixture

Capacitance 

Bridge

LCR Meter

Scaling Fixture

Device Under Test

Computer with

Control Software

Figure 4. Photograph of the measurement system.

scaling process. The fixed ratio IVD limits the scaling system to the measurement of standards 

that are nominally 1, 10, or 100 in value.

The scaling fixture controls the circuitry used in switching measurement instruments, 

measurement ratios (1:1 and 10:1), high-frequency and low-frequency IVDs, offset 

correction configurations, and signal levels. The LCR meter and scaling fixture combination 

is self-calibrated by comparing calibrated capacitors. This essentially establishes the ratio of 

the scaling fixture as a 1:1 and 10:1 interface. 

2.1. Calibration of the Scaling Ratio

Four measurements are required to calibrate the 10:1 scaling ratio of the scaling fixture circuitry. 

The measurement process is controlled by software that prompts the user to set the appropriate 

switches on the measurement interface to achieve the proper settings during measurement.

First, a 100 pF standard capacitor, Z100 pF, is measured using a 3T capacitance bridge, Z100 pF_CB,

shown in Eq. (1). The capacitor measured is a four-terminal-pair device and the capacitance 

bridge is 3T, so the scaling fixture provides an appropriate 4TP-to-3T connection switch (see the 

discussion associated with Figure 6). 

The scaling fixture switches are then configured so that the capacitance standard is connected to 

the LCR meter. The impedance of this standard is then measured using the LCR meter with the 

10:1 IVD switched into the circuit, yielding Z100 pF_LCR, as in Eq. (2). The IVD makes the 100 pF 

capacitor appear to the LCR meter to have the impedance of a 1 nF capacitor.
5

As an example, a 100 pF capacitor measured at 1 kHz has an impedance of 

1591.55kΩ

2 (10 )(10 )

1

Z100 pF 3 10

. A 1 nF capacitor has an impedance of 159.155 k at 

1 kHz. 

The 100 pF capacitor is measured with the scaling fixture set to the ratio of 0.1 and the 1 nF 

capacitor is measured with the scaling fixture set to the ratio of 1.0. Connecting the 100 pF 

capacitor to the LCR meter through the 10:1 IVD reduces the voltage across the voltage ports of 

the LCR meter and thus causes the LCR meter to see the same nominal impedance value as is 

produced by the 1 nF capacitor connected to the LCR meter without the IVD. In the derivation of 

the scaling ratio, below, we use the notation K1 nF_LCR to describe the scale factor of the LCR 

meter when measuring 1 nF.

The impedance of the 1 nF capacitor, Z1nF, is measured using the LCR meter with the scaling 

ratio set to 1.0, yielding Z1 nF_LCR, shown in Eq. (3). And finally, the impedance of the 1 nF 

standard is measured using the capacitance bridge, yielding Z1 nF_CB, shown in Eq. (4). 

The scaling ratio, K, is calculated from the described measurements. The ratio of impedances 

measured using the LCR meter is equal to the ratio of impedances derived from the capacitance 

bridge measurements multiplied by the scaling ratio, represented in Eq. (5). This provides the 

basis to compute the scaling ratio, as shown in Eq. (6).

Z100 pF _CB Z100 pF

(1)

Z100 pF_ LCR K Z100 pF K1 nF _ LCR

(2)

Z1 nF _ LCR 0 Z1 nF K1 nF _ LCR 1.

(3)

Z1 nF _CB Z1 nF

(4)

CB

CB

LCR

LCR

Z

Z

K

Z

Z

1 nF _

100 pF _

1 nF _

100 pF _

(5)

LCR

LCR

CB

CB

Z

Z

Z

Z

K

1 nF _

100 pF _

100 pF _

1 nF _

(6)

Equations (7) and (8) represent formulas used to estimate the in-phase and quadrature 

components of the scaling ratio. Empirically, the quadrature component is on the order of a few 

μrad. Since the angle is small, the imaginary component represents the phase of the scaling ratio. 

The in-phase component is simply called the scaling ratio.

Kratio Re K

(7)
6

K Im K

(8)

A detailed explanation of scaling ratio measurements will be presented in the Section 6.

2.2. Calibration of 10 nF Capacitor

Once the scaling ratio, K, is determined, the system can be used to calibrate capacitors. The 

calibration of the 10 nF capacitor will be presented in sufficient detail here. More information 

will be presented in the section 6.

A 1 nF reference capacitor, Z1 nF, is measured using the capacitance bridge, yielding Z1 nF_CB,

shown in Eq. (9), and then is measured using the LCR meter with the scaling ratio, K, set to 0.1, 

yielding Z1 nF_LCR, shown in Eq. (10). Then, a 10 nF device under test, ZDUT_10 nF, is measured 

using the LCR with the scaling ratio, K, set to 1.0, yielding Z10 nF_LCR, represented in Eq. (11). 

These measurements are used to find the impedance of the device under the test, as in Eqs. (12) 

and (13).

Z1 nF _CB Z1 nF

(9)

Z1 nF _ LCR K Z1 nF K10 nF _ LCR

(10)

10 nF _ 0 10 nF _ _10 nF 1. Z LCR K LCR ZDUT

(11)

CB

DUT

LCR

LCR

K Z

Z

Z

Z

1 nF_

_10 nF

1 nF_

10 nF_

(12)

CB

LCR

LCR

DUT Z

Z

Z

Z K 1 nF _

1 nF _

10 nF _

_10 nF

(13)

The goal of this calibration process is to calibrate the 10 nF capacitor by using a known 1 nF 

capacitor. Comparison of the measurement of the 1 nF standard with the scaling ratio and the 

measurement of the 10 nF standard without the scaling ratio, factoring the scaling calibration 

back into the equation, allows the accurate determination of the 10 nF standard capacitor.

2.3. Calibration of 100 nF Capacitor

The next capacitor, 100 nF, is measured by using the result of the 10 nF, and making the same 

two sets of measurements mentioned above (Eqs. (11) and (12)). The 10 nF capacitor, ZDUT_10 nF, 

is measured using the LCR meter, yielding Z10 nF_LCR, and then the 100 nF capacitor, ZDUT_100 nF, 

is measured using the LCR meter, yielding Z100 nF_LCR. The result is obtained using Eq. (14).

_10 nF

10 nF _

100 nF _

_100 nF DUT

LCR

LCR

DUT Z

Z

Z

Z K

(14)
7

This same process is repeated to characterize all of the other capacitors in the set. It is important 

to mention that all of the errors are accumulated during this bootstrap process. 

There are several measuring methods that are available for calibrating 1 nF capacitors. They 

include calibration using a vector network analyzer in combination with a capacitance bridge. 

This method covers a very wide frequency range (up to 10 MHz) and is complex to perform. Its 

expanded uncertainty with k = 2 is on the order of 25 parts in 106at 100 kHz. A multi-frequency 

automatic capacitance bridge may be used to measure a 1 nF capacitor up to 20 kHz with an 

expanded uncertainty on the order of 10 parts in 106at 10 kHz. In the field of metrology, it is 

important to independently verify results. Since the proposed capacitance scaling method derives 

the values from a known standard that is traceable to the U.S. representation of the farad, it is 

considered a valid and reliable measurement procedure. Another important property of the 

described method is that it uses the ratio of measurements to establish the result. The LCR meter 

is an instrument that produces repeatable results with good linearity but has significant 

systematic errors. Since the impedances measured using the LCR meter are compared in a ratio 

format, the effects of systematic errors are reduced, providing a lower measurement uncertainty 

than taking absolute measurements. 

This concludes the theoretical presentation of the capacitance scaling method. In practice, there 

are several additional test method considerations that are necessary to improve the estimation of 

the scaling factor, K, and the estimation of the capacitance of the device under test. These will be 

discussed in section 3.

3. Description of Graphical Control Software

A control software program was written with an extensive set of on-line instructions in order to 

make the calibration process user-friendly (see Figure 5). The program will be presented step by 

step. 

The first step is the scaling ratio calibration. It is recommended that this calibration be performed 

before each calibration of a capacitor set, in order to minimize the measurements errors. 

The program creates one file to store the measurements and another file where the report 

summarizing the calibration results will be stored. All of the file names are unique and include 

the capacitor value and the date and time of calibration. Extra care is introduced in order to keep 

the serial numbers of the calibrated devices in order. This is done in order to organize the 

measurements in unique storage and report files.

It is not possible to accurately provide a scaling ratio over the frequency range from 100 Hz to 

100 kHz with a single transformer. Therefore, the system is implemented using a low-frequency 

transformer operating from 100 Hz to 10 kHz, and a high-frequency transformer operating from 

10 kHz to 100 kHz. The software program prompts the user to select the appropriate transformer

by setting switches on the interface.

The user is guided through the calibration process with prompts on the computer screen. The 

procedure pauses until the user responds to the prompt. 
8

Figure 5. Example of a user-friendly interactive window.

All instrumentation control is performed via software. The sequencing of the measurements 

taken, the specific values for the voltages applied, and the appropriate frequencies are all 

prescribed and controlled without user intervention. Also, data averaging and formatting is 

performed, with averages and standard deviations recorded for further analysis.

Another program has been developed to perform the actual capacitor calibration. It requires 

selection of a file containing the scaling ratio information.

3.1. Offset Measurements

There are several additional details associated with the capacitance scaling procedure. For higher 

values of capacitance at higher frequencies, the offset voltage of the LCR meter high potential

port measurement circuitry is measured by shorting the high and low potential ports. 

Without offset corrections, capacitance scaling calculation consists of combining four quantities:

the known standard impedance, ZSTD, the scaling ratio, K, and the LCR meter measurements of

the device under the test and the standard impedance, ZDUT_LCR and ZSTD_LCR. In equation form, 

the impedance of the device under test, ZDUT, is

STD

STD LCR

DUT LCR

DUT Z

Z

Z

Z K

_

_

. (15)
9

Equation (15) is the general representation of the capacitance scaling method and straightforward 

modification is applied to incorporate offset corrections. The modified equation is

.

(0.1)

_ _

_ _

STD

STD LCR STD OFFSET

DUT LCR DUT OFFSET

DUT Z

Z Z

Z Z

Z K

(16)

The scaling fixture allows a direct measurement of the offset with the LCR meter in the same 

measurement state as for the measurement of the standard and device under test. The (0.1) 

multiplier is an approximation of the scaling ratio, K, and is used to scale the offset 

measurement.

The software prompts the user to throw the necessary switches on the scaling fixture at 

appropriate times to measure the offset (see Table 1). The software also incorporates the offset 

corrections into the calculation of the impedance. 

Table 1. Offset switch operation for the standard capacitor (STD) and the device under 

test (DUT) for different capacitance values over the frequency range of interest.

Frequency

Capacitance 100 Hz 1 kHz 10 kHz 100 kHz

STD 1 nF off off off off

DUT 10 nF off off off on

STD 10 μF off off off on

DUT 100 nF off off on on

STD 100 nF off off on on

DUT 1 μF off on on on

STD 1 μF off on on on

DUT 10 μF on on on on

STD 10 μF on on on N/A

DUT 100 μF on on on N/A

3.2. Selecting Voltage Ranges for the Capacitance Scaling Calibration

The LCR meter has limited current and voltage capability when measuring the impedance of the 

device under test (DUT). In practice, the LCR meter limits the test voltage applied to the DUT to 

1 Vrms and the current to 20 mArms. The ALC function (Automatic Level Control) allows the 

control of the amplitude of the applied voltage. The reason for using this command is that 

without it, even when the user selects a 1 Vrms test voltage, the instrument lowers that voltage to 

optimize the current. This change is performed without specifically informing the user. This 

results in uncontrolled application of the applied voltage to the capacitor under test. Proper 

determination of the capacitance requires knowledge of the test voltage. When ALC is ON, the 

requested voltage is guaranteed. If the instrument can not achieve the requested voltage, the 

information is sent to the user. 

For higher values of capacitance, the LCR meter allows the ALC to operate only as long as 

AUTO is selected for the impedance range. Table 2 shows the test voltages used in the scaling 
10

procedure. The goal for selection was to apply as high a test voltage as possible while 

maintaining a small standard deviation of measurement and allowing ACL to be ON.

Table 2. Selected test voltages for the capacitance scaling system.

Frequency

Capacitance 100 Hz 1 kHz 10 kHz 100 kHz

10 nF 600 mV 600 mV 600 mV 600 mV

100 nF 600 mV 600 mV

A(10k)

600 mV

A(1k)

200 mV

A(100)

1 F

600 mV

A (1k)

600 mV 

A (100)

200 mV

A (100)

20 mV

A(10)

10 F

600 mV 

A (100)

200 mV 

A (100)

20 mV

A (10)

ALC OFF 5mV 

A (10)

100 F

200 mV 

A (100)

20 mV

A (10)

ALC OFF 5 mV 

A (10) N/A

Letter A in the table shows when AUTO impedance range must be selected, allowing the meter 

to select the impedance measurement range. The impedance range actually used is shown in 

parentheses. This information is necessary because the offset measurements must be performed 

at the same operating range and applied voltage to minimize linearity errors of the meter. The 

ALC feature on the LCR meter is not operational for the 5 mV voltage range. Such a small 

voltage is applied due to the fact that the measured impedance at high frequencies and high 

values of capacitance is less than 1 Ω.

Tables 3-7 present the combinations of frequencies, capacitance values and voltage settings 

incorporated into the control software routines that calibrate each capacitor. For example, the 

100 nF capacitor calibration is performed in three steps (see Table 4). The first step is the 

calibration of the scaling ratio at 600 mV for frequencies of 100 Hz, 1 kHz, and 10 kHz, and at 

200 mV for 100 kHz. The second step is the calibration of the 10 nF capacitor at the same 

voltage settings. The final step is the calibration of the 100 nF capacitor at the same settings. 

Data is collected at each intermediate step in the scaling process as well as at the final step.

Table 3. Selected test voltages for the 10 nF capacitor calibration.

Frequency

Step 100 Hz 1 kHz 10 kHz 100 kHz

1. Scaling ratio for 10 nF 600 mV 600 mV 600 mV 600 mV 2. 10 nF calibration

Table 4. Selected test voltages for the 100 nF capacitor calibration.

Frequency

Step 100 Hz 1 kHz 10 kHz 100 kHz

1. Scaling ratio for 100 nF

2. 10 nF calibration for 100 nF 600 mV 600 mV 600 mV 200 mV

3. 100 nF calibration
11

Table 5. Selected test voltages for the 1 μF capacitor calibration.

Frequency

Step 100 Hz 1 kHz 10 kHz 100 kHz

1. Scaling ratio for 1 μF

600 mV 600 mV 200 mV 20 mV 2. 10 nF calibration for 1 μF

3. 100 nF calibration for 1 μF

4. 1 μF calibration

Table 6. Selected test voltages for the 10 μF capacitor calibration.

Frequency

Step 100 Hz 1 kHz 10 kHz 100 kHz

1. Scaling ratio for 10 μF

600 mV 200 mV 20 mV 5 mV

2. 10 nF calibration for 10 μF

3. 100 nF calibration for 10 μF

4. 1 μF calibration for 10 μF

5. 10 μF calibration

Table 7. Selected test voltages for the 100 μF capacitor calibration.

Frequency

Step 100 Hz 1 kHz 10 kHz

1. Scaling ratio for 100 μF

200 mV 20 mV 5 mV

2. 10 nF calibration for 100 μF

3. 100 nF calibration for 100 μF

4. 1 μF calibration for 100 μF

5. 10 μF calibration for 100 μF

6. 100 μF calibration

This set of tables shows all the intermediate steps for which data are collected and the same 

organization will be followed in section 6. Separate determinations of the scaling ratio, K, are 

necessary for the different calibrated capacitance values because the applied voltages are 

different for the different capacitance values. 

4. Uncertainty Analysis for the Capacitance Scaling Measurement Procedure

4.1. Impedance Parameter Definitions

Impendence, Z, is defined as the ratio of voltage, V, to current, I,

R j

X

j

R jX R X e Me

I

V

Z

arctan( )

2 2

, (17)

where R and X are the real part (resistance) and imaginary part (reactance) of the impedance, 

respectively. M is the magnitude and is the phase of the impedance vector.
12

Impedance requires two parameters for full characterization. In this presentation, the capacitors 

are modeled as the parallel combination of resistance, RP and capacitance, CP. These definitions 

are related in the following manner. 

,

1 1

1

1

1 1

1

2

2

2

2

R jX

C R

C R

j

C R

R

C R

R j C R

j C R

R

j C

R

j C

R

Z

P P

P P

P P

P

P P

P P P

P P

P

P

P

P

P

(18)

where is the angular frequency. 

In the capacitance scaling measurement procedure, the LCR meter is used to measure R and X, 

and the capacitance bridge is used to measure capacitance, CP, and dissipation factor, DF. The 

following formulas are used to find capacitance and dissipation factor based on the LCR meter 

readings.

sin 1

1

Im

1 1

M

Z

j C C

Z R

P P

P

(19)

X RPCP

R

DF 1

(20)

To convert the capacitance bridge readings to impedance, Eq. (21) is used

DF

j

C DF

Z

P

1

1

1

. (21)

4.2. Capacitance Error

Equations (22) through (25) are used to derive the capacitance uncertainties from the measured 

impedance represented in Eq. (19). Partial derivatives of the parallel capacitance are used to 

determine the uncertainty components. Symbol ∆ denotes the variable with respect to which the 

derivative is taken.

sin 1sin 1sin 1

M

M

M M M

CP

(22)

cos

1

sin 1sin 1

2 2 M

M

M M

CP

(23)
13

Partial derivatives are used throughout this document to derive uncertainty. Equation (24) shows 

how uncertainties are determined from Eq. (23).

2

2

2

2

2

2

2

2

cos

1

sin 1sin 1u

M

u M

M

u

M

uc CP

(24)

Relative uncertainty for the parallel capacitance measurement circuit is expressed as

2 2 2

sin

cos

u

M

u u M

C

u C

P

c P

(25)

Since we are measuring capacitors, phase is very close to 90°. The largest deviations from 

90° measured for this investigation are on the order of 0.05°. The multiplier 

sin

cos

is then 

0.001, leading to a negligible phase contribution to the capacitance uncertainty. 

The relative frequency error 

u

is insignificant since the instrumentation used incorporates a 

crystal oscillator with a drift rate of less than 0.1 part in 106over the observation time. Since the 

capacitance scaling procedure uses a ratio of measurements, the systematic error in frequency 

will not have significant impact. 

Therefore the relative error in capacitance is approximated with

M

u M

C

u C

P

c P

. (26)

In the capacitance scaling procedure, the impedance of a capacitor is measured as

STD

STD LCR

DUT LCR

DUT Z

Z

Z

Z K

_

_

. (27)

When partial derivatives are applied to this formula, the following result is obtained:

STD

STD

STD LCR

DUT LCR

STD LCR

DUT LCR

DUT

DUT

Z

Z

Z

Z

Z

Z

K

K

Z

Z

_

_

_

_

(28)
14

The factor 

STD LCR

DUT LCR

Z

Z

_

_

is treated as a single contribution since the ratio of measured impedances is 

very close to one and they are both measured using the same instrument. Equation (29) shows 

that the relative errors subtract.

STD LCR

STD LCR

DUT LCR

DUT LCR

STD LCR

DUT LCR

STD LCR

DUT LCR

Z

Z

Z

Z

Z

Z

Z

Z

_

_

_

_

_

_

_

_

(29)

This component remains for consideration because the standard and DUT measurements are not 

exactly the same and the uncertainty contribution due to this factor will be proportional to the 

measured difference.

Partial derivatives are investigated to find the uncertainty of the scaling ratio, K (see Eq. (6)). 

C B

C B

C B

C B

LCR

LCR

C B

C B

LCR

LCR

C B

C B

LCR

LCR

LCR

LCR

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

K

100 pF _

1 nF _

100 pF _

1 nF _

1 nF _

100 pF _

100 pF _

1 nF _

1 nF _

100 pF _

100 pF _

1 nF _

1 nF _

100 pF _

1 nF _

100 pF _





(30)

CB

CB

CB

CB

LCR

LCR

LCR

LCR

Z

Z

Z

Z

Z

Z

Z

Z

K

K

100pF _

1nF _

100pF _

1nF _

1nF _

100pF _

1nF _

100pF _

(31)

Equation (31) will be used to estimate the uncertainty of the scaling ration, K.

4.3. Dissipation Factor Error

The derivation of the dissipation factor error shown below follows from Eq. (20). The dissipation 

factor is given in radians and is commonly converted to μrad (10-6×radians).

X

X

R

R

X

X

X

R

X

R

X

R

R

DF 2

1

(32)

X

X

DF

R

R

DF DF

(33)
15

The measured dissipation factor is very small for the capacitors addressed in this investigation. 

The values range from about 1 μrad at low frequency for small capacitors, to 1000 μrad at higher 

frequencies for larger capacitors. Since for this capacitance calibration procedure the dissipation 

factor is very small, the following approximation holds:

X

R

X

R

arctan

and DF≈ (see

Eq. (17)). This notion provides for simple phase calculations. From Eq. (27), the phase and 

dissipation factor estimates for the device under test are given by Eqs. (34) and (35).

Z STD

Z

Z DUT K Z

STD LCR

_ DUT LCR _

_

_

(34)

Z STD

Z

DFZ DUT DFK DFZ DF

STD LCR

_ DUT LCR _

_

_

(35)

5. Instrumentation Errors

5.1. LCR Meter Uncertainty

An LCR meter is used to measure the impedance of the capacitors. Table 8 presents the nominal 

measured impedance, |Z|. This information is important to establish the range of values and relate 

the uncertainty of the instrument to the measurements. 

Table 8. Nominal measured impedances given in ohms.

Frequency

Capacitance 100 Hz 1 kHz 10 kHz 100 kHz

1 nF 1591549.431 159154.943 15915.494 1591.549

10 nF 159154.943 15915.494 1591.549 159.155

100 nF 15915.494 1591.549 159.155 15.915

1 μF 1591.549 159.155 15.915 1.592

10 μF 159.155 15.915 1.592 0.159

100 μF 15.915 1.592 0.159 N/A

Information for the LCR meter error analysis is taken from the LCR meter manual [7]. The 

manual provides information only on measurement accuracy. The accuracy refers to the 

manufacturer specifications and will be used as an uncertainty component for the LCR meter 

throughout this document. As given in the LCR meter manual, absolute measurement accuracy, 

AT, when measuring the magnitude of the impendence of a capacitor (in %), is 

AT AE ACAL

, (36)

where AE is the relative accuracy and ACAL is the calibration accuracy. The absolute measurement 

accuracy, AT, will be included as a factor in the computation of the capacitance and phase 

measurement uncertainties when using the LCR meter. 
16

Relative accuracy, AE , given in %, is defined as

AE A Ka Kaa KbKbb Kc Kd Ke[ ( )100 ] , (37)

where the terms of the equation are

A Basic accuracy

Ka High impedance proportional factor 

Kaa Cable length factor for high impedance proportional factor

Kb Low impedance proportional factor 

Kbb Cable length factor for low impedance proportional factor

Kc Calibration interpolation factor 

Kd Cable length factor

Ke Temperature factor

Table 9 shows the basic accuracy of the LCR meter [7]. The capacitance scaling measurement 

procedure uses the MEDIUM option for the measurement time parameter (this information is 

necessary to determine basic accuracy contribution for each capacitance value and frequency). 

Table 9. Basic accuracy, A, obtained from manufacturer specifications, given in %.

DUT 

Capacitor

Capacitance

Scaling

Measurements

Frequency

100 Hz 1 kHz 10 kHz 100 kHz

10 nF 1 nF 0.2 0.05 0.05 0.05

10 nF 0.1 0.05 0.05 0.05

100 nF

1 nF 0.2 0.05 0.05 0.07

10 nF 0.1 0.05 0.05 0.07

100 nF 0.1 0.05 0.05 0.07

1 μF

1 nF 0.2 0.05 0.07 0.1

10 nF 0.1 0.05 0.07 0.1

100 nF 0.1 0.05 0.07 0.1

1 μF 0.1 0.05 0.07 0.1

10 μF

1 nF 0.2 0.07 0.1 0.6

10 nF 0.1 0.07 0.1 0.6

100 nF 0.1 0.07 0.1 0.6

1 μF 0.1 0.07 0.1 0.6

10 μF 0.1 0.07 0.1 0.6

100 μF

1 nF 0.14 0.1 0.6

N/A

10 nF 0.14 0.1 0.6

100 nF 0.14 0.1 0.6

1 μF 0.14 0.1 0.6

10 μF 0.14 0.1 0.6

100 μF 0.14 0.1 0.6
17

The software program that controls the LCR meter is set up to perform MEDIUM integration, so 

the following formulas apply for the impedance proportional factors.

m s

a

Z V

K

200 10.001

and

s

b m

V

K Z x

70 1 10 1

9

, (38)

where Vs is the test signal voltage given in mV. |Zm| is the impedance of the measured capacitor 

in ohms and is equal to

fC

Zm

2

1

.

Ka is negligible for impedances above 500 Ω and Kb is negligible for impedances below 500 Ω. 

The results are summarized in the following tables. The Ka values are dominant in the case of 

high frequencies and high values of capacitance.

Table 10. Impedance proportional factor, Ka, values.

Frequency

Capacitance 100 Hz 1 kHz 10 kHz 100 kHz

1 nF 0 0 0 0

10 nF 0 0 0 0

100 nF 0 0 0 0.0001

1 μF 0 0 0.0001 0.0057

10 μF 0 0.0001 0.0057 0.2576

100 μF 0.0001 0.0057 0.2576 N/A

Table 11. Impedance proportional factor, Kb, values.

Frequency

Capacitance 100 Hz 1 kHz 10 kHz 100 kHz

1 nF 0.0018 0.0002 0 0

10 nF 0.0002 0 0 0

100 nF 0 0 0 0

1 μF 0 0 0 0

10 μF 0 0 0 0

100 μF 0 0 0 N/A

The cable length factor, Kaa = 1, and the cable length factor, Kbb = 1 + 5f. Parameter f is the 

frequency in kHz. In our case, the capacitor is connected to the LCR meter via a 0.61 m cable.

The calibration interpolation factor, Kc = 0, because all of the measurements are taken at direct 

calibration frequencies. 

The cable length factor, Kd = 0.00025(1 + 50fm), for the calibration interpolation factor is given 

for a cable length of 1 m. Frequency, fm, is entered in units of MHz, so the cable length factor is 

approximately 0.00025.
18

The temperature factor, Ke = 1. In our case, the temperature in the laboratory is 23 °C and Ke = 1 

for the temperatures from 18 °C to 28 °C. When all of the manufacturer accuracy factors are 

combined, the LCR meter relative accuracy is obtained (see Table 12). Table 13 shows the LCR 

meter calibration accuracy.

Table 12. LCR meter relative accuracy, AE, obtained from

manufacturer specifications, given in %.

DUT 

Capacitor

Capacitance

Scaling

Measurement

Frequency

100 Hz 1 kHz 10 kHz 100 kHz

10 nF 1 nF 0.3781 0.0681 0.0522 0.0519

10 nF 0.1180 0.0521 0.0506 0.0524

100 nF

1 nF 0.3781 0.0681 0.0522 0.0719

10 nF 0.118 0.0521 0.0506 0.0728

100 nF 0.102 0.0505 0.0512 0.0841

1 μF

1 nF 0.3781 0.0681 0.0726 0.1030

10 nF 0.1180 0.0521 0.0707 0.1072

100 nF 0.1020 0.0505 0.0717 0.1581

1 μF 0.1005 0.0511 0.0829 0.6670

10 μF

1 nF 0.3781 0.0919 0.1068 0.6077

10 nF 0.118 0.0724 0.1016 0.6276

100 nF 0.102 0.0706 0.1061 0.8591

1 μF 0.1005 0.0715 0.1569 3.1776

10 μF 0.1011 0.0828 0.6659 26.3626

100 μF

1 nF 0.3552 0.161 0.6257

N/A

10 nF 0.1617 0.1064 0.6055

100 nF 0.1424 0.1014 0.6264

1 μF 0.1405 0.106 0.858

10 μF 0.1411 0.1568 3.1765

100 μF 0.1528 0.6657 26.3614

The factor, f, in Table 13 is frequency given in kHz. The magnitude of the calibration accuracy is 

given in percent. Note that only amplitude calibration accuracy is taken into consideration since it 

has the predominant influence on the uncertainty of the capacitors calibrated, as seen in Eq. (26).

Table 13. Calibration accuracy, ACAL, obtained from manufacturer specifications, given in %.

Frequency

Capacitance 100 Hz 1 kHz 10 kHz 100 kHz

1 nF 0.03+1×10-3f 0.03+1×10-3f 0.03+1×10-3f 0.05

10 nF 0.03+1×10-3f 0.03+1×10-4f 0.03 0.05

100 nF 0.03+1×10-4f 0.03 0.05 0.05

1 μF 0.03 0.03 0.05 0.05

10 μF 0.03 0.03 0.05 0.05

100 μF 0.03 0.03 0.05 N/A
19

LCR meter measurements are used in ratios in the capacitance scaling calculations. Because the 

scaling fixture allows the LCR meter to measure two very similar impedances, ZDUT and ZSTD, the 

manufacturer specified accuracy was modified in the following way to include differential 

nonlinearity, which comes into play when the two measurements are not the same magnitude, as 

given in Eq. (39). Throughout the rest of this document, the differential nonlinearity given in 

Eq. (39) will be included when estimating the measurement uncertainty. It will be called the 

LCR meter nonlinearity, u(LCR_NL). The LCR meter nonlinearity is the dominant factor in the 

measurement uncertainty of low impedance capacitors.

STD

E CAL DUT STD

Z

A A Z Z

u LCR NL

3

( )

_ (39)

Treating the LCR meter maximum uncertainty as a Type B uncertainty, the standard uncertainty 

estimate must be divided by 

3

to comply with the guidelines for the expression of 

measurement uncertainty [9].

5.2. Automatic Capacitance Bridge Uncertainty

The capacitance scaling system uses a commercial automatic capacitance bridge [8]. This 

instrument was calibrated NIST and that information was used in the uncertainty analysis in this 

report. The manufacturer specifications were used in the cases where NIST calibration was not 

available. 

Table 14 summarizes the uncertainty of the capacitance bridge determined at NIST when 

measuring 100 pF and 1 nF capacitors.

Table 14. Combined standard uncertainty of the capacitance bridge 

as calibrated at NIST. The values are given in parts in 106.

Capacitance

Frequency 100 pF 1 nF

100 Hz 0.7 1.0

1 kHz 0.3 0.6

10 kHz 6.0 6.3

6. Capacitance Uncertainty

6.1. 10 nF Capacitance

The impedance of the 10 nF capacitor is measured as

CB

LCR

LCR

CAL Z

Z

Z

Z K 1 nF_

1 nF_

10 nF_

10 nF_

. (40)
20

In order to find the uncertainty of the 10 nF capacitor, the uncertainty of the scaling ratio, K,

must be determined. The scaling ratio is measured using the equation

LCR

LCR

CB

CB

Z

Z

Z

Z

K

1 nF_

100 pF_

100 pF_

1 nF_

. (41)

From Eq. (41), the uncertainty of the scaling ratio is calculated as

u LCR NL u CB NL

Z

Z

s

Z

Z

u K s

LCR

LCR

C B

C B

c

( ) _ _

2 2

2

1 nF _

100 pF _

2

1 nF _

100 pF _ , (42)

where s(x) represents the standard deviation of measurement x. u(LCR_NL) is the uncertainty 

contribution from the nonlinearity of the LCR meter and scaling fixture, and u(CB_NL) is the 

uncertainty contribution from the nonlinearity of the capacitance bridge.

While the 100 pF and 1 nF air capacitors have insignificant dissipation factors over the 

frequency range of interest, their capacitance is not particularly stable (typically 30 parts in 106

per degree C). Over the course of several months of measurements using the capacitance scaling 

system, we observed fluctuations in the 1 kHz capacitance of standard air capacitors on the order 

of 10 parts in 106. In order to discriminate between the uncertainty of the scaling system and the 

instability of the 4TP air standards, more stable 3T nitrogen capacitors were examined. At 1 kHz, 

the nitrogen capacitors varied less than 0.1 part in 106. The nitrogen capacitors are connected to 

the 4TP scaling fixture using a coaxial adapter. 

A 3T capacitance bridge is used to measure 3T capacitors of values 100 pF and 1 nF at 

frequencies of 100 Hz, 1 kHz and 10 kHz. The only instance where the 3T capacitance bridge is 

used to measure 4TP capacitors is at 1 kHz. These measurements are used in calculations to 

extrapolate the 4TP capacitance value at 100 kHz, [4, 5].

In order to characterize the effects of the leads on capacitance measurements when a 3T 

capacitance bridge is used in the measurement procedure, the circuit shown in Figure 6 is 

analyzed. The ratio 

1

2

V

V

is used to estimate the lead influence, as analyzed in Eq. (43). 

cont

j C j C

R j L

j C j C j C C

R j L

j C

R j L

j C j C C

R j L

j C j C

R j L

j C j C j C C

V

V

SF

C C

SF C DUT

SF

C C

SF C DUT

SF SF

SF C DUT 

 (4 3)

1 1 1 1 1

1 1 1 1 1

)

1 1 1

2 2

1

2
21

Figure 6. Measurement setup for 4TP 1 nF capacitor using 3T capacitance bridge.

This ratio is nominally one. When the 4TP 1 nF air capacitor is measured at 1 kHz, the deviation 

of the ratio from one is computed to be less than 1 part in 109. Assuming similar circuit 

parameters for the 1 nF nitrogen capacitor, the computed ratio deviation at 10 kHz is less than 1 

part in 107. The lead influence is minimized further due to the fact that the 3T capacitance 

bridge manufacturer automatically compensates for the lead impedances of the 1-meter 

measurement cables.

During previous work characterizing 4TP air standard capacitors using a network analyzer, we 

observed minimal change between the capacitance values at 1 kHz and 100 kHz, when 

calibrating the 100 pF and 1 nF capacitors [5, 6]. The differences between the capacitances

measured at 1 kHz and those measured at 100 kHz are less than 1 part in 106, with an expanded 

uncertainty (k = 2) of 25 parts in 106. Measured dissipation factors for the same 100 pF and 1 nF 

air capacitors are less than 1 μrad from 1 kHz to 100 kHz, with an expanded uncertainty (k = 2) 

of 24 μrads, for both capacitors. Based on the uncertainty analysis in section 6, we observe that 

the measurement noise from the scaling fixture and LCR meter have significantly more effect on 

the overall measurement uncertainty than the 1 kHz-to-100 kHz extrapolation factor. 

Additionally, a 1 nF 4TP air capacitor was measured at 1 kHz and 10 kHz using the 3T

capacitance bridge. The difference between the recorded values was on the order of 1 part in 106. 

The combined standard uncertainty of the capacitance bridge, when measuring 1 nF at 10 kHz, is 

R=0.015 O

L=3 n H

C=107 p F

ZDUT 1000 pF

H

L

RS F LS F

CS F

RS F=0.015 O

LS F=2 n H

RS F LS F

RS F CS F

LCUR

LPOT

HPOT

HCUR

3T 

Capacitance

Bridge

CS F=62 pF

CS F

1 m cable corrected 

for by manufacturer

Scaling fixture

reference frame

0.61 m cable connecting scaling fixture to capacitor

LS F

Measured capacitor

V

V2

1

RC=0.024 O

LC=11 n H

CC=

17 pF

RCLC CC

RCLC CC

RCLC CC

.
22

on the order of 6 parts in 106. The same capacitor was characterized at the same frequencies

using the network analyzer approach, and the differences were less than 1 part in 106. The 

combined standard uncertainty of the network analyzer approach to characterizing capacitance 

and dissipation factor for 4TP air standard capacitors is 13 parts in 106, and 12 μrads, 

respectively, at 10 kHz. 

Therefore, the previous work allows for 1 kHz 3T measurements of the 100 pF and 1 nF air 

capacitors to be used at 100 kHz. The capacitance measurements at 1 kHz are used to estimate 

the impedance of the capacitors at 100 kHz, according to

1 kHz

100 kHz

1

measured at

at j C

Z .

Measurements were performed to determine the nonlinearity of the LCR meter. Using capacitors 

that varied from nominal as much as 100 parts in 106, the LCR meter readings were in error by 

less than 1 part in 106, supporting previous studies [2]. The LCR meter nonlinearity is not the 

dominant error when measuring a 10 nF capacitor, but it becomes more significant for higher 

valued capacitors at higher frequencies. The nonlinearity of the capacitance bridge was found to 

be less than 1 part in 10

6

. 

To investigate the stability of the scaling fixture, we replaced the capacitance scaling IVD with a 

commercial IVD of known stability and observed the same level of variations in the scaling ratio, 

K. To determine whether this instability is time dependent, we temporarily modified the software 

routine, using only one measurement point, to shorten the measurement process from 30 minutes 

to 5 minutes. The standard deviation of the scaling ratio was reduced by nearly an order of 

magnitude to approximately 0.3 part in 106. This indicates that the stabilities of the interface 

circuitry, the LCR meter, and the capacitors are the main contributors to the uncertainty of the 

scaling ratio calibration. 

Two sets of capacitors are used in the measurement procedure. Measurements at 100 Hz, 1 kHz 

and 10 kHz are performed using 3T nitrogen capacitors and the capacitance and dissipation 

factor are measured using a multi-frequency capacitance bridge. The nitrogen capacitors used in 

this part of the measurement procedure are stable to better than 0.5 part in 106. The nitrogen 

capacitors have significant frequency dependence at 100 kHz, so the 100 kHz measurement is 

done with 4TP air capacitors exhibiting smaller dissipation factor. An extrapolation method is 

used to determine the performance of the air capacitors at 100 kHz [3, 4]. The air capacitors are 

kept in an insulated case so that user handling of the capacitors is eliminated and temperature 

changes are minimized during measurements. 

The LCR meter measures the same nominal impedance while measuring the 100 pF capacitor

and the 1 nF capacitor with the scaling ratio of 0.1. However, the two measurements are not 

exactly the same. The contribution from this difference is labeled LCR nonlinearity and is given 

in Eq. (39).

Table 15 shows the LCR meter uncertainties and the measured differences. The measurement 

uncertainty contribution from the differences is the product of the two columns, as shown in 

column 4 of Table 16, which shows all uncertainty contributions from Eq. (42).
23

Table 15. Actual LCR meter deviations when measuring 100 pF 

and 1 nF capacitors for 10 nF scaling ratio calibration.

Frequency

LCR Meter

Uncertainty

3

( ) AE ACAL

[parts in 106]

LCR Meter

Deviations

measured

measured measured

Z

Z Z

1 nF_

100 pF_ 1 nF_

100 Hz 2362 0.000020

1 kHz 624 0.000030

10 kHz 532 0.000040

100 kHz 588 0.000090

The maximum LCR meter linearity uncertainty component over all measurements performed 

during this investigation was not more 0.1 part in 106. However, for another set of calibrated 

capacitors with values further from nominal, the measured differences will be larger. The LCR

meter nonlinearity contributions should reflect the new measured differences. 

Table 16. Combined standard uncertainty of scaling ratio for 10 nF capacitor, in parts in 106.

Frequency

LCR

LCR

Z

Z

s

1 nF_

100 pF_

data obtained 

from a set of 

measurements

CB

CB

Z

Z

s

1 nF_

100pF_

data obtained from 

NIST capacitance 

bridge calibration

LCR Meter 

Nonlinearity

for K

calculated 

from Equation 

(39) and 

Table 14

uc K10 nF

calculated 

from Equation 

(42)

100 Hz 8 1 0.0 9

1 kHz 6 0.6 0.1 6

10 kHz 5 6.3 0.0 8

100 kHz 5 13* 0.1 13

(*) The asterisk indicates a measurement using 4TP air capacitors and its capacitance is obtained 

from measurement at 1 kHz and extrapolation to 100 kHz. The uncertainty of the measurement 

procedure is 13 parts in 106.

At 100 kHz, the uncertainty of the scaling ratio, K, is computed using

u LCR NL u CB NL

Z

Z

s

Z

Z

u K u

LCR

LCR

C B

C B

c

( ) _ _

2 2

2

1 nF_

100 pF_

1 nF_

2 100 pF_

, (44)

where the first term is contribution of the capacitance bridge using an extrapolation to 100 kHz 

and the second term is the contribution of the LCR meter measurements at 100 kHz. The 

capacitance bridge nonlinearity for measuring 100 pF and 1 nF is less than 1 part in 106for all 

measurements addressed in this document.
24

The next step is to calculate the uncertainty in measuring the 10 nF capacitor. Based on the 

formula used to perform the measurements, we have

CB

LCR

LCR

C Z

Z

Z

Z K 1 nF_

1 nF_

10 nF_

10 nF_ AL

. (45)

The uncertainty of the 10 nF capacitance is computed using

u LCR NL u CB NL

Z

Z

u C u Z s

LCR

LCR

c

( ) C B _ _

2 2

2

1 nF_

10 nF_

1 nF_

2

10 nF

, (46)

where u(Z1 nF_CB) is the Type B uncertainty of the capacitance bridge derived from measuring

the 1 nF capacitor as calibrated by NIST. Table 17 presents the uncertainty components for the 

10 nF capacitor. Note that the nonlinearity of the capacitance bridge is insignificantly small and 

is not included.

Table 17. Uncertainty contributions for 10 nF capacitor, in parts in 106.

Frequency

LCR

LCR

Z

Z

s

1 nF_

100 pF_

data obtained 

from a set of 

measurements

uc K10 nF uc Z1 nF_CB

data obtained 

from NIST 

capacitance 

bridge 

calibration

LCR Meter 

Nonlinearity

for 10 nF

data obtained 

from a set of 

measurements 

and LCR meter

uncertainty 

specifications

100 Hz 8 9 1 0.0

1 kHz 10 6 0.6 0.0

10 kHz 11 8 6.3 0.0

100 kHz 5 13 13* 0.1

*At 100 kHz, the uncertainty of the 10 nF capacitor is modified using

u LCR NL u CB NL

Z

Z

u C u Z u K s

LCR

LCR

c

( ) C B _ _

2 2

2

1 nF_

2 100 pF_

1 nF_

2

10 nF

. (47)

The calculated uncertainty of a 10 nF capacitor is given in Table 18. For comparison, the 

standard deviation of the measured capacitance is also presented. The agreement is acceptable. 
25

Table 18. Combined standard uncertainty of 10 nF capacitance.

Frequency Applied voltage

[mV]

uc(C10 nF)

[parts in 106]

10 nF

10 nF

C

s C

[parts in 106]

data obtained 

from a set of 

measurements

100 Hz 600 12 12

1 kHz 600 12 13

10 kHz 600 15 10

100 kHz 600 18 9

6.1.1. Additional Tests of the 10 nF Capacitor

The improved stability of the nitrogen dielectric standard capacitors comes at the expense of 

increased frequency dependence. The frequency dependence of the 1 nF standard was measured 

using the NIST 4TP Bridge designed by Cutkosky [10]. The results, taken from [6], are given in 

Table 19.

Table 19. Frequency dependence of the 1 nF standard capacitor (nitrogen 

dielectric) measured using the NIST 4TP Bridge, in parts in 106.

Frequency [kHz] 0.1 1 10

1 nF Deviation from

Nominal Value 7.28 8.74 9.83

A 10 nF nitrogen dielectric capacitor with short-term stability of better than 1 part in 107 was 

measured using the NIST 4TP Bridge and the capacitance scaling system. A summary of the 

results is shown in Table 20. 

Table 20. 10 nF capacitor measurements at several frequencies, in parts in 106. The * denotes the 

Type A uncertainty as calculated based on the observed measurement standard deviations.

Deviation from NIST 4TP 

Bridge Measurement

Measurement Uncertainty

Coverage Factor (k = 1)

Frequency

[kHz]

0.1 1 10 0.1 1 10

NIST 4TP 

Bridge

0 0 0 0.3 <0.1 0.2

Capacitance 

Scaling

System

19 1.9 1.2 23* 4* 6*
26

The results in Table 20 show excellent agreement between the NIST manual bridge and the 

capacitance scaling system at 1 kHz and 10 kHz using stable 3T nitrogen capacitors. The

capacitance scaling measurements were done before the voltage levels were controlled using the

ALC implementation and the agreement at 100 Hz is not as good due to the LCR meter

performance. The latest measurements have much improved stability. 

6.2. 100 nF Capacitance

Derivations similar to those of the 10 nF capacitor are used for the 100 nF capacitor. The formula 

to find the 100 nF impedance is given in Eq. (48) and its uncertainty is calculated in Eq. (49).

CAL

LCR

LCR

CAL Z

Z

Z

Z K 10 nF

10 nF_

100 nF_

100 nF

(48)

u C u LCR NL

Z

Z

u C u K s c

LCR

LCR

c

( )

c

( ) _

2

10 nF

2

2

10 nF_

2 100 nF_

100 nF

(49)

The LCR meter nonlinearity contributions for K and 10 nF are negligible and the data for 100 nF 

is given in Table 21. Table 22 provides data necessary for the calculation of the uncertainty for 

the 100 nF capacitor. The notation C10 nF/100 nF refers to the intermediate calibration of the 10 nF 

capacitor used in the calibration of the 100 nF capacitor. Similar notation will be used to 

reference other intermediate steps in the calibration process. The capacitance uncertainty for the 

100 nF capacitor is given in Table 23.

Table 21. LCR meter deviations when measuring the 10 nF and 

100 nF capacitors for the 100 nF calibration.

Frequency

LCR Meter 

Uncertainty

3

( ) AE ACAL

[parts in 106]

LCR Meter Deviations

measured

measured measured

Z

Z Z

100 nF_

10 nF_ 100 nF_

LCR Meter

Nonlinearity

for 100 nF

[parts in 106]

100 Hz 763 0.000100 0.1

1 kHz 465 0.000120 0.1

10 kHz 584 0.000080 0.0

100 kHz 774 0.002230 1.7
27

Table 22. Uncertainty contributions for the 100 nF capacitor, in parts in 106. Note 

that the 10 nF capacitor is calibrated as part of the 100 nF calibration. 

Frequency

LCR

LCR

Z

Z

s

1 nF_

100 pF_

data obtained 

from a set of 

measurements

uc K100 nF LCR

LCR

Z

Z

s

1 nF_

10 nF_

data obtained 

from a set of 

measurements

uc C10 nF/100 nF LCR

LCR

Z

Z

s

10 nF_

100 nF_

data obtained 

from a set of 

measurements

100 Hz 6 6 2 7 2

1 kHz 7 7 11 13 14

10 kHz 9 11 5 13 3

100 kHz 6 10 13 18 4

Table 23. Combined standard uncertainty of 100 nF capacitance.

Frequency Applied voltage

[mV]

uc(C100 nF)

[parts in 106]

100 nF

100 nF

C

s C

[parts in 106]

data obtained 

from a set of 

measurements

100 Hz 600 10 12

1 kHz 600 21 19

10 kHz 600 18 23

100 kHz 200 23 18

6.3. 1 μF Capacitance

The 1 μF capacitor is characterized using the formula

CAL

LCR

LCR

CAL Z

Z

Z

Z K 100 nF_

100 nF_

1 μF _

1 μF _

. (50)

The uncertainty for the 1 μF capacitance is given as

u C u LCR NL

Z

Z

u C u K s c

LCR

LCR

c

( )

c

( ) _

2

100 nF

2

2

100 nF_

2 1μF_

1μF

. (51)

The LCR meter nonlinearity contributions for K, 10 nF, and 100 nF are small and the data for 

100 nF is given in Table 24. Note the high value of the uncertainty at 100 kHz. Table 25 

provides data necessary for the calculation of the uncertainty for the 1 μF capacitor. Table 26 
28

gives the capacitance uncertainty for the 100 nF capacitor calibration for the 1 μF capacitor.

Table 27 gives the capacitance uncertainty for the 1 μF capacitor.

Table 24. LCR meter deviations when measuring the 100 nF and 

1 μF capacitors for the 1 μF calibration.

Frequency

LCR Meter

Uncertainty

3

( ) AE ACAL

[parts in 106]

LCR Meter Deviations

measured

measured measured

Z

Z Z

1 μF _

100 nF_ 1 μF_

LCR Meter

Nonlinearity

for 1 μF

[parts in 106]

100 Hz 753 0.000008 0.0

1 kHz 468 0.000008 0.0

10 kHz 767 0.000518 0.4

100 kHz 4139 0.022370 92.6

Table 25. Uncertainty contributions for the 1 μF capacitor, in parts in 106. Note 

that there is a 10 nF calibration step as part of the 1 μF calibration. 

Frequency

LCR

LCR

Z

Z

s

1 nF_

100 pF_

data obtained 

from a set of 

measurements

uc K1 μF LCR

LCR

Z

Z

s

1 nF_

10 nF_

data obtained 

from a set of 

measurements

uc C10 nF/1 μF

100 Hz 8 8 1 8

1 kHz 7 7 6 10

10 kHz 19 20 10 23

100 kHz 3 13 7 19

Table 26. Capacitance uncertainty for the 100 nF capacitor, in parts in 106. 

Frequency

LCR

LCR

Z

Z

s

10 nF_

100 nF_

data obtained 

from a set of 

measurements

uc C100 nF/1 μF LCR

LCR

Z

Z

s

100 nF_

1μF _

data obtained 

from a set of 

measurements

100 Hz 3 12 2

1 kHz 7 14 5

10 kHz 7 32 2

100 kHz 8 24 7
29

Table 27. Combined standard uncertainty of 1 μF capacitance.

Frequency Applied voltage

[mV]

uc C1 μF

[parts in 106]

1 μF

1 μF

C

s C

[parts in 106]

data obtained 

from a set of 

measurements

100 Hz 600 15 25

1 kHz 600 16 31

10 kHz 200 38 63

100 kHz 25 97 9

Note the significant discrepancy between the low standard deviation of measurements at 

100 kHz and the high uncertainty. The main contribution comes from the LCR nonlinearity 

component.

6.4. 10 μF Capacitance

The 10 μF capacitor is characterized using Eqs. (52) and (53).

CAL

LCR

LCR

CAL Z

Z

Z

Z K 1 μF _

1 μF_

10 μF_

10 μF_

(52)

u C u LCR NL

Z

Z

u C u K s c

LCR

LCR

c

( )

c

( ) _

2

1μF/10μF

2

2

1μF_

2 10 μF_

10μF

(53)

The 10 μF LCR meter nonlinearity contribution was similar to that for the 1 μF capacitor. The 

data for 10 μF is given in Table 28. Note the large contribution at 100 kHz. The reason for this 

inflation is that the impedance of the measured capacitor is 0.16 Ω and the measured impedance 

for 10 μF and 1 μF differ by 2.2%. The optimal impedance to be measured by the LCR meter is 

500 Ω. Table 29 provides data necessary to calculate of the uncertainty for the 10 μF capacitor. 

Table 30 gives the uncertainty contributors to the 10 μF capacitance measurement. The 

combined standard 10 μF capacitance uncertainty is given in Table 31. Note the discrepancy 

between the low standard deviation of measurements at 100 kHz and the high uncertainty.
30

Table 28. LCR meter deviations when measuring the 1 μF and 

10 μF capacitors for the 10 μF calibration.

Frequency

LCR Meter

Uncertainty

3

( ) AE ACAL

[parts in 106]

LCR Meter Deviations

measured

measured measured

Z

Z Z

10 μF _

1 μF _ 10 μF _

LCR Meter

Nonlinearity

for 10 μF

[parts in 106]

100 Hz 757 0.000030 0.0

1 kHz 651 0.000040 0.0

10 kHz 4133 0.000170 0.7

100 kHz 152493 0.022000 3354.9

Table 29. Uncertainty contributions for the 10 μF capacitor, in parts in 106. Note 

that there is a 10 nF calibration step as part of the 10 μF calibration. 

Frequency

LCR

LCR

Z

Z

s

1 nF_

100 pF_

data obtained 

from a set of 

measurements

uc K10 μF LCR

LCR

Z

Z

s

1 nF_

10 nF_

data obtained 

from a set of 

measurements

uc C10 nF/10 μF

100 Hz 12 12 3 13

1 kHz 35 35 13 37

10 kHz 43 44 3 44

100 kHz 15 20 15 28

Table 30. Uncertainty contributions for the 10 μF capacitor, in parts in 106. Note 

that there are 100 nF and 1 μF calibration steps as part of the 10 μF calibration. 

Frequency

LCR

LCR

Z

Z

s

10 nF_

100 nF_

data obtained 

from a set of 

measurements

uc C100 nF/10 μF LCR

LCR

Z

Z

s

100 nF_

1 μF _

data obtained 

from a set of 

measurements

uc C1 μF/10 μF LCR

LCR

Z

Z

s

1 μF _

10 μF _

data obtained 

from a set of 

measurements

100 Hz 22 28 2 31 3

1 kHz 7 51 4 62 2

10 kHz 10 63 7 77 26

100 kHz 8 35 25 416 679
31

Table 31. Combined standard uncertainty of 10 μF capacitance.

Frequency Applied voltage

[mV]

uc C10 μF

[parts in 106]

10 μF

10 μF

C

s C

[parts in 106]

data obtained 

from a set of 

measurements

100 Hz 600 33 44

1 kHz 200 71 116

10 kHz 25 92 190

100 kHz 5 3448 645

6.5. 100 μF Capacitance

The 100 μF capacitor is characterized using equations (54) and (55).

CAL

LCR

LCR

CAL Z

Z

Z

Z K 10 μF_

10 μF_

100 μF_

100 μF_

(54)

u C u LCR NL

Z

Z

u C u K s c

LCR

LCR

c

( )

c

( ) _

2

10 μF

2

2

10 μF_

2 100 μF_

10 μF

(55)

The LCR meter nonlinearity contribution was similar to the data presented for the 10 μF 

capacitor. The data for the 100 μF capacitor is given in Table 32. Note the large contribution at 

10 kHz. The reason for this inflation is the fact that the impedance of the measured capacitor is 

0.16 Ω and the measured impedance for 100 μF and 10 μF differ by 4.5%. The optimal 

impedance to be measured by the LCR meter is 500 Ω. Table 33 provides data necessary for the 

calculation of the uncertainty for the 100 μF capacitor. Table 34 gives the uncertainty 

contributors to the 100 μF capacitor calibration. Table 35 shows the 100 μF combined standard 

uncertainty. Table 36 gives a summary of the expanded capacitance uncertainties (k = 2) for the 

scaling procedure. Note that the expanded uncertainties are approximately twice the combined 

standard uncertainties reported in Tables 18, 23, 27, 31, and 35.
32

Table 32. LCR meter deviations when measuring the 100 μF and 

10 μF capacitors for the 100 μF calibration.

Frequency

LCR Meter 

Uncertainty

3

( ) AE ACAL

[parts in 106]

LCR Meter Deviations

measured

measured measured

Z

Z Z

100 μF _

10 μF _ 100 μF _

LCR Meter 

Nonlinearity

for 100 μF

[parts in 106]

100 Hz 1055 0.000259 0.3

1 kHz 4016 0.000062 0.2

10 kHz 152486 0.004489 684.5

Table 33. Uncertainty contributions for the 100 μF capacitor, in parts in 106. Note that there are 

10 nF and 100 nF calibration steps as part of the 100 μF calibration. 

Frequency

LCR

LCR

Z

Z

s

1 nF_

100 pF_

data obtained 

from a set of 

measurements

uc K100 μF LCR

LCR

Z

Z

s

1 nF_

10 nF_

data obtained 

from a set of 

measurements

uc C10 nF/100 μF LCR

LCR

Z

Z

s

10 nF_

100 nF_

data obtained 

from a set of 

measurements

uc C100 nF/100 μF

100 Hz 15 15 41 44 105 115

1 kHz 114 114 189 221 82 261

10 kHz 262 262 214 339 73 435

Table 34. Uncertainty contributions for the 100 μF capacitor, in parts in 106, cont’d. Note that 

there are 1 μF and 10 μF calibration steps and as part of the 100 μF calibration. 

LCR

LCR

Z

Z

s

100 nF_

1 μF _

data obtained 

from a set of 

measurements

uc C1 μF/100 μF LCR

LCR

Z

Z

s

1 μF _

10 μF _

data obtained 

from a set of 

measurements

uc C10 μF/100 μF LCR

LCR

Z

Z

s

10 μF _

100 μF _

data obtained 

from a set of 

measurements

4 116 23 119 257

12 285 42 310 62

513 722 455 893 962
33

Table 35. Combined standard uncertainty of 100 μF capacitance.

Frequency

Applied voltage

[mV]

uc C100 μF

[parts in 106]

100μF

100μF

C

s C

[parts in 106]

data obtained 

from a set of 

measurements

100 Hz 200 284 101

1 kHz 25 336 577

10 kHz 5 1503 806

Table 36. Summary of expanded capacitance uncertainty (k = 2), in parts in 106.

Frequency Capacitance

10 nF 100 nF 1 μF 10 μF 100 μF

100 Hz 25 20 35 70 570

1 kHz 25 43 40 150 680

10 kHz 30 40 90 200 3000

100 kHz 35 43 200 7000 n/a

7. Dissipation Factor Uncertainty

7.1. 10 nF Dissipation Factor

The phase uncertainty of the scaling ratio is calculated using the uncertainty of the capacitance 

bridge and the LCR meter as shown in Eq. (56).

C B

C B

LCR

LCR

LCR

LCR

C B

C B

Z

Z

Z

Z

Z

Z

Z

Z

K

100 pF_

1 nF_

1 nF_

100 pF_

1 nF_

100 pF_

100 pF_

1 nF_

(56)

As a reminder, Eqs. (7) and (8) showed that the scaling ratio is represented in terms of magnitude 

and phase. Since the phase uncertainty is presented in μrad, the contribution from the LCR meter 

to the scaling ratio phase uncertainty is multiplied by the factor 

CB

CB

Z

Z

100 pF_

1 nF_

that has a nominal 

value of 0.1. The factor 

LCR

LCR

Z

Z

1 nF_

100 pF_

has a nominal value of 1.

Equation (57) presents the scaling ratio phase uncertainty calculation.
34

u CB NL u LCR NL

s u DF u DF

u K

C B C B

Z

Z

c

LCR

LCR

_

100

0.1 _

0.1

( )

2

2

1 nF_

2

100 pF_

2

2

1 n F_

100 p F_





(57)

The uncertainty in measuring the dissipation factor using the capacitance bridge is derived from 

the bridge manufacturer specifications [8]. The uncertainty in measuring the dissipation factor 

using the LCR meter is produced using the meter manufacturer specifications [7]. The meter 

specifications define dissipation factor uncertainty as 

100

AE

where AE is the relative uncertainty, as 

given in Eq. (37). Since we are measuring a ratio of impedances using the LCR meter, the factor 

100

u LCR _ NL

is the appropriate estimate of the contribution. The Type A contribution from the 

LCR meter measurements is given using the standard deviations of the measurements, 

LCR

LCR

Z

Z

s

1 nF_

100 pF_

0.1

. The results are presented in Table 37. Note that since the measured angles are 

small, dissipation factor and phase are used interchangeably. In the discussions below, phase ( ) 

is used to represent the imaginary part of the scaling ratio and dissipation factor (DF) is used for 

capacitive impedance. 

Table 37. Uncertainty contributions for the scaling ratio phase, in μrad.

Frequency

Measured 

DF for 

100pF and 

1 nF 

capacitors

u C100 pF_CB

data obtained 

from the 

manufacturer 

manual

u C1 nF_CB

data obtained 

from the 

manufacturer 

manual

LCR

LCR

Z

Z

s

1 nF _

100pF _

0.1

data obtained 

from a set of 

measurements

uc K 10 nF

100 Hz 2 3.7 3.7 1 5

1 kHz 2 1.9 0.3 1 2

10 kHz 10 10.5 10.3 1 15

100 kHz *Air capacitor dissipation factor negligible 0 18

*At 100 kHz the measurements are performed using air capacitors. The measurement 

procedure is slightly different compared to the rest of the frequencies. The air capacitors are 

measured at 1 kHz and their impedance is extrapolated to 100 kHz. The discussion in section 

6.1 argued the assumption that the air capacitors have negligible dissipation factor. The 

uncertainty in measuring the dissipation factor of the 1 nF and 100 pF capacitors at 100 kHz 

is 12 μrad. These contributions are labeled in Eq. (58) as 

u DF100 pF_CB_extrapolated _to_100 kHz

and 

u DF1 nF_CB_extrapolated _to_100 kHz

.

At 100 kHz, the uncertainty of measuring the scaling ratio phase,

K , is 
35

u CB NL u LCR NLu DF

s u DF

u K

C B extrapolated t o

C B extrapolated t o

Z

Z

c

LCR

LCR

_

100

0.1 _

0.1

( )

2

2

1 nF_ _ _ _100 kHz

2

100 pF_ _ _ _100 kHz

2

2

1 n F_

100 p F_





(58)

The uncertainty of measuring the dissipation factor of a 10 nF capacitor is given as

u CB NL u LCR NLu DF u K s DF u DF C B

Z

c c Z

LCR

LCR

_

100

_ 2

2

1 nF_

2

2

2

10 nF

1 n F_

1 0 n F_

(59)

Table 38 shows the calculations of the dissipation factor uncertainty for the 10 nF capacitor. All 

of the contributions from Equation 59 are previously given except for the Type A factor

LCR

LCR

Z

Z

s

1 nF _

10 nF _

. This factor has a dominant role in determining the uncertainty. 

Table 38. Combined standard uncertainty of 10 nF capacitor dissipation factor, in μrad.

Frequency

LCR

LCR

Z

Z

s

1 nF _

10 nF _

data obtained from a set of 

measurements

uc DF10 nF

100 Hz 13 15

1 kHz 6 6

10 kHz 12 21

100 kHz 17 27

7. 2. 100 nF Dissipation Factor

The dissipation factor uncertainty of the 100 nF capacitor is given in Eq. (60) and Table 39

provides the uncertainty component values and the combined standard uncertainty.

2

10 nF

2

2

2

100 nF 100

_

1 0 n F_

100 n F_

u LCR NL

u DF u K s DF uc DF

Z

c c Z

LCR

LCR

(60)
36

Table 39. Combined standard uncertainty of 100 nF capacitor dissipation factor, in μrad.

Frequency

u K 100 nF u DF10 nF/100 nF

LCR

LCR

Z

DFZs

10 nF _

100 nF _

data obtained 

from a set of 

measurements

uc DF100 nF

100 Hz 5 13 25 29

1 kHz 2 4 8 9

10 kHz 15 22 23 35

100 kHz 18 33 25 45

The fourth column presents the standard deviations of the measured dissipation factor for the 

100 nF capacitor. The same approximation will be used throughout the report. This is a 

conservative estimate. 

7.3. 1 μF Dissipation Factor

The dissipation factor uncertainty for the 1 μF capacitor is given in Eq. (61) and Table 40

provides the uncertainty component values and the combined standard uncertainty.

2

100 nF

2

2

2

1μF 100

_

100 n F_

1 μF _

u LCR NL

u DF u K s DF u DF

LCR

LCR

Z

c c Z

(61)

Table 40. Combined standard uncertainty of 1 μF capacitor dissipation factor, in μrad.

Frequency

uc K 1 μF u DF10 nF/1μF u DF100 nF/1μF

LCR

LCR

Z

DF Zs

100 nF _

1 μF _

data obtained 

from a set of 

measurements

uc DF1 μF

100 Hz 5 14 31 47 57

1 kHz 2 6 13 13 19

10 kHz 15 22 39 38 57

100 kHz 18 27 26 47 57

7.4. 10 μF Dissipation Factor

The dissipation factor uncertainty of the 10 μF capacitor is calculated using (62) and is presented 

in Table 41.
37

2

1μF

2

2

2

10 μF 100

_

1 μF _

1 0 μF _

u LCR NL

u DF u K s DF uc DF

Z

c c Z

LCR

LCR

(62)

Table 41. Combined standard uncertainty of 10 μF capacitor dissipation factor, in μrad.

Freq.

uc K 10 μF u DF10 nF/10 μF DF100 nF/10 μF u u DF1μF /10 μF LCR

LCR

Z

DF Zs

100 nF _

1 μF _

data obtained 

from a set of 

measurements

uc DF10μF

100 Hz 5 8 14 199 29 201

1 kHz 3 18 51 87 86 122

10 kHz 9 80 95 158 86 180

100 kHz 18 61 138 234 545 593

7.5. 100 μF Dissipation Factor

The dissipation factor uncertainty for the 100 μF capacitor is calculated using (63) and is 

presented in Table 42. A summary of expanded uncertainties (k = 2) is shown in Table 43.

2

10 μF

2

2

2

100 μF 100

_

1 0 μF _

100μF _

u LCR NL

u DF u K s DF uc DF

Z

c c Z

LCR

LCR

(63)

Table 42. Combined standard uncertainty of 100 μF capacitor dissipation factor, in μrad.

Freq.

uc K 100 μF u DF10 nF/100 μF DF100 nF /100 μF u DF1μF/100μF u DF10μF / 100μF u

LCR

LCR

Z

DF Zs

100 nF _

1 μF _

data obtained 

from a set of 

measurements

uc DF100 μF

100 Hz 6 33 66 105 149 131 198

1 kHz 14 134 291 484 712 636 955

10 kHz 17 48 72 124 172 1049 1063

Table 43. Summary of expanded dissipation factor uncertainties (k = 2), in μrad.

Frequency Capacitor Value

10 nF 100 nF 1 μF 10 μF 100 μF

100 Hz 30 60 120 400 200

1 kHz 15 20 50 250 2000

10 kHz 45 70 120 380 2200

100 kHz 55 90 120 1200 n/a
38

8. Summary

Throughout this document, the Type A components of the uncertainties were derived from stepup impedance measurements using the LCR meter. The Type B components were derived from 

nonlinearity coefficients from the measurement instruments used. The standard deviations of the 

measured capacitances were compared with the uncertainty bounds (see Tables 45 and 36). In 

some instances, the standard deviations in the measured capacitances were higher than the 

uncertainty bounds. This discrepancy may be due to environmental effects on the measured 

reference capacitors. 

When the standard deviations of the measured reference capacitors are used as the Type A 

uncertainty contributions, the resulting expanded uncertainties are given in Table 44. Table 45 

shows the standard deviations of the measured reference capacitors. It is important to note that 

Table 45 shows only standard deviations of the final measurement results. In calculating the 

values in Table 44, all of the described uncertainty components in the step up measurement 

procedures are taken into account. Table 36 is duplicated here for convenient comparison.

Table 44. Expanded capacitance uncertainties (k = 2), when the standard deviations of the 

measured reference capacitors are used as the Type A component, in parts in 106.

Frequency Capacitor Value

10 nF 100 nF 1 μF 10 μF 100 μF

100 Hz 30 33 70 140 300

1 kHz 30 50 80 340 1810

10 kHz 30 63 175 540 4030

100 kHz 40 60 200 7000 n/a 

Table 45. Standard deviations of the measured reference capacitors, in parts in 106.

Frequency Capacitor Value

10 nF 100 nF 1 μF 10 μF 100 μF

100 Hz 12 12 25 44 101

1 kHz 13 19 31 116 577

10 kHz 10 23 63 190 806

100 kHz 9 18 9 645 n/a

Table 36. Summary of expanded capacitance uncertainty (k = 2), in parts in 106.

Frequency Capacitor Value

10 nF 100 nF 1 μF 10 μF 100 μF

100 Hz 25 20 33 70 570

1 kHz 25 43 37 150 680

10 kHz 30 40 80 200 3000

100 kHz 37 47 200 7000 n/a

In the case of dissipation factor, the LCR meter measures the phase so that the standard 

deviations used in the analysis are the standard deviations of the measured dissipation factor of 

the reference standards.
39

Reported uncertainties for Special Tests of customer standard capacitors will reflect the values 

from Table 44, using the standard deviations of the measured reference capacitors as the Type A 

component.

9. Conclusion

The capacitance scaling system is described and calculations of uncertainties for each capacitor 

are presented in detail. Where appropriate, both Type A and Type B uncertainty components are 

considered. The capacitance scaling method uses ceramic capacitors with temperature 

coefficients on the order of 35 parts in 106per °C (values 10 nF, 100 nF, 1 μF and 10 μF). 

Precautions were made to minimize the temperature changes during measurements, but the 

uncertainty of the capacitor was not separated from that of the capacitance scaling measurement 

system due to a lack of more stable capacitance standards. The presented analysis of the 

capacitance scaling method includes capacitor instability and, therefore, provides conservative 

uncertainties.

This measurement system will allow NIST to provide customers with measurement services for 

four-terminal-pair capacitance and dissipation factor. The capacitance range covered is from 

10 nF to 100 μF. The frequencies measured are 100 Hz, 1 kHz, 10 kHz, and 100 kHz.

Acknowledgments: The authors wish to thank Katsumi Yokoi, Akiu Yamazaki, and the group 

at Agilent Technologies Japan, Measurement Standards Center, who developed this 

measurement approach. They provided critical assistance during this project; specifically, 

providing the scaling fixture, assistance in developing the software used in the measurement 

system, and numerous communications.

10. References

[1] T. Aoki and K. Yokoi, “Capacitance Scaling System”, IEEE Transactions on 

Instrumentation and Measurement, Vol. 46, No. 2 April 1997, pp. 474-476. 

[2] K. Suzuki, A. Yamazaki, and K. Yokoi, “Non-Linearity Evaluation Method of FourTerminal-Pair (4TP) LCR Meter,” Proceedings of the 2001 NCSL International 

Workshop & Meas., 49, No. 2, pp. 398-404.

[3] S. Avramov-Zamurovic, B. Waltrip, A. Koffman and G. Piper, “Standard Capacitor 

Calibration Procedure Implemented Using Control Software”, American Society of 

Engineering Educators Conference Proc., June 20-23, 2004, Salt Lake City, UT.

[4] S. Avramov-Zamurovic, A. D. Koffman, N. M. Oldham, and B. C. Waltrip, "The 

Sensitivity of a Method to Predict a Capacitor's Frequency Characteristics", IEEE 

Trans. Instrum. Meas., 49, No. 2, pp. 398-404 (Apr 2000).

[5] A. D. Koffman, S. Avramov-Zamurovic, B. C. Waltrip, and N. M. Oldham, 

"Uncertainty Analysis for Four Terminal-Pair Capacitance and Dissipation Factor 

Characterization at 1 MHz and 10 MHz", IEEE Trans. Instrum. Meas., 49, No. 2, pp. 

346-348 (Apr 2000).
40

[6] A. D. Koffman, S. Avramov-Zamurovic, B. C. Waltrip, and Y. Wang, “Evaluation of 

a Capacitance Scaling System” to be published in December 2007.

[7] Agilent Technologies (HP) 4284A Precision LCR Meter 20 Hz – 1 MHz Operation 

Manual, Tokyo Japan, December 1991.

[8] Andeen-Hagerling AH 2700A 50 Hz – 20 kHz Ultra-Precision Capacitance Bridge, 

Operation and Maintenance Manual, Cleveland, OH, June 2003.

[9] B. N. Taylor and Chris E. Kuyatt, “Guidelines for Evaluating and Expressing the 

Uncertainty of NIST Measurement Results,” NIST Technical Note 1297, September 

1994.

[10] R. D. Cutkosky, “Techniques for comparing four-terminal-pair admittance 

standards,” J. Res. NBS, vol. 74C, pp. 63-78, 1970.
41

Appendix A 

Measurement of Reference Standard Capacitors

A set of Agilent Technologies116380C ceramic capacitors (including a specially-made 100 μF

standard) was measured several times over the course of approximately one month and Tables 46

and 47 present the average values obtained.

Table 46. Average measured capacitance values for the set of ceramic standards.

Frequency Capacitor Value

10 nF 100 nF 1 μF 10 μF 100 μF

100 Hz 9.99991 100.0111 1.000124 10.00133 99.9827

1 kHz 9.99909 99.9992 0.999988 9.99970 99.9841

10 kHz 9.99879 100.0037 1.000546 9.99940 100.2650

100 kHz 10.00057 100.2269 1.024391 10.42616 n/a

Table 47. Average measured dissipation factor values for the set of ceramic standards, in μrad.

Frequency Capacitor Value

10 nF 100 nF 1 μF 10 μF 100 μF

100 Hz 16 6 36 31 153

1 kHz 33 33 55 21 438

10 kHz 62 135 797 232 93

100 kHz 305 2154 20268 1035 n/a

 

1

Identification of commercial products does not imply endorsement by the U.S. Government, nor does it imply that such 

products are necessarily the best available.