NBS

Reference ; publications

NBSIR 84-3016

NATL INST. OF STAND & TECH

AllilDb EEflflM2

A GENERALIZED METHOD FOR THE

CALIBRATION OF FOUR-TERMINALPAIR TYPE DIGITAL IMPEDANCE

METERS

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

August 1984

“•QC

100

U56

84-3016

1984
f'l
JfATIONAt BUREAU OF STANDARDS

LIBRARY

A GENERALIZED METHOD FOR THE ^ CALIBRATION OF FOUR-TERMINALPAIR TYPE DIGITAL IMPEDANCE

METERS

R.M. Judish

R.N. Jones

Electromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

August 1984

Prepared for: Sandia National Laboratories

Albuquerque, New Mexico

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

1. Introduction and Background 1 2. Purpose and Scope 2 3. Four-Termi nal -Pai r Interface and Adapters to

Other Connector Types 3

4. Theoretical Approach 5 4.1 Impedance Relationship through the Adapter 5 4.2 Transformati on to Reflection Coefficients 7 5. Calibration Experiment 8

5.1 Calibration Curves 8

5.2 Statistical Model for the Calibration Curve 10

5.3 Least-Squares Solution for the Calibration Parameters 11

6. Measurement of Unknowns 12

6.1 Measurement of an Unknown r 12

6.2 Uncertainty in T2u 13

6.3 Transformati on to Impedance 14

7. Example 14

7.1 Standards Used in Calibration 14

7.2 Least-Squares Estimation of the Calibration Parameters 19

7.3 Measurement of Unknown r 26

7.4 Transformation to Z 26

7.5 Calibration Parameters with aim Cable 31

8. Summary 31

9. References 33

Appendix A. FORTRAN Program Listing 34

Appendix B. Printout from Program for Data at 1 MHz 42

Appendix C, Printout from Program for Data at 10 MHz 49

A Generalized Method for the Calibration of

Four-Termi nal -Pai r Type Digital Impedance Meters

R. M. Judish and R. N. Jones*

National Bureau of Standards

Boulder, Colorado 80303

Since the introduction of automated, four-termi nal -pai r type

digital impedance meters, there has been a continuing interest in

the development of calibration techniques which would satisfactorily verify the accuracy capabilities of these instruments. Various

attempts have been made and all have helped to provide a certain

degree of confidence in instrument performance, but until now, a generalized approach with a good mathematical and statistical background has been lacking. This paper describes a calibration procedure having such a background and illustrates its use. The

calibration is accomplished through the use of impedance standards

which relate instrument readings to the values of the standards

through a known functional relationship. The calibration procedure

described estimates the parameters associated with the functional

relationship and requires the use of a computer. Calibration is

accomplished at the reference plane of the impedance standards and

any adapter required to connect the standards to the instrument is

assumed to be an integral part of the impedance meter.

Keywords: calibration; digital impedance meter; impedance; leastsquares; measurement; reflection coefficient; uncertainty

1. Introduction and Background

Until about a decade ago, most impedance measurements in the rf range were made by passive methods utilizing either null or resonance principles.

Other methods such as the vector impedance meter were available but, in

general, where the best accuracy was required, they were not as satisfactory.

*Electromagneti c Technology Division.

1
With the introduction of the digital impedance meters utilizing a constant

current source in conjunction with an internal resistance standard and a com- plex voltage ratio detector, came a new era in the measurement of impedance

[1], These new instruments have capabilities which make them far more useful.

Their versatility, measurement speed, convenience, accuracy, and relatively

low cost have made the older methods obsolete in many situations.

With the rapidly increasing acceptance and use of these instruments, we are encountering the problem of accuracy verification. Early approaches, for

the most part, utilized spot checks with available standards and adapters, but

at best such methods lack any significant degree of mathematical generality.

In other words, even if there is a standard available to provide a check of

the instrument accuracy at one point, there is no real assurance that the

instrument is correct at some other point where a standard is not available.

Thus, there exists a need for improved calibration methods to use with these

instruments.

Although there are several different models of the four-termi nal -pai

(4TP) LCR (inductance, capacitance, resistance) meters in current use, it is

believed that the procedure to be described is applicable for each one with

the single provision that appropriate standards of impedance are available.

The data presented in this report were generated using a Hewlett-Packard

Model 4275A LCR meter, serial number 1851J00419. This particular instrument

is the property of the Boulder Laboratories of the National Bureau of Standards (NBS).

2. Purpose and Scope

The project is an initial effort to demonstrate the feasibility of the

proposed calibration approach. To further develop the procedure to the point

where all of the meter character! sti cs are analyzed and integrated into a

^Certain commercial equipment, instruments, or materials are identified in

this paper in order to adequately specify the experimental procedure. Such

identification does not imply recommendation or endorsement by the National

Bureau of Standards, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

2
complete calibration package is beyond the scope of this initial phase. Within this initial effort we will concentrate on analyzing data taken at frequencies of 1 and 10 MHz. At each of these frequencies two adapter conditions

have been studied. The first condition is with a 4TP to 14 mm coaxial adapter

connected directly to the front panel of the meter. The second condition

utilized the same adapter, but with aim long cable harness inserted between

the front panel of the meter and the 4TP to 14 mm coaxial adapter. The latter

is a configuration used in remote measurement applications. While there are many other situations and conditions that could have been undertaken, such as temperature, warm-up time, etc., they have been left for future effort if

deemed necessary. The purpose here is primarily to learn how to implement the

procedure, to study the error sources, and to gain some insight into how it can best be implemented in the future to solve the many and varied calibration

needs for impedance meters of this general type.

3. Four-Termi nal -Pai r Interface and Adapters to Other Connector Types

In the background discussion, mention was made of the fact that the new generation digital LCR meters employ a measurement principle which is different from the older, more traditional methods. One result of the new method is

that the measurement interface (the manner for connecting an unknown impedance

for measurement) is a 4TP arrangement of coaxial connectors consisting of a "Hi“ and a "Lo" coaxial connector for current and a "Hi" and a "Lo" coaxial

connector for voltage. These are arranged in the manner shown in figure 1.

Such an arrangement is necessary to facilitate the measurement process, but it

does have some important practical disadvantages which include the following:

1. For most measurement applications, it is necessary to provide an adapter to either a single coaxial (a one-port) or to a double coaxial connector (a two-port ).^ Adapters for the measurement of

leaded components are also provided by the manufacturer as well as remote probes for in situ measurements of circuit components. Calibration with these in use is not addressed in this effort. To do so would require the development of special standards.

^A two-port is often called a three-terminal measurement, especially for some capacitance or admittance measurements.

3
Figure 1. Measurement interface for 4TP-type LCR meter.

2. The calibration problem with these meters presents some difficulties

mainly because the measurement interface is a coaxial 4TP configuration for which there are no established standards at NBS. Existing

two-terminal (one-port) or three-terminal (two-port) impedance standards cannot be connected directly to the 4TP interface of these

meters without some type of an adapter which will itself affect the

measurement result. ^ The Hewlett-Packard Company does market standards made specifically for instruments with 4TP-type measurement

interfaces. These are resistance and capacitance elements whose

values can be measured directly at dc or low frequencies and the high

frequency values are then derived from a lumped circuit model by calculation using best estimates for residual impedance contained in the

^Because many LCR meters of the 4TP variety have a feature which allows for

calibration at any measurement interface utilizing a procedure employing both

short- and open-circuit references, the reader may take exception to the

statements in 2, above. However, a simple experiment utilizing the 1 m

remote measurement feature of the instrument will serve to illustrate the

problem. This simple experiment is performed by precalibrating with an adapter connected directly to the instrument and then measuring an unknown

device. Following this the 1 m remote measuring harness is inserted between

the meter and the adapter and the calibration routine is again executed.

When again measuring the same unknown device as before a significantly

different result may be observed. This will illustrate the importance of

generating correction factors for each measurement configuration.

4
model. To date, no concentrated attempt has been made at NBS to

either duplicate or confirm this approach except for 4TP capacitors

[2]. To do so hardly seems appropriate because even with a good

absolute calibration at the 4TP interface, there is still a question

of the calibration accuracy at the reference plane of an added

adapter. It is to be noted, however, that over large portions of the

frequency and impedance ranges of these meters, wel 1 -desi gned

adapters do not seriously degrade the measurement accuracy.

In this effort, two specific measurement configurations were

studied, each employing an adapter from the 4TP interface at the

instrument panel to the GR900, 14-mm-type precision coaxial con- nector. In one configuration the 4TP to 14 mm adapter was attached

directly to the instrument and in the second configuration aim

remote measurement harness was inserted between the front panel of

the meter and the 4TP to 14 rmi adapter. In these two configurations

the "cable length" switch on the meter panel was set to "0" and "Im",

respecti vely

4. Theoretical Approach

4.1 Impedance Relationship through the Adapter

The measurement of impedance for devices with 14 mm coaxial connectors

requires the use of a 4TP to 14 mm coaxial adapter. The adapter becomes part

of the device under test (OUT) and the LCR meter measures the impedance of

this DDT in combination with the adapter. What is required is the impedance

of the OUT. This is illustrated in figure 2.

In figure 2 the meter measures the impedance at reference plane 1 while

the impedance at reference plane 2 is desired. If we view the adapter as a general network with terminal variables v and i, then we can express the relationship between these terminal variables at the two reference planes by the

following equations [3].

v^ = Av^ - Bi^

il = ‘^''2 - “’2

( 4 - 1 )

5
Reference

Plane 1 Reference

Plane 2 — '2

Figure 2. Measurement configuration and symbolism

for meter calibration procedure.

Taking the ratio of to ip and dividing the numerator and denominator of

the right side by -Di

2 gives the following:

V

i

1

1

A \

D i_

C ^

D i.

+ 1

(4-2)

Noting that the ratio v/i is equal to the impedance Z, and letting A' = -A/D,

B' = B/D, C = -C/D we get:

A'Z + B'

2 , . .

In theory, if there were no measurement error, we could obtain the

unknown parameters A', B', C by measuring the impedance 1-^ for three known

values of and solving three complex equations like eq (4-3) for the three

complex unknowns.

6
However, in practice, Zi is error the following statistical

measured with error,

model is used:

A'Z + B' 2i

To account for this

(4-4)

where

is the measured impedance for the ith standard,

Z

2 ] is the known impedance of the ith standard,

e^ is a complex random error in the ith measured impedance, and

A', B‘, C are the unknown parameters.

One approach to estimate the parameters in eq (4-4) is via the method of

least squares. However, one assumption for the valid use of ordinary least

squares is that the e-j have a constant variance. Because the relationship in

eq (4-4) is assumed valid for all values of Z in the complex plane, the constant variance assumption means the effect of the random errors is independent

of Z. The application of least squares to eq (4-4) resulted in large residuals for large values of impedance. This suggests that the variance of the e-j

is not constant for all values of Z but is a function of Z. A common procedure used to accommodate nonconstant variance of the error term e^ is to

transform the data so that the assumption of equal variance is valid. In many

instances this transformation has to be obtained empirically. Fortunately,

there is a t ransformati on based on physical relationships that achieves the

required result.

4.2 Transformati on to Reflection Coefficients

If we choose a, b as terminal variables (see fig. 2) instead of v, i, we

are led to the following network equations [3]

^1

= ''11^2 ^12^2

" '^21^2 ''22^2*

Taking the ratio of b^ to a^ and dividing the numerator and denominator of the

right side by r22^2

7
(4-6)

If we let a = ^ ^

~ ratio b^/a^

is the reflection coefficient at reference plane 1 in figure 2 and

the reflection coefficient at reference plane 2, then we can rewrite

eq (4-6) as follows

The model in eq (4-7) transforms the reflection coefficient T

2 to a reflection

coefficient Ti through the parameters a, 3, and y. Thus, the model is a linear fractional transformation of T

2 to and the estimation of the parameters a, 3, Y can be thought of as "calibrating" the adapter. The statistical

methods to estimate these unknown parameters are discussed in the next section.

5. Calibration Experiment

5.1 Calibration Curves

Calibration is a process of intercomparing an unknown with a standard and

assigning a value to the unknown based on the value of the standard. If the

calibration is desired over an extended regime of interest, and a functional

relationship can be shown to exist between the standard and unknown, then a calibration curve can be used to assign values to the unknown based on the

values of the standard. For example, the functional relationship between y

and X mi ght be

where y is a reading or measurement from an instrument, and x is the known

value of the standard.

a + 3

r, (4-7) =

1 Y + 1

y

= a + 3x (5-1)

8
The calibration of unknowns is affected by a two-step process. First,

the calibration experiment produces data on n standards, say x^-

, y^; i = 1,

. . ,n where x^- is the known value of the ith standard and y^ is the corresponding measurement made by the instrument on the ith standard. In the

example, these data are used to estimate the parameters a and 6 in eq (5-1).

After obtaining the estimates for the parameters, denoted by a and e, and

measuring a future unknown, y^, the correspondi ng estimated value for x is

found by solving eq (5-1), This gives

3 .

where x^ is the estimated value. This procedure is illustrated in figure 3.

The dots are the coordinates of the x,y data obtained in the calibration

experiment. The solid line is the estimated calibration curve. A future

reading, yf, is related to the standards by drawing a line horizontally from

this value and "reading" the x coordinate of the point of intersection. This

gives the estimated value x^.

Standards

Figure 3. Schematic diagram of a calibration curve.

9
A final requirement is an assessment of the uncertainty in x^. There are several statistical issues concerning the construction of interval estimates

for Xf, but these are beyond the scope of this paper. One approach is to use propagation of error formulas to estimate the variance of in eq (5-2).

This procedure is used to access the uncertainty for the calibration of LCR

impedance meters and is discussed in a later section.

5.2 Statistical Model for the Calibration Curve

The model in eq (4-7) which relates the two reflection coefficients is

analogous to the simple linear model presented in the last section. Geometrically, it is more difficult to visualize the relationship in eq (4-7)

because the variables and T 2 as well as the parameters a, 3, y are complex

quantities. However, this does not pose significant problems analytically.

In section 4.2 we defined T

2 to be the reflection coefficient at refer- ence plane 2 in figure 2, while is the reflection coefficient at reference

plane 1. If we choose a device for which we know the reflection coefficient

T2 and connect it to the adapter we can measure which is the reflection

coefficient obtained by the LCR meter. We assume that the relationship in

eq (4-7) holds for all values of r inside the unit circle, however due to

measurement error in Ti the relationship will not be exact. This leads to the

following statistical model:

li Y r^. + 1 (5-3)

where

is the LCR meter reading when the ith standard is connected to the

adapter,

T2 -j is the known reflection coefficient for the ith standard,

e^ is a complex random error of measurement in and

a, 3, and y are unknown complex parameters.

In the remainder of this paper it is assumed that the errors e^- are random in nature and that any errors in the standard values are negligible

compared to the e^ . This assumption will be discussed in connection with the

evaluation of the procedures to be presented in a later section.

10
5.3 Least-Squares Solution for the Calibration Parameters

The least-squares solution for the parameters in eq (5-3) is one that

minimizes the following quantity:

n S = 5, 1=1

(5-4)

where

oc r^l + 6

^i

” ^li ” r„. +

1^’ (5-5)

2i

and n is the number of standards.

Since the e^ are complex numbers, the least-squares solution is implemented in the following manner. S in eq (5-4) can be rewritten as:

S = I [Re(e.)^ + Im(e. )^],

i=l

^ ^ (5-6)

where we denote the real and imaginary parts of a complex number by Re and Im,

respectively. Substitution of eq (5-5) into eq (5-6) gives

“^21 "

®,.2

“

^21 " ^.2

s = I {[Re(r ) - Re e-—-—)] + [Im(r ) - )] }. (5-7)

1=1 ^

'^Zl ^ ^ ^ *^21 ^

If we now make the following notational changes:

y. = Re(r^.)

(x^ ,1) = Re(-

e. = Re(e.

1 1

^

^2i " ^

Y F2i + 1

y. = Im(r^.

a r„. + 6

f^-(x^. ,1) = + 1

e. = Im (e.

1

' V

i=l, 3, 5, • • *,2n-l

]

i = 1, 3, 5, • • »,2n - 1

i=l, 3, 5, • ••,2n-l

i = 2, 4, 6 • • *2n,

)

i = 2, 4, 6 • • »2n,

i = 2, 4, 6 • • *2n,

11
and

e = [Re(a), Im(a), Re(a), Im(3), Re(y), ImCy)].

We can rewrite the model in eq (5-3) as

(5-8)

The least-squares solution for_9 in eq (5-8) is one that minimizes

S = I (Yi - U (x,-

, 1)) (5-9)

i=l

and it can be shown that the expressions in eqs (5-7) and (5-9) are equivalent. It should be noted that the expressions f^-(x^,j^) are nonlinear in the

parameters, i.e., a, g, and y» thus eq (5-9) has to be solved by nonlinear

iterative least squares techniques. This poses no great difficulty as soft- ware is readily available that can solve nonlinear least-squares problems [4].

The nonlinear least-squares solution has been implemented at NBS and

tested using data obtained at 1 and 10 MHz. The results are discussed in

section 7.

6. Measurement of Unknowns

6.1 Measurement of an Unknown r

After obtaining estimates for the calibration parameters a, g, and y, it

is possible to estimate the reflection coefficient for an unknown OUT.

Letting ^lu be the LCR reading for an unknown OUT, then the estimated reflection coefficient, T2

, is obtained by inverting eq (4-7) for T

2 which gives:

r.lu

r2u

(6-1)

12
where

is the measured r when the DDT is connected to the adapter,

a, 3, Y 3re the estimates of the calibration parameters, and

^2^J is the estimated r for the OUT.

6.2 Uncertainty in F2 ^

In order to estimate the uncertainty in T

2 ^j we need to know the uncertainty in the parameter estimates a, 3, y and in the meter reading

These are the quantities which appear on the right side of eq (6-1). The

least-squares solution described in s£ction 5.3 provides an estimate of the

standard deviation of This is given by

a

= S/[2(n - p)], (6-2)

where

S = sum of squares in eq (5-7) at the solution,

n = number of standards in the calibration experiment, and

p = number of parameters estimated.

The estimated standard deviations for a, 3, y are also obtained from the

least-squares solution. We will not describe how we obtain these estimates,

but will denote these by S , S^, and S . ^

An approximation for the standard deviation of r

2 n can be obtained by

using propagation of error formulas [5]. Application of this procedure to

eq (6-1) gives this standard deviation, denote this by S(r

2 u). The details of

these procedures are beyond the scope of this report but will be demonstrated

with sample data in the next section.

^The quantities r, a, 3, X are complex numbers. We can treat these analytically as two real variables. Thus, each complex quantity actually has two

standard deviations, one associated with the real part and one associated

with the imaginary part. We use the notation Sa, for example, to represent

the standard deviation for either the real or imaginary part of a for

rotational convenience.

13
6.3 Transformation to Impedance

It was mentioned in section 4.1 that the objective was to obtain the

impedance for the DUT. The calibration procedure described so far gives the

reflection coefficient for the DUT and its associated standard deviation.

These are and S(r2 y). The impedance is obtained from the reflection coefficient by the transformation

•2u - ^2u>

(6-3)

Finally, we require an estimate of the uncertainty in obtain an estimate of the standard deviation of by the application of propagation of error formulas. We have the necessary quantities to accomplish this, namely

T2 u and S(r2 u). We will now present the application of the procedures

described to sample data obtained at 1 and 10 MHz.

7. Example

7.1 Standards Used in Calibration

The calibration experiment discussed in section 5.2 required known values

for n standards. At 1 MHz the standards used to calibrate the meter were a short-circuit, open-circuit, 50 and 100 ^ terminations, a 1000 pF capacitor,

and inductors of 1, 2.5, 5, 10, and 25 yH. These values relative to 50 Q are displayed in figure 4. The standards used at 10 MHz were a short-circuit, an open-circuit, 50 and 100 terminations, a 1 yH inductor, and capacitors of

200 and 1000 pF. These values relative to 50 are displayed in figure 5.

At each frequency the standard was connected to the adapter and the

impedance was measured‘s by the LCR meter. These were then transformed to

reflection coefficients by:

Z - 50

^ ' Z + 50*

These data are presented in tables 1 and 2.

(7-1)

‘sprior to the measurements the manufacturer's suggested calibration procedure

was not used. The meter was turned on and allowed to warm up as specified in

the operating manual. For these tests, the LCR meter was not "zeroed" on open and short circuit terminations. This procedure was used because it was felt that the stored set of default values provided a more stable reference

for the measurements than the daily measurements of opens and shorts.

14
Figure 4. The reflection coefficients of the various

impedance standards used at 1 MHz.

15
Figure 5. The reflection coefficients

impedance standards used at

of the various

10 MHz.

16
Table 1. Calibration data, 1 MHz; with adapter connected to LCR meter

Standard Impedance

Standard value

h

{^)

LCR reading

Refl ecti on Standard valuecoeff i ci ent

LCR reading

^1

Short Re 0.00000 0.00646 -1.00000 -0.99973

Im 0.00000 0.11945 0.00000 0.00478

50 Q Re 50.02500 50.06500 0.00025 0.00065

Im 0.08730 0.05400 0.00087 0.00054

100 n Re 99.83000 99.93900 0.33258 0.33306

Im -0.19790 -Ol.16900 -0.00088 -0.00075

Open Re 0.00000 -25.72000 1.00000 1.00000

Im -159000.00000 73680.00000 -0.00063 -0.00136

1000 pF Re 0.00000 0.00000 0.82016 0.81944

Im -159.06700 -158.72000 -0.57214 -0.57316

1 uH Re 0.07970 0.09770 -0.96789 -0.96620

Im 6.07200 6.17770 0.23860 0.24246

2.5 pH Re 0.16980 0.20430 -0.81709 -0.81431

Im 15.62000 15.70530 0.56574 0.56757

5 yH Re 0.26180 0.28000 -0.44740 -0.44829

Im 30.76200 30.71800 0.88586 0.88482

10 yH Re 0.47440 0.50000 0.17351 0.17365

Im 59.66100 59.67400 0.97691 0.97646

25 yH Re 1.41370 1.39000 0.79400 0.79421

Im 149.38000 149.43600 0.59853 0.59841

17
Table 2. Calibration data, 10 MHz; with adapter connected to LCR meter

Standard Impedance

Standard value LCR reading

h h

Reflection coefficient

Standard value LCR reading

F2

Short Re 0.00000 0.03240 -1.00000 -0.99767

0.00000 1.13870 0.00000 0.04547

50 a Re 50.06400 49.87800 0.00064 -0.00114

Im 0.00294 0.90600 0.00003 0.00908

100 a Re 99.93000 99.52100 0.33307 0.33122

m -1.25200 -0.87550 -0.00557 -0.00392

Open Re 0.00000 -21.96000 0.99998 0.99995

-15915.00000 -72730.00000 -0.00628 -0.01375

1000 pF Re 0.00000 0.05569 -0.82755 -0.84936

m -15.35900 -14.18000 -0.56139 -0.52390

1 yH Re 0.03070 0.28747 0.22774 0.23130

In 63.05100 63.34900 0.97323 0.96835

200 pF Re 0.00000 0.06723 0.43364 0.40692

'm

-79.55100 -77.04000 -0.90108 -0.91259

Table 3. Least -squares estimates of calibration parameters;

with adapter connected to LCR meter.

1 MHz 10 MHz

Real Imagi nary Real Imagi nary

a 0.99983 -0.00218 0.99823 -0.02415

Sa 0.00040 0.00040 0.00127 0.00127

3 -0.00065 0.00066 -0.00511 0.00852

S

0.00036 0.00036 0.00110 0.00110

Y -0.00120 -0.00111 -0.00716 -0.00973

0.00041 0.00041 0.00130 0.00130

Residual standard

deviation a 0 .00096 0.00285

Degrees of freedom 14 8

18
7.2 Least-Squares Estimation of the Calibration Parameters

A Fortran program, written on a CDC CYBER 750, solves the least-squares

formulation presented in section 5.3. This program was then run using the

data in tables 1 and 2 to test the procedure. The program listing is presented in appendix A, and program printouts for data at 1 and 10 MHz are given

in appendices B and C, respectively. The estimated parameters and standard

deviations using these data are presented in table 3.

Table 3 gives the real and imaginary parts of the estimated parameters

and their associated standard deviations. For example, at 1 MHz the real part

of a was estimated to be 0.99983 with a standard deviation of 0.0004. The

A residual standard deviation, a’ is computed from eq ( 6 - 2 ).

If the model in eq (5-3) is adequate then the residuals are estimates of

the random errors, e^ , given in eq (5-5). The residuals r^- are computed by

A A

C. r + 6

Re(rp = Re(FjP - Re ) (7-1)

Y r„. + 1 21

Im(rp = Im(r^p - Im (-7

-^^

). (7-2)

Y r^i ^ 1

These residuals are displayed in figures 6 through 9. In these figures

the residuals computed in eqs (7-1) and (7-2) are plotted on the z axis

against the standard values which are on the x-y plane. These plots illustrate the magnitude of the residuals as a function of location of the standard

values for r in the unit circle. Other graphs of these residuals are presented in figures 10 and 11. In these graphs the imaginary part of the residual is plotted on the y axis against the real part of the residual on the

X axis. The residuals appear as vectors in these graphs and should be uniformly distributed around the origin. An outlier, or abnormally large vector,

would need to be investigated before the estimation results could be accepted.

It is not within the scope of this paper to discuss all the diagnostic procedures associated with the examination of these residuals. These will be

developed further in later work.

19
LCR

CRLIBRflTION 1

MHZ

20

Figure

6.

The

real

part

of

the

residuals

at1

MHz.
LCR

CRLIBRRTION 1

MHZ

r

21

Figure

7.

The

imaginary

part

of

the

residuals

at1

MHz.
LCR

CRLIBRRTION

10

^^HZ

22

Figure

8.

The

real

part

of

the

residuals

at

10

MHz.
LCR

CnLIBRRTION

10

MhZ

23

Figure

9.

The

imaginary

part

of

the

residuals

at

10

MHz.
LCR

CALIBRATION 1

MHz

24

Figure

10.

Plot

of

residuals:

Imaginary

versus

real

at1

MHz
LCR

CALIBRATION

10

MHz

25

Plot

of

residuals:

Imaginary

versus

real

at

10

MHz.
7.3 Measurement of Unknown r

Having obtained estimates of the calibration parameters we can now apply

the methods presented in section 6.1 and 6.2 to obtain the reflection coefficient and the associated standard deviation for an unknown device. To demonstrate the procedure a program was written which evaluates eq (6-1) to obtain

r2 u and its standard deviation S(r

2 u).

The data used for are the original

LCR meter readings which are presented in tables 1 and 2 as The "estimated values," r2 (j, and the known standard values T

2 which they estimate are presented in tables 4 and 5. The difference between V

2 ^ and T

2 is also tabulated.

In these tables the real and imaginary part of r

2 n and T2 are presented

and the numbers right underneath these are estimates of the errors in these

quantities. For T

2 u these are the quantities S(r2 u) computed by propagation

error formulas. For T 2 these are estimates of the systematic errors in the

standards. The differences between and T2 can be compared to the standard

deviation of ^

2 ^J, S(^2 ^J) to get a feel for the adequacy of the proposed procedures. Examination of these differences indicates that the assumption of

the error e^ being mostly random is not likely. The e^- contain a systematic

component due to the large systematic errors in F 2

. Therefore, the standard

deviations S(r

2 ^j) are not really estimates of the random error in ^

2 ^J since a significant systematic error is present.

7.4 Transformation to Z

Having obtained V

2 ^ we can now transform these to impedances by eq (6-3),

and estimate the standard deviation from propagation of error techniques.

This was done using the values for V

2 ^ in tables 4 and 5. The results, l

2 ^ corresponding to V2yx presented in tables 6 and 7, along with the known

impedances, Z

2

. The entries in these tables are in the same format as those

in tables 4 and 5, except the values are for impedance rather than reflection

coeff ici ent

26
Table 4 . Calibration of r at

adapter connected to

1 MHz; with

LCR meter.

Devi ce "Estimated"

meter readings

Known standard

val ues

Di fference

«eC2u) R,(r2> Re(r2„) - ReCFj)

Short -1.00046 0.00084

0.00114* 0.00114*

-1.00000

o.ooooo’’’ 0.00000

0.00000"^

-0.00046 0.00084

50 n 0.00130

0.00103

-0.00012

0.00103

0.00025

0.00050

0.00087

0.00040

0.00105 -0.00099

100 0.33363

0.00103

-0.00081

0.00103

0.33258

0.00045

-0.00088

0.00045

0.00105 0.00007

Open 0.99961

0.00111

-0.00094

0.00111

1.00000

0.00000

-0.00063

0.00000

-0.00039 -0.00031

1000 pF 0.82002

0.00110

-0.57138

-0.00110

0.82016

0.00004

-0.57214

0.00006

-0.00014 0.00076

1 pH -0.96781

0.00119

0.23932

0.00119

-0.96789

0.00025

0.23860

0.00027

0.00008 0.00072

2.5 pH -0.81647

0.00126

0.56596

0.00126

-0.81709

0.00061

0.56574

0.00065

0.00062 0.00022

5 pH -0.44983

0.00133

0.88492

0.00133

-0.44740

0.00103

0.88586

0.00091

-0.00243 0.00093

10 pH 0.17369

0.00132

0.97696

0.00132

0.17351

0.00108

0.97691

0.00081

0.00018 0.00005

25 pH 0.79441

0.00120

0.59814

0.00120

0.79400

0.00072

0.59853

0.00075

0.00041 -0.00039

*These

errors entries are

• S(T

2 l|)

which are the standard deviations obtained from propagation of

^These entries are estimated limits to the systemati

error in the standard values.

27
Table 5. Calibration of r at 10 MHz

Device "Estimated" Known standard Difference

meter readings values

«e(r2u> y’’2u) Re(r2> '^e('’2u)

- Ip (I’2u) - W

Short -1.00204

0.00354*

0.00338

0.00354*

-1.00000

o.ooooo"^

0.00000^

0.00000^

-0.00204 0.00338

50 n 0.00396

0.00305

0.00066

0.00305

0.00064

0.00049

0.00003

0.00008

0.00332 0.00063

100 0.33623

0.00305

-0.00539

0.00305

0.33307

0.00045

-0.00557

0.00045

0.00316 0.00018

Open 0.99965

0.00343

-0.00771

0.00343

0.99998

0.00000

-0.00628

0.00001

-0.00033 -0.00143

1000 pF -0.82613

0.00363

-0.56385

0.00363

-0.82755

0.00123

-0.56139

0.00182

0.00142 -0.00246

1 yH -0.22398

0.00359

0.97246

0.00359

0.22774

0.00092

0.97323

0.00030

-0.00376 -0.00077

200 pF 0.43176

0.00359

-0.90048

0.00359

0.43364

0.00061

-0.90108

0.00030

-0.00188 0.00054

*These

errors

entries are

• S(r2 u) which are the standard deviations obtained from propagation of

^These entries are estimated limits to the systemati

error in the standard values

28
Table 6. Calibration of Z at 1 MHz

Devi ce "Estimated" Known standard Ui fference

meter readi ngs values

Re(Z2u) W w Re(Z 2u)

- v^2u) - y^2)

Short -0.01155 0.02090 0.00000 0.00000^ -0.01155 0.02090

0.02849* 0.02849* 0.00000^ o.ooooo'^

50 Q 50.13004 -0.01198 50.02500 0.08730 0.10504 -0.09928

0.10309 0.10309 0.05000 0.04000

100 fJ 100.06759 -0.18219 99.83000 -0.19790 0.23759 0.01571

0.23120 0.23120 0.10000 0.10000

Open 3.72E+4 -9.05E+4 0.00000 -1.&9E+5 3.74E+4 6.85E+4

10.62E+5 10.62E+5 0.00000 100.00000

1000 pF 0.15153 -159.21656 0.00000 -159.06700 0.15153 -0.14956

0.30698 0.30698 0.00000 0.02000

1 pH 0.07729 6.09026 0.07970 6.07200 -0.00241 0.01826

0.03018 0.03018 0.00500 0.00600

2.5 uH 0.18061 15.63470 0.16980 15.62000 0.01081 0.01470

0.03477 0.03477 0.01000 0.01500

5 yH 0.25240 30.67236 . 0.26180 30.76200 -0.00940 -0.08964

0.04606 0.04606 0.02000 0.03000

10 yH 0.46987 59.67123 0.47440 59.66000 -0.00453 0.01023

0.08058 0.08058 0.04000 0.06000

25 yH 1.39296 149.52158 1.41370 149.38000 -0.02074 0.14158

0.29879 0.29879 0.11000 0.15000

*These entries are S(r2^j) which are the standard deviations obtained from propagation of errors •

^These entries are estimated limits to the systematic error in the standard values.

29
Table 7. Calibration of Z at 10 MHz with adapter connected

directly to LCR meter.

Devi ce "Cal i brated" Known standard Di fference

meter readi ngs val ues «e(Z2u) m(Z2u) w W '*e(Z2u ) - Re<Z2> ‘m(Z2u> - W

Short -0.05119 0.08415 0.00000 0.00000 -0.05119 0.08415

0.08826 0.08826 0.00000 0.00000

50 a 50.39772 0.06657 50.06400 0.00294 0.33372 0.06363

0.30834 0.30834 0.05000 0.00080

100 n 100.64417 -1.22179 99.93000 -0.12520 0.71417 0.03021

0.69511 0.69511 0.10000 0.10000

Open 539.75235 -12.94E+3 0.00000 -1.5915E+4 539.75235 29.70E+2

5.78E+2 5.78E+2 0.00000 15.00000

1000 pF -0.00564 -15.43647 0.00000 -15.35900 -0.00564 -0.07747

0.09924 0.09924 0.00000 0.06000

1 yH 0.13374 62.82566 0.3070 63.05100 -0.10304 -0.22534

0.23198 0.23198 0.00600 0.06000

200 pF 0.11461 -79.42231 0.00000 -79.55100 0.11466 0.12869

0.31685 0.31685 0.00000 0.06000

*These entries are S(rp,.) which are the standard deviations obtai ned from propagation of errors •

^These entries are estimated limits to the systematic error in the standard values.

30
Table 8. Least-squares estimates of calibration

parameters with 1 m cable inserted.

1 MHz 10 MHz

Real Imagi nary Real Imagi nary

a 0.99965 -0.00286 0.99883 -0.02638

S a 0.00033 0.00033 0.00089 0.00089

3 0.00055 0.00033 0.00972 0.00967

0.00034 0.00034 0.00068 0.00068

Y 0.00052 -0.00144 0.00802 -0.00779

0.00044 0.00044 0.00849 0.00849

Residual standard

deviation a 0,.00053 0.00145

Degrees of freedom 4 8

7.5 Calibration Parameters with aim Cable

To further demonstrate the estimation procedure, data were obtained at 1 and 10 MHz with aim remote measurement harness inserted between the front

panel of the meter and the adapter. Only the resulting parameter estimates

are given. These are presented in table 8. At 1 MHz data were not obtained

for the inductors of 1, 2.5, 5, 10, and 25 yH, therefore only five standards

were used.

8. Summary

A method has been presented which permits a calibration of the LCR meter

when an adapter is connected to the test port. This allows the measurement of

impedance for devices with 14 mm precision coaxial connectors.

The estimation procedures for the parameters and associated uncertainties

described in this report assume that the errors are random (see sect. 5.2).

However, application of the method on actual data indicates that this is probably not the case. The propagation of error formula used to estimate the

various standard deviations; i.e., S(^

2 ^J) and S(Z2 ^j) depend on the nature of

31
randomness in these errors. While in actuality these uncertainty estimates

are reasonable bounds based on practical considerations, they do not have the

same interpretation statistically, as if the errors were random.

The major contribution to the systematic error is from the uncertainty in

the values of the standards. Recall in section 5.2 it was assumed this error

is negligible. Examination of the error limits for the values of the standard

indicate that this error is not negligible as in many cases it is the same order of magnitude as the residual standard deviation computed from eq (6-2)

and given in table 3.

For example, at 1 MHz the computed residual standard deviation given in

table 3 is 0.00096. In the event the errors are random this is an estimate of

the "average" error in or the measured reflection coefficient. However,

we see from table 4 that the uncertainties for the real and imaginary parts of

the standard reflection coefficient values for the inductor are the same order

of magnitude (0.00025 to 0.00108). It is beyond the scope of this paper to

address how to accomodate the errors in the standards. This will be done in

future work.

At this point it is not clear how the procedure, even if successful,

might be utilized and made practical to the measurement community. A requirement to use the procedure would be computer capable of solving nonlinear systems of equations. Additionally, future work should be done to detrmine the

minimum number of standards needed at each frequency to carry out a reliable

cal i brati on.

We wish to acknowledge the help given us by Cletus A. Hoer who suggested

the theoretical approach that was used in developing this procedure and also

the assistance of Stanley R. Booker of the Sandia Corporation, Albuquerque,

New Mexico, for a great deal of technical support and advice during the work.

Finally, we acknowledge the support of the Sandia Corporation which funded the

project and thank them for it.

32
9. References

[1] Maeda, K.; Narimatsu, Y. Multi frequency LCR meters test components under

realistic conditions. Hewlett-Packard J. pp. 24-31; 1979 February.

[2] Jones, R. N. Evaluation of three-termi nal and four-terminal pair

capacitors at high frequencies. Nat. Bur. Stand. (U.S.) Tech. Note 1024;

1980 September.

[3] Kerns, D. M.; Beatty, R. W. Basic theory of waveguide junctions and

introductory microwave network analysis. New York; Pergamon Press, pp.

36, 37; 1967.

[4] Donaldson, J. R.; Tryon, P. V. Nonlinear least squares regression using

STARPAC. Nat. Bur. Stand. (U.S.) Tech. Note 1068-2; Oct. 1983.

[5] Ku, H. H. Notes on the use of propagation of error formulas. Nat. Bur.

Stand. (U.S.) J. Res. 70C(4); 1966. Reprinted in KU; 1969.

33
Appendix A. FORTRAN Program Listing

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40
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41
Appendix B. Printout from Program for Data at 1 MHz

42
STEP

SHE

EOR

OBSERVATIONS

FAILING

STEP

SIZE

SELECTION

CRITERIA

ARPanXIH

ATING

parameter

starting

value

scale

oerivativf

count

notes

row

number

INOEX

FIXEO

(PAP)

(SCALE)

(STP) FC

<0X o ^ ^ m ooo «\J

O ^ ^ CV

INi fKt ^

00O 1 I I %u UJ UJ O^ IT

fSi 00m O' ^ ^ o

in QDO

OX

l/> z 0 0 c •H (SI (SI 4 UJ 4 z X

LU UJ >— a.

w HH Z 0 UJ X X X Z X 0 UI UJ z 4

31 l-H 0 orUJ

u -4- 4- 4 4 1- 0 c 0 c 0 C 0 c. t 1 1 1 1 1 U. or Ui

UJ u. LU UJ 4X. UJ 0 (/) 4 4* 4 4 4 UJ 3 mm

(SI fS) (SI (Si fs (SjX z 4 or <v (V (SJ (SI <NJ (SJ 0 0 UJ

N. fK f*- mmK 4

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fSl 0 UJ >- LUU Ui >-

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u ft

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43

RESinUAL

SHNPARO

DEVIATION

FOR

INPUT

PAPAMFTFR

VALUES

(RSO)

.9i27E-03
N0NL1NE4P

LEAST

SQUABFS

FSTIHATTn*^

WTTH

NIJMEOI^AILY

APPOnx

ATEO

DEPIVATIVES.

CnSTISiJEO

STARPAC

1.03S

-

MARCH

6»

1904 o (/I ^ fSI OD O O o rH «n o o- ^ lo m (*> « (M O

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Ui 1 1 1 •< o X

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46
RE^LFCTinN

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m CM CM o o o o o o fSI CM cn cn cr> #n CM CM o f-4 ^ #4 z o o o o o o o o o o o o o o o o o o o o < o o o o o o o o o o o o o o o o o o o o >o

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47
PREOTCTEn

ESTIHATEO

STANDARD

LCR

READING

LCP

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STANDARD

STANDARD

DEVIATION

DIFFERENCE

m o OD ^ «o •-t o o ^ ®

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NBS-114A (REV. 2-BC)

4.

U.S. DEPT. OF COMM. 1. BIBLIOGRAPHIC DATA

SHEET (See instructions) |

PUBLICATION OR 2 REPORT NO.. Performing Organ. Report No. 3. Publication Date

NBSIR 84-3016 August 1984

TITLE AND SUBTITLE

A Generalized Method for the Calibration of Four-Terminal-Pai r Type Digital

Impedance Meters

5. AUTHOR(S)

R. M. Judish and R. N. Jones

6. PERFORMING ORGANIZATION (If joint or other than MBS, see in struction s) 7^. Conftact/Grant No. Sandia document

national bureau of standards No. b2-/86b

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

4-19-83 to 9-30-83

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Sandia National Laboratories

Albuquerque, New Mexico

10.

SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS So ftware Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Since the introduction of automated, four-termi nal -pai r type digital impedance

meters, there has been a continuing interest in the development of calibration techniques which would satisfactorily verify the accuracy capabilities of these instruments. Various attempts have been made and all have helped to provide a certain

degree of confidence in instrument performance, but until now, a generalized approach

with a good mathematical and statistical background has been lacking. This paper

describes a calibration procedure having such a background and illustrates its use. The calibration is accomplished through the use of impedance standards which relate

instrument readings to the -values of the standards through a known functional relationship. The calibration procedure described estimates the parameters associated

with the functional relationship and requires the use of a computer. Calibration is

accomplished at the reference plane of the impedance standards and any adapter

required to connect the standards to the instrument is assumed to be an integral part

of the impedance meter.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) calibration; digital impedance meter; impedance; least-squares; measurement;

reflection coefficient; uncertainty

13. AVAILABILITY

U n I i m i ted

[_ For Official Distribution. Do Not Release to NTIS

I I Order From Superintendent of Documents, U.S. Government Printing Office, Washington,

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

60

15. Price

$ 10.00

USCOMM-DC 6043-P80 * U.S. GOVERNMENT PRINTING OFFICE; 1984—780-812 9458 REGION NO 8
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