THE SCIENCE AND ENGINEERING REVIEW OF DOSHISHA UNIVERSITY, VOL. 55, No. 4 JANUARY 2015

( 55 )

Effect of Measuring System on Small-Capacitance Measurement using

Transient Waveforms

Diah PERMATA*and Naoto NAGAOKA*

(Received November 4, 2014)

An impedance measurement using transient voltage and current waveforms with Fourier transforms is a practical

method especially for a power system because steady-state measuring using a variable frequency source is difficult to apply.

Since the transient method is an indirect measuring method, its accuracy depends on the measuring condition. This paper

investigates the effect of the measurement system on the measured results. A capacitive circuit, which represents the impedance

between electrodes implanted into a piece of wood, is used as a circuit under test (CUT). Three variables affecting the measured

result are identified: a stray capacitance between CUT and ground, a capacitance of a voltage probe, and a sheath surge

impedance of a current injected cable. These parameters are investigated and solved by theoretical calculation. Finally, the

capacitance obtained from measured result is confirmed by a theoretical calculation.

differential mode, common mode, stray capacitance, probe capacitance, sheath surge impedance

The impedances of wooden poles or wooden

cross arms have been investigated by many researches.

Some of them are concentrated on the leakage current

against a low-frequency voltage 1, 2). The impedance

model for studying the impulse strength of a

wood-porcelain combination also has been proposed by

the references 3 and 4. The authors have studied the

impedance between electrodes implanted into a piece of

wood from a steady state measurement, and the

equivalent circuit have been derived 5). Although the

steady-state measurement is theoretically accurate, it is

difficult to obtain an impedance of a practical wooden

pole in a high frequency region. On the other hand, a

measurement using transient voltage and current

waveforms with Fourier transform is practical because

no high frequency voltage or current source is required.

In this paper, the measurement method is investigated

using a model circuit prior to a real measurement.

Three variables are identified affecting the

measured result; a stray capacitance between electrodes

and the ground Cg, a capacitance of a voltage probe Cp,

and a sheath surge impedance of a current injected cable

Rsh. These parameters are important for an investigation

on a high impedance measurement.

Two measurements are carried out in this study;

differential and common mode tests. A circuit-under-test

(CUT) are represented by an equivalent circuit for a

theoretical analysis. The circuit with the measurement

system is theoretically calculated using a nodal analysis

method. The capacitance of the parameters Cg and Cp

are solved by the theoretical calculation.

A discrete Fourier transform is used to convert

the transient voltage and current from time to frequency

domain. The impedance is defined as a ratio between

* Department of Electrical and Electronic Engineering, Graduate School of Science and Engineering, Doshisha University, Kyoto Telephone: +81-774-65-6337, E-mail: etl1104@mail4.doshisha.ac.jp

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Diah PERMATA and Naoto NAGAOKA356

the transformed voltage and current. It is numerically

calculated using Maple in this paper. The capacitances

are taken from the imaginary part of the impedance.

2.1 Measurement and theoretical study

The CUT is a capacitive circuit, which simulates

the impedance between the electrodes implanted into a

piece of wood 5). The CUT consists of a ceramic

capacitor between the terminals Cel of 3 pF and two

capacitors Cs of 10 pF which simulates the stray

capacitance in the wood specimen as illustrated in Fig. 1.

When practical electrodes are used, the investigation of

the stray capacitance to ground is impossible because

there is no common terminal in the practical electrodes

implanted into a piece of wood. The CUT enables the

investigation. The capacitor to ground Cg in Fig. 1

expresses the stray capacitance to ground. However the

capacitance is unknown. The capacitor Cga is an

additional ceramic capacitor for the investigation in

Section 3.1. The CUT is tested in a differential and a

common mode as shown in Fig. 1.

The equivalent circuits for the two modes are

illustrated in Fig. 2. The resistances Rc and Rsh express

the characteristic impedances of the coaxial and sheath

modes of the cable, respectively. The stray capacitance

to ground Cg and an input capacitance of a voltage

probe Cp are taken into account. The voltage difference

between the electrodes cannot be directly measured,

because the stray capacitance to ground of the

oscilloscope affects the voltage measurement. The fast

transient voltage has to be measured by a “grounded”

instrument. The voltages to ground of the electrodes are

measured by an oscilloscope and the voltage between

the electrodes is indirectly measured as the difference

between the voltages. Nodal analysis is applied to

analyze the equivalent circuits. The circuit equation for

the differential and common modes are shown in Eqs.

(1) and (2), respectively.

(a) Differential Mode

(b) Common Mode

Fig. 1. Measurement setup.

(a) Differential Mode

(b) Common Mode

Fig. 2. Equivalent circuit.

IV

3D2V Cable of 100 m

PG

Rs

CsCelCs

CgaCg

Cel

3D2V Cable of 100 m

PG

VI

Rs

CsCs

SC

CgaCg

12

I Rc

RsCel

Cs Cs

CpCpRsh

Cg3

4

Cg

23

2Cs

1

I Rc

Rs

CpRsh

Effect of Measuring System

( 57 )

357

I

-I

VVVV

cRsRcRsR

gCsCsssCssCcRssC

cRshRpCsCelCselsCsRssCelsC

sRpCsCelCs

0

0

1101102

111

11

4

3

2

1

(1)

II

VVV

RRRR

RRR

RRCCCC

Cps

cscs

ccsh

ssgs

gs

0

1111

1110

10122

3

2

1

(2) 2.2 Transformation from time domain to frequency domain

To obtain the frequency characteristics of the

impedances of the CUT from the transient voltage and

current waveforms, a Discrete Fourier Transform (DFT)

is used to convert into a frequency domain 6). A time

interval t in this study is 0.2 ns. The transient

waveforms, which are used as input data for DFT

calculation, are truncated before a round trip time for

the travelling wave (2 ) on the current injected cable to

neglect the effect of the impedance of the source (P.G.).

To minimize the truncation errors, an exponential

window function with damping coefficient is applied

to the time domain waveforms 7).

)exp( tiW (3)

In this paper, the damping coefficient of

= =2 f is employed. N

ittfikjtitiffkF

02expexp2)(

N

itikjtift

02exp2

N

itsitift

0exp2 (4)

The DFT with the exponential window function

(modified DFT) is called as Discrete Laplace Transform

(DLT) because the equation is identical to the definition

of Laplace transform with operator s.

Fig. 3 shows measured voltage and current

waveforms. The transient voltage and current

waveforms are truncated at 2 = 1 s, i.e., before

arriving a reflected travelling wave from the end of the

cable. Since the length of the current injected cable is

100 m and the traveling velocity of the coaxial mode is

198 m/ s, the travelling time becomes 0.5 s. The

window function is indispensable to reduce the

truncation error on the voltage waveform which is

converges to a non-zero voltage.

The impedances of the modes are obtained by the

voltage and current in frequency domain as shown in

Eqs. (5) and (6).

)(

)()(

fI

fVfZ

diff

diffdiff (5)

)(

)()(

fI

fVfZ

comm

commcomm (6)

Fig. 4 illustrates the impedances and their angles.

The impedances in a frequency range between 1 MHz

and 10 MHz are inversely proportional to the frequency

and show a capacitive characteristic.

Fig. 3. Measured voltage and current waveforms.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Curr

ent [

A]

Volta

ge [V

]

Time [ s]

Voltage - common mode

Current - differential mode

Current - common mode

Voltage - differential mode

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Diah PERMATA and Naoto NAGAOKA358

(a) Impedance

(b) Angle

Fig. 4. Impedance and their angle obtained by DLT.

There is a series resonance at around 20 MHz and

some oscillations are observed in the region higher than

10 MHz. It is assumed that the resonance is due to the

stray inductance of the circuit and the oscillation is due

to noise induced on the circuit.

The load capacitances seeing from the source

resistor Rs are shown in Fig. 5. The capacitances shown

by bold lines are obtained from the imaginary parts of

impedances, which are determined from the measured

results. It shows that the capacitance affected by noise

in a frequency range above 10 MHz. These values go to

negative in the region.

Fig. 5 shows the theoretical capacitances by the

broken lines. The theoretical capacitances of the

differential modes are easily obtained by applying Y-

conversion to the circuit consisting of Cs and Cg in

Fig. 2 (a). By the conversion a capacitance C1

= CsCg/(2Cs+Cg) is connected in parallel with Cp, and a

capacitor C2 = Cs2/(2Cs+Cg) is connected in parallel with

Cel.

Fig. 5. The numerical and theoretical capacitances in

differential and common modes.

The characteristic of the circuit is divided into

two regions by the time constant of the parallel circuit

of the capacitive branch (C1//Cp) and the resistance Rsh

( = (C1+Cp)Rsh. In this study, a frequency region up to

10 MHz is defined as a low frequency region (f <<

1/2 ), and a high frequency region becomes a band

over 100 MHz.

In a low frequency region, the parallel capacitive

branch (C1//Cp) can be neglected since the resistance Rsh

is far lower than the impedance of the branch. The node

2 is grounded via the resistor Rsh. The equivalent

capacitance is given by a parallel connected two

branches, as shown in Eq. (7).

21 CCCCC elpLFdiff

gs

sel

gs

gsp CC

CC

CCCC

C22

2

2s s g

p els g

C C CC C

C C (7)

In a high frequency region, the resistor Rsh can be

neglected due to the low impedance of the parallel

connected capacitive branch (C1//Cp). The differential

mode-capacitance in the high frequency region is shown

in Eq. (8).

0.1

1

10

100

1 10 100

Impe

danc

e [kΩ

]

Frequency [MHz]

|Zdiff| |Zcomm|

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

1 10 100

Angl

e [d

egre

e]

Frequency [MHz]

comm diff

1

10

100

1 10 100

Capa

cita

nce

[pF]

Frequency [MHz]

Cdiff-numerical Cdiff-theoretical

Ccomm-numerical

Ccomm-theoretical

Effect of Measuring System

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359

1 2

2

1212 2 2

12

diff HF p el

s g sp el

s g s g

p s el

C C C C C

C C CC CC C C C

C C C

(8)

In the common mode, the capacitance is

independent of the frequency, because the effect of the

sheath surge impedance Rsh of the current injection

cable is isolated by the high resistance Rs.

gs

gspcomm CC

CCCC

22

(9)

Since the measured results are affected by noise

at a frequency above 10 MHz, the circuit parameters

aforementioned are analyzed in the frequency range

from 1 MHz to 10 MHz. From the theoretical

calculation, the effect of the sheath surge impedance Rsh

is only observed in the differential mode. However, in a

low frequency region below 10 MHz, Eq. (7) shows that

the capacitance is independent of Rsh. The curve of the

differential-mode capacitance Cdiff in Fig. 5 also shows

the capacitance is independent of the frequency. The

sheath surge resistance Rsh, which affects the frequency

dependent of capacitance, is negligible below frequency

10 MHz. The other parameters will be discussed in the

following sections.

3.1 Stray capacitance between CUT and ground Cg

To investigate the stray capacitance between the

CUT and ground Cg, an additional capacitance Cga of

47 pF is connected between the common terminal to

ground, i.e., in parallel with the unknown stray

capacitance Cg as shown in Fig. 1. The additional

capacitance Cga is expressed by dotted lines. The effect

of Cg is discussed using a capacitance difference C

between the capacitance with and without Cga. The

equivalent capacitances of the modes with the additional

capacitance Cga are obtained as shown in Eqs. (10) and

(11) by substituting Cg+Cga into Cg in Eqs. (7) and (9).

The difference in the differential mode Cdiff is given by

the difference between Eqs. (10) and (7) as shown in Eq.

(12). As the same manner, the difference Ccomm is

obtained as Eq. (13) by subtracting Eq. (9) from Eq.

(11). The stray capacitance to ground Cg is obtained by

solving Eqs. (12) or (13) and substituting the measured

C. The theoretical calculation by Eqs. (12) and (13)

shows the ratio of the capacitance difference C in

common and differential mode is four at a low

frequency. The measured ratio converges to four with a

frequency decrease as shown in Fig. 6. The Cg for both

modes are shown in Fig. 6. The very small stray

capacitance of 1 to 2 pF can be estimated by the

proposed method.

2s

diff LH withCga p els

C C C Cs g gaC C C

C C Cg ga (10)

gags

gagspwithCgacomm CCC

CCCCC

22

(11)

)2)(2(

2

gCsCgaCgCsCgaCsC

Cdiff (12)

diffcomm CgCsCgaCgCsC

gaCsCC 4

)2)(2(

24 (13)

Fig. 6. C and Cg and the ratio between both C,

common and differential modes.

3.2 Voltage probe capacitance Cp

Equations (14) and (15) show a capacitance of a

voltage probe Cp obtained from Eqs. (7) and (9). These

results are calculated by substituting the capacitances

obtained from the results shown in Fig. 5 into Eqs. (14)

0123456789

101112131415

1 10

Capa

cita

nce

[pF]

Frequency [MHz]

delta diffCg-diffdelta comCg-commratio

( 60 )

Diah PERMATA and Naoto NAGAOKA360

and (15). The voltage probe capacitances Cp of the

differential and common modes are shown in Fig. 7. The

specification of the probe capacitance provided by the

manufacturer is 8 pF. The results shows that the

accuracy of the proposed method. Even if the probe

capacitance is unknown, it can be estimated by the

method.

gs

gsseldiffdiffp

CC

CCCCCC

2 (14)

gs

gscommcommp

CC

CCCC

2

2 (15)

Fig. 7. Cp in differential and common modes.

3.3 Comparison between numerical and theoretical

result

The driving point capacitance can be

theoretically calculated using measured parameters. The

parameters Cg and Cp are taken at the lowest frequency

of 1 MHz. The difference of those in differential and

common modes are within 1 pF, and hence the Cg and

Cp are taken to be 1 pF and 8 pF, respectively for both

modes. The comparisons of the capacitances between

the theoretical and numerical results illustrated in Fig. 5

show satisfactory accuracy in a frequency range below

10 MHz. The range is enough to analyze a lightning

surge simulation.

A measuring method of the equivalent circuit

parameters of impedance between electrodes implanted

into a wood is investigated using a pi-type capacitive

circuit model. From a common and differential mode

measurement, the effects of a sheath surge impedance of

the current injection cable, a voltage probe and a stray

capacitance are clarified.

The pi-type capacitive circuit model of the

electrodes will be used for a fast transient simulation.

However, the sheath surge impedance, the stray

capacitance and the voltage probe capacitance have to

be taken into account for the first transient. This study

has shown the effect of these parameters on the transient

in each frequency region. The sheath surge impedance

of the current injection cable determines the frequency

region.

1) K. L. Wong and M. F. Rahmat, “Study of Leakage Current Distribution in Wooden Pole Using Ladder Network Model,” IEEE Trans. Power App. Syst., 25 [2], 995-1000 (2010).

2) K. L. Wong, S. Pathak and M. F. Rahmat, “Aging Effect on Leakage Current Flow in Wooden Poles,” IEEE Trans. Dielectr. Electr. Insul., 16 [1], 133-138 (2009).

3) M. Darveniza, G. J. Limbourn, and S. A. Prentice, “Line Design and Electrical Properties of Wood,” IEEE. Trans. Power App. Syst., 86 [11], 1344-1356 (1967).

4) M. Darveniza, B. C. Holcombe, and R. H. Stillman, “An Improved Method for Calculating the Impulse Strength of Wood Porcelain Insulation,” IEEE Trans. Power App. Syst., 98 [6], 1909-1915 (1979).

5) D. Permata, N. Nagaoka and A. Ametani, “A Modeling Method of Impedance between Electrodes Implanted into Wood,” Proc. the Technical Meeting of HV-IEEJ, HV-13-035, 691-698 (2013).

6) E. Oran Brigham, The Fast Fourier Transform, (Prentice Hall, New Jersey, 1974), p.91-99.

7) P. Moreno and A. Ramirez, “Implementation of the Numerical Laplace Transform: A Review,” IEEE Trans. Power Del., 23[4], 2599-2608 (2008).

0

1

2

3

4

5

6

7

8

9

10

1 10

Capa

cita

nce

[pF]

Frequency [MHz]

Cp-diff Cp-comm