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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 6, DECEMBER 2007
High-Frequency Model of the Rogowski Coil
With a Small Number of Turns
Valentinas Dubickas and Hans Edin
Abstract—In this paper, the Rogowski coil is modeled as a
distributed-element transmission line. The elements in the distributed transmission line are modeled using physically based
equations. It allows self and transfer impedances of the Rogowski
coil to be modeled using the physical dimensions and the material
properties of the coil. The models of the impedances were verified
by measurements in the frequency domain on three Rogowski
coils. The coils mounted on the power cable were modeled and
simulated in the time domain and afterward compared with the
measurements.
Index Terms—Distributed element model, high-frequency
measurements, Rogowski coil, simulations.
I. I NTRODUCTION
Fig. 1.
R
OGOWSKI COILS used for high-frequency measurements usually have a small number of turns to minimize
self-inductance. Applications of the high-frequency Rogowski
coils range from measuring high pulsed currents [1]–[3] to
partial discharge measurements on power cables [4]. In this
paper, Rogowski coils would be used for online extraction of
high-frequency power cable parameters. As the coil’s use is
widespread, theoretical investigations and models can be found
[1], [2], [5], [6], where the Rogowski coil is modeled as a
distributed-element transmission line. However, a direct comparison of the model and the measurements in the frequency
domain is seldom found, whereas a direct comparison of the
model and the measurements in the time domain could not
be found in the literature by the authors. In this paper, three
screened Rogowski coils are modeled in the frequency and
time domains, and afterward, the models are verified by the
measurements. Due to the lack of space, the results only of one
coil are presented in this paper.
Rogowski coil schematics.
TABLE I
DIMENSIONS AND PARAMETERS OF THE INVESTIGATED ROGOWSKI
COILS. DIMENSIONS ARE IN MILLIMETERS
Fig. 2.
Rogowski coil on the power cable.
tance H from the cores to a shield, the winding wire radius rw ,
and the number of turns N are presented in Table I.
II. S CHEMATICS
Three Rogowski coils, namely Rog1, Rog2, and Rog3,
were built and investigated (Fig. 1). The Rogowski coils have
polyvinyl chloride cores for mechanical support of the windings. The coils were shielded by copper enclosures. This provides shielding from noise interference [6] and forms a constant
capacitance to ground. A circular cross-sectional copper wire
was used for windings. The dimensions of the cores, the disManuscript received September 29, 2005; revised August 22, 2007.
The authors are with the School of Electrical Engineering, Royal Institute of Technology, 100 44 Stockholm, Sweden (e-mail: valentinas.
[email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIM.2007.907965
III. M ODEL
A Rogowski coil mounted on a power cable (see Fig. 2) is
modeled by distributed elements, and the model is presented in
Fig. 3, where
characteristic impedance of the cable;
Zc
Zload impedance of the cable load;
distributed mutual inductance between the cable’s
M
conductor and windings of the Rogowski coil;
distributed windings’ self inductance;
L
Zskin
distributed windings’ wire internal impedance due to
the skin effect;
distributed windings’ stray capacitance to the shield;
C
Cstr
distributed stray capacitance between the turns;
0018-9456/$25.00 © 2007 IEEE
DUBICKAS AND EDIN: HIGH-FREQUENCY MODEL OF THE ROGOWSKI COIL WITH A SMALL NUMBER OF TURNS
2285
line equations are used for modeling, i.e.,
dV (x)
dx = −ZS I(x) + jωM I2
dI(x)
dx = −jωC V (x)
(3)
where
ZS =
1
(Zskin
+ jωL ) jωC
str
Zskin
+ jωL +
1
jωCstr
.
(4)
The general solution of the coupled transmission line (3) is
V (x) = V0+ e−γx + V0− e+γx
I(x) =
V0+ e−γx
V − e+γx
jωM I2
− 0
+
Z0
Z0
ZS
(5)
where
Propagation constant:
(Zskin
+ jωL )
γ=
1
Zskin
+ jωL + jωC
str
·
C
.
Cstr
Wave impedance:
(Z + jωL )
skin
Z0 = −
C Z
ω 2 Cstr
skin + jωL +
Fig. 3.
Rm
lc
lw
1
jωCstr
(6)
.
(7)
Model of the cable and the Rogowski coil system.
Applying boundary conditions
resistance of measuring equipment;
length of the cable;
length of the windings wire.
V (x = 0) = 0
V (x = lw ) = Rm I(x = lw )
yields
A. Transfer Function (TF) of the Cable
A TF of the cable was modeled using an ABCD matrix.
The ABCD matrices are convenient as they allow the model
to be expanded by simply multiplying the matrices of different
components, i.e.,
1 Zc
cosh(γc lc )
Zc sinh(γc lc )
ABCD =
.
1
0 1
cosh(γc lc )
Zc sinh(γc lc )
(1)
The first ABCD matrix in (1) represents the series impedance Zc that is connected to the cable. It models a matched
connection between the cable, the pulse generator, and the
oscilloscope. The ABCD matrix of the modeled system can
be converted to a Z matrix using (25) to express it as a
TF, i.e.,
Gcbl (ω) =
V2
Zload z21
=
.
V1
z11 Zload + z21 z12 − z22 z11
(8)
V0+ = −V0−
V0+ = −
(9)
The output voltage of the Rogowski coil V3 = V (x = lw ), i.e.,
V3 =
jωM I2 Rm Z0 sinh(γlw )
.
ZS (Z0 sinh(γlw ) + Rm cosh(γlw ))
(10)
TF is obtained using V2 = I2 Zload in (10), i.e.,
Grog (ω) =
=
(2)
jωM I2 Rm Z0
.
2ZS (Z0 sinh(γlw ) + Rm cosh(γlw ))
V3
V2
jωM Rm Z0 sinh(γlw )
.
Zload ZS (Z0 sinh(γlw ) + Rm cosh(γlw ))
(11)
B. TF of the Rogowski Coil
C. Theoretical Expressions of the Rogowski Coil
Model Elements
At high frequencies, wave propagation in the Rogowski coil
becomes considerable. Therefore, time–harmonic transmission
The internal impedance of the Rogowski coil windings is
modeled as a circular cross-sectional wire impedance including
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 6, DECEMBER 2007
the skin effect, i.e.,
Zskin =
lw γs I0 (γs rw )
π rw σw 2 I1 (γs rw )
(12)
where
γs =
jωµw µ0 σw
σw
conductivity of the windings wire;
permeability of the windings wire;
µw
I0 , I1 modified Bessel functions.
The self-inductance of the Rogowski coil is modeled as the
inductance of an ideal toroid, i.e.,
D
µ0 hN 2
ln
LT =
.
(13)
2π
d
In addition to the inductance of the ideal toroid, there is the
inductance due to the windings leakage flux. The leakage flux
will be present in the Rogowski coils with a small number of
turns as the flux is not fully enclosed in the toroid and the flux
linkage around the individual windings appears. As the leakage
flux is enclosed between the windings wire and the shield, the
leakage inductance LL is approximated as a circular conductor
at the distance H over the ground plane. Thus
2H
µ0 lw
ln
LL =
.
(14)
2π
rw
Fig. 4.
Impedance of Rog2.
TABLE II
COMPARISON OF THE THEORETICAL ROGOWSKI COIL ELEMENTS
WITH THE MEASURED AND ESTIMATED ONES
The total distributed self-inductance is then given by
L =
LT + LL
.
lw
(15)
The distributed mutual inductance of the Rogowski coil is
D
µ0 hN
M =
ln
.
(16)
2πlw
d
The distributed stray capacitance from the windings to the
shield is also approximated as a circular conductor at the
distance H over the ground plane, i.e.,
C =
2πε l
0s ,
ln 2H
r w lw
ls = lw − N h.
(17)
The stray capacitance between the turns was neglected for pure
theoretical simulations. However, it is possible to estimate it
from the measurements of the Rogowski coil impedance ZR .
D. Rogowski Coil Impedance ZR
To verify the theoretical element values, the impedances of
the Rogowski coils ZR were measured and compared with the
theoretical values. The Rogowski coil can be seen as a transmission line with shortened far end. During the measurements,
a female–female N-type connector was used to directly connect
the Rogowski coil to Port1 of the network analyzer. Port2
was left disconnected. The female–female N-type connector
was modeled by a series inductance LN = 1 nH and a shunt
capacitance CN = 1.5 pF [7]. Therefore, the theoretical impedance of the Rogowski coil is
ZR_theor = jωLN +
Z0 tanh(lw γ)
1 + jωCN Z0 tanh(lw γ)
(18)
where γ and Z0 are obtained from (6) and (7). ZR_meas was
extracted from the one-port measurements of the scattering
parameter S11 with a network analyzer, which has an input
impedance ZNA = 50 Ω, i.e.,
ZR_meas = ZNA ·
1 + S11
.
1 − S11
(19)
To verify the theoretical values of the elements, it is possible to
estimate them by fitting ZR_theor (18) to ZR_meas . As the initial point, the inductances of the Rogowski coils were measured
with an LCR meter. The theoretical impedance expressions due
to the skin effect were used. The capacitances C and Cstr were
estimated by finding the best fit of the ZR_estm and ZR_meas
curves (see Fig. 4 and Table II).
The capacitances Cstr of Rog1 and Rog2 are small and did
not affect the measurements.
E. Rogowski Coil Transfer Impedance Zt
A parameter that is usually used to describe the qualitative
properties of the Rogowski coil is the transfer impedance
DUBICKAS AND EDIN: HIGH-FREQUENCY MODEL OF THE ROGOWSKI COIL WITH A SMALL NUMBER OF TURNS
2287
Fig. 7. Rogowski coil on the power cable.
Fig. 5.
Transfer impedance measurement setup.
Fig. 8. Comparison of Rog2 time-domain measurements and the models.
Fig. 6.
Transfer impedance of Rog2.
Zt = V3 /I2 . The transfer impedance provides data about the
sensitivity and the bandwidth of the Rogowski coil. The theoretical transfer impedance Zt_theor can be directly obtained
from (10), i.e.,
Zt_theor =
jωM Rm Z0 sinh(γlw )
.
ZS (Z0 sinh(γlw ) + Rm cosh(γlw ))
(20)
Zt_theor was modeled using the theoretical values of the elements. The transfer impedances Zt_meas were extracted from
the measurements with the network analyzer, i.e.,
Zt_meas =
S21 ZNA
.
(1 − S11 )
(21)
The measurement setup is depicted in Fig. 5. Port1 of the
network analyzer is connected through the circular conductor
in the center of the Rogowski coil to Zterm . The shield of
the Rogowski coil serves as a return path for current I2 . The
transfer impedance Zt of Rog2 measurements and model are
compared in Fig. 6.
IV. T IME -D OMAIN M ODELING
The time-domain measurements were performed with the
Rogowski coils mounted on the power cable (see Fig. 7). The
coils were placed on the cable after the folded back screen
wires. To provide the propagation path for high frequencies,
copper tape was wound on the unshielded cable section and
connected to the shield of the coil. To reduce the reflections, the
cable was terminated with its characteristic impedance. A pulse
V1 (t) of 0.25 V and 13 ns length was injected to the cable and
was measured by an oscilloscope Ch1 . The oscilloscope Ch2
measured the output signal V3 (t) of the Rogowski coil. The
model of the cable and the Rogowski coil system is presented
in Fig. 2. Therefore, the output signal V3 (t) is simulated using
(2), (11), and the Fourier transform of the measured input signal
V1 (t), i.e.,
V3 (t) = F −1 {F {V1 (t)} · Gcbl (ω) · Grog (ω)} .
(22)
The model is simulated with the theoretical element values. It
can be improved by using the measured and estimated values in
the simulation. The measurements and the models in the time
domain of coil Rog2 are compared in Fig. 8.
V. T OOL FOR THE R OGOWSKI C OIL D ESIGN
The model of the Rogowski coil transfer impedance (20)
together with the theoretical expressions of the Rogowski coil
elements could be used as a tool for designing the coils. As an
example, the transfer impedance magnitude |Zt (f, N )| of Rog2
is plotted as a function of the number of turns N and frequency
f (see Fig. 9). |Zt (f, N )| is a maximum when N = 6. It is due
to the fact that M ∝ N , whereas L ∝ N 2 [see (13) and (16)].
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 6, DECEMBER 2007
whereas the Z matrix is defined as
z11 z12
I1
V1
=
.
V2
z21 z22
I2
(24)
The ABCD to Z matrix conversion is given as follows:
1 A T
Z=
C 1 −D
T = BC − AD.
(25)
R EFERENCES
Fig. 9. Transfer impedance magnitude |Zt (f, N )| as a function of frequency
and number of turns N .
Therefore, if N is higher than the optimum value, L damps
|Zt (f, N )| more than M increases. Higher N also reduces the
bandwidth of the coil. However, coils with a higher number
of N have higher |Zt (f, N )| at lower frequencies. Therefore,
when designing the Rogowski coil, the number of turns could
be adjusted to meet the desired specifications.
VI. C ONCLUSION
A high-frequency model of the Rogowski coil with a small
number of turns was developed by modeling the coil as a
distributed-element transmission line. The physically based
equations of the element values allow self and transfer impedances to be modeled using the physical dimensions and the
material properties of the coil.
The model can be further fine-tuned by measuring and estimating the values of the coil’s elements.
The transfer impedance model of the Rogowski coil with the
theoretical values of the elements could be used as a tool for the
Rogowski coil design.
A PPENDIX
The ABCD matrix is defined as follows:
V1
A B
V2
=
C D
I1
I2
(23)
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Mar. 1982.
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in Proc. IEEE 7th Int. Conf. Properties Appl. Dielectr. Mater., Jun. 1–5,
2003, vol. 1, pp. 215–219.
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probe,” in Proc. Int. Symp. EMC, Aug. 21–25, 2000, vol. 2, pp. 569–573.
[6] C. Hewson and W. Ray, “The effect of electrostatic screening of Rogowski
coils designed for wide-bandwidth current measurement in power electronic applications,” in Proc. IEEE 35th Annu. Power Electron. Spec. Conf.,
Jun. 20–25, 2004, vol. 2, pp. 1143–1148.
[7] R. Papazyan, P. Pettersson, H. Edin, R. Eriksson, and U. Gäfvert, “Extraction of high frequency power cable characteristics from S-parameter
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470, Jun. 2004.
Valentinas Dubickas received the B.Sc. degree in
electrical engineering from Kaunas University of
Technology, Kaunas, Lithuania, in 2001 and the
M.Sc. degree in electrical engineering from the
Royal Institute of Technology, Stockholm, Sweden,
in 2003. He is currently working toward the Ph.D.
degree at the Royal Institute of Technology.
His current research involves online power cable
diagnostics.
Hans Edin was born in Sandviken, Sweden, on April
16, 1971. He received the M.Sc. and Ph.D. degrees
in electrical engineering from the Royal Institute
of Technology (KTH), Stockholm, Sweden, in 1995
and 2001, respectively.
He is currently an Assistant Professor with the
KTH. He is involved in research concerning the development of insulation diagnostic methods for highvoltage equipment, particularly partial discharge
analysis, dielectric spectroscopy, and high-frequency
techniques for localization of defects.