아힌베나드
(Bernard Arhin)
1iD
생차야
( seng Chhaya)
1iD
차한주
(Hanju Cha)
†iD
-
(Dept. of Electrical Engineering, Chungnam National University, Korea )
Copyright © The Korean Institute of Electrical Engineers(KIEE)
Key words
Characteristic impedance, LCR meter, Network analyzer, Propagation delay, series parameters, shunt parameters, short-Open test.
1. Introduction
Electrical cables are designed to ensure efficient energy transfer from generation
point to consumers over a long transmission line and distribution network. Extracting
the rights cable impedance parameters is one of the most essential operational task
in power network and communication lines. Although applications of cables in both
electrical network and communication lines have become more advanced as compared to
other wired communication media such as optic fiber and coaxial cables. Low voltage
(LV) power cables plays a major role in energy transfer and information transmission.
The LV network for example is composed of several elements which determines its overall
behavior. In order to obtain a very accurate information of electrical cables, it
is necessary to understand the effect of the cable’s configuration and model. The
features of the most commonly used cable technologies which are the underground cable
and the overhead line varies. A numbe r of research about cable characterization have
been carried out by various authors. In (1) an electrical power cable was modeled using analytical formulations, taking into
account the physical and geometrical characteristics of the power cable components.
While in (2-3) modeling and characterization of electrical cable was carried out by using full wave
simulation software based on the finite element method. However, the analytical formulations
described do not correspond exactly to the geometries of the cables studied. Therefore,
taking experimental analysis of these cables have become a necessary approach to understand
the nature of the cables actual features and behavior. A short-open method is utilized
in this paper to study and estimates electrical power cable impedance parameters.
Network analyzer and an LCR meter are used for this analysis. The primary and secondary
parameters obtained from LCR meter and network analyzer are compared to an experimental
value to assess the effectiveness of the short-open method. The experimental test
is carried out for a 16m power cable at a room temperature of $25^{\circ}{C}$.
2. Transmission Line Model
The characteristics of a cable can be analyzed using transmission line theory. Equations
deduced from transmission line theory is used to extract cable parameters. The type
of cable considered in this study is the two wire cable. The effect of coupling between
the cable conductors is neglected in this study. For easy analysis of the cable the
elementary section of the cable is utilized instead of distribution network model.
The transmission line model as shown in Fig. 1 and 2 consist of series parameters and shunt parameters (4).
Fig. 1. Transmission line equivalent circuit
Fig. 2. Elementary transmission line section
Fig. 3. Voltage reflection mechanism
The cable when examined from transmission line theory can also be define by its secondary
parameters as characteristic impedance $Z_{c}(\Omega)$ and the propagation constant
$\gamma\left(m^{-1}\right)$ (Fig. 3). The propagation constant defines phase shift and attenuation of voltage and current
waveforms of the cable. As shown in Fig. 2. the elementary line section of the cable is made up of primary parameters $R_{P}$,
$L_{P}$, $G_{P}$ and $C_{P}$ which are the per unit length series resistance $(\Omega
/m)$, series inductance $(H/m)$, shunt conductance $(S/m)$ and shunt capacitance $(F/m)$
(5). The secondary parameters which are the characteristic impedance and propagation
constant are related to the primary parameters as follows:
$\omega$ is the angular frequency (rad/s).
The primary parameters of the cable can be extracted from the secondary parameters
by using the following expressions.
A transmission line can be modeled as a lumped or distributed. A transmission line
model is considered distributed after it crosses a certain length. When the length
is less than the specified value, it is then considered as a singular lumped element.
The lumped values can be obtained after multiplying the distributed values by the
cable’s length [Rp(Ω/m), Lp(H/m), Cp(F/m) and Gp(S/m)]. Lumped transmission line can
be modelled as a pi or T model as shown in Fig. 4. Pi model has the shunt capacitance of each line i.e. phase to neutral divided into
two equal parts. One part is lumped at the sending end while the other is lumped at
receiving end. In a T model of a medium transmission line, the series impedance is
divided into two equal parts, while the shunt admittance is concentrated at the center
of the line. Pi model was selected for the cable characterization and modeling.
Fig. 4. Lumped transmission line model (a) Pi model (b) T model
2.1 Lossless Transmission Line
Secondary parameters of the cable is simplified by considering the cable as a lossless
transmission line model. The propagation constant $\gamma\left(m^{-1}\right)$ of the
cable is a complex parameter which is composed of attenuation constant $\alpha(Np/m)$
as the real part and phase constant $\beta(rad/m)$ as the imaginary part. For lossless
transmission line model the effect of cable series resistance $R_{P}$ and shunt conductance
$G_{P}$ are regarded to be negligible, therefore the attenuation constant becomes
zero (6-9). Using the lossless transmission line condition the propagation delay time and characteristic
impedance of the can be deduce as
propagation velocity $v(m/s)$ is determined by finding ratio of angular velocity and
phase shift.
propagation delay time $t_{d}(s)$ is deduced from the ratio of cable length $l(m)$
and propagation velocity.
characteristic impedance is expressed as
3. Short-Open Method
Cable primary and secondary parameters can be extracted by using a method known as
the short-open method. As shown in Fig. 5 series impedance are determined by shorting the end terminal of the cable while the
shunt capacitance and shunt conductance obtained by leaving the end terminal open
with no load. Since cable parameters are affected by frequency and length the series
impedance and shunt admittance are expressed to include frequency effect as given
in equation (5). The effect of length is also considered in the propagation constant equation which
is detailed in equation (15) and Fig. 6.
The primary parameters per unit length are determined from the secondary parameters.
In the short circuit state the LCR meter and network analyzer measurement for series
resistance
Fig. 5. Cable equivalent circuit for short-open circuit condition
Ra(Ω) and series inductance La(Ω) are used to determine the short circuit series impedance
$Z_{SC}$ (5,8). In the open circuit state the measured shunt conductance Ga(S) and shunt capacitance
Ca(F) from LCR meter and network analyzer are also used to find the cables shunt admittance
$Y_{OC}$. characteristic impedance Zc(Ω) of the cable is computed by finding square
root of the ratio of series impedance $Z_{SC}$ and shunt admittance $Y_{OC}$. Propagation
constant $\gamma\left(m^{-1}\right)$ can be deduce by computing for the inverse hyperbolic
tangent of the square root of the product of the series impedance $Z_{SC}$ and shunt
admittance $Y_{OC}$. The cable series resistance per unit length $R_{P}$(Ω/m) is extracted
from the real part of the complex product of characteristic impedance and propagation
constant while the cable series inductance per unit length $L_{P}$(H/m) is obtained
from the ratio of the imaginary part of the product of Zc(Ω) and $\gamma\left(m^{-1}\right)$
and angular frequency ω(rad/s). Shunt conductance of the cable per unit length $G_{P}$(S/m)
can be obtained from real part of the complex ratio of Zc(Ω) and $\gamma\left(m^{-1}\right)$.
While the shunt capacitance per unit length $C_{P}$(F/m) of the cable is extracted
from the ratio of the imaginary part of the ratio of Zc(Ω) and $\gamma\left(m^{-1}\right)$,
and angular frequency ω(rad/s). Cable parameter extraction from the short-open method
is summarized in Fig. 6.
Network analyzer allows impedance measurement to be carried out for higher frequency
values. They are calibrated at the calibration plane of the auto-balancing bridge
and the measurement accuracy is specified at the calibrated reference of the network
analyzer. However, an actual measurement cannot be made directly at the calibration
plane because the unknown terminals do not geometrically fit to the shapes of components
that are to be tested such as the additional connector leads added to the electrical
power cable. In most network analyzer measurement, device under test (DUT) are placed
across the test fixture’s terminals and not at the calibration plane (10). As a result, a variety of error sources such as residual impedance, admittance,
electrical length, etc. are involved in the circuit
Fig. 6. Flow chart for cable parameter extraction from short-open method
between the DUT and the unknown terminals. Therefore, instrument compensation functions
are required in most measurements instrument. The instrument’s compensation function
eliminates measurement errors due to these error sources.
Fig. 7. Application of LCR meter and network analyzer in short-open method
The standard compensation function used for the network analyzer are the open, short,
load compensation and cable length correction function. The open, short compensation
function eliminate the effects of the test fixture and also allows complicated errors
to be removed. The cable length correction offsets the error due to the test leads
and removes any induced errors. Induced errors are dependent upon test frequency,
test fixture, test leads, device under test (DUT) connection configuration, and surrounding
conditions of the DUT. Therefore compensation functions are performed in network analyzers
before measurement to obtain accurate measurement results.
3.1 Verification of Cable Parameters
Cable parameters extracted from the short-open method is verified by determining the
cables propagation delay ($t_{d}$). Propagation delay, or delay, is a measure of the
time required for a signal to propagate from one end of the circuit to the other.
As shown in equation (15) propagation delay depends on the series inductance, shunt capacitance and length
of the cable for a lossless cable. Therefore to verify the obtained cable parameters
the input and output voltage of the cable is compared on oscilloscope. The propagation
delay is determined by finding the time difference between the initial rising points
of the input and output voltage of the cable. The value is then compared to the computed
delay in equation (15).
Fig. 8. Measurement method for cable propagation delay (a) Propagation delay for 2-wire
caable (b) PWM inverter voltage and cable output voltage
4. Experimental Analysis for Cable Parameter Extraction
The short-open circuit measurement is illustrated through experimental setup in Fig. 9 which consist of a 16m two wire cable, network analyzer and an LCR meter. The 16m
cable is used for the experiment due to recent change in cable length for inverter
drives in the industries. In the mid 1990s, long cables such as several hundreds meter
cable was used for studies in the industries. This was as a result of the Si based
semiconductor switches such as IGBT and BJT devices with low dv/dt. However, in recent
years wide band-gap semiconductor switches such as SiC and GaN are mostly used in
high dv/dt converters which allows short cables such as 10m or 15m cable to be utilized.
Therefore, 16m cable was good enough to carry out
Fig. 9. Experimental setup for short-open measurement
cable analysis and verification on cable characteristics and dv/dt. Network analyzer
and LCR meter measurements values are compared to check for consistency and repeatability
of the short-open circuit method.
Table 1. Parameters for short-open method
|
Cable Length
(m)
|
Temparature
(℃)
|
Minimum Frequency
(Hz)
|
Maximum
Frequency
(kHz)
|
|
16
|
25
|
50
|
200
|
The frequency range for the measurements was 50 Hz to 200 kHz at a room temperature
of $25^{\circ}{C}$. As shown in Fig. 9 the short-open method is carried out by first shorting one end of the cable and using
an LCR meter to measure the series resistance Ra(Ω) and series inductance La(H) at
the other end of the cable at different frequency levels. While keeping the LCR meter
at the same location of the cable the cable end terminal is then left open to measure
the shunt capacitance Ca(F) and shunt conductance Ga(S) of the cable. The entire measurement
procedure is repeated with a network analyzer. The complex short circuit series impedance
$Z_{SC}$ and shunt admittance $Y_{OC}$ for both LCR meter and network analyzer measurements
are computed which are then used to determine characteristic impedance Zc(Ω) and propagation
delay $\gamma\left(m^{-1}\right)$ of the cable. As indicated in the flow chart (Fig. 6) the secondary values are used to determine the primary parameters per unit length.
The measured values from LCR meter and the network analyzer are documented in an excel
file from which the computed primary and secondary parameters are compared and plotted.
The plotted values for primary parameters are the resulting values after multiplying
the computed distributed values ($R_{P}$(Ω/m), $L_{P}$(H/m), $C_{P}$(F/m) and $G_{P}$(S/m))
by cable length (16m).
Fig. 10. Cable primary parameters (a) Cable series resistance (b) Cable series inductance
(c) Cable shunt capacitance (d) Cable shunt conductance
As illustrated in Fig. 10 and 11 cable characteristics changes as frequency increases. High frequency causes skin
effect which limit currents flow through the cable. Proximity effect also affects
cable characteristics as frequency increases. Skin effect and proximity effect cause
cable primary parameters i.e as per unit length series resistance and shunt conductance
to increase and series inductance and shunt capacitance to decrease. Secondary parameters
which are the propagation constant and characteristic impedance depend on cable’s
distributed impedance (Rp, Lp, Cp, Gp) and length. Variation in distributed impedance
from high frequency causes the characteristic impedance and propagation constant to
change. The influence of series resistance and shunt conductance on propagation constant
and propagation delay is mostly considered negligible. Therefore, the effect of primary
parameters considered are the series inductance and shunt capacitance. Series resistance
and shunt conductance are mostly considered negligible therefore decrease in cable
series inductance and shunt capacitance cause propagation constant and characteristic
impedance to decrease.
Fig. 11. Cable secondary parameters (a) Cable characteristic impedance (b) Cable propagation
delay
There is a high level of similarity for the computed primary and secondary values
from LCR meter and network analyzer. The resulting primary parameters for the 16m
power cable from network analyzer measurement are given as $R=145m ohm$, $L=8.71u
H$,$C=792p F$ and $G=5.6u S$ at 30kHz. With LCR meter the resulting primary parameters
were computed as $R=147m ohm$, $L=8.9u H$, $C=795p F$ and $G=5.9u S$ at 30kHz. Computed
secondary values which are the cable characteristic impedance and propagation delay
were also compared for both LCR meter and network analyzer in Fig. 10. Distributed transmission line model is determined by dividing lumped primary parameters
by the cable length. Network analyzer measurements gave characteristic impedance
magnitude and propagation delay values of $\left | Z_{C}\right | =105 ohm$ and $t_{d}=83.1ns$
while LCR meter had $\left | Z_{C}\right | =106 ohm$ and $t_{d}=84ns$ at 30 kHz respectively.
4.1 Experimental Verification of Cable Parameters
Accuracy for cable primary and secondary parameters are verified by measuring the
cable’s propagation delay for PWM inverter source. As shown in Fig. 12 the experimental setup which is used for determining cable propagation delay are
constituted of a DSP board, DC power supply with dc link voltage of 150 V, and a buck
converter. The cable end terminal is left open in this experimental test. Open circuit
terminal of the cable also correspond to a voltage peak which is almost twice the
DC link voltage of the buck converter (7). Therefore for a 150V DC link voltage a voltage peak close to 300V is expected at
the cable open end terminal. The buck converter is an SiC controlled converter having
a gate driver with a switching frequency of 30 kHz and a dead time of 1us. The gate
driver has maximum and minimum operating voltages of 15 V and -5V. As shown in Fig. 13 the buck converter generates a voltage level of 150 V with rising time from 0V to
150V of 75ns.
Fig. 12. Experimental setup for finding actual cable propagation delay
Fig. 13. Output voltage of buck converter
Fig. 14. Cable propagation delay measurement
As illustrated in Fig. 14 the cable output terminal gives output peak voltage of 303V with a propagation delay
of 84ns. The propagation delay value from the experimental results confirms the computed
cable parameters for especially the LCR meter. Network analyzer had a value of 83ns
which is close enough to the experimental value. Therefore, the primary cable parameters
for the given frequency (30kHz) can be trusted as the near perfect values for cable
parameter modeling especially measurement from LCR meter.
5. Conclusion
Application of short-open method in measuring cable parameters have been discussed
in this paper. LCR meter and network analyzer was used to carry out the short-open
method with frequency range from 50 Hz to 200 kHz. The computed propagation delay
from series and shunt parameters for LCR meter was the same to the measured propagation
delay of the power cable after being subjected to a PWM inverter source. Both LCR
meter and network analyzer gave a very similar values for cable primary parameters
which shows consistency and repeatability of the short-open method. The short-open
method was also simple and proved to be effective for estimating the cable secondary
parameters. Extracted cable parameters from the short-open method can be utilized
in surge filter design for motor drive system and voltage peak estimation in electrical
power cables.
Acknowledgements
This research was supported by Korea Electric Power Corporation. (Grant number : R21XO01-3)
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저자소개
He received the B.S. degree in electrical and electronic engineering from University
of Energy and Natural Resources, Ghana, in 2018.
He is currently pursuing M.S degree at the Department of Electrical Engineering,
Chungnam National University, Daejeon, Korea.
He received B.S. degree in Electrical Engineering from National Polytechnic Institute
of Cambodia, Cambodia, in 2019, and is currently pursuing his M.S. and Ph.D. at Chungnam
National University.
He received the B.S. degree from Seoul National University, Seoul, Korea, in 1988;
the M.S. degree from the Pohang Institute of Science and Technology, Pohang, Korea,
in 1990;
and the Ph.D. degree from Texas A&M University, College Station, TX, USA, in 2004,
all in electrical engineering.
From 1990 to 2001, he was at LG Industrial Systems, Anyang, Korea, where he was engaged
in the development of power electronics and adjustable speed drives.
Since 2005, he has been with the Department of Electrical Engineering, Chungnam National
University, Daejeon, Korea.