International Journal of Electronics
Vol. 97, No. 4, April 2010, 457–473
Transfer function-based modelling for voltage oscillation phenomena
in PWM motor drives with long feeding cables
Downloaded By: [2007-2008 Yeungnam University Medical Center Medical Library] At: 10:19 2 April 2010
Sang-Choel Leea and Ju H. Parkb*
a
Public and Original Technology Research Center, Daegu-Gyungbuk Institute of Science and
Technology (DGIST), 711 Hosan-dong, Taegu 790-784, Republic of Korea; bRobust Control
and Nonlinear Dynamics Laboratory, Department of Electrical Engineering, Yeungnam
University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea
(Received 9 March 2009; final version received 19 August 2009)
In this article, a transfer function-based modelling is proposed to investigate
voltage oscillation phenomena, i.e. over-voltage at the motor terminal, associated
with pulse-width modulation (PWM) inverter-fed motor drives with long feeding
cables. As such, the long feeding cable is assumed to be a distortionless
transmission line; then, a bounce diagram and time-harmonic method are utilised
to derive a simple model with a minimum computational burden that is easy to
realise using the Matlab/Simulink software package. Furthermore, the model
takes account of the inverter output and the motor terminal filters, which are
commonly used to suppress the motor terminal over-voltage. The model accuracy
is verified by a comparison with the circuit-oriented software, OrCAD/PSpice,
simulation results.
Keywords: over-voltage; long feeding cable; PWM motor drive; transmission line;
reflection coefficient; inverter output filter; motor terminal filter
1. Introduction
Modern variable speed drives utilise high-speed semiconductor switching devices, and
fast switching is essential to achieve a highly efficient drive. Yet fast switching with a
long motor feeding cable encounters problems due to reflected wave phenomena,
where the reflections produce a voltage oscillation at the motor terminal and a current
oscillation at the inverter output, mainly as a result of an impedance mismatch
between the cable and the load, i.e. motor and inverter. The problems caused by this
ringing are twofold: the voltage oscillation can lead to over-voltages and a degradation
of the motor’s insulation (Persson 1992; von Jouanne, Rendusara, Enjeti and Gray
1996; von Jouanne and Enjeti 1997) and the current oscillation can interfere with the
control system of the motor (Skibinski, Kerkman, Leggate, Pankau and Schlegel
1998), especially when using a modern vector control algorithm.
Various approaches have already been suggested for voltage oscillation
modelling at the motor terminal or inverter output. For example, the single
lumped-parameter circuit-based model (Aoki, Satoh and Nabae 1999; Mutoh and
Kanesaki 2008), the three-phase concentrated parameter model (Rodriguez et al.
*Corresponding author. Email: jessie@ynu.ac.kr
ISSN 0020-7217 print/ISSN 1362-3060 online
Ó 2010 Taylor & Francis
DOI: 10.1080/00207210903325211
http://www.informaworld.com
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458
S-C. Lee and J.H. Park
2006) and the cascade-associated ‘N-branch’ – ‘PI’ circuits (Paula, de Andrade,
Chaves, Domingos and de Freitas 2008) were proposed to design a passive overvoltage suppression filter. Although these models are very simple to apply when
designing the filter parameter, the physical meaning is lost with regard to
understanding the voltage reflection phenomena. As such, the lumped-parameter
cable model can be split into multiple equivalent segments, i.e. ladder circuit.
However, this also creates inaccuracies in the reflected wave modelling. Moreover,
the time-steps must be very small to obtain accurate results, and when more
segments are added, convergence problems appear, as well as the need for longer
time simulations (Skibinski et al. 1998).
For the power cable between the inverter and the motor, it is well known that
distributed-parameter representation provides more accurate results in the study
of high-frequency transients than the lumped-parameter model (Moreira, Lipo,
Venkataraman and Bernet 2002). Yet an accurate and complex model utilising the
general transmission line characteristic requires a considerable amount of calculation
time. Thus, a recently proposed simple lossless line model is able to shorten the
computational time (von Jouanne et al. 1996; von Jouanne and Enjeti 1997), and a
distortionless line model power cable representation yields more accurate results
(Skibinski et al. 1998). Moreover, the inclusion of distortion has been demonstrated
to be important for an accurate analysis of the over-voltage, resulting in the
development of a practical multiple segment model (Moreira et al. 2002).
However, most of the proposed models are based on a lumped circuit model with
single or multiple segments, and for computer simulations usually implemented
using circuit-oriented tools, e.g. OrCAD/PSpice, although some of the complexity
could also be implemented using equation-based computer simulation tools, e.g.
Matlab/Simulink. Accordingly, this article proposes a transfer function-based model
for the phenomena resulting in voltage oscillation, thereby allowing an accurate
prediction of the transmission delay, oscillation frequency and over-voltage level. In
addition, the proposed modelling method has no convergence problems and enables
a relatively fast simulation, plus the motor and inverter output filter can be modelled
with impedance transfer functions of an arbitrary degree. In Section 3, a bounce
diagram is utilised to describe the effects of the transmission line, such as the
transmission delay, attenuation and reflections at the inverter output and motor
terminal, while in Section 4 the use of a time harmonic method allows explicit forms
of the terminal voltages on the motor and inverter sides to be the same as the infinite
sum of the forward-travelling and backward-travelling waves in Section 3. In
Sections 5 and 6, the Matlab/Simulink software package is used to simulate the
proposed model and also take account of an inverter output filter or motor terminal
filter, which are commonly used to suppress the inverter pulse rise time. Finally, the
proposed model is verified in comparison with the circuit-oriented software OrCAD/
PSpice.
2. Reflected wave phenomena analysis
The PWM pulses travelling through long feeding cables between the inverter and the
motor behave similarly to travelling waves on transmission lines. If only one of
the inverter legs is switched at a time, the other leg remains in a steady state, while
the third keeps switching. In this case, the three-phase system can be reduced to a
two-wire system (Lee and Nam 2002; Moreira et al. 2002). Generally, the
International Journal of Electronics
459
transmission line is characterised by partial differential equations, known as
Telegrapher’s equations,
ð1Þ
where R, L, G and C are the distributed resistance (O/m), inductance (H/m),
conductance ( /m) and capacitance (F/m) of the transmission line, respectively, and
the position and time-dependent v(x, t) and i(x, t) are the voltage and current at
position (x) and time (t), respectively. Although the unit cell equivalent circuit shown
in Figure 1 is only exactly within the limit of Dx ! 0, it is still an excellent
approximation when Dx is small enough. Hence, a transmission line of any length
can be modelled as a cascaded chain of unit cells with a large number of cells to
improve the accuracy. By applying the Laplace transform, Equation (1) can be
transformed with a Laplace variable s ¼ jo
O
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@vðx; tÞ
@iðx; tÞ
¼ Riðx; tÞ L
;
@x
@t
@iðx; tÞ
@vðx; tÞ
¼ Gvðx; tÞ C
@x
@t
@Vðx; sÞ
¼ ðR þ sLÞIðx; sÞ;
@x
@Iðx; sÞ
¼ ðG þ sCÞVðx; sÞ
@x
ð2Þ
where V(x, s) and I(x, s) are the Laplace transform of v(x, t) and i(x, t), respectively.
Equation (2) can be decoupled by differentiating one with respect to x and
substituting it into the other, resulting in the wave equations
@ 2 Vðx; sÞ
¼ g2 ðsÞVðx; sÞ;
@x2
@ 2 Iðx; sÞ
¼ g2 ðsÞIðx; sÞ
ð3Þ
@x2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where gðsÞ ¼ ðR þ sLÞðG þ sCÞ is called the propagation constant and is a
function of the transmission-line parameters, as well as the frequency. The solutions
of Equation (3) are
Vðx; sÞ ¼ Vþ ðsÞegðsÞx þ V ðsÞegðsÞx ;
Iðx; sÞ ¼ Iþ ðsÞegðsÞx þ I ðsÞegðsÞx
Figure 1.
The unit cell of an arbitrary transmission line.
ð4Þ
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S-C. Lee and J.H. Park
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where Vþ(s), V7(s), Iþ(s) and I7(s) are all constants independent of position
and possibly complex numbers. Note that the terms with e7g(s)x represent
forward-propagating waves, whereas the terms with eg(s)x represent backwardpropagating waves. As V and I are not independent quantities, the general
solutions (4) are substituted into Equation (3) to obtain the relationship between
the current
and the voltage, Vþ(s) ¼ Z0(s)Iþ(s) and V7(s) ¼ 7Z0(s)I7(s), where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z0 ðsÞ ¼ ðR þ sLÞ=ðG þ sCÞ is the characteristic impedance. The behaviour of
the current wave is similar to the voltage wave, yet the amplitude is divided by Z0 (s).
As such, the general solutions (4) can be rewritten as
Vðx; sÞ ¼ Z0 ðsÞIþ ðsÞegðsÞx Z0 ðsÞI ðsÞegðsÞx ;
Iðx; sÞ ¼
Vþ ðsÞ gðsÞx V ðsÞ gðsÞx
e
e
:
Z0 ðsÞ
Z0 ðsÞ
ð5Þ
As shown in Figure 2, the motor terminal impedance ZL(s) involves modelling the
motor at the forward-propagating end, while ZG(s) involves modelling the output
impedance of the PWM voltage source inverter at the backward-propagating end. A
voltage reflection occurs at the each end of transmission line as a result of an
impedance mismatch between the characteristic impedance of the cable and the
terminated load impedance. The reflection coefficient is defined by the ratio of the
forward-propagating wave and backward-propagating wave and denoted by KL(s) at
the forward end (i.e. motor) and KG(s) at the backward end (i.e. inverter):
2.1.
KL ðsÞ ¼
Vðl; sÞ ZL ðsÞ Z0 ðsÞ
;
¼
Vðl; sÞþ ZL ðsÞ þ Z0 ðsÞ
KG ðsÞ ¼
Vð0; sÞþ ZG ðsÞ Z0 ðsÞ
:
¼
Vð0; sÞ ZG ðsÞ þ Z0 ðsÞ
ð6Þ
Case of distortionless transmission line
The assumption of a distortionless transmission line model implies that the ratios
R/L and G/C are equal, although the substitution of measured data from a real cable
shows that this unique condition can, not occur. Yet, for a normal motor cable, the
R 6¼ 0 and G ¼ 0 conditions are satisfied, and the pulse distortion is low for a large
gauge wire and dominated by high-frequency alternating current (AC) skin effect
losses during the pulse transition time. Also, experience has shown that this
approximation is acceptable for investigating motor over-voltages (Skibinski et al.
Figure 2.
PWM inverter-fed motor drive system with long feeding cable.
461
International Journal of Electronics
1998). Thus, by assuming a distortionless line, the model is simplified and easily
simulated, allowing the previously derived propagation constant and characteristic
impedance to be rewritten as follows,
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
gðsÞ ¼ s LC ½m1 ; Z0 ¼ L=C ½O:
ð7Þ
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The phase velocity of the wave, which is the rate at which the constant phase fronts
move, the travelling time (or transmission delay), which is the propagation time to
reach the end of the line length l, and the attenuation factor are then calculated as
pffiffiffiffiffiffiffi
n ¼ 1= LC ½m=s;
3.
pffiffiffiffiffiffiffi
t ¼ l LC ½s;
R
a ¼ e2l
pffiffiC
L
:
ð8Þ
Inverter–Cable–Motor modelling (I): bounce diagram approach
To show a graphical analysis, Persson (1992), von Jouanne et al. (1996) and von
Jouanne and Enjeti (1997) used snapshots of the voltages on the line at various
points in time, which provide a global picture of how the waves reflect and re-reflect
off the motor or the inverter. Another analysis method called a bounce diagram is
also often utilised to illustrate the voltage wave propagation through the cable.
Figure 3 shows a bounce diagram in which the progression of the leading edges of
the incident and reflected voltage waves are displayed as functions of both time t and
position x. Here, it is assumed that the length of the cable is equal to l, and the
inverter and motor are located at x ¼ 0 and x ¼ l, respectively. The two time axes
are drawn vertically at x ¼ 0 and x ¼ l.
Owing to the PWM inverter output impedance, the voltage exciting the
transmission line is different from the injected PWM voltage V(s) applied as the
input to the cable. The relationship between these two voltages is calculated using
the voltage division rule between the PWM inverter output impedance and the
Figure 3.
Bounce diagram explanation – schematic.
462
S-C. Lee and J.H. Park
characteristic impedance of the cable. In the sequel, the transfer function denoted by
G(s) is
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GðsÞ ¼
V1 ð0; sÞþ
Z0 ðsÞ
1 KG ðsÞ
:
¼
¼
ZG ðsÞ þ Z0 ðsÞ
2
VðsÞ
ð9Þ
It should be noted that V1 (0, s)þ is the first voltage launched into the cable, where the
subscript ‘1’ denotes the first propagating wave and the superscript ‘þ’ denotes
the forward propagation from the inverter to the motor. Moreover, V1 (x, s)þ denotes
the propagating voltage at distance x from the inverter. Since the time delay is assumed
to be (x/l)t at position x, then V1(x, s)þ ¼ G(s)V(s)e7(x/l)ts. Note that the incident
voltage is reflected at the motor terminal due to an impedance mismatch. Hence,
the reflected voltage at the motor terminal is given by V1 (l, s)7 ¼ KL(s)V1 (l, s)þ,
where the superscript ‘–’ denotes the backward propagation from the motor to
the inverter. At a distance x from the inverter, the reflected voltage is given by
V1(x, s)7 ¼ KL(s)G(s)V(s)e72tse(x/l)ts.
The reflection then re-occurs on the inverter side. The reflected voltage at the
inverter output is given by V2(0, s)þ ¼ KGV1(0, s)7. Note that the voltage reflection
at the inverter terminal initiates another forward propagation and the subscript ‘2’
is used to denote the second propagation. Further, at position x, V2(x, s)þ ¼
KGKL(s)G(s)V(s)e72tse7(x/l)ts. The bouncing process occurs infinitely, as shown in
the bounce diagram in Figure 3. Hence, the voltages in the cable are represented
as the sum of the infinite reflections. To facilitate the computation of the power
series, the forward-travelling voltages and backward-travelling voltages are
added separately. Thus, V(x, s)þ and V(x, s)7 denote the sum of the forwardtravelling component and sum of the backward-travelling component at position x,
respectively. Since 0 5 jKLj,jKGj 5 1, it follows that
Vðx; sÞþ ¼
1
X
Vn ðx; sÞþ ¼ V1 ðx; sÞþ þ V2 ðx; sÞþ þ V3 ðx; sÞþ þ n¼1
¼ GðsÞVðx; sÞ½eðx=lÞts þ KG ðsÞKL ðsÞeðð2lþxÞ=lÞts þ K2G ðsÞK2L ðsÞeðð4lþxÞ=lÞts þ ¼
1 KG ðsÞ
eðx=lÞts
VðsÞ
2
1 KG ðsÞKL ðsÞe2ts
ð10Þ
and
Vðx; sÞ ¼
1
X
Vn ðx; sÞ ¼ V1 ðx; sÞ þ V2 ðx; sÞ þ V3 ðx; sÞ þ n¼1
¼ GðsÞVðx; sÞ½KL ðsÞeððx2lÞ=lÞts þ KG ðsÞK2L ðsÞeððx4lÞ=lÞts
þ K2G ðsÞK3L ðsÞeððx6lÞ=lÞts þ 1 KG ðsÞ KL ðsÞeððx2lÞ=lÞts
VðsÞ:
¼
2
1 KL ðsÞKG ðsÞe2ts
ð11Þ
Hence, the Laplace transform for the voltage at any point x on the cable is given by
Vðx; sÞ ¼ Vðx; sÞþ þ Vðx; sÞ
¼
1 KG ðsÞ eðx=lÞts þ KL ðsÞe2ts eðx=lÞts
VðsÞ:
2
1 KL ðsÞKG ðsÞe2ts
ð12Þ
International Journal of Electronics
463
For example, the inverter output voltage and motor terminal voltage are obtained
using x ¼ 0 and x ¼ l, respectively, as follows
1 KG ðsÞ 1 þ KL ðsÞe2ts
VðsÞ
2
1 KL ðsÞKG ðsÞe2ts
ð13Þ
1 KG ðsÞ ð1 þ KL ðsÞÞets
VðsÞ:
2
1 KL ðsÞKG ðsÞe2ts
ð14Þ
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Vð0; sÞ ¼ Vðx; sÞjx¼0 ¼
Vðl; sÞ ¼ Vðx; sÞjx¼l ¼
4. Inverter–cable–motor modelling (II): time harmonic approach
A different approach is used to derive the explicit form of the voltage response.
A high-frequency model of the inverter–cable–motor system is utilised as shown in
Figure 2, where the inverter is modelled using an ideal PWM voltage source, plus a
series high-frequency resistor, ZG(s) (Lee and Nam 2002), while the motor is simply
modelled using a complex number ZL(s) (Skibinski et al. 1998). The transmission
lines transform the impedance of the load, i.e. motor, into a different value when it is
viewed through the cable. According to the impedance transformation formula
(Demarest 1998), the input impedance seen at the inverter terminal is given by
pffiffiffiffiffiffiffi
ZL ðsÞ þ Z0 ðsÞ tanhð jol LCÞ
1 þ KL ðsÞe2ts pffiffiffiffiffiffiffi ¼ Z0 ðsÞ
Zin ðsÞ ¼ Z0 ðsÞ
1 KL ðsÞe2ts s¼jo
Z0 ðsÞ þ ZL ðsÞ tanhð jol LCÞ
ð15Þ
where o is the operating frequency. Once Zin(s) is obtained, then the voltage at the
inverter output is given by
Vð0; sÞ ¼
Zin ðsÞ
VðsÞ:
ZG ðsÞ þ Zin ðsÞ
ð16Þ
It then follows from Equations (15) and (16) that
Vð0; sÞ ¼
1 KG ðsÞ 1 þ KL ðsÞe2ts
VðsÞ:
2
1 KL ðsÞKG ðsÞe2ts
ð17Þ
The voltage V (l, s) is represented as the sum of the forward-travelling component
Vþ(l, s) and the backward-travelling component V7(l, s), i.e. V (l, s) ¼ Vþ(l, s) þ
V7(l, s). Note that, at the motor terminal, the forward and backward travelling
voltages are related by V7(l, s) ¼ KL(s)Vþ(l, s). Hence,
Vðl; sÞ ¼ Vþ ðl; sÞ þ V ðl; sÞ ¼ ð1 þ KL ðsÞÞVþ ðl; sÞ:
ð18Þ
Meanwhile, when including the transmission delay, the voltage at the inverter
terminal becomes
Vð0; sÞ ¼ Vþ ð0; sÞ þ V ð0; sÞ ¼ Vþ ðl; sÞets þ V ðl; sÞets
¼ ð1 þ KL ðsÞe2ts Þets Vþ ðl; sÞ:
ð19Þ
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S-C. Lee and J.H. Park
Therefore, a closed loop expression is obtained from Equations (17), (18) and (19)
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Vðl; sÞ ¼
ð1 þ KL ðsÞÞets
1 KG ðsÞ ð1 þ KL ðsÞÞets
Vð0;
sÞ
¼
VðsÞ:
2
1 þ KL ðsÞe2ts
1 KL ðsÞKG ðsÞe2ts
ð20Þ
Normally, ZG(s) is very small compared to Z0(s). Then, the reflection coefficient
KG(s) is approximated to 71 and thus j(1–KG(s)/2j 1, which is the same
approximation shown by Lee and Nam (2002). Note, from Equations (17) and (20),
the closed loop expressions of V (0, s) and V (l, s) have the same results in Equations
(13) and (14), respectively.
5.
Block diagram representation
5.1. Infinite summation form
The infinite summation form can be realised by a block diagram, as shown in
Figure 4, based on the bounce diagram in section 3. The extent of the repeated wave
bouncing can be determined depending on the compromise between the calculation
time and the accuracy. This block diagram has a similar meaning to the cascadeconnected LC ladder unit cells usually implemented by circuit-oriented simulation
tools, e.g. OrCAD/PSpice.
Yet, the advantage of the block diagram form is that the reflected wave shaping
can be observed at every reflection to enable the design of adequate filter parameters.
Thus, while the voltage measured by the oscilloscope is the resulting sum of infinitely
many reflected waves, the block diagram provides information on how the reflection
coefficients affect the waveform and the dominant factor that makes the resultant
voltage waveform.
5.2. Closed-loop form
When compared with the infinite summation form, the closed form is realised
directly from the closed form of transfer function V (0, s) and V (l, s). The closed
loop form block diagram realisation is shown in Figures 5 and 6(a) for the inverter
output voltage and motor terminal voltage, respectively.
Figure 4.
voltage.
Simulink implementation schematic for inverter output and motor terminal
International Journal of Electronics
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Figure 5.
465
Block diagram representation for inverter output voltage.
Figure 6. (a) Block diagram representation for motor terminal voltage and (b) its simplified
representation.
Figure 7. Unified block diagram representation for voltages and currents at inverter output
and motor terminal.
The inverter output voltage V (0, s) has two terms: the forward-travelling
component V þ(0, s) and the backward-travelling component V7(0, s). First, the
forward-travelling component proceeds directly from the voltage source V (s), which
is only filtered by G(s); then the following one is delayed through the transmission
line (e7ts), reflected at the motor terminal (KL(s)), delayed through the transmission
line (e7ts) again and reflected at the inverter terminal (KG(s)) to reach the initial
position. Second, the backward-travelling component is delayed through the
transmission line (e7ts), reflected at the motor terminal (KL(s)), delayed through
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466
S-C. Lee and J.H. Park
Figure 8. Complete schematic circuits in OrCAD/Spice; (a) without filter, (b) with a motor
terminal resistor-capacitor (RC) filter and (c) with an inverter output resistor-inductorcapacitor (RLC) filter.
the transmission line (e7ts ) again and reaches the inverter output; then the following
one is reflected at the inverter terminal (KG(s)) to reach the initial position.
Similarly, the motor terminal voltage V(l, s) also has two terms: the forwardtravelling component Vþ(l, s) and the backward-travelling component V7(l, s).
A more simple realisation is shown in Figure 6(b). Using Equation (14) or (20), the
transmission line is modelled as a feedback loop, whereby the denominator of the
transfer function is a closed loop gain, and the nominator is an open loop gain.
Although a simple block diagram sacrifices some of the physical meaning, the
burden of the simulation time and complexity is minimal.
Figure 9. Simulation results by Matlab/Simulink; (a) without filter, (b) with motor terminal RC filter, (c) with inverter output RLC filter and (d) with
Vdc/2 level 2t delay method.
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International Journal of Electronics
467
Figure 10. Simulation results by OrCAD/PSpice (a) without filter, (b) with motor terminal RC filter, (c) with inverter output RLC filter and (d) with
Vdc/2 level 2t delay method.
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S-C. Lee and J.H. Park
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International Journal of Electronics
Figure 11.
469
Simulation settings in OrCAD/Spice.
5.3. Unified form for voltage and current
A block diagram describing the voltages at the beginning (i.e. inverter) and end (i.e.
motor) of the transmission line at the same time is shown in Figure 7. This single
block diagram is a unified model that enables the feedback loop to be constructed
and specific components to be summed to calculate the motor terminal or inverter
output voltage. When compared with the previous methods, the unified model is very
simple. At this stage, the corresponding currents at each end of the transmission line
can be calculated using Equation (21), as shown in Figure 7
Ið0; sÞ ¼
6.
VðsÞ Vð0; sÞ
;
ZG ðsÞ
Iðl; sÞ ¼
Vðl; sÞ
:
ZL ðsÞ
ð21Þ
Simulation studies
To examine the model accuracy, the transmission line system shown in Figure 2
was simulated based on the transfer function-based model, Equation (13)
and (14), using the equation-based simulator Matlab/Simulink and the circuitoriented simulator OrCAD/PSpice. The complete schematic circuits simulated in
OrCAD/Psipe are shown in Figure 8, which are captured in real-simulation
environment.
The parameters for the cable were as follows: length l ¼ 100 m, inductance per
meter L ¼ 0.97 mH/m, capacitance per meter C ¼ 45 pF/m, and attenuation factor
a ¼ 0.9754 , when R ¼ 75 mO/m. The PWM voltage source inverter was modelled
with a charged electrolytic DC-link capacitor VDC ¼ 300 V, along with a series stray
resistance RG ¼ 2O . The source was passed through a step function with a slope of
3000 V/ms, i.e. trise ¼ 0.1 ms, from zero to a DC-link voltage that is typical in an
insulated gate bipolar transistor (IGBT) device, normally from 40 ns to 400 ns. The
high-frequency motor model from Skibinski et al. (1998) was utilised with Rhf ¼ 300
O, Chf ¼ 750 pF, Rlf ¼ 2.5 O and Llf ¼ 180 mH. Also, according to Skibinski et al.
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S-C. Lee and J.H. Park
Figure 12. Comparison of the different incident and reflected voltages to constitute inverter
output voltage; (a) without filter, (b) with inverter output RLC filter, (c) with motor terminal
RC filter and (d) with Vdc/2 level 2t delay method by Matlab/Simulink.
(1998), the improvements obtained from more detailed models were often marginal.
There were four important (dynamic) phenomena in the distortionless transmission
line system, i.e. the characteristic impedance Z0 ¼ 146.8 O, transmission delay
2
þ2:984106 s2:393109
t ¼ 0.66 ms, reflection coefficient at the motor KL ¼ 0:3428s
; and
s2 þ2:985106 sþ2:475109
reflection coefficient at the inverter KG ¼ 70.9731.
The simulation results from Matlab/Simulink and OrCAD/PSpice are presented
in Figures 9 and 10, respectively. A strong voltage oscillation at the motor
terminals was clearly visible in Figure 9(a) without any suppression scheme.
Thus, conventionally well-known over-voltage suppression filters were introduced,
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International Journal of Electronics
471
Figure 13. Comparison of the different incident and reflected voltages to constitute motor
terminal voltage; (a) without filter, (b) with inverter output RLC filter, (c) with motor terminal
RC filter and (d) with Vdc/2 level 2t delay method by Matlab/Simulink.
a motor terminal RC filter in Figure 9(b) and an RLC inverter output filter in
Figure 9(c). The motor terminal RC filter parameters were Rmotor ¼ 75 O and
Cmotor ¼ 50 nF, while the inverter output RLC filter parameters were Rinverter ¼
150 O, Linverter ¼ 750 mH, and Cinverter ¼ 220 nF. The results of the more recent
over-voltage suppression method (Lee and Nam 2002) are shown in Figure 9(d). The
proposed transfer function-based model produced practically the same results as
shown in Figure 10, which were simulated by OrCAD/PSpice utilising the LOSSY
transmission line model, thereby verifying the model accuracy. The simulation
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472
S-C. Lee and J.H. Park
settings are captured in Figure 11, where a maximum time step can be left blank or
with default value.
When utilising the proposed modelling, as shown in Figure 5, each incident and
reflected voltage wave was observed, then infinitely summed to constitute the
measurable voltages at the inverter output and motor terminal. For a practical filter
design, examining the incident and reflected wave responses is also very helpful.
Meanwhile, with the conventional inverter output RLC filter in Figures 12(b)
and 13(b), the pulse slope was reduced through the use of the filter inductor.
However, with the conventional motor terminal RC filter, the first reflected wave
was cancelled by the second incident wave in Figures 12(c) and 13(c), v1(0, t)7 þ
v2(0, t)þ 0 and v1(l, t)7 þ v2(l, t)þ 0, respectively. Moreover, the recent overvoltage suppression method (Lee and Nam 2002) in Figures 12(d) and 13(d) utilised
a Vdc/2 voltage level and 2t time delayed PWM pulses to cancel reflected waves.
7.
Conclusions
This article introduced a transfer function-based model for the voltage and current
oscillation at the ends of a long motor cable, i.e. inverter output and motor terminal.
If a distortionless transmission line is assumed, then the transfer function describing
a PWM inverter-fed motor drive system through a long feeding cable is reduced to
linear time-invariant transfer functions with pure time delays. As such, the transfer
function is a combination of reflection coefficients and transmission delay time.
Moreover, the advantage of a linear time invariant model is that it can be easily
implemented using equation-based simulation packages, such as Matlab/Simulink.
The assumption of a distortionless transmission line is demonstrated as acceptable
based on comparing simulation results with the commercial circuit simulator
OrCAD/PSpice. Furthermore, the proposed transfer function-based model has been
demonstrated to be of almost the same result as an OrCAD/PSpice circuit simulator
when utilising the LOSSY transmission line model, thereby verifying the accuracy of
the proposed model. The proposed model can also be incorporated into a complete
model of an electric drive to study how the cable oscillation affects the motor
control. In addition, since the proposed model takes into account the effect of the
filter, this allows a more detailed secondary effects analysis and design than a normal
motor model and conduction power cable. Finally, the proposed model can be
analysed using various methods developed for linear control systems and simulated
using numerous software packages developed for linear differential equations.
Acknowledgement
This work is the outcome of a DGIST R/D Program, funded by the Ministry of Education,
Science and Technology (MEST), Korea.
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