energies
Article
Partial Stray Inductance Modeling and Measuring of
Asymmetrical Parallel Branches on the Bus-Bar of
Electric Vehicles
Chengfei Geng
ID
, Fengyou He *, Jingwei Zhang and Hongsheng Hu
School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221116, China;
ge-cumt-001@163.com (C.G.); hangjingwei@cumt.edu.cn (J.Z.); hhssgd@126.com (H.H.)
* Correspondence: hfy_cumt@263.net; Tel.: +86-516-8013-9898
Academic Editors: Joeri Van Mierlo and Omar Hegazy
Received: 16 August 2017; Accepted: 26 September 2017; Published: 1 October 2017
Abstract: In order to increase the power rating of electric vehicles, insulated gate bipolar translator
(IGBT) modules with multiple power terminals are usually adopted. The transient current sharing of
the same polarity power terminals is related to the stray inductance in the branches of the bus-bar.
Based on the laminated bus-bar of a three-phase inverter in the electric vehicles that consists of
asymmetrical parallel branches, this paper investigates the transient current imbalance sharing
caused by the asymmetrical stray inductance in the parallel branches of the bus-bar from the view of
energy storing and releasing of stray inductance for the first time. Besides, the partial self-inductance
and mutual-inductance model of the parallel branches is set up. Finally, a high-precision partial stray
inductance measurement method is proposed, and the accuracy of the partial stray inductance model
for asymmetrical parallel branches is verified by experimental tests.
Keywords: laminated bus-bar; electric vehicles; partial self-inductance; partial mutual inductance;
asymmetrical branches
1. Introduction
In recent years, electrical vehicles (EVs) have gained unprecedented evolution in consideration
of the current energy crisis and environmental protection [1]. For an EV inverter, high efficiency,
high power density, low harmonics, small volume, and low cost are essential [2–4]. Due to the
low stray inductance, the laminated bus-bar has now been widely used in power electronics and
converter products [5–7]. With power devices continuing to move upwards in current levels and
switching frequency, laminated bus-bars have been attracting increasing interest from both industry
and academia for the system benefits they exhibit [8]. The over-voltage spikes across the semiconductor
during switching directly results from the commutation loop stray inductance. The high di/dt values
produced during commutations by these new fast switching devices and diodes are currently the main
source of conducted and radiated noise emissions, which usually interferes with nearby equipment
through conduction or radiation [9,10]. For an EV inverter, the position of the small signal control
circuit is often very close to high voltage, large current components such as the bus-bar or the power
devices considering the limited space in an enclosure. The strength of Electro Magnetic Interference
(EMI) emission is associated with transient current distribution on the bus-bar. So, for reliable operation
of the control circuit in an EV inverter, the analysis, extraction, and evaluation of the stray inductance
for a laminated bus-bar are very important.
Due to the increasing power of EV inverters, IGBT modules with multiple power terminals
are often adopted. Moreover, the transient current sharing performance between power terminals
of the same polarity is related to the stray inductance of the branches. In prior research, the most
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Energies 2017, 10, 1519
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focus has been on optimizing the bus-bar structure based on the circuit topology [6,7,11]. Partial
inductance theory is applied to build the model of bus-bar stray inductance [12–14]. The Partial
Element Equivalent Circuit (PEEC) method [15,16] is adopted to construct the model and investigate
the transient current distribution in the laminated bus-bar. Furthermore, with consideration of the
current path, a partial self-inductance and mutual-inductance model of a H-bridge bus-bar has been
built [17]. All the modeling and analysis are based on a single power electronics topology. But, the
transient imbalance current sharing caused by the asymmetrical stray inductance in bus-bar parallel
branches is comparatively scarce. This paper deeply investigates the transient imbalance current
sharing arising from the asymmetrical stray inductance in bus-bars from the view of stray energy
storing and releasing for the first time. The partial self-inductance and mutual-inductance model of
the asymmetrical parallel branches is also set up to facilitate transient current analysis.
In order to evaluate the stray inductance of the bus-bar and verify the modeling accuracy, many
measuring methods have been proposed by scholars and engineers. These include gap voltage
measurement based on the IGBT turn-on process [18], time domain reflectometry (TDR) method [19],
resonant capacitance measurement methods [6], stray inductance integral arithmetic extraction method
for bus-bar based on turn-on/off transient process [20], network analyzer measurement [17], etc.
However, all these methods mentioned above focus on the total stray inductance of the current
communication loops rather than the partial inductance. This paper proposes a high precision
method to measure the partial inductance in the bus-bar, and the measuring accuracy is also analyzed
considering the spatial transient electromagnetic field. Experimental evaluation of partial stray
inductance extraction including direct and indirect measurement verifies the proposed model and
measuring method.
2. Inverter Layout and Equivalent Circuit of the Bus-Bar
2.1. Inverter Layout Consideration
The main drivers for the development of EVs can be summarized to be cost, power density,
specific volume, efficiency, and reliability. In this design, the power rating of the EV inverter is 200 kW.
The inverter is assembled using individual building blocks serving the system, including switch
(power device IGBT), energy storage (DC link capacitor), current distributor (copper bus-bar structure),
and cooling system (liquid cooled heat sink). In addition, sensor functions (Hall elements) and control
circuits are integrated. This setup is connected using screws and is supported by mechanical parts
and protected by an enclosure, which is shown in Figure 1a. The inner structure of the inverter is
shown in Figure 1b, in which the 1.2-KV/1400-A half bridge IGBT module is used. A low-inductance,
high-frequency film capacitor is used to replace the conventional electrolytic bulk capacitors to reduce
size and enhance the reliability.
High-speed switching devices produce lots of EMI noise during the fast on-off process, which
usually interferes with nearby equipment through conduction or radiation [8–10]. Typical problems
caused are measurement errors in sensors, false control signals, and interference in the gate voltages of
switching devices [8]. Excessive noise can deteriorate control, reduce efficiency, and cause malfunctions
of system. In this design, the control circuit is on the topside of the bus-bar. High impedance circuits
are sensitive to EMI, such as the signal conditioning circuit of analog to digital (AD) acquisition.
The source of EMI is associated with the transient current changes in the bus-bar. Simulation plays an
important role in understanding the operating principle of the whole system and the functioning of
some key components because it is more instructive and can save time and cost. Previous simulation
platforms only concern analysis of the control algorithm. Parasitic elements are usually omitted to
save simulation time [21]. So, in order to predict and evaluate the EMI/Electro Magnetic Compatibility
(EMC) performance accurately and to further guide the layout of some key components in a limited
space for EV inverters, a precise simulation platform that includes the main parasitic components
should be built up.
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DC capacitor
DC capacitor
Current
sensor
Current
DSP Control
sensor
Board
DCC
DSP Control
Board
DCB
C
A
B
DC+ A
DC+
Water cooled
output
Water cooled
output
enclosure
Water cooled input
enclosure
Water cooled input
(a)
(a)
DC capacitor
DC capacitor
Lamiated bus bar
Lamiated bus bar
DC-
N
DC-
N
Gate Driver
Gate Driver
C
P
C
P
B
b
B
b
A
A
DC+
DC+
IGBT
IGBTmodules
modules
(b)
(b)
Figure
The
structure
thethe
EV motor
motor
driverdriver
system
(MDS).
(a)
structure;
(b)(b)
Internal
Figure
1.
structure
ofofthe
EV
system
(MDS).(MDS).
(a)External
External
structure;
Internal
Figure
1. 1.The
The
structure
of
EV
motor
system
(a) External
structure;
(b)
structure.
structure.
Internal structure.
EquivalentCircuit
CircuitofofBus-bar
Bus-barConsidering
Considering Stray
Stray Inductance
2.2.2.2.
Equivalent
Inductance
2.2.
Equivalent
Circuit of
Bus-bar Considering
Stray Inductance
Thestructure
structureand
andcircuit
circuitprinciple
principle of
of the half
bridge
IGBT
module
adopted
in this
design
is is
The
bridge
IGBT
module
adopted
this
design
The
structure
and
circuit
principle
ofterminals
the the
halfhalf
bridge
IGBT
module
adopted
in represent
thisin
design
is shown
shown
in
Figure
2.
There
are
5
power
on
the
IGBT,
terminals
9
and
11
collector
shown
in 2.
Figure
2.are
There
are 5 terminals
power terminals
on theterminals
IGBT, terminals
9 and
11 represent
collector
in
Figure
There
5 power
on the IGBT,
9 and
11
represent
collector
of the
of
the
upper
IGBT,
terminals
10
and
12
represent
emitter
of
the
lower
IGBT
and
terminal
8
represents
of the upper
IGBT,
terminals
1012and
12 represent
emitter
oflower
the lower
IGBT
and terminal
8 represents
upper
IGBT,
terminals
10
and
represent
emitter
of
the
IGBT
and
terminal
8
represents
the
the middle terminal.
the middle
terminal.
middle
terminal.
9/11
5
12
12 11
11 10
10 9
9 8 5
8 5 4
3
6
4
3
67
7
1
2
1
2
4
6
9/11
5
4
3
6
8
3
NTC
8
NTC
7
7
1
2
1
2
10/12
10/12
Figure 2. The structure and circuit principle of the IGBT module.
Figure
Figure 2.
2. The
The structure
structure and
and circuit
circuit principle
principle of
of the
the IGBT
IGBT module.
module.
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detailed structure
structureof
ofthe
thelaminated
laminatedbus-bar
bus-barisisshown
shownininFigure
Figure
The
power
terminal
P and
The detailed
3. 3.
The
power
terminal
P and
N
N are
connected
to anode
the anode
and cathode
bus capacitor.
A,CB,are
and
are the
output
are
connected
to the
and cathode
of theof
DCthe
busDC
capacitor.
A, B, and
theCoutput
terminals
terminals
of the
inverter.
The power
terminals
9 and
11 onbus
theare
positive
bus to
arethe
connected
of
the inverter.
The
power terminals
9 and
11 on the
positive
connected
Collector to
of the
Collector
of the
IGBT.
The power
terminals
and 12 on
the
negative
bus to
arethe
connected
to the
upper
IGBT.
Theupper
power
terminals
10 and
12 on the10negative
bus
are
connected
Emitter of
Emitter
of the lower IGBT.
lower
IGBT.
This paper focuses on the stray inductance of parallel branches of the bus-bar, without the
stray inductance
inductance in
in the
the capacitor
capacitor and
and IGBT
IGBT module.
module. The simplified stray
consideration of the stray
inductance model of the three phase inverter is shown
shown in
in Figure
Figure 4a.
4a. The detailed
detailed stray
stray inductance
inductance
model with the consideration of the asymmetrical parallel branch of every phase on the bus-bar is
Figure 4b.
4b. Lσ1
shown in Figure
and LLσ2σ2represent
representstray
stray inductance
inductance from
from positive
positive DC
DC bus
bus terminal
terminal P to
σ1 and
and 11
11 respectively.
respectively. LLσ3
σ3 and Lσ4
terminals 9 and
represent the
the stray
stray inductance
inductance from negative DC bus
σ4 represent
respectively.
and 10
10 respectively.
terminal N to terminals 12 and
P
N
12
11
10
9
Structure of the bus-bar.
Figure 3. Structure
bus-bar.
Lσ1
LσB1
LσA1
Lσ2
LσC1
9
P
SA1
SB1
11
SC1
c
SA1
Udc
Udc
SA2
SB2
SC2
SA2
N
LσA2
8
LσB2
LσC2
(a)
Lσ3
12
Lσ4
10
e
(b)
Figure 4. Equivalent circuit of every phase leg. (a) Simplified circuit; (b) Detailed circuit.
Figure 4. Equivalent circuit of every phase leg. (a) Simplified circuit; (b) Detailed circuit.
3. Transient Analysis and Modeling of Asymmetrical Branches
3. Transient Analysis and Modeling of Asymmetrical Branches
3.1. Transient
Analysis
3.1.
Transient Analysis
There are
are eight
eight switch
switchstates
statesfor
foraathree-phase
three-phaseinverter
inverterand
andonly
only
one
power
device
is permitted
There
one
power
device
is permitted
to
to
change
state
at
any
time
under
normal
operation
conditions.
The
double
pulse
test
is
used
to test
change state at any time under normal operation conditions. The double pulse test is used
to test
the
the IGBT
over-voltage
and stray
inductance
the bus-bar,
meanwhile,
every shares
phase shares
thetesting
same
IGBT
over-voltage
and stray
inductance
of theofbus-bar,
meanwhile,
every phase
the same
testing
principle,
and
this
paper
defines
phase
B
as
the
example.
Figure
5
is
the
basic
principle
of
the
principle, and this paper defines phase B as the example. Figure 5 is the basic principle of the test
test
circuit.
The
i
1, i2, i3, and i4 represent the current in the branches of Lσ1, Lσ2, Lσ3, and Lσ4. Furthermore,
circuit. The i1 , i2 , i3 , and i4 represent the current in the branches of Lσ1 , Lσ2 , Lσ3 , and Lσ4 . Furthermore,
taking Proximity
ProximityEffect
Effect
into
consideration,
coupled
inductance
exists among
fourbranches.
current
taking
into
consideration,
coupled
inductance
exists among
the fourthe
current
branches.
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i1
Lσ1
i1
Lσ1
i2
i2
P
P
Mσ1_2
Mσ1_2
Lσ2
Lσ2
S1
Mσ1_3
Mσ1_3
Udc
Udc
S1
Mσ2_3
Mσ2_3
Mσ1_4
Mσ1_4
Mσ2_4
Mσ2_4
Lσ3
Lσ3
N
N
Mσ3_4
Mσ3_4
Lσ4
Lσ4
S2
S2
i3
i3
Load
Load
iload
iload
i4
i4
Figure 5. The test circuit of every phase leg.
Figure
ofevery
everyphase
phaseleg.
leg.
Figure5.5.The
The test
test circuit
circuit of
The transient analysis is based on the two branches of Lσ1 and Lσ2, and the waveforms of the
The transient
analysis
basedon
on thetwo
two branches
branches of
σ1 and
Lσ2,σ2and
thethe
waveforms
of the
The
analysis
is isbased
of LLof
, and
waveforms
of the
σ1t0and
doubletransient
pulses test
is depicted
in Figurethe
6. The time
intervals
–t1, tL3–t
4, and t7–t8 represent steadydouble
pulses
test
is
depicted
in
Figure
6.
The
time
intervals
of
t
0–t1, t3–t4, and t7–t8 represent steadydouble
pulses
test
is
depicted
in
Figure
6.
The
time
intervals
of
t
–t
,
t
–t
,
and
t
–t
represent
0
1
3
4
7
8
state of conducting and blocking. This paper focuses on the transient process of turn-on and turn-off
state of conducting
and blocking.
This paper
focuses on
the transient
process
of turn-on and
turn-off and
steady-state
of conducting
andinterval
blocking.
focuses
transient
of turn-on
which refers
to the time
of This
t1–t3 paper
and t4–t
7, and on
thethe
current
in process
t8–t9 shares
the same
which refers to the time interval of t1–t3 and t4–t7, and the current in t8–t9 shares the same
turn-off
which refers
the time
interval
of tinterval.
characteristics
withto
current
in the
t1–t3 time
1 –t3 and t4 –t7 , and the current in t8 –t9 shares the same
characteristics with current in the t1–t3 time interval.
characteristics with current in the t1 –t3 time interval.
VVgg
t t
½·
itotal
½·
itotal
ii11
t t
i2i2
tt00
t1t1t2tt23t3
t4t45t65t67 t7
t8 t8t9 t9
Figure 6. The waveforms of the double pulse test.
Figure
pulse
test.
Figure6.6.The
Thewaveforms
waveformsofofthe
thedouble
double
pulse
test.
3.1.1. Turn-Off Transient
3.1.1. Turn-Off Transient
3.1.1.
Transient
At tAt
,
S1
receives
the turn-off
signal and
and thebus-bar
bus-bar can
be equivalent
the circuit
depicted
1
S1 receives
to to
theto
circuit
depicted
in in
At tt11,, S1
receives the
the turn-off
turn-offsignal
signal andthe
the bus-barcan
canbebeequivalent
equivalent
the
circuit
depicted
in Figure
7,7,ininwhich
itotal
represents the turn-off current. According to Kirchhoff’s Current Law, the
Figure
which
i
total represents the turn-off current. According to Kirchhoff’s Current Law, the
Figure 7, in which itotal represents the turn-off current. According to Kirchhoff’s Current Law, the
branch
current
satisfies
i1 +i1i+2 i=2 =i3i3++i4i4 == iitotal
, meanwhile,
the voltage
in stray
inductance
of each
branch
branch
current
satisfies
inin
stray
inductance
of of
each
branch
branch
current
satisfies
i1 + i2 = i3 + i4 = total
itotal, ,meanwhile,
meanwhile,the
thevoltage
voltage
stray
inductance
each
branch
satisfies
u
=
u
and
u
=
u
.
satisfies
u
L1
=
u
L2
and
u
L3
=
u
L4
.
L1uL1 =L2
L3L3 = uL4
satisfies
uL2 and u
L4.
The turn-off process of an IGBT can be regarded as the stray inductance energy releasing process,
so the following analysis is based on magnetic field energy stored in the stray inductance. At t1 , the
energy stored in branches Lσ1 and Lσ2 are given as
(
WLσ1 (t1 ) = 12 Lσ1 i12 (t1 )
WLσ2 (t1 ) = 21 Lσ2 i22 (t1 )
(1)
WLσ1 (t1 ) and WLσ2 (t1 ) are determined by the current and inductance in the two branches at t1 ,
meanwhile, S1 starts to turn off and itotal starts to decrease. Hence, the energy stored in Lσ1 and
Energies 2017, 10, 1519
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Lσ2 begins to release, moreover, the speed of releasing is determined by the stray inductance in the
two branches and the speed of turn-off. If the energy stored in Lσ1 and Lσ2 is released completely
at t3 simultaneously, that means WLσ1 (t3 ) = 0 and WLσ2 (t3 ) = 0, and the current direction in the two
branches depicted in Figure 7 does not change from t1 to t3 . If the energy stored in Lσ1 and Lσ2 does
not release completely at t3 simultaneously, that means WLσ1 (t2 ) 6= 0 or WLσ2 (t2 ) 6= 0 and one of the
two branches must be released completely prior to the other. This study supposes the energy stored in
Lσ2 releases completely at t2 , namely i2 (t2 ) = 0. From t1 to t2 , the current i1 and i2 decrease gradually
shown in Figure 8 (t1 –t2 ) but the current direction does not change. At t2 , i1 (t2 ) = itotal (t2 ) and Lσ1 can
be equivalent to a power source, and the direction of potential is shown in Figure 7, so the branch
of Lσ2 provides the path of energy release for Lσ1 that means the current i2 changes direction from t2
and increases at the same slope as i1 shown in Figure 8 (t2 –t3 ), but they share the contrary direction.
At t3 , the current i1 and i2 are equal in magnitude and opposite in direction and itotal = 0, moreover,
neglecting the influence of tail current, the IGBT is closed completely. Due to the asymmetrical stray
inductance in branches Lσ1 and Lσ2 , a loop current is caused after turning off the IGBT.
At the turn-off transient, due to uL1 = uL2 , the variation ratio of i1 and i2 can be calculated by
di1
L
= σ2
di2
Lσ1
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(2)
uL1
Lσ1
i1
Mσ1_2
Lσ2
uL2
i2
Mσ1_3
Mσ2_3
Mσ1_4
Mσ2_4
i3
uL3
uL2
Energies 2017, 10, 1519
itotal
Mσ3_4
Lσ4
Lσ3
i4
Figure 7. The equivalent circuit of the turn-off transient.
Figure 7. The equivalent circuit of the turn-off transient.
7 of 17
i1 IGBT can be regarded
The turn-off process of an
i1 as the stray inductance energy releasing
process, so the following analysis is based on magnetic field energy stored in the stray inductance.
itotal
itotal
At t1, the energy stored in branches Lσ1 and Lσ2 are given as
i2
(t1-t2)
i1
1 i22

WL 1 (t1 ) 2 L 1i1 (t1 )

(t2)
1
WL (t1 ) L 2i22 (t1 )
2
2

i1
(1)
WLσ1(t1) and WLσ2(t1) are determined iby
the current and inductance
total
itotal in the two branches at t1,
meanwhile, S1 starts to turn off and itotal starts to decrease. Hence, the energy stored in Lσ1 and Lσ2
i2
begins to release, moreover, the ispeed
of releasing is determined
by the stray inductance in the two
2
branches and the speed of turn-off. If the energy stored in Lσ1 and Lσ2 is released completely at t3
(t3the
) current direction in the two branches
simultaneously, that means W(t
Lσ12
(t-t
3)3=) 0 and WLσ2(t3) = 0, and
depicted in Figure 7 does not change from t1 to t3. If the energy stored in Lσ1 and Lσ2 does not release
8. The current
direction
branches at the turn-off transient.
completely at t3 Figure
simultaneously,
that
meansof
Wparallel
Lσ1(t2) ≠ 0 or WLσ2(t2) ≠ 0 and one of the two branches
Figure
8. The current
direction
of
parallel
branches at the turn-off transient.
must be released completely prior to the other. This study supposes the energy stored in Lσ2 releases
completely
at t2Transient
, namely i2(t2) = 0. From t1 to t2, the current i1 and i2 decrease gradually shown in Figure
3.1.2. Turn-On
8 (t1–t2) but the current direction does not change. At t2, i1(t2) = itotal(t2) and Lσ1 can be equivalent to a
At t4, S1 receives the open signal, due to the reverse recovery characteristic of the freewheeling
power source, and the direction of potential is shown in Figure 7, so the branch of Lσ2 provides the
diode inside S2, and S2 is equivalent to a short-circuit. The equivalent circuit is depicted in Figure 9.
path of energy release for Lσ1 that means the current i2 changes direction from t2 and increases at the
i2
i2
(t2-t3)
Energies 2017, 10, 1519
(t3)
Figure 8. The current direction of parallel branches at the turn-off transient.
7 of 16
3.1.2.
Turn-On
Transient
3.1.2.
Turn-On
Transient
, S1receives
receivesthe
theopen
opensignal,
signal, due
due to
to the
the reverse
the
freewheeling
AtAt
t4 ,t4S1
reverse recovery
recoverycharacteristic
characteristicofof
the
freewheeling
diode
inside
S2,
and
S2
is
equivalent
to
a
short-circuit.
The
equivalent
circuit
is
depicted
inin
Figure
9. 9.
diode inside S2, and S2 is equivalent to a short-circuit. The equivalent circuit is depicted
Figure
uL1
Lσ1
i1
Mσ1_2
uL2
i2
Lσ2
Mσ1_3
itotal
Mσ2_3
Mσ1_4
Mσ2_4
uL3
i3
Lσ4
uL4
Lσ3
Mσ3_4
i4
Figure
9. The equivalent circuit of the turn-on transient.
Figure 9. The equivalent circuit of the turn-on transient.
i1 (ti14(t)4and
associatedwith
withthe
thecommutation
commutation
between
Lσ2 during
t3 –t4 .
) andi2i(t
2(t
are mainly
mainly associated
between
Lσ1 Land
Lσ2 during
t3–t4. At
σ1 and
44)) are
and
Lσ2Lis
released
completely,
that
means
W
Lσ1
(t
4
)
=
0
and
W
Lσ2
(t
4
)
=
0,(t4 )
Attt44, ,ififthe
theenergy
energystored
storedininLσ1
Lσ1
and
is
released
completely,
that
means
W
(t
)
=
0
and
W
σ2
Lσ1 4
Lσ2
also
i
1
(t
4
)
=
0
and
i
2
(t
4
)
=
0.
Starting
from
t
4
,
i
1
and
i
2
increase
rapidly
from
zero
to
the
peak
of
reverse
= 0, also i1 (t4 ) = 0 and i2 (t4 ) = 0. Starting from t4 , i1 and i2 increase rapidly from zero to the peak
current, the
rising the
slope
is related
strayto
inductance
the branch,inand
magnitude
of recovery
reverse recovery
current,
rising
slopetoisthe
related
the stray in
inductance
thethe
branch,
and the
and
direction
of
current
is
shown
in
Figure
9.
If
the
energy
stored
in
L
σ1
and
L
σ2
does
not
release
magnitude and direction of current is shown in Figure 9. If the energy stored in Lσ1 and Lσ2 does not
completely
duringduring
t3–t4, that
some loop current exists and i1(t4) = − i2(t4), as depicted in Figure
release
completely
t3 –tmeans
4 , that means some loop current exists and i1 (t4 ) = − i2 (t4 ), as depicted
10
(t
4). Starting from t4, due to the external voltage, i1 increases rapidly and i2 decreases to zero quickly,
in Figure 10 (t4 ). Starting from t4 , due to the external voltage, i1 increases rapidly and i2 decreases to
as depicted in Figure 10 (t4–t5), until i1(t5) = itotal(t5), as depicted in Figure 10 (t5). Then, starting from t5,
zero
quickly, as depicted in Figure 10 (t4 –t5 ), until i1 (t5 ) = itotal (t5 ), as depicted in Figure 10 (t5 ). Then,
i1 and i2 keep the same direction and rise to maximum value at the same time. During the period of
starting from t5 , i1 and i2 keep the same direction and rise to maximum value at the same time. During
t6–t7, the reverse recovery current is at the vanishing stage and is equivalent to the load current iload.
the period of t6 –t7 , the reverse recovery current is at the vanishing stage and is equivalent to the load
Similar to the turning-off process, i1 and i2 may change direction, but the differential of the two
current iload . Similar to the turning-off process, i1 and i2 may change direction, but the differential of
currents
can10,
also
Energies 2017,
1519be described by Formula (2).
8 of 17
the two currents can also be described by Formula (2).
i1
i1
itotal
itotal
i2
i2
(t4)
(t4-t5)
i1
i1
itotal
i1
itotal
itotal
i2
i2
i2
(t5)
(t5-t6)
(t6-t7)
Figure 10. The current direction of parallel branches at the turn-on transient.
Figure 10. The current direction of parallel branches at the turn-on transient.
3.2. Model of Stray Inductance
3.2. Model of Stray Inductance
Based on the above analysis, although the direction of current changes at the turn-on transient
Based on the above analysis, although the direction of current changes at the turn-on transient
or the turn-off transient, the position of the bus-bar does not change, and the dotted terminals of
or the turn-off transient, the position of the bus-bar does not change, and the dotted terminals of the
the branch equivalent inductance remains the same. Ultimately, the over-voltage caused by mutual
branch equivalent inductance remains the same. Ultimately, the over-voltage caused by mutual
inductance
also
stays
the
same.
inductance
also
stays
the
same.
The partial stray inductance model for Lσ1, Lσ2, Lσ3, and Lσ4 is as follows:
 u L1   Ls 1
u L 2  M 2 1
u    M
 L3   3 1
u
M
M1 2
Ls 2
M32
M
M1 3
M 23
Ls 3
M
M1 4 
 i1 
M 2 4  d i2 
 
M 3 4  dt i3 

 
i
L
(3)
Energies 2017, 10, 1519
8 of 16
The partial stray inductance model for Lσ1 , Lσ2 , Lσ3 , and Lσ4 is as follows:







u L1
u L2
u L3
u L4
Lsσ1
M21
M31
M41
 
 
=
 
M12
Lsσ2
M32
M42
M13
M23
Lsσ3
M43

M14
M24
M34
Lsσ4

 d 


·
·
 dt 

i1
i2
i3
i4




(3)
The uL1 , uL2 , uL3 , and uL4 represent induced voltage of stray inductance Lσ1 , Lσ2 , Lσ3 , and Lσ4 ,
respectively, at the switching transient. The Lsσ1 , Lsσ2 , Lsσ3 , and Lsσ4 represent self-inductance in
branches Lσ1 , Lσ2 , Lσ3 , and Lσ4 . Meanwhile the M12 (M21 ), M13 (M31 ), M14 (M41 ), M23 (M32 ), M24 (M42 ),
and M34 (M43 ) represent the mutual-inductance for each branch.
While measuring the bus-bar stray inductance, only the transient voltage and current of the
bus-bar are available, so the measured values are all of equivalent stray inductance, and the
mathematical model is as follows.



 

i1
u L1
Lσ1
0
0
0


 u   0
Lσ2
0
0 
 L2  
 d  i2 
(4)
·


=
·
 u L3   0
0
Lσ3
0  dt  i3 
i4
u L4
0
0
0
Lσ4
According to Equation (3) and (4), the bus-bar equivalent stray inductance is expressed as:

h


L
=
Lsσ1
σ1



h


 Lσ2 = M21
h


L
=
M31

σ3


h


 L = M
σ4
41
M12
M13
M14
Lsσ2
M23
M24
M32
Lsσ3
M34
M42
M43
Lsσ4
i h
· 1
i h
1
· di
di
i h 2
1
· di
di
i h 3
1
· di
di
4
di2
di1
di3
di1
di4
di1
1
di3
di2
di4
di2
di2
di3
1
di4
di3
di2
di4
di3
di4
1
iT
iT
iT
(5)
iT
For Equation (5), the differential is difficult to obtain from measurement at the turn-off transient.
Because of the excellent linearity at the turn-on transient, the differential can be replaced by difference.
 h


1



h


di1

di
h 2

di
1


di3


h


di1

di2
di1
di3
di1
di4
di1
1
di3
di2
di4
di2
di2
di3
1
di4
di3
di2
di4
di3
di4
di4
1
iT
iT
iT
iT
∆i2
∆i1
∆i3
∆i1
∆i4
∆i1
∆i1
∆i2
1
∆i3
∆i2
∆i4
∆i2
h
∆i1
∆i3
∆i2
∆i3
1
∆i4
∆i3
h
∆i1
∆i4
∆i2
∆i4
∆i3
∆i4
=
h
=
h
=
=
1
1
iT
iT
iT
(6)
iT
Based on Figures 7 and 9, the simplified equivalent circuit is shown in Figure 11.
The induced voltage in the switching transient of the stray inductance LσPc and LσNe meets:
"
u Pc
u Ne
#
"
=
LsσPc
MPN
MPN
LsσNe
#
d
·
dt
"
itotal
itotal
#
(7)
where, uPc = uL1 = uL2 , uNe = uL3 = uL4 . LsσPc and LsσNe represent the self inductance and mutual
inductance of LσPc and (between) LσNe . The current matrices meet:

"
itotal
itotal
#
"
=
1
0
1
0
0
1
0
1
#


·

i1
i2
i3
i4





(8)
Energies 2017, 10, 1519
Energies 2017, 10, 1519
9 of 16
9 of 17
Based on Equations (7) and
inductance matrix
T
T meets as:
 (8), the simplified
"
LsσPc
MPN
where:
 di di di 
 i i3 i4 
 1 2 3 4   1 2

  di#1 di1 "di1 
 i1 #i1 i1 
T
T


M
0 i1 0 i3 −1i4 

di1 =
di3 di41 1  
PN
1   
1 ·L ·

 0 0 1i 1 i i 
L
sσNe
2
2
 di2 di2 di2 
 2

T
T



i1 i2 i4 
 di1 di2 di4
1

1



  di di
di3 
 i3 i3 i3
  3 3
TM
T
M
M
 di1 di2 Ldisσ1
 12 i1 13
i2 14
i3 
3
  M23 M24 1

 M 1 Lsσ2
L4 =di4 di21
 di
4

 i4 i4 i4 
1
1
0
0
0
0
1
1
  −1




 M31 M32 Lsσ3 M34 
M41 M42equivalent
M43 Lsσ4
Based on Figure 7 and Figure 9, the simplified
circuit is shown in Figure 11.
LσPc
uPc
LσPc
MPN
(6)
(10)
uPc
itotal
MPN
itotal
(9)
Energies 2017, 10, 1519
10 of 17
LσNe uNe
LσNe uNe
 Ls1 M 12 M 13 M 14 
M
Ls 2 M 23 M 24 
 21

L

(a)
(10)
 M 31 M 32 Ls 3 M 34  (b)


Mswitching
M 43 transient.
Ltransient.
Figure
(a)(a)
Turn
42
s 4 
 Min41inthe
Figure11.
11.The
Thesimplified
simplifiedequivalent
equivalentcircuit
circuit
the
switching
Turnoff
offtransient;
transient;(b)
(b)turn
turn
onontransient.
transient.
4. Measuring
Method
forinPartial
Stray Inductance
The induced
voltage
the switching
transient of the stray inductance LσPc and LσNe meets:
4. Measuring Method for Partial Stray Inductance
4.1. Basic Principle
uPc   LsPc M PN  d itotal
4.1. Basic Principle
(7)
u Ne   M PN LsNe   dt itotal
The measuring principle for partial
stray inductance is based on the inductance voltage integral
The measuring principle for partial stray inductance is based on the inductance voltage integral
method
stray
is determined
by the
inductance
outside the
where,
uPc[20].
= uL1Under
= uL2, uhigh
Ne = ufrequency,
L3 = uL4. LsσPcthe
and
LsσNeinductance
represent the
self inductance
and
mutual inductance
method [20]. Under high frequency, the stray inductance is determined by the inductance outside the
tending to
be. The
constant.
The
voltagemeet:
and current waveform of the stray inductance at the
ofconductor,
LσPc and (between)
LσNe
current
matrices
conductor, tending to be constant. The voltage and current waveform of the stray inductance at the
turn-off process of an IGBT is shown in Figure 12.
turn-off process of an IGBT is shown in Figure 12.
 i1 
ito ta l 1 1 0 0 i2 
(8)
ito ta l  0 0 1 1  i3 
 
i4 
i t 
Based on Equations (7) and (8), the simplified inductance matrix meets as:
u t 
A
 LsPc
M
 PN

1


M PN
1
1
0
0
 1 1
B   

L

Lt sNe   0 0 1 1
0


u t
0

t1
i


1
0 

0 

1 

1 t
2
(9)
where:
Figure 12. The current, voltage waveform of inductance at the turn-off transient.
Figure 12. The current, voltage waveform of inductance at the turn-off transient.
Furthermore, the voltage on stray inductance depends on the changing rate of current, expressed
by following equation:
u t   L 
di
dt
(11)
Energies 2017, 10, 1519
10 of 16
Furthermore, the voltage on stray inductance depends on the changing rate of current, expressed
by following equation:
di
u(t) = Lσ ·
(11)
dt
By the calculation of the definite integral for Equation (11), the bus-bar partial stray inductance is
obtained and t1 –t2 represents the time of the turn-on transient or turn-off transient.
R t2
Lσ =
t1
u(t)dt
(12)
i ( t2 ) − i ( t1 )
4.2. Measurement Error Analysis
Based on Equation (12), the measured value of partial stray inductance is determined by the
integral value of voltage during the period of t1 –t2 and the values of i(t2 ) and i(t1 ). So, the precise
partial stray inductance can be obtained once the integral value of the voltage and the two transient
current values are available.
The measurement diagram of stray inductance is shown in Figure 13. Because the bus-bar partial
stray inductance is very small, the voltage on the stray inductance at the switching transient is also very
small, so a passive probe is needed to raise the bandwidth and precision of the voltage measurement.
The probe of the oscilloscope is a high impedance circuit, vulnerable to the interference of an external
magnetic field, and the interference source is mainly the voltage on the measuring lead-wire caused
by
Energies 2017, 10, 1519
11 of 17
the spatial transient magnetic field.
Busbar
Measuring
wire
i
A
cl2
Oscilloscope
c13
Lσ
Oscilloscope
probe
cl1
Figure 13.
13. The
The measuring
measuring method
method of
of partial
partial stray
stray inductance.
inductance.
Figure
The
is divided
divided into
into ccl1l1,, ccl2l2,, and
The measuring
measuring wire
wire is
and ccl3l3, ,shown
shownin
inFigure
Figure 13.
13. According
According to
to the
the theory
theory
of
partial
inductance,
the
induced
voltage
on
the
measuring
lead-wire
caused
by
the
spatial
transient
of partial inductance, the induced voltage on the measuring lead-wire caused by the spatial transient
magnetic
by:
magnetic field
field can
can be
be expressed
expressed by:
d(
R
Verr =Verr 
R
R
dA
(  ·Adl d+l   AA d· ldl
+
 A  dlA) · dl)
cl1
ccl2l 2
cl 1
dt
dt
cl 3 cl3
(13)
(13)
represents the magnetic vector potential caused by the bus-bar
bus-bar current
current and the
the magnitude
magnitude and
and
A represents
direction is decided by the Biot–Savart law [14]
AA=
µ
4π
4
JJ
vv RR ·dvdv
Z
(14)
(14)
J represents
Based
on on
Formula
(14),(14),
A shares
the same
direction
with
representsthe
thecurrent
currentdensity
densityvector.
vector.
Based
Formula
A shares
the same
direction
the
and isand
inversely
proportional
to thetodistance
fromfrom
the bus-bar
in magnitude.
So, the
withcurrent
the current
is inversely
proportional
the distance
the bus-bar
in magnitude.
So,
measuring
precision
can be
raised,
guaranteeing
the lead-wire
cl1 andccl1l2 are
to the busthe measuring
precision
can
be raised,
guaranteeing
the lead-wire
andperpendicular
cl2 are perpendicular
to
bar, and cl3 is very far from the bus-bar. If the distance between cl3 and bus-bar exceeds a certain value,
the measuring voltage remains constant, which stands for the measured value with high precision.
5. Simulation and Experimental Results
Energies 2017, 10, 1519
11 of 16
the bus-bar, and cl3 is very far from the bus-bar. If the distance between cl3 and bus-bar exceeds a
certain value, the measuring voltage remains constant, which stands for the measured value with
high precision.
5. Simulation and Experimental Results
5.1. Parameter Extraction
Ansys Q3D software was used to extract the partial stray inductance of the laminated bus-bar
model in this study to verify the measured values of partial stray inductance. For extraction of the
partial stray inductance, a current source and sink must be set up. For a certain conductor, only one
current sink is allowed, but multiple current sources are permitted in the Q3D software. Taking the
proximity effect of the bus-bar into consideration, excitation is demanded for both a positive bus-bar
and a negative bus-bar to simulate the model. The positive bus-bar terminals 9 and 11 are set as
Source2 and Source1 and the capacitor anode is set as Sink1. Similarly, the negative bus-bar terminals
10 and 12 are set as Source4 and Source3 and the capacitor cathode is set as Sink2. The excitation source
settings and simulated current density of the bus-bar is shown in Figure 14. Because the excitation has
the opposite direction of the actual current, the simulated mutual inductance in the branches of the
Energies 2017,
10, 1519
12 of of
17
positive
bus-bar
and negative bus-bar must be negative. The self inductance and mutual inductance
the asymmetrical parallel branches are described in Table 1.
Sink1
Sink2
Source3
Source1
Source4
Source2
Figure 14. The excitation source settings and simulated current density of the bus-bar.
Figure 14. The excitation source settings and simulated current density of the bus-bar.
Table 1. Simulated partial stray inductance matrix.
Table 1. Simulated partial stray inductance matrix.
nH
Lσ 1
Lσ 2
Lσ 3
Lσ 4
nH
Lσ1
Lσ1
Lσ2 Lσ1
Lσ3 Lσ2
Lσ4 Lσ3
22.72
22.72
23.95
23.95
−
1.56
−
7.56
−1.56
Lσ4
−7.56
Lσ2
23.95
23.95
37.42
37.42
−2.18
−14.5
−2.18
−14.5
Lσ3
−1.56
−1.56
−2.18
−2.18
17.18
17.48
17.18
17.48
Lσ4
−7.56
−7.56 −14.5
−14.5 17.48
17.48 29.37
29.37
In order to verify the accuracy of the simplified model, the stray inductance LσPc and LσNe should
In orderbased
to verify
accuracy The
of the
simplified
model,
theas
stray
inductance
LσPcpositive
and LσNebus-bar
should
be extracted
on the
simulation.
capacitor
anode
is set
Source1,
and the
be
extracted
based
on
simulation.
The
capacitor
anode
is
set
as
Source1,
and
the
positive
bus-bar
terminals 9 and 11 are set as Sink1. Similarly, the negative bus-bar terminals 10 and 12 are set as
terminalsand
9 and
are set cathode
as Sink1.isSimilarly,
the The
negative
bus-bar
terminals
10 and
12 are
set as
Source2
the 11
capacitor
set as Sink2.
simplified
model
excitation
source
settings
Source2
and
the
capacitor
cathode
is
set
as
Sink2.
The
simplified
model
excitation
source
settings
and
and current density of the bus-bar are shown in Figure 15. The simulated self inductance and mutual
current
density
of
the
bus-bar
are
shown
in
Figure
15.
The
simulated
self
inductance
and
mutual
inductance of the simplified branch shown in Figure 10 are described in Table 2.
inductance of the simplified branch shown in Figure 10 are described in Table 2.
source1
sink2
source2
sink1
In order to verify the accuracy of the simplified model, the stray inductance LσPc and LσNe should
be extracted based on simulation. The capacitor anode is set as Source1, and the positive bus-bar
terminals 9 and 11 are set as Sink1. Similarly, the negative bus-bar terminals 10 and 12 are set as
Source2 and the capacitor cathode is set as Sink2. The simplified model excitation source settings and
current density of the bus-bar are shown in Figure 15. The simulated self inductance and mutual
Energies 2017, 10, 1519
12 of 16
inductance of the simplified branch shown in Figure 10 are described in Table 2.
source1
sink2
source2
sink1
source2
sink1
Figure 15. The simplified model excitation source settings and current density of the bus-bar.
Figure 15. The simplified model excitation source settings and current density of the bus-bar.
Table 2. Simulated stray inductance matrix.
Table 2. Simulated stray inductance matrix.
nH nH
LσPcLσPc
LσNeLσNe
LσPc
LσPc
17.81
17.81
−
1.44
−1.44
LσNe
LσNe
−1.44
−1.44
16.46
16.46
The simplified inductance matrix calculated based on Equation (9) is shown in Table 3. Calculation
results are compared with the simulated results in Table 2. They have only 1 nH difference. That means
the calculated values can match the simulated values very well, and the simplified method is accurate.
Table 3. Calculated stray inductance matrix.
nH
LσPc
LσNe
LσPc
LσNe
19.14
−1.51
−1.51
17.15
5.2. Experimental Measurement
In order to measure the branch stray inductance, the inverter testing platform is constructed as
shown in Figure 16. The measuring equipment and power device are shown in Table 4. It must be
noted that the oscilloscope must have the function of multi-channel digital integral that can measure
the stray inductance of the two parallel branches under the same conditions. The experimental test
is classified as the direct measuring method and the indirect measuring method. For the indirect
measuring method, a Rogowski coil is adopted to measure the transient current variation ratio in
the four branches. Meanwhile, using the simulation value of the stray inductance and Equation (6),
the equivalent stray inductance can be obtained. The direct measuring method can be realized by the
method proposed by the paper.
Table 4. The experimental equipment.
Equipment
Parameter
IGBT module
IGBT driver
Rogowski coil
Voltage probe
Oscilloscope
Inductive load
FF1400R12IP4
2SP0320V2A0
CWT30B
Agilent-10073C
Agilent-MSO8064A
About 7uH
the stray inductance of the two parallel branches under the same conditions. The experimental test is
classified as the direct measuring method and the indirect measuring method. For the indirect
measuring method, a Rogowski coil is adopted to measure the transient current variation ratio in the
four branches. Meanwhile, using the simulation value of the stray inductance and Equation (6), the
equivalent
stray inductance can be obtained. The direct measuring method can be realized by
the
Energies 2017, 10, 1519
13 of 16
method proposed by the paper.
Laminated
busbar
Busbar capacitor
IGBT module
And driver
Measuring
wire
Rogowski coil
Figure
Figure 16.
16. The
The experimental
experimental platform.
platform.
Table 4. The experimental equipment.
5.3. Indirect Measurement
Equipment
Due to the excellent linearity
of the transient currentParameter
of the IGBT module at the turn-on transient,
IGBT
the indirect measurement takes
themodule
turn-on transient asFF1400R12IP4
an example. In order to verify the consistency
Energies 2017, 10, 1519
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IGBT
2SP0320V2A0
of the transient current’s change
ratedriver
under different testing
conditions, a double pulse test is applied
Rogowski
coil leg testing circuit
CWT30B
on S1 and S2. At first, according
to the phase
shown in Figure 5, S1 is tested with a
double pulse, then, upon switching the position of the load, S2 is tested also with the double pulse.
Voltage the
probe
Agilent-10073C
double pulse, then, upon switching
position of the
load, S2 is tested also with the double pulse.
The transient current i1, i2, i3, and i4 shown in Figure 17a,b, corresponding to the second turn-on of
Agilent-MSO8064A
The transient current i1 , i2 , i3Oscilloscope
, and i4 shown in Figure
17a,b, corresponding to the second turn-on
lower module S2 and upper module
S1, and their main difference is the current magnitude and
Inductive
load
About
7uH
of lower module S2 and upper
module
S1, and their main
difference
is the current magnitude and
direction before turn on. The variation in the transient current is basically consistent after turning on.
direction before turn on. The variation in the transient current is basically consistent after turning
Choosing Δt = t1 − t2 = 100ns and recording the difference of the two moments, because the
5.3.
Measurement
on. Indirect
Choosing
∆t = t1 − t2 = 100 ns and recording the difference of the two moments, because the
oscilloscope has sampling
error, tests are made many times for each testing method to compute the
oscilloscope
has
sampling
error, tests
made many
times
forIGBT
eachmodule
testing at
method
to compute
the
Due
to
the
excellent
linearity
of theare
transient
current
of
the
the turn-on
arithmetic mean value to raise measurement
accuracy.
Table
5 shows
the average
valuestransient,
for fifty
arithmetic
mean
value to raise
measurement
accuracy.asTable
5 shows In
theorder
average
valuesthe
for fifty repeat
the
indirect
measurement
takes
the turn-on
transient
an example.
verify
repeat
measurements
of the
upper
IGBT module
and lower
IGBT module
atto
the
turn-on consistency
transient.
measurements
of
the
upper
IGBT
module
and
lower
IGBT
module
at
the
turn-on
transient.
of the transient current’s change rate under different testing conditions, a double pulse test is applied
on S1 and S2. At first, according to the phase leg testing circuit shown in Figure 5, S1 is tested with a
(a)
(b)
Figure 17.
17. The
The current
current waveform
waveform at
at the
the turn-on
turn-on transient.
transient. (a)
Second turning-on
turning-on of
S2; (b)
(b) Second
Second
Figure
(a) Second
of S2;
turning-on of
of S1.
S1.
turning-on
Table
5. Measured
Measured current
current values.
values.
Table 5.
Δi(A)
S1
S1 S2
∆i(A)
S2
Δi1
∆i1
104.57
104.57
106.25
106.25
Δi2
∆i2
76.71
76.71
75.21
75.21
Δi3
∆i
66.363
66.36
70.35
70.35
Δi4
∆i
108.46 4
108.46
110.36
110.36
According to Equations (4), (6), and Table 1, the partial stray inductance can be calculated, as
shown in Table 6. The difference of the two measured methods is within 5 nH, mainly deriving from
the sampling error of the oscilloscope.
Table 6. Calculated inductance based on Tables 1 and 3.
Energies 2017, 10, 1519
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According to Equations (4), (6), and Table 1, the partial stray inductance can be calculated, as
shown in Table 6. The difference of the two measured methods is within 5 nH, mainly deriving from
the sampling error of the oscilloscope.
Table 6. Calculated inductance based on Tables 1 and 3.
nH
Lσ 1
Lσ 2
Lσ 3
Lσ 4
Turn on transient of S1
Turn on transient of S2
Difference
31.45
30.69
0.76
47
47.9
0.9
40.68
39.8
0.88
28.11
24
4.11
5.4. Direct Measurement
The direct measurement is based on the inductance voltage integral method regardless of the
current linearity. Therefore, it can be realized at the turn-on transient or the turn-off transient.
The voltage sampling error is very small, because the voltage is based on digital integration function.
However, the current is sampled at t1 and t2 moments, so the current sampling error is relatively large.
In order to raise measurement precision, tests are made many times at both the turn-on transient and
the turn-off transient.
Due to the limited channel amounts of the oscilloscope, Lσ1 and Lσ2 are extracted firstly, as shown
in
Figure
18a,b, in which the i1 and i2 are the current waveforms at the switching transient in branches
Energies 2017, 10, 1519
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Lσ1 and Lσ2 . Also, uL1 and uL2 are the voltage waveforms at the switching transients on branches
L
is observed
voltage
variation
is basically
consistent
at the
switching
transient.
σ1 and
and
Lσ2. LItσ2is. It
observed
thatthat
the the
voltage
variation
is basically
consistent
at the
switching
transient.
f1
fand
and
f
represent
the
time
integral
of
u
and
u
and
the
integral
time
is
100
ns.
1
2
L1 uL2 and
L2 the integral time is 100 ns.
f2 represent
the time integral of uL1 and
(a)
(b)
Figure
transient.
Figure 18.
18. The
The test
test waveform
waveform at
at the
the turn-off
turn-off transient.
transient. (a)
(a) Turn-on
Turn-on transient;
transient; (b)
(b) Turn-off
Turn-off transient.
The partial stray inductance for Lσ1 and Lσ2, based on the turn-on transient and the turn-off
The partial stray inductance for Lσ1 and Lσ2 , based on the turn-on transient and the turn-off
transient, are listed in Table 7. According
to the same testing conditions, Lσ3 and Lσ4 are measured, as
transient, are listed in Table 7. According to the same testing conditions, Lσ3 and Lσ4 are measured, as
shown in Table 7. It is observed that the measurement difference is within 3 nH at both the turn-on
shown in Table 7. It is observed that the measurement difference is within 3 nH at both the turn-on
transient and the turn-off transient, so the turn-on transient and the turn-off transient are valid for
transient and the turn-off transient, so the turn-on transient and the turn-off transient are valid for the
the extraction of partial stray inductance.
extraction of partial stray inductance.
Table 7. The measured values of Lσ1 and Lσ2.
Table 7. The measured values of Lσ1 and Lσ2 .
nH
Lσ1
Lσ2
L
σ
2
30.56
45.6
Turn-onTurn-off
transienttransient
30.56 31.37 45.6
42.77
Turn-off transient
Difference 31.37 0.81 42.77
2.83
nH
Lσ 1
Turn-on transient
Difference
0.81
2.83
Lσ3
Lσ 3
39.56
39.56
41.37
41.37
1.81
1.81
Lσ4
26.6 Lσ 4
27.57 26.6
0.97 27.57
0.97
The stray inductance mean values obtained from indirect measurement and direct measurement
are shown in Table 8. The measurement difference is within 5 nH so the consistency is very high.
Table 8. Comparison of stray inductance based on indirect and direct measurements.
nH
Lσ1
Lσ2
Lσ3
Lσ4
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The stray inductance mean values obtained from indirect measurement and direct measurement
are shown in Table 8. The measurement difference is within 5 nH so the consistency is very high.
Table 8. Comparison of stray inductance based on indirect and direct measurements.
nH
Lσ1
Lσ2
Lσ3
Lσ4
Indirect measurement
Direct measurement
Difference
31.56
30.9
0.66
47.4
44.19
3.21
40.24
40.46
0.22
26.01
27.1
1.01
LσPc and LσNe are measured based on the proposed direct measuring method. As shown in
Table 9, the measurement values are almost the same as the simulation values and the calculated
values. So, simulation, calculation, and measurement can validate each other, which can further prove
the accuracy of the modeling and measuring method.
Table 9. Comparison of stray inductance based on simulation, calculation, and measurement.
Method
LσPc
LσNe
simulation
calculation
measurement
16.37
17.6
17.2
15.02
15.6
16.1
6. Conclusions
In this paper, the laminated bus-bar of a three-phase inverter in an electric vehicle with asymmetric
branching is investigated. For the first time, the imbalanced transient current on the bus-bar is deeply
analyzed from the view of stray inductance energy storage and release. The partial self-inductance
and mutual-inductance model of the asymmetrical parallel branches on the bus-bar is built and
the technique to reduce the equivalent inductance circuit models is developed based on matrix
calculation. Furthermore, a high-precision partial inductance measuring method considering spatial
electromagnetic transient field is proposed. At last, both the simulation values based on Q3D and
the calculation values based on matrix calculation are verified using the proposed direct and indirect
measurement methods.
For future work, frequency-domain simulations will be conducted to investigate the effects of
the asymmetrical stray inductance on the inverter’s EMI performance. This will provide insight for
effective component layout inside the enclosure of EV inverters.
Acknowledgments: The project is supported by the National Key Research and Development Program of China
(2016YFC0600906).
Author Contributions: Fengyou He and Chengfei Geng conceived and designed the experiments. Chengfei Geng
designed and performed the simulations. Jingwei Zhang performed the experiments. Chengfei Geng and
Hongsheng Hu wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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