Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
6,
No.
4,
August
2016,
pp.
1529
–
1533
ISSN:
2088-8708,
DOI:
10.11591/ijece.v6i4.10704
1529
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
A
Criterion
on
Existence
and
Uniqueness
of
Beha
vior
in
Electric
Cir
cuit
T
akuya
Hirata
*
,
Ek
o
Setiawan
*
,
Kazuya
Y
amaguchi
**
,
and
Ichijo
Hodaka
***
*
Interdisciplinary
Graduate
School
of
Agriculture
and
Engineering,
Uni
v
ersity
of
Miyazaki
**
Department
of
Control
Engineering,
National
Institute
of
T
echnology
Nara
Colle
ge
***
Department
of
En
vironmental
Robotics,
F
aculty
of
Engineering,
Uni
v
ersity
of
Miyazaki
Article
Inf
o
Article
history:
Recei
v
ed
Feb
5,
2016
Re
vised
May
22,
2016
Accepted
Jun
8,
2016
K
eyw
ord:
beha
vior
of
electric
circuit
switching
circuit
circuit
analysis
ABSTRA
CT
Beha
vior
of
electric
circuits
can
be
observ
ed
by
solving
circuit
equations
symbolically
as
well
as
numerically
.
In
general,
symbolic
computation
for
circui
ts
with
certain
number
of
circuit
elements
needs
much
more
time
than
numerical
computation.
It
is
reasonable
to
check
the
e
xistence
and
uniqueness
of
the
solution
to
circuit
equations
beforehand
in
order
to
a
v
oid
computation
for
the
case
of
no
solution.
Indeed,
some
circuits
ha
v
e
no
solution;
in
that
case,
one
should
notice
it
and
a
v
oid
to
w
ait
meaningless
computation.
This
paper
proposes
a
ne
w
theorem
to
check
whether
gi
v
en
circuit
equations
ha
v
e
a
solution
and
their
v
oltages
and
currents
of
all
circuit
elements
are
uniquely
determined
or
not.
The
theorem
is
suitable
for
de
v
eloping
a
computer
algorithm
and
helps
quick
symbolic
computation
for
electric
circuits.
Copyright
c
2016
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Ichijo
Hodaka
Department
of
En
vironmental
Robotics,
F
aculty
of
Engineering,
Uni
v
ersity
of
Miyazaki
1-1,
Gakuen
Kibanadai
Nishi,
Miyazaki,
889-2192,
Japan
hijhodaka@cc.miyazaki-u.ac.jp
1.
INTR
ODUCTION
An
y
simulation
of
beha
vior
of
an
actual
circuit
is
based
on
modelling
of
the
circuit
with
idealization.
The
modelling
is
a
crucial
step
of
analysis
and
design;
it
depicts
the
actual
c
ircuit
as
a
circuit
diagram,
determines
a
w
orking
point,
and
linearizes
characteristics
of
electric
components
in
the
actual
circuit
with
a
w
orking
frequenc
y
.
Beha
vior
of
the
actual
circuit
is
represented
by
a
solution
to
circuit
equations
deri
v
ed
from
the
circuit
diagram
and
the
fundamental
la
ws
such
as
Kirchhof
f
’
s
la
ws,
Ohm’
s
la
w
,
and
the
electric
characteristics
of
inductor
and
capacitor
.
SPICE[1],
a
defacto
standard
circuit
simulator
,
simulates
and
plots
the
beha
vior
by
numerical
calculation.
That
is,
we
there
assume
e
xistence
and
uniqueness
of
beha
vior
.
W
e
also
assume
them
if
we
measure
beha
vior
of
an
actual
circuit.
Hence,
a
modelling
process
should
not
sacrifice
e
xistence
and
uniqueness
of
beha
vior;
v
oltages
and
currents
at
components
in
the
actual
circuit
or
in
the
simulation
model
should
be
single-v
alued
functions
defined
on
all
time.
Some
studies
address
circuit
diagrams
which
ha
v
e
lost
e
xistence
and
uniqueness
of
beha
vior
.
F
or
e
xam-
ple,
some
of
inductor
currents
and
capacitor
v
oltages
in
the
circuit
diagram
are
not
eligible
for
a
member
of
state
v
ariables[2]-[5].
This
can
be
translated
into
a
problem
to
find
a
spanning
tree
in
the
circuit
by
graph
theory[3]-[5].
Ho
we
v
er
,
little
is
kno
wn
about
a
direct
procedure
to
check
a
gi
v
en
circuit
diagram.
In
this
paper
,
we
propose
a
quick
criterion
to
check
whether
a
solution
to
linear
circuit
equations
e
xists
and
is
uniquely
determined
or
not.
The
criterion
enables
us
to
v
alidate
complicated
circuits
such
as
a
switching
circuit
with
computers,
and
thus
dra
w
a
pr
oper
circuit
diagram
-
a
diagram
which
has
e
xistence
and
uniqueness
of
beha
vior
-
with
help
of
computers.
Notice
that
the
criterion
is
writt
en
in
a
symbolic
equation.
That
means
our
result
w
ould
be
a
fundamental
result
to
contrib
ute
the
field
of
symbolic
calculation[6][7]
of
electric
circuits.
J
ournal
Homepage:
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x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Evaluation Warning : The document was created with Spire.PDF for Python.
1530
ISSN:
2088-8708
2.
LA
WS
OF
ELECTRIC
CIRCUITS
AND
CIRCUIT
EQ
U
A
TIONS
F
or
a
gi
v
en
circuit
diagram,
circuit
equations
are
defined
by
the
fundamental
la
ws,
Kirchhof
f
’
s
v
oltage
la
w(KVL),
Kirchhof
f
’
s
current
la
w(KCL),
Ohm’
s
la
w
,
and
the
electric
characteristics
of
inductor
and
capacitor
,
where
all
v
ariables
in
the
la
ws
and
the
characteristics
are
assumed
to
be
single-v
alued
functions
of
time
t
.
In
general,
we
ha
v
e
Aw
+
B
y
+
C
z
=
0
(1)
H
(
x
(
t
)
x
(
t
0
))
=
Z
t
t
0
z
1
(
p
)
dp
(2)
where,
A;
B
;
C
;
and
H
are
constant
matrices
whose
entries
are
determined
by
the
la
ws
and
the
characteristics
of
electric
circuit
and
resistance
of
resistors,
x
=
i
L
v
C
;
u
=
v
V
i
I
;
w
=
u
x
;
y
:
the
v
ector
of
node
v
oltages
;
z
1
=
v
L
i
C
;
z
=
2
6
6
6
6
6
6
6
6
4
i
V
v
I
v
R
i
R
z
1
v
S
i
S
3
7
7
7
7
7
7
7
7
5
;
(3)
and
v
V
;
v
I
;
v
C
;
v
L
;
v
R
,
and
v
S
are
v
oltages
of
v
oltage
sources,
current
sources,
capacitors,
inductors,
resistors
and
switches,
and
i
V
;
i
I
;
i
C
;
i
L
;
i
R
,
and
i
S
are
currents
of
v
oltage
sources,
current
sources,
capacitors,
inductors
,
resistors
and
switches,
and
v
N
is
potential
of
nodes,
respecti
v
ely
.
This
paper
assumes
that
all
the
v
ariables
are
Riemann
inte
grable,
and
thus
the
notation
R
in
the
equation
(2)
is
Riemann
inte
gral
(for
definition
of
inte
gral,
see
e.g.
[8]).
N
1
N
3
N
0
N
2
Figure
1.
A
boost
con
v
erter
-
CCM
(
S
1
:
ON,
S
2
:
OFF)
Example
Figure
1
and
2
represent
so-called
continuous
conduction
mode
(CCM)
and
discontinuous
conduction
mode
(DCM)
of
a
boost
con
v
erter
[9],
respecti
v
ely
.
Circuit
equations
of
the
circuit
diagram
in
Figure
1
are
represented
IJECE
V
ol.
6,
No.
4,
August
2016:
1529
–
1533
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1531
in
the
form
(1)
and
(2),
where
A
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
B
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1
1
0
0
1
0
1
0
0
1
0
1
0
1
0
1
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
C
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
1
0
0
0
1
0
1
R
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
(4)
H
=
L
0
0
C
;
x
=
i
L
v
C
;
z
1
=
v
L
i
C
;
w
=
2
4
v
V
i
L
v
C
3
5
;
y
=
2
6
6
4
v
N
1
v
N
3
v
N
0
v
N
2
3
7
7
5
;
and
z
=
2
6
6
6
6
6
6
6
6
6
6
6
6
4
i
V
v
R
i
R
v
L
i
C
v
S
1
i
S
1
v
S
2
i
S
2
3
7
7
7
7
7
7
7
7
7
7
7
7
5
:
(5)
N
1
N
3
N
0
N
2
Figure
2.
A
boost
con
v
erter
-
DCM
(
S
1
:
OFF
,
S
2
:
OFF)
Circuit
equations
of
Figure
2
are
represented
in
the
form
(1)
and
(2),
where
A
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
B
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1
1
0
0
1
0
1
0
0
1
0
1
0
1
0
1
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
C
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
1
0
0
0
1
0
1
R
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(6)
A
Criterion
on
Existence
and
Uniqueness
of
Behavior
in
Electric
Cir
cuit
(T
akuya
Hir
ata)
Evaluation Warning : The document was created with Spire.PDF for Python.
1532
ISSN:
2088-8708
and
the
other
notations
are
already
defined
as
(5).
The
v
ariable
v
L
cannot
be
uniquely
determined
on
the
algebraic
circuit
equations
(6)
of
Figure
2
for
an
y
v
V
;
i
L
,
and
v
C
,
which
is
understood
by
looking
at
the
circuit
diagram.
W
e
will
propose
a
theorem
to
capture
the
abo
v
e
situation
by
rank
calculations
of
matrices
A;
B
and
C
in
the
algebraic
equation
(1).
3.
THE
EXISTENCE
AND
UNIQ
UENESS
OF
SOLUTION
OF
ALGEBRAIC
CIRCUIT
EQ
U
A
TIONS
If
for
an
y
w
,
a
solution
(
y
;
z
)
to
t
he
algebraic
circuit
equation
(1)
of
a
circuit
e
xists
and
z
is
unique,
then
the
circuit
is
said
to
be
pr
oper
in
this
paper
.
A
circuit
is
said
to
be
impr
oper
if
it
is
not
proper
.
Theor
em
1.
A
circuit
is
proper
if
and
only
if
the
coef
ficient
matrices
of
its
algebraic
equation
in
the
form
(1)
satisfy
rank
[
A
B
C
]
=
rank
[
B
C
]
=
rank
B
+
m:
where
m
is
the
number
of
column
of
the
matrix
C
.
The
proof
of
Theorem
1
is
gi
v
en
in
the
appendix
A.
Rank
calcul
ation
in
Theorem
1
is
performed
on
general
purpose
symbolic
computation
system
widely
used.
This
helps
us
check
whether
a
circuit
is
proper
or
not,
e
v
en
if
it
is
lar
ge
and
complicated,
and
then
dif
ficult
to
check
by
hand.
Example
(A
pr
oper
cir
cuit)
W
e
recall
the
circuit
in
Figure
1
and
obtain
rank
[
A
B
C
]
=
12
;
rank
[
B
C
]
=
12
;
rank
B
=
3
;
m
=
9
for
(4).
Therefore
the
circuit
is
proper
by
Theorem
1.
Example
(An
impr
oper
cir
cuit)
W
e
recall
the
circuit
in
Figure
2
and
obtain
rank
[
A
B
C
]
=
12
;
rank
[
B
C
]
=
11
;
rank
B
=
3
;
m
=
9
for
(6).
Therefore
the
circuit
is
improper
by
Theorem
1.
The
whole
beha
vior
of
a
proper
circuit
is
gi
v
en
by
Corollary
1
in
the
appendix
A.
4.
CONCLUSION
Beha
vior
of
an
actual
electric
circ
u
i
t,
a
set
of
v
oltages
and
currents
of
all
circuit
components,
is
assumed
to
be
e
xpressed
as
a
single-v
alued
function
defined
on
all
time,
in
general.
Designers
depict
the
actual
circuit
as
a
schematic
diagram
to
simulate
beha
vior
of
the
actual
circuit.
Careful
designers
model
the
circuit
as
a
diagram
with
a
decision
of
including
parasitic
elements
or
not
and
reflect
a
w
orking
point
and
a
w
orking
frequenc
y
of
the
circuit
on
the
diagram.
The
y
write
circuit
equations
equi
v
alent
to
the
diagram,
and
then,
solv
e
the
equations
to
simulate
beha
vior
of
the
actual
circuit.
Ho
we
v
er
,
e
xistence
and
uniqueness
which
are
assumed
to
beha
vior
of
the
actual
circuit
are
not
necessarily
inherited
to
the
solution
because
the
diagram
and
the
equations
are
a
reduction
of
the
actual
circuit.
This
paper
has
proposed
a
criterion
for
the
e
xistence
and
uniqueness
to
be
guaranteed.
The
proposed
criterion
is
e
xpressed
as
equality
between
ranks
of
coef
ficient
matrices
in
the
circuit
equations.
Ef
fecti
v
e
and
quick
calculation
of
matrix
rank
is
a
v
ailable
in
general
purpose
symbolic
computing
tools.
Therefore
our
result
contrib
utes
to
the
field
of
symbolic
computation
of
electric
circuits.
A
LEMMA,
COR
OLLAR
Y
AND
PR
OOF
Cor
ollary
1.
If
a
circuit
is
proper
,
the
equations
(1)
and
(2)
are
uniquely
solv
ed
as:
z
=
K
1
x
+
K
2
u
(7)
x
(
t
)
=
e
F
(
t
t
0
)
x
(
t
0
)
+
e
F
t
Z
t
t
0
e
F
p
Gu
(
p
)
dp:
(8)
Pr
oof
1.
W
e
immediately
obtain
the
e
xpression
(7)
by
Theorem
1.
This
includes
z
1
=
K
11
x
+
K
21
u
.
Notice
that
the
matrix
H
is
block-diagonal
whose
blocks
ha
v
e
capacitances
C
’
s
and
self
or
mutual
inductances
(
L
’
s
and
M
’
s
respecti
v
ely)
in
their
entries
and
is
in
v
ertible.
If
we
put
F
=
H
1
K
11
and
G
=
H
1
K
21
and
apply
inte
gration
by
parts[8]
with
a
matrix
e
xponential
function,
we
ha
v
e
(8).
W
e
remark
that
the
solution
(8)
is
obtained
without
assuming
x
(
t
)
to
be
dif
ferentiable.
IJECE
V
ol.
6,
No.
4,
August
2016:
1529
–
1533
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1533
Lemma
1.
Let
B
and
C
be
matrices
wi
th
a
common
number
of
ro
ws.
The
equation
B
y
+
C
z
=
0
has
a
solution
z
for
an
y
y
if
and
only
if
rank
[
B
C
]
=
rank
C
.
Pr
oof
2.
(only
if)
There
is
a
matrix
Z
such
that
B
=
C
Z
.
Then
[
B
C
]
=
[
C
Z
C
]
=
C
[
Z
U
m
]
,
where
U
m
is
an
identity
matrix.
rank
[
B
C
]
=
rank
C
,
because
[
Z
U
m
]
is
ro
w
full
rank.
(if)
Since
rank
[
B
C
]
=
rank
C
and
Im
[
B
C
]
Im
C
,
Im
[
B
C
]
=
Im
C
.
Hence
Im
B
Im
[
B
C
]
=
Im
C
.
This
means
that
for
e
v
ery
fix
ed
y
,
there
is
a
z
such
that
B
y
+
C
z
=
0
.
Pr
oof
3
(F
or
Theorem
1)
.
(only
if)
By
the
assumption
and
Lemma
1,
rank
[
A
B
C
]
=
rank
[
B
C
]
.
Let
w
=
0
.
The
solution
space
Q
of
the
equation
(1)
is
Q
=
k
er[
B
C
]
.
Let
y
1
z
1
2
Q
and
y
2
2
k
er
B
.
Since
y
2
0
2
Q
,
z
1
=
0
.
Hence,
dim(k
er[
B
C
])
=
dim(k
er
B
)
.
By
rank-nullity
theorem,
we
obtain
rank
[
B
C
]
=
rank
B
+
m:
(if)
Let
Q
(
w
)
=
f
y
z
j
Aw
+
B
y
+
C
z
=
0
g
.
By
Lemma
1,
for
e
v
ery
fix
ed
w
,
Q
(
w
)
6
=
.
Since
rank
[
B
C
]
rank
B
+
rank
C
and
rank
C
m
in
general,
rank
[
B
C
]
=
rank
B
+
m
follo
ws
m
=
rank
[
B
C
]
rank
B
rank
C
m
.
So,
rank
C
=
m
=
rank
[
B
C
]
rank
B
:
(9)
Use
a
general
equality
rank
[
B
C
]
=
rank
B
+
rank
C
dim(
Im
B
\
Im
C
)
[10]
and
(9)
sho
w
dim(
Im
B
\
Im
C
)
=
0
:
(10)
Let
y
1
z
1
and
y
2
z
2
2
Q
(
w
)
.
Then
B
y
1
+
C
z
1
=
B
y
2
+
C
z
2
.
By
(10),
C
(
z
1
z
2
)
=
0
.
The
first
equality
in
(9)
implies
z
1
=
z
2
.
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W
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Nagel
and
D.
O.
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w
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ork
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IRE
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uh
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ariable
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R.
A.
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ariable
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49,
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K.
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R.
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ismenetsk
y
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A
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in
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cuit
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akuya
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ata)
Evaluation Warning : The document was created with Spire.PDF for Python.