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In-Depth Information
2 Converting from MNA to State Variable Form
State Variable analysis (SV) is a concise way to describe the behaviour of circuits,
but SV formulation is non-trivial compared with MNA, and this has limited the
use of SV in circuit simulation. Nevertheless, a method of transforming MNA to
SV [10] addressed this problem. SV is formed by eliminating excess voltage and
current variables in MNA. Hence, only capacitor voltages and inductor currents
are preserved. Here, an example of MNA formulation and SV formulation shows
the idea before and after conversion by the method described in [10]. The fol-
lowing process has been implemented in MATLAB. For a simple RLC circuit,
MNA gives
Ax
=
B
:
⊛
⊝
⊞
⊠
v
n
+1
L
v
n
+1
R
v
n
+1
C
i
n
+1
1
i
n
+1
L
A
=
B
(1)
where
A
is a matrix of conductances,
B
is a vector of sum of currents through
nodes and values of the independent sources at current step
n
which
n
is the
number of calculated time steps.
v
n
+1
L
,v
n
+1
R
,v
n
+1
C
are the voltages across ele-
ments at the next time step and
i
n
+
1
,i
n
+
L
are values of the currents of the
supply and inductor at the next time step. In the SV form, the circuit equation
iswritteninthegeneralform:
dx
dt
=
f
(
t, x
(
t
)
,u
(
t
))
(2)
u
is the vector of input sources and state variables stored in vector
x
. All variables
are time dependent.
t
is time. To transform equation (1) to equation (2), rewrite
equation (1) as:
A
d
A
s
1
0
A
s
2
X
n
+1
1
X
n
+1
2
=
B
1
B
2
(3)
in which
A
s
is a sub-matrix containing conductances of only static elements and
A
d
is a sub-matrix of conductances of other elements - dynamic and static. The
minimum number of state variables,
r
o
, is the number of rows of sub-matrix
A
d
,A
s
1
,X
1
and
B
1
.
X
1
is the vector of state variables.
R
is the number of rows
and columns of the whole matrix, therefore,
R
r
o
is the number of internal nodes
which are not part of the state variables, and called excess variables, represented
by vector
X
2
.Furthermore,
A
d
is with the size
r
o
∗
−
r
o
where
X
2
and
B
2
are vectors
with size
r
0
∗
1.The matrix is reordered by converting
A
s
2
to (
A
21
I
)and
I
is
the unity matrix with size (
R
−
r
o
)
∗
(
R
−
r
o
):
A
d
A
12
A
21
I
X
n
+1
1
X
n
+1
2
=
B
1
B
2
(4)
The transformation is summarized in Algorithm 1. Excess variables, need to
be eliminated as well using (5).
