§ 17-5 State variables and normal-form equations
The set of variables is a hybrid set that may include both
currents and voltages,They are the inductor currents and the
capacitor voltages,Each of these quantities may be used directly
to express the energy stored in the inductor or capacitor at any
instant of time,They collectively describe the energy state of the
system,They are called the state variables.
State variables--,,,
21 CCLi ??
L--KVL C--KCL
02 ??? sCLdtdiL ??
0)(1 1211 ????? sCsCC iRdtdC ????
0)(1 1222 ????? CsCLC RidtdC ????
s?
Li
?
?
2C?
si
Ri
2C
1C
?? 1C?
L
??
R
The state equations are said to be in normal form when the
derivative of each state variables is expressed as a linear
combination of all the state variables and forcing functions,The
ordering of the equations defining the derivatives and the order in
which the state variables appear in every equations must be the
same.
02 ??? sCLdtdiL ??
0)(1 1211 ????? sCsCC iRdtdC ????
0)(1 1222 ????? CsCLC RidtdC ????
sCL LLdt
di ?? 11
2 ???
ssCCC iCRCRCRCdt
d
11
2
1
1
1
1 1111 ????? ????
sCCLC RCRCRCiCdt
d ????
2
2
2
1
22
2 1111 ????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
s
s
C
C
L
C
C
L
i
RC
CRC
Li
RCRCC
RCRC
L
dt
d
dt
d
dt
di
?
?
?
?
?
0
1
11
0
1
111
11
0
1
00
2
11
2
1
222
11
2
1
The normal form equations:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
s
s
C
C
L
C
C
L
i
RC
CRC
Li
RCRCC
RCRC
L
dt
d
dt
d
dt
di
?
?
?
?
?
0
1
11
0
1
111
11
0
1
00
2
11
2
1
222
11
2
1
?????
?? fbqaqor '
v e c to rs ta teq ??
?
v e c t ors t at et h eofde r i v at i v eq ??
?
'
v e c t orf u n c t i o nf or c i n gf ??
?
m a t r i xs y s t e mba ????,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
s
s
L
C
C
i
L
RC
CRC
i
L
CRCRC
RCRC
?
?
?
0
1
0
1
11
0
1
0
111
0
11
2
11
2
1
222
11
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
dt
di
dt
d
dt
d
L
C
C
2
1
?
?