3.3 State Equations
123
Assuming the incremental capacitance C 1 (v C 1 ) = 0 for any v C 1 , and the incremental inductance L 2 (i L 2 ) = 0 for any i L 2 , we obtain the SEs in normal form
dv C 1
dt
= −
1
C 1 (v C 1 )
ˆ
i 1 (v C 1 , i L 2 , t)
(3.50)
di L 2
dt
= −
1
L 2 (i L 2 )
ˆ
v 2 (v C 1 , i L 2 , t)
(3.51)
where once more the explicit dependence on t accounts for time-varying sources.
The initial conditions are v C 1 (t 0 ) = v C 1 0 and i L 2 (t 0 ) = i L 2 0 .
Suppose now that there is a charge-controlled capacitor
v C 1 = ˆ
v C 1 (q C 1 ) = v 1
and a flux-controlled inductor
i L 2 = ˆ
i L 2 (ϕ L 2 ) = i 2 .
Let us use q C 1 and ϕ L 2 as state variables. Suppose to replace the capacitor by
a voltage source v 1 and the inductor by a current source i 2 and consider under the
unique solvability assumption the same representation (3.49) as before for N. Then,
noting that i C 1 = dq C 1 /dt = −i 1 and v L 2 = dϕ L 2 /dt = −v 2 we obtain the SEs in
normal form
dq C 1
dt
= − ˆ
i 1 ( ˆ
v C 1 (q C 1 ), ˆ
i L 2 (ϕ L 2 ), t)
(3.52)
dϕ L 2
dt
= −ˆ v 2 ( ˆ
v C 1 (q C 1 ), ˆ
i L 2 (ϕ L 2 ), t).
(3.53)
The initial conditions are q C 1 (t 0 ) = q C 1 0 and ϕ L 2 (t 0 ) = ϕ L 2 0 .
Example 3.5 Consider the network in Fig. 3.13a with a linear capacitor C 1 , a linear
inductor L 2 , and two nonlinear resistors i R 1 = e
v R 1 , v R 2 = i 4
R 2
. By replacing
C 1 with a voltage source and L 2 with a current source, we obtain the network in
Fig. 3.13b.
By the KCL at the cut-set C we obtain
i 1 = ˆ
i 1 (v 1 , i 2 ) = v 1 + i R 1 − i 2 − 1 = v 1 + e
v 1 − i 2 − 1.
Applying KVL at the loop formed by L 2 , C 1 and R 2 we obtain
v 2 = ˆ
v 2 (v 1 , i 2 ) = v R 2 + v 1 = (1 + i 2 )
4
+ v 1 .
123
Assuming the incremental capacitance C 1 (v C 1 ) = 0 for any v C 1 , and the incremental inductance L 2 (i L 2 ) = 0 for any i L 2 , we obtain the SEs in normal form
dv C 1
dt
= −
1
C 1 (v C 1 )
ˆ
i 1 (v C 1 , i L 2 , t)
(3.50)
di L 2
dt
= −
1
L 2 (i L 2 )
ˆ
v 2 (v C 1 , i L 2 , t)
(3.51)
where once more the explicit dependence on t accounts for time-varying sources.
The initial conditions are v C 1 (t 0 ) = v C 1 0 and i L 2 (t 0 ) = i L 2 0 .
Suppose now that there is a charge-controlled capacitor
v C 1 = ˆ
v C 1 (q C 1 ) = v 1
and a flux-controlled inductor
i L 2 = ˆ
i L 2 (ϕ L 2 ) = i 2 .
Let us use q C 1 and ϕ L 2 as state variables. Suppose to replace the capacitor by
a voltage source v 1 and the inductor by a current source i 2 and consider under the
unique solvability assumption the same representation (3.49) as before for N. Then,
noting that i C 1 = dq C 1 /dt = −i 1 and v L 2 = dϕ L 2 /dt = −v 2 we obtain the SEs in
normal form
dq C 1
dt
= − ˆ
i 1 ( ˆ
v C 1 (q C 1 ), ˆ
i L 2 (ϕ L 2 ), t)
(3.52)
dϕ L 2
dt
= −ˆ v 2 ( ˆ
v C 1 (q C 1 ), ˆ
i L 2 (ϕ L 2 ), t).
(3.53)
The initial conditions are q C 1 (t 0 ) = q C 1 0 and ϕ L 2 (t 0 ) = ϕ L 2 0 .
Example 3.5 Consider the network in Fig. 3.13a with a linear capacitor C 1 , a linear
inductor L 2 , and two nonlinear resistors i R 1 = e
v R 1 , v R 2 = i 4
R 2
. By replacing
C 1 with a voltage source and L 2 with a current source, we obtain the network in
Fig. 3.13b.
By the KCL at the cut-set C we obtain
i 1 = ˆ
i 1 (v 1 , i 2 ) = v 1 + i R 1 − i 2 − 1 = v 1 + e
v 1 − i 2 − 1.
Applying KVL at the loop formed by L 2 , C 1 and R 2 we obtain
v 2 = ˆ
v 2 (v 1 , i 2 ) = v R 2 + v 1 = (1 + i 2 )
4
+ v 1 .