11.7 Examples Concerning Class LME
421
where q MC 0 = q MC (t 0 ). Hence, O
(−1,−1)
C q
= 1, the state variable is ϕ C (t; t 0 ) and
the IC is ϕ C (t 0 ; t 0 ) = 0. It is worth noting that the CR depends upon q C 0 .
In the (v, i)-domain we have
v C (t) = f C (q C (t))
where
˙
q C (t) = i C (t).
Hence, O
(0,0)
C q
= 1, the state variable is q C (t) with corresponding ICs q C 0 . There is
no reduction of order in passing from the (v, i)-domain to the (ϕ, q)-domain.
11.7.2 Flux-Controlled Nonlinear Inductor L ϕ
A flux-controlled nonlinear inductor L ϕ is defined by the algebraic CR i L (t) =
f L (ϕ L (t)) in the (ϕ, i)-domain, i.e., α = −1 and β = 0. The corresponding CR in
the (ϕ, q)-domain is
˙
q L (t; t 0 ) = f L (ϕ L (t; t 0 ) + ϕ L 0 )
where ϕ L 0 = ϕ L (t 0 ). Hence, O
(−1,−1)
L ϕ
= 1, the state variable is q L (t; t 0 ), and the
IC is q L (t 0 ; t 0 ) = 0. It is worth noting that the CR depends upon ϕ L 0 .
In the (v, i)-domain we have
i L (t) = f L (ϕ L (t))
where
˙
ϕ L (t) = v L (t).
Hence, O
(0,0)
L ϕ
= 1, the state variable is ϕ L (t) with corresponding IC ϕ L 0 . There is
no reduction of order in passing from the (v, i)-domain to the (ϕ, q)-domain.
Remark 11.4 Note that the CR in the (ϕ, q)-domain depends upon the IC at t 0 for
the state variable of the element in the (v, i)-domain. Since the state variable in the
(ϕ, q)-domain is an incremental electric variable, then the IC at t 0 is by construction
null.
Remark 11.5 Table 11.1 summarizes the results thus obtained for a nonlinear
capacitor and inductor. Note that the chief result on order reduction in Remark 11.2
continues to hold even in the presence of nonlinear inductors and capacitors.
421
where q MC 0 = q MC (t 0 ). Hence, O
(−1,−1)
C q
= 1, the state variable is ϕ C (t; t 0 ) and
the IC is ϕ C (t 0 ; t 0 ) = 0. It is worth noting that the CR depends upon q C 0 .
In the (v, i)-domain we have
v C (t) = f C (q C (t))
where
˙
q C (t) = i C (t).
Hence, O
(0,0)
C q
= 1, the state variable is q C (t) with corresponding ICs q C 0 . There is
no reduction of order in passing from the (v, i)-domain to the (ϕ, q)-domain.
11.7.2 Flux-Controlled Nonlinear Inductor L ϕ
A flux-controlled nonlinear inductor L ϕ is defined by the algebraic CR i L (t) =
f L (ϕ L (t)) in the (ϕ, i)-domain, i.e., α = −1 and β = 0. The corresponding CR in
the (ϕ, q)-domain is
˙
q L (t; t 0 ) = f L (ϕ L (t; t 0 ) + ϕ L 0 )
where ϕ L 0 = ϕ L (t 0 ). Hence, O
(−1,−1)
L ϕ
= 1, the state variable is q L (t; t 0 ), and the
IC is q L (t 0 ; t 0 ) = 0. It is worth noting that the CR depends upon ϕ L 0 .
In the (v, i)-domain we have
i L (t) = f L (ϕ L (t))
where
˙
ϕ L (t) = v L (t).
Hence, O
(0,0)
L ϕ
= 1, the state variable is ϕ L (t) with corresponding IC ϕ L 0 . There is
no reduction of order in passing from the (v, i)-domain to the (ϕ, q)-domain.
Remark 11.4 Note that the CR in the (ϕ, q)-domain depends upon the IC at t 0 for
the state variable of the element in the (v, i)-domain. Since the state variable in the
(ϕ, q)-domain is an incremental electric variable, then the IC at t 0 is by construction
null.
Remark 11.5 Table 11.1 summarizes the results thus obtained for a nonlinear
capacitor and inductor. Note that the chief result on order reduction in Remark 11.2
continues to hold even in the presence of nonlinear inductors and capacitors.