200
5 Flux-Charge Analysis Method of Memristor Circuits
where C = diag(C 1 , . . . , C n C ) and L = diag(L 1 , . . . , L n L ) are nonsingular
diagonal matrices and q C (t; t 0 ), ϕ L (t; t 0 ) are the vectors of incremental capacitor
charges and inductor fluxes, respectively.
Under the assumption of unique solvability of the (n C + n L )-port N R , we can
write the hybrid representation (cf. Chap. 3)
−q C (t; t 0 ) = q a (t; t 0 ) = h a (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
−ϕ L (t; t 0 ) = ϕ b (t; t 0 ) = h b (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
(5.35)
where ϕ C (t; t 0 ), q L (t; t 0 ) are the vectors of incremental capacitor fluxes and
incremental inductor charges, respectively. Moreover, u(t) = (ϕ E (t), q A (t)) is the
vector of fluxes and charges of the ideal independent sources within N R . Note that
h a (·) and h b (·) depend on the independent sources ϕ E , q A , and also ϕ M 0 , since the
CR of a memristor in the (ϕ, q)-domain depends upon the initial flux in the same
memristor. 8
By substituting (5.35) in (5.34), and considering that ϕ C (t; t 0 ) = ϕ a (t; t 0 ) and
q L (t; t 0 ) = q b (t; t 0 ), the following SE formulation for t ≥ t 0 in the (ϕ, q)-domain
for a circuit in LM is obtained:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
C
d
dt
ϕ C (t; t 0 )
= q C 0 − h a (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
L
d
dt
q L (t; t 0 )
= ϕ L 0 − h b (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
ϕ C (t 0 ; t 0 ) = 0
q L (t 0 ; t 0 ) = 0.
(5.36)
The structure of the SEs (5.36) suggests the following observations:
• if there are n C capacitors and n L inductors, the state variables of the SE
representation in the (ϕ, q)-domain are the n C + n L incremental variables
ϕ C (t; t 0 ) and q L (t; t 0 );
• the order of the SE representation in the (ϕ, q)-domain is n C + n L ;
• the evolution of the incremental variables ϕ C (t; t 0 ) and q L (t; t 0 ) is influenced via
q C 0 and ϕ L 0 by the initial conditions for the state variables in the (v, i)-domain
and, for n M memristors, the n M memristors initial conditions ϕ M 0 as well. The
initial conditions q C 0 , ϕ L 0 and ϕ M 0 indeed appear as constant inputs in the r.h.s.
of (5.36);
• the IVP for the SEs (5.36) has, by definition, zero initial conditions for the
incremental variables ϕ C (t; t 0 ) and q L (t; t 0 );
8 Functions h a (·) and h b (·) depend also on q M 0 if charge-controlled memristors are included in
LM.
5 Flux-Charge Analysis Method of Memristor Circuits
where C = diag(C 1 , . . . , C n C ) and L = diag(L 1 , . . . , L n L ) are nonsingular
diagonal matrices and q C (t; t 0 ), ϕ L (t; t 0 ) are the vectors of incremental capacitor
charges and inductor fluxes, respectively.
Under the assumption of unique solvability of the (n C + n L )-port N R , we can
write the hybrid representation (cf. Chap. 3)
−q C (t; t 0 ) = q a (t; t 0 ) = h a (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
−ϕ L (t; t 0 ) = ϕ b (t; t 0 ) = h b (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
(5.35)
where ϕ C (t; t 0 ), q L (t; t 0 ) are the vectors of incremental capacitor fluxes and
incremental inductor charges, respectively. Moreover, u(t) = (ϕ E (t), q A (t)) is the
vector of fluxes and charges of the ideal independent sources within N R . Note that
h a (·) and h b (·) depend on the independent sources ϕ E , q A , and also ϕ M 0 , since the
CR of a memristor in the (ϕ, q)-domain depends upon the initial flux in the same
memristor. 8
By substituting (5.35) in (5.34), and considering that ϕ C (t; t 0 ) = ϕ a (t; t 0 ) and
q L (t; t 0 ) = q b (t; t 0 ), the following SE formulation for t ≥ t 0 in the (ϕ, q)-domain
for a circuit in LM is obtained:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
C
d
dt
ϕ C (t; t 0 )
= q C 0 − h a (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
L
d
dt
q L (t; t 0 )
= ϕ L 0 − h b (ϕ C (t; t 0 ), q L (t; t 0 ), u(t; t 0 ), ϕ M 0 )
ϕ C (t 0 ; t 0 ) = 0
q L (t 0 ; t 0 ) = 0.
(5.36)
The structure of the SEs (5.36) suggests the following observations:
• if there are n C capacitors and n L inductors, the state variables of the SE
representation in the (ϕ, q)-domain are the n C + n L incremental variables
ϕ C (t; t 0 ) and q L (t; t 0 );
• the order of the SE representation in the (ϕ, q)-domain is n C + n L ;
• the evolution of the incremental variables ϕ C (t; t 0 ) and q L (t; t 0 ) is influenced via
q C 0 and ϕ L 0 by the initial conditions for the state variables in the (v, i)-domain
and, for n M memristors, the n M memristors initial conditions ϕ M 0 as well. The
initial conditions q C 0 , ϕ L 0 and ϕ M 0 indeed appear as constant inputs in the r.h.s.
of (5.36);
• the IVP for the SEs (5.36) has, by definition, zero initial conditions for the
incremental variables ϕ C (t; t 0 ) and q L (t; t 0 );
8 Functions h a (·) and h b (·) depend also on q M 0 if charge-controlled memristors are included in
LM.