7.3 Structure of Memristor Network and Differential Algebraic Equation. . .
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Fig. 7.4 Circuit for finding the CR of an element D
q
M ∈ N D
ϕ(t; t 0 ) + ϕ L (t; t 0 ) + ϕ M (t; t 0 ) = 0.
Moreover, KqL yields
q(t; t 0 ) = q L (t; t 0 ) = q M (t; t 0 ).
The CR of the inductor is (Chap. 5)
ϕ L (t; t 0 ) = L ˙
q L (t; t 0 ) − ϕ L (t 0 )
and that of the memristor is
ϕ M (t; t 0 ) = h(q M (t; t 0 ) + q M (t 0 )) − h(q M (t 0 )).
By substitution, we obtain that the considered element D
q
M ∈ N D has a CR in terms
of q(t; t 0 ) and ϕ(t; t 0 ) given by
−ϕ(t; t 0 ) = L ˙
q(t; t 0 ) − ϕ L (t 0 ) + h(q(t; t 0 ) + q M (t 0 )) − h(q M (t 0 )).
The CRs in the (ϕ, q)-domain of elements D ϕ and D q in N D can be expressed
in vector form as follows. 2 We leave to the reader the rather straightforward
verification of these formula using KqL, KϕL and the CRs of basic circuit elements
obtained in Chap. 5.
2 A negative resistor in parallel with a flux-controlled memristor or in series with a chargecontrolled memristor can be embedded in the memristor without modifying the results here
provided.