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7 Pulse Programming of Memristor Circuits
• D
ϕ
M ∈ N D . Incremental fluxes and charges of D
ϕ
M define the vectors ϕ γ M (t; t 0 )
and q γ M (t; t 0 ), respectively. By FCAM, the CRs of D
ϕ
M are easily expressed in
the following vector form: 3
− q γ M (t; t 0 ) = C γ M ˙
ϕ γ M (t; t 0 ) − q C γ M (t 0 )
+f(ϕ γ M (t; t 0 ) + ϕ M γ M
(t 0 ))
−f(ϕ M γ M
(t 0 ))
(7.2)
where C γ M is a diagonal matrix having the capacitance of capacitors in D
ϕ
M and
q C γ M (t 0 ) are their ICs, whereas ϕ M γ M
(t 0 ) are the ICs of memristors in D
ϕ
M . The
memristor fluxes in D
ϕ
M are ϕ M γ M
(t) = ϕ γ M (t; t 0 ) + ϕ M γ M
(t 0 ) and the capacitor
voltages are v C γ M (t) = ˙
ϕ γ M (t; t 0 ).
• D
ϕ
G ∈ γ G . Fluxes and charges in such D
ϕ
G define the vectors ϕ γ G (t; t 0 ) and
q γ G (t; t 0 ), respectively. The CRs of the elements D
ϕ
G are
− q γ G (t; t 0 ) = C γ G ˙
ϕ γ G (t; t 0 ) − q C γ G (t 0 ) + G γ G ϕ γ G (t; t 0 )
(7.3)
where C γ G is a diagonal matrix having the capacitance of capacitors in D
ϕ
G and
q C γ M (t 0 ) are their ICs.
• D
ϕ
C ∈ N D . Fluxes and charges in such capacitors define the vectors ϕ γ C (t; t 0 )
and q γ C (t; t 0 ), respectively. The CRs of the elements D
ϕ
C are
− q γ C (t; t 0 ) = C γ C ˙
ϕ γ C (t; t 0 ) − q C γ C (t 0 )
(7.4)
where C γ C is a diagonal matrix having the capacitance of capacitors in D
ϕ
C and
q C γ C (t 0 ) are their ICs.
We can use dual notations for D
q
M ∈ N D as follows.
• D
q
M ∈ N D . The CRs are
− ϕ λ M (t; t 0 ) = L λ M ˙
q λ M (t; t 0 ) − ϕ L λ M
(t 0 )
+h(q λ M (t; t 0 ) + q M λ M (t 0 ))
−h(q M λ M (t 0 ))
(7.5)
where ϕ L λ M
(t 0 ) (resp., q M λ M (t 0 )) are the ICs for inductors (resp., memristors) in
D
q
M . The memristor charges in D
q
M are q M λ M (t) = q λ M (t; t 0 ) + q M λ M (t 0 ) and
the inductor currents are i L λ M (t) = ˙
q λ M (t; t 0 ).
3 The active sign convention is used for any element D ϕ and D q in N D .
7 Pulse Programming of Memristor Circuits
• D
ϕ
M ∈ N D . Incremental fluxes and charges of D
ϕ
M define the vectors ϕ γ M (t; t 0 )
and q γ M (t; t 0 ), respectively. By FCAM, the CRs of D
ϕ
M are easily expressed in
the following vector form: 3
− q γ M (t; t 0 ) = C γ M ˙
ϕ γ M (t; t 0 ) − q C γ M (t 0 )
+f(ϕ γ M (t; t 0 ) + ϕ M γ M
(t 0 ))
−f(ϕ M γ M
(t 0 ))
(7.2)
where C γ M is a diagonal matrix having the capacitance of capacitors in D
ϕ
M and
q C γ M (t 0 ) are their ICs, whereas ϕ M γ M
(t 0 ) are the ICs of memristors in D
ϕ
M . The
memristor fluxes in D
ϕ
M are ϕ M γ M
(t) = ϕ γ M (t; t 0 ) + ϕ M γ M
(t 0 ) and the capacitor
voltages are v C γ M (t) = ˙
ϕ γ M (t; t 0 ).
• D
ϕ
G ∈ γ G . Fluxes and charges in such D
ϕ
G define the vectors ϕ γ G (t; t 0 ) and
q γ G (t; t 0 ), respectively. The CRs of the elements D
ϕ
G are
− q γ G (t; t 0 ) = C γ G ˙
ϕ γ G (t; t 0 ) − q C γ G (t 0 ) + G γ G ϕ γ G (t; t 0 )
(7.3)
where C γ G is a diagonal matrix having the capacitance of capacitors in D
ϕ
G and
q C γ M (t 0 ) are their ICs.
• D
ϕ
C ∈ N D . Fluxes and charges in such capacitors define the vectors ϕ γ C (t; t 0 )
and q γ C (t; t 0 ), respectively. The CRs of the elements D
ϕ
C are
− q γ C (t; t 0 ) = C γ C ˙
ϕ γ C (t; t 0 ) − q C γ C (t 0 )
(7.4)
where C γ C is a diagonal matrix having the capacitance of capacitors in D
ϕ
C and
q C γ C (t 0 ) are their ICs.
We can use dual notations for D
q
M ∈ N D as follows.
• D
q
M ∈ N D . The CRs are
− ϕ λ M (t; t 0 ) = L λ M ˙
q λ M (t; t 0 ) − ϕ L λ M
(t 0 )
+h(q λ M (t; t 0 ) + q M λ M (t 0 ))
−h(q M λ M (t 0 ))
(7.5)
where ϕ L λ M
(t 0 ) (resp., q M λ M (t 0 )) are the ICs for inductors (resp., memristors) in
D
q
M . The memristor charges in D
q
M are q M λ M (t) = q λ M (t; t 0 ) + q M λ M (t 0 ) and
the inductor currents are i L λ M (t) = ˙
q λ M (t; t 0 ).
3 The active sign convention is used for any element D ϕ and D q in N D .