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7 Pulse Programming of Memristor Circuits
• λ M two-terminal elements D
q
M ∈ N D made of an inductor in series with a chargecontrolled memristor
• λ R two-terminal elements D
q
R ∈ N D made of an inductor in series with a negative
resistor
• λ L = n L − λ M − λ R two-terminal elements D
q
L ∈ N D made of just an inductor
Note that:
• D q = D
q
M
D
q
R
D
q
L and n L = λ M + λ R + λ L
• D ϕ = D
ϕ
M
D
ϕ
G
D
ϕ
C and n C = γ M + γ G + γ C .
Subnetwork N A denotes the nonlinear adynamic (n C + n L )-port network with
no capacitors and inductors. The conceptual decomposition of N into N A and N D
is shown in Fig. 7.2. Note that N A
N D results to be N.
Concerning the structure of N A , let us assume that it contains:
• μ F flux-controlled memristors A
ϕ
M ∈ N A
• μ Q charge-controlled memristors A
q
M ∈ N A
• ρ G negative conductances A
ϕ
G ∈ N A
• ρ R negative resistors A
q
R ∈ N A .
The structure of N A is essential to develop a general methodology to study DAEs
and SEs. In this regard, it is convenient to further decompose N A , i.e., to extract
from N A the linear negative resistors and the memristors. Such negative resistors
and memristors are not in parallel to a capacitor or in series with an inductor.
Note that
n M = μ F + μ Q + γ M + λ M
is the total number of memristors in N. The extraction of A
ϕ
M , A
ϕ
G , A
q
M , and A
q
R
from N A yields a linear network N R with (n C + n L + μ F + μ Q + ρ G + ρ R )-ports
having only positive (linear) resistors and independent sources. We suppose that
there are:
• n E ideal independent voltage sources e(t) such that ϕ e (t; t 0 ) =
t
t 0
e(τ )dτ
• n A ideal independent current sources a(t) such that q a (t; t 0 ) =
t
t 0
a(τ )dτ .
7.3.1 Constitutive Relations of Two-Terminal Elements
Here, we provide a systematic description of the different types of two-terminal
elements in N D , N A , and N R . For each two-terminal element connected to N R we
use coordinated reference directions for incremental flux and charge as shown in
Fig. 7.2.
We start with simple examples illustrating how we can find the CR of an element
D
ϕ
M ∈ N D and D
q
M ∈ N D .
7 Pulse Programming of Memristor Circuits
• λ M two-terminal elements D
q
M ∈ N D made of an inductor in series with a chargecontrolled memristor
• λ R two-terminal elements D
q
R ∈ N D made of an inductor in series with a negative
resistor
• λ L = n L − λ M − λ R two-terminal elements D
q
L ∈ N D made of just an inductor
Note that:
• D q = D
q
M
D
q
R
D
q
L and n L = λ M + λ R + λ L
• D ϕ = D
ϕ
M
D
ϕ
G
D
ϕ
C and n C = γ M + γ G + γ C .
Subnetwork N A denotes the nonlinear adynamic (n C + n L )-port network with
no capacitors and inductors. The conceptual decomposition of N into N A and N D
is shown in Fig. 7.2. Note that N A
N D results to be N.
Concerning the structure of N A , let us assume that it contains:
• μ F flux-controlled memristors A
ϕ
M ∈ N A
• μ Q charge-controlled memristors A
q
M ∈ N A
• ρ G negative conductances A
ϕ
G ∈ N A
• ρ R negative resistors A
q
R ∈ N A .
The structure of N A is essential to develop a general methodology to study DAEs
and SEs. In this regard, it is convenient to further decompose N A , i.e., to extract
from N A the linear negative resistors and the memristors. Such negative resistors
and memristors are not in parallel to a capacitor or in series with an inductor.
Note that
n M = μ F + μ Q + γ M + λ M
is the total number of memristors in N. The extraction of A
ϕ
M , A
ϕ
G , A
q
M , and A
q
R
from N A yields a linear network N R with (n C + n L + μ F + μ Q + ρ G + ρ R )-ports
having only positive (linear) resistors and independent sources. We suppose that
there are:
• n E ideal independent voltage sources e(t) such that ϕ e (t; t 0 ) =
t
t 0
e(τ )dτ
• n A ideal independent current sources a(t) such that q a (t; t 0 ) =
t
t 0
a(τ )dτ .
7.3.1 Constitutive Relations of Two-Terminal Elements
Here, we provide a systematic description of the different types of two-terminal
elements in N D , N A , and N R . For each two-terminal element connected to N R we
use coordinated reference directions for incremental flux and charge as shown in
Fig. 7.2.
We start with simple examples illustrating how we can find the CR of an element
D
ϕ
M ∈ N D and D
q
M ∈ N D .