Memory Elements: A Paradigm Shift in
Lagrangian Modeling of Electrical Circuits
Dimitri Jeltsema
Delft Institute of Applied Mathematics
Delft University of Technology
The Netherlands
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Classical Circuit Theory
Classical (linear) circuit theory defines three basic elements:
I Resistor R
I Inductor L
I Capacitor C
each described by a relationship between two of the four basic variables
I Voltage V
I Current I
I Flux
φ
I Charge q
Rt
Rt
where φ(t ) = −∞ V (τ )dτ , and q (t ) = −∞ I (τ )dτ.
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Are We Missing Something?
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The Memristor
‘Missing’ relationship between φ and q defines the memristor.*
I Charge-controlled:
φ = φ̂(q ) or by using φ̇ = V and q̇ = I
V = RM (q )I
where RM (q ) :=
dφ̂
dq
(q ) is the incremental memristance.
I The memory-effect stems from the fact that
Z
t
q (t ) =
I (τ )dτ,
−∞
i.e., a memristor bookkeeps the current flowing through it!
*L.O. Chua. Memristor—the missing circuit element. IEEE Trans. Circ.
Theory, CT–18(2):507–519, September 1971.
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Mechanical Memristor
Perhaps more transparent in the mechanical domain. By analogy:
=
Rt
I Displacement: x (t )
=
I Momentum: p (t )
−∞ F (τ )dτ
Rt
−∞ v (τ )dτ
Example: tapered dashpot. Displacement dependent ⇒ p = p̂(x )
F
F
x
I Diff. wrt time yields F
= RM (x )v with RM (x ) :=
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dp̂
dx
(x ).
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Mechanical Memristor
Blueprint for memristors are pinched hysteresis loops:
I Sinusoidal excitation: v (t )
= sin(t )
I Two-valued everywhere (except at origin)
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Mechanical Memristor
Underlying single-valued characteristic: p(t ) = 13 x 3 (t )
h
i2
Rt
⇒ F (t ) = x (0) + 0 v (τ )dτ v (t )
|
{z
}
RM (x (t ))
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Hodgkin-Huxley Model
In 1952, Nobel laureates Hodgkin and Huxley suggested a circuit model (left) that
represents represent the biophysical characteristic of cell membranes. However,
they (mis-)interpreted the conductances as time-varying.
It is shown∗ that GNa and GK actually correspond to memristors (right).
*L.O. Chua and S.M. Kang. Memristive devices and systems,
Proc. IEEE, 64:209-223, 1976.
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Improved Josephson Junction Circuit Model
More rigorous analysis has revealed
presence of additional small current
I = g cos(ko φM )V , for some constants
g , ko ⇒ flux-controlled memristor:
qM = q̂M (φM ) =
g
ko
sin(ko φM ),
with q̇M = I and φ̇M = V .
∗
T.P. Orlando and K.A. Delin, Foundation of Superconductivity.
Reading, MA: Addison-Wesley, 1991.
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HP’s (Hewlett-Packard) Memristor
Meanwhile, researchers at HP are trying to create alternative for flash memory
using a two-layer semiconductor constructed from layers of titanium oxide.
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HP’s (Hewlett-Packard) Memristor
*D.B. Strukov, G.S. Snider, D.R. Stewart, and R.S. Williams. The missing
memristor found. Nature, 453:80–83, May 2008.
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Other Interesting Applications
I Non-volatile nano memory
I Crossbar latches as transistor replacements
I Analog computing
I Programmable logic and signal processing
I Learning circuits
I Models for biological phenomena
I Quantum computing
I ???
Check June 2012 special issue on Memristor Technology in Proc IEEE.
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But the story does not end... It actually just started...
Rt
Let us also consider: σ(t ) = −∞ q (τ )dτ,
I Define
ρ(t ) =
Rt
−∞ φ(τ )dτ
ρ = ρ̂(q ) . Then, differentiation wrt time yields
φ = LM (q )I ⇒ Meminductor
where LM (q ) :=
I Define
dρ̂
dq
(q ) is the incremental meminductance.
σ = σ̂(φ) . Then, differentiation wrt time yields
q = CM (φ)V
where CM (φ) :=
dσ̂
dφ
⇒ Memcapacitor
(φ) is the incremental memcapacitance.
*M. Di Ventra et al. (2009). Circuit elements with memory: memristors,
memcapacitors and meminductors. Proc. of the IEEE, 97(10), 1717-1724.
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Family of Memory Elements (Mem-elements)
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Btw: memcapacitor? Or is it FLUX Capacitor?
Remember this guy...
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Classical Circuit-Theory at its Limits
I Straightforward application of conventional Lagrangian modeling to circuits
with memory elements may yield the wrong equations.
I We provide a novel variational method to overcome these problems using a
different set of configuration coordinates
I The new Lagrangian equations do not correspond to KVL (Kirchhoff voltage
law) or KCL (Kirchhoff current law) anymore, but represent conservation of
flux and charge, i.e., iKVL and iKCL.
I Furthermore, we show that our configuration coordinates are electrical
analogies of a mechanical quantity called “absement”.
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Classical Lagrangian Approach
I If the inductor is current-controlled, we need to
consider loop analysis and define
q̇
Z
φ̂(I )dI −
L(q , q̇ ) =
0
1
2C
q2,
which upon substitution into
d ∂L
dt ∂ q̇
−
∂L
q
= 0, yields φ̂0 (q̇ )q̈ + = 0.
∂q
C
I Or a node analysis in case of a flux-controlled inductor, i.e.,
1
∗
2
Z
L (φ, φ̇) = C φ̇ −
2
φ
Î (φ0 )dφ0 ,
0
which together with
d ∂L∗
dt ∂ φ̇
−
∂L∗
= 0, yields C φ̈ + Î (φ) = 0.
∂φ
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Classical Lagrangian Approach
Now, replace conventional inductor by a meminductor...
The Lagrangian becomes (q̇ = I)
1
1
2
2C
L(q , q̇ ) = LM (q )q̇ 2 −
q2,
generating the equation of motion
LM (q )q̈ +
q
1 0
LM (q )q̇ 2 +
= 0.
2
C
However, if we just apply KVL, we obtain (recall φ = LM (q )I)
φ̇ +
q
C
= LM (q )İ + L0M (q )I 2 +
q
C
= 0.
The Lagrangian framework generates an erroneous factor 12 !
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Two Problems
(1) Path-dependence: magnetic (co-)energy
1
L
2 M
(q )q̇ 2 depends on q.
(2) Self-adjointness: Necessary condition to find L such that
d ∂L
dt ∂
ẋ i
−
∂L
≡ Aij (x , ẋ )ẍ j + Bi (x , ẋ ) = 0,
∂x i
is given by the integrability conditions (see Santilli (1978))
Aij = Aji ,
∂ Aik
∂ Ajk
=
,
j
∂ ẋ
∂ ẋ i
∂ Bj
1 ∂
∂ Bi
∂ Bj k
∂ Bi
− i =
− i ẋ ,
∂x j
∂x
2 ∂ x k ∂ ẋ j
∂ ẋ
∂ Bi
∂ Bj
∂ Aij
+ i = 2 k ẋ k .
∂ ẋ j
∂ ẋ
∂x
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Not Self-Adjoint...
I It is directly verified that, with A(q )
= LM (q ) and B (q , q̇ ) = L0M (q )q̇ 2 + q /C
⇒ not self-adjoint ⇒ no Lagrangian formulation as such.
I Possible solution: introduce Rayleigh dissipation function
1
D(q , q̇ ) = L0M (q )q̇ 3 ,
6
so that
LM (q )q̈ +
|
1 0
q
∂D
LM (q )q̇ 2 +
=−
.
2{z
C}
∂ q̇
d ∂L
− ∂L
dt ∂ q̇
∂q
However, variational principle is lost :-(
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Conservation of Flux and Charge
The conventional Lagrangian formalism essentially codes the KVL and KCL in
terms of energy storage via the Lagrangian.
Lets consider iKVL and iKCL
X
φi (t ) =
j
Vi (τ )dτ,
−∞
i
X
t
Z
φi (t ) = 0,
Z
qj (t ) = 0,
t
qj (t ) =
Ij (τ )dτ,
−∞
respectively (flux and charge can neither be created nor be destroyed).
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Memory State Functions
Recall that fundamental relationship of a meminductor reads ρ = ρ̂(q )
For that reason, let us instead of the magnetic (co-)energy define
a state function of the form
∗
Z
T̄ (q ) :=
ρ̂(q )dq
Furthermore, since a linear memcapacitor reduces to a conventional linear
capacitor, i.e., σ = C φ, we may associate it with a function of the form
Ū (σ) :=
1
2C
σ2
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An Alternative Variational Principle
Then select, instead of a loop charge q, the integrated loop charge σ , with σ̇ = q,
as the configuration variable, and define the Lagrangian
σ̇
Z
L̄(σ, σ̇) =
ρ̂(q )dq −
0
1
2C
σ2.
Indeed, invoking Hamilton’s principle by considering variations in terms of σ , we
get the Lagrangian type of equation
d ∂ L̄
dt ∂ σ̇
−
∂ L̄
= 0,
∂σ
which, in turn, generates ρ̂0 (σ̇)σ̈ + σ/C = 0 (where ρ̂0 (σ̇) =: LM (q )).
Differentiating the latter with respect to time yields
...
ρ̂0 (σ̇) σ + ρ̂00 (σ̇)σ̈ 2 +σ̇/C = 0
|
{z
}
≡ φ̇ +
q
C
.
φ̇
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Integrated Charge = Electrical “Absement”
Classical Analogy: Charge Displacement
In mechanics, displacement x and its various derivatives define an ordered
hierarchy of meaningful concepts:
I 1st derivative of displacement is velocity
I 2nd derivative is acceleration
I 3rd derivative is jerk
I 4th derivative is jounce
I etc.
Recently, also -1st, -2nd, -3rd, etc. derivatives introduced to model flow-based
musical instruments*.
*S. Mann et al. (2006). Hydraulophone design considerations: absement, displacement, and
velocity-sensitive music keyboard in which each key is a water jet. In Proc. 14th annual
ACM int. conf. on Multimedia, Santa Barbara, 519-528.
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Integrated Charge = Electrical “Absement”
I Absement = contraction of ‘absence’ and ‘displacement’
I In SI units measured in meter times seconds [ms]
I One meter-second corresponds to being absent one meter from an origin or
other reference point for a duration of one second
I integrated charge
absement
I absement of a capacitor is a measure for the amount of charge that is needed
in a particular time interval to charge the capacitor to a certain level
I For a memcapacitor absement is as a measure of how much charge and time
is needed to write its memory.
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Concluding Remarks
I Memory elements become increasingly important.
I Memory elements mimic behavior of synapse.
I Create biologically-inspired electronic circuits.
I Various interesting applications.
I HP announced that memristive random access memory may reach the
market as early as 2013.
I Teaching memory elements is a sine qua non for the training of the next
generation of scientists and engineers.
I Classical circuit- and systems theory needs to be modified and/or extended.
I Absement a natural and important variable in this context.
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THANKS! And never forget that...
“The only things you are going to regret in this life are
the risks you didn’t take.”
Grumpy Old Men (1993)
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