Dr. Chua postulated the existence of memristance in 1971 using the idea that the basic circuit elements are relationships between voltage, current, flux, and charge. There was no circuit element relating charge and flux, so Chua proposed memristance to fill this hole.
Image: J. J. Yang/HP Labs via IEEE SpectrumTrying to understand this, I looked at the math for the circuit elements we know well and then for memristors.
Resistor:
V=IR (ohm’s law).
Capacitor:
dq = C dv.
I is charge per unit time, so I = dq/dt.
Substituting for I dt for dq, I dt/dv = C.
Rearranging, dv/dt = I/C .
When I use a capacitor, I typically think of it this way. The rate of voltage change across the cap is I / C.
=
Inductor:
dφ = L di.
Because dφ = v dt, v dt = L di.
So di/dt = V/L.
When I use an inductor, I think of it as analogous to a capacitor. The rate of current change through the inductor is V/L.
Memristor:
dφ = M dq.
I should still be able to use dφ = v dt, so v dt = M dq.
dq/dt = I = V/M.
That is ohms law, except the value of M is a function of the amount of charge that has flowed through the memristor. Some websites refer to M as a function of q, M(q).
(I do not have access to Chua’s original paper on memristance or the paper last year on HP’s work, so all I have to go on articles referencing the papers.)
The other three circuit elements are constants, not functions. This makes M fundamentally different from the other circuit elements. I would love for someone to explain in a comment why I am wrong and why memristance is mathematically analogous to the other three circuit elements.
Leon Chua recently published an article describing memcapacitors and meminductors as additional circuit elements which applies memristive effects to capacitors and inductors. To me this indicates the label of a fourth fundamental circuit element may not be entirely accurate and memristive systems are more likely to be simply a generalization of the known equations of resistors, capacitors, and inductors to dynamic systems having memory effects.
ReplyDeleteFor example, the 2008 Nature article "The missing memristor found" uses the following form of the memristor equation:
v = R(w)i
dw/dt = f(i)
where v is the voltage, i is the current, w is a state variable, dw/dt is the rate of change of the state variable with respect to time, and
f() is a function of current (which is defined in terms of film thickness and mobility of oxygen vacancies in the Nature article). These equations reduce to Ohm's law for dw/dt = 0 so in this sense memristors are only generalizations of resistors.