Ideal Memristor
Memristor
Considered the missing electrical components, memristors map the relationship between flux \(\phi\) and charge \(q\).
Formally, a memristor is defined as:
Any two-terminal black box is called a memristor if, and only if, it exhibits a pinched hysteresis loop for all bipolar periodic input current signals (resp., input voltage signals) which result in a periodic voltage (resp., current) response of the same frequency, in the voltage–current (v–i) plane.
There are generally 3 kinds of memristors:
- Extended Memristor
- Generic Memristor
- Ideal Memristor
Summary of Formalism
| Current-Controlled | Voltage-Controlled |
|---|---|
| \(v = R(\mathbf{x},i)\cdot i\) | \(i = G(\mathbf{x},v)\cdot v\) |
| \(R(\mathbf{x},0) \neq \infty\) | \(G(\mathbf{x},0) \neq \infty\) |
| \(d\mathbf{x}/dt = \mathbf{f}(\mathbf{x},i)\) | \(d\mathbf{x}/dt = \mathbf{g}(\mathbf{x},v)\) |
Table 1. Extended Memristor
| Current-Controlled | Voltage-Controlled |
|---|---|
| \(v = R(\mathbf{x})\cdot i\) | \(i = G(\mathbf{x})\cdot v\) |
| \(d\mathbf{x}/dt = \mathbf{f}(\mathbf{x},i)\) | \(d\mathbf{x}/dt = \mathbf{g}(\mathbf{x},v)\) |
Table 2. Generic Memristor
| Current-Controlled | Voltage-Controlled |
|---|---|
| \(\phi = \hat{\phi}(q)\) | \(q = \hat{q}(\phi)\) |
| \(v = R(q)i\) | \(i = G(\phi)v\) |
| \(\frac{dq}{dt} = i\) | \(\frac{d\phi}{dt} = v\) |
| \(M(q) \triangleq \frac{d\hat{\phi}(q)}{dq}\) | \(G(\phi) \triangleq \frac{d\hat{q}(\phi)}{d\phi}\) |
| \(\hat{\phi}(q) = \phi_{0} + \int R(q) dq\) | \(\hat{q}(\phi) = q_{0} + \int G(\phi)d\phi\) |
Table 3. Ideal Memristor
Memory in Ideal Memristor
The word Memristor is a combination of Memory + Resistor. Naturally, one would ask the question of what the memory of the component means. By looking at the above formulations, we realize first that:
\begin{equation}
\begin{split}
v &= R(q)\cdot i \\
&=\frac{d\hat{\phi}(q)}{dq} \cdot i \\
& = \frac{d\hat{\phi}(q)}{dq} \frac{dq}{dt} \\
&= \frac{d\hat{\phi}(q)}{dt} \\
\end{split}
\end{equation}
Therefore at any given time \(t^{-}\) with corresponding values \(q(t^{-}),\phi(t^{-})\), if we shutdown the power supply, meaning \(v(t) = 0, i(t) = 0\), we would have the following for the Current-Controlled version:
\begin{equation}
\begin{split}
\frac{dq}{dt} &= i = 0 \\
\frac{d\phi}{dt} &= \frac{d\hat{\phi}(q)}{dt} = v = 0 \\
\end{split}
\end{equation}
Therefore, the values of \(q, \phi\) are “frozen in time” until the power supply is turned back on again. Consequently, the value of the memristance \(R(q)\), which characterizes this circuit element, is also frozen in time as the gradient of the \(\phi-q\) curve at this particular value pair \((\phi,q)\). Essentially, assuming no leakage of any sort, an ideal memristor can hold the memristance value indefinitely and this value can be read out by turning back on the power supply and observing it.
The similar reasoning can be easily applied to the Voltage-Controlled case as well.
Generic Memristor
Generic Memristor has the following characterisitics:
- Pinched Hysterisis loop degenerate to a straight line for sufficiently high frequency (asympototic linear v-i relationship)
- As \(f \rightarrow \infty\): The slope of the limiting straight lines is not constant, but changes with the amplitude \(A\) of the sinusoidal input voltage \(v(t) = A \sin(\omega t)\)
- As \(f \rightarrow (DC)\): Just like a nonlinear resistor
Extended Memristor
Extended Memristor has the following characterisitcs:
- As \(f \rightarrow \infty\): Pinched Hysterisis loop degenerate to single-value for sufficiently high frequency (asympototic linear v-i relationship)
- As \(f \rightarrow (DC)\): Just like a nonlinear resistor