6.2  Memristive Sensing Principles

The memristive sensing scheme explained below can be classified as a novel charge-discharge (time domain) technique in which the capacitance or the inductance is calculated by measuring the memristance (memductance) and thus eliminates the need for time (frequency) measurement. In fact, the memristance (in the unit of Ω  ) or memductance (in the unit of S      - 1
=  Ω   ) of a memristor can be simply determined by Ohm’s law, which makes the memristive sensing straightforward and fast. Furthermore, because a memristor holds the information even if its voltage/current is turned off (non-volatility), it allows for measurement circuits with low leakage power consumption and new functionalities.

6.2.1  Charge-Controlled Memristors Capacitance Sensing

Consider a charge-controlled memristor connected in series with a capacitor as shown in Fig. 6.1. According to Eq. 2.4 and Eq. 2.10 for the capacitance and memristance, respectively, we have

      dqC--   dqC--∕dt    ---iC---
C  =  dv   =  dv   ∕dt =  dv   ∕dt
         C       C           C


dM--(q)-   dM---(q)dqM--    dM--(q-)
        =                =          iM.
  dt         dqM    dt        dqM

As iM  =  iC   , by substituting iM   from Eq. 6.2 into Eq. 6.1 we obtain

     (  dM  (q))  - 1 dM ∕dt    (  dM  (q) ) - 1 dM
C  =    --------     -------- =    --------     ----.
         dqM         dvC ∕dt        dqM         dvC

The term dM  ∕dq
        M   is related to the intrinsic properties of the memristor. For a linear resistor this term is zero. However, it is nonzero for a memristive device or system as it is supposed to memorize the history of current flowing through the modulation in its memristance. A charge-controlled memristor with the term dM  ∕dqM   being constant is suited for charge-based capacitance sensing as it can reduce the measurement to a simple memristance measurement. In fact, on one hand, as dv   ∝ dq
  C       C   and dq   =  dq
   C       M   , we have dvC  ∝  dqM   . On the other hand, as for the memristor the term dM  ∕dqM   is constant, we have             dqM  ∝  dM  . Therefore, dv   ∝  dM
   C  and thus dM  ∕dv
        C   is equal exactly to ΔM   ∕Δv
          C   . As a result, Eq. 6.3 is written as

      (         )             (         )
        dM  (q )  - 1 ΔM         dM  (q)  - 1 M  (t = t0) -  M (t =  0)
C  =    --------     ------=     --------     --------------------------.
           dq        ΔvC           dq         vC (t = t0) -  vC(t =  0)

This means that one can calculate the capacitance by measuring the modulation of the memristance and the voltage across the capacitor (or the memristor). Therefore, unlike other time domain capacitance measurement methods, memristive sensing does not need extra hardware for time/frequency measurement since the time parameter has been implicitly taken into account in the memristance modulation. Furthermore, to measure the memristance modulation, a simple circuit can be employed to switch the memristor from the MC circuit (Fig. 6.1) to a basic readout circuit employing an arbitrary resistance read method [96], without loosing the information during the switching due to the non-volatility of the memristor. During the readout, the memristance and the charge in the capacitor can be reset to the initial state for which the M  and vC   are equal to M  (t = 0 )  and vC (t = 0 )  , respectively. Then, for the next measurement, one should measure only M  (t =  t0)  and probably vC(t =  t0)  if the capacitor is not fully charged (vC  <  vs   ).


Figure 6.1.: A diagram of the charge-based memristive capacitance sensing circuit. Power Monitoring

The unique ability of a memristor to record the historic profile of the voltage/current applied makes it suitable for power measurement [96]. The total energy generated by an electric power supply       ∫
E  =    Vs(t)Is (t)dt  , where Vs (t)  is the voltage across the source and Is(t )  is the current flowing through it. For a circuit powered by a DC voltage source (Fig. 6.2) [96], the energy is given by:

        ∫  t2                               ( dM  (q )) - 1
E =  Vs      Is(t)dt  = Vs ΔqM   =  VsΔM      --------
          t1                                    dqM

Therefore, a charge-controlled memristor with a constant term dM   (q)∕dq
            M   reduces the power measurement to a memristance measurement. To minimize the impact of the memristor on measurements, the memristance has to be much smaller than the circuit resistance.


Figure 6.2.: A diagram of the memristive power monitoring circuit with a charge-controlled memristor.

6.2.2  Flux-Controlled Memristors Inductance Sensing

According to Eq. 2.5 and Eq. 2.11 for the inductance and memductance of a flux-controlled memristor, respectively, we have

      dφ       dφ  ∕dt       v
L  =  ---L-=   ---L-----=  ---L----
      diL      diL ∕dt     diL∕dt


dW   (φ)     dW  (φ )d φ      dW   (φ)
--------- =  -----------M- =  ---------vM.
   dt         d φM     dt        dφ

When a flux-controlled memristor is connected in parallel to an inductor (v   = v
 M      L   in Fig. 6.3), the inductance is obtained as

      (         )                (          )
        dW  (φ )  - 1 dW  ∕dt      dW   (φ )  - 1 dW
L  =    ---------     --------=    ---------     -----.
          dφM         diL∕dt         dφM         diL

A flux-controlled memristor with a constant dW  ∕d φM   term is thus suited for flux-based inductance measurement. For a common conductor the term dW  ∕d φM   is zero. However, it is nonzero for a memristive device and is related to the intrinsic properties of the memristor. Based on a similar discussion presented above for the MC circuit, in the ML circuit with a memristor with dW  ∕d φM  =  const.  , we get             diL ∝  dW  . Therefore, Eq. 6.8 is simplified to

     (          ) - 1         (          ) - 1
L =     dW--(φ-)-    ΔW----=    dW---(φ)-     W--(t-=--t0-) --W--(t-=-0-).
         d φM        ΔiL          dφM          iL(t =  t0 ) - iL(t =  0)

Hence, the flux-controlled memristor in the ML circuit reduces the inductance measurement to a straightforward memductance measurement.


Figure 6.3.: Basic memristor-inductor (ML) circuit for flux-based sensing. Power Monitoring

Similar to the charge-based power monitoring, in a circuit powered by a DC current source (Fig. 6.4) [96], a flux-controlled memristor with the term dW  (φ )∕d φ  being constant reduces the power measurement to a memductance measurement.

        ∫                                 ( dW  (φ )) - 1
E =  Is    Vs(t)dt  = Is Δ φM  =  IsΔW      ---------

To minimize the impact of the memristor on measurements, the memductance must be much smaller than the circuit conductance. In the following section we study different spintronic memristors which are suited for both charge- and flux-based measurements.


Figure 6.4.: A diagram of the memristive power monitoring circuit using a flux-controlled memristor.