IEEE Circuits and Systems Magazine - Q2 2018 - 30

t4

t5
-1

Expt.
Expt. Avg.
Ideal Model

0
iFM (mA)
(a)

-0.8

0

t2

0.0 0.4
νFM (V)
(b)
t3

t4

0.8

0.0
φFM (mV.s)

-0.5
-1.0

1

0.5

t3

-0.5

0.0
φFM (mV.s)

0.5

(c)
t5

PFML
PD

t6

t1

1.5

t2

t3

t4

t5

2
Time (ms)
(f)

4

t6

EFML
ED

0.5
0.0

0
-0.5

t2

1.0
P (mW)

(diFM /dt)/vFM (A /V.s)

-0.4

t1

t5

0.0

t4

t1

5

t6

-70

1

Ideal Model
Expt. (Raw)
Expt. (Smoothed)

t1 t5
t4
t2

t3

0

dφFM /dt (V)

t6

t6

0.5

E (µJ)

t3

70
diFM /dt (A /s)

φFM (mV.s)

t1

0.0

-0.6

1.0

t2

0.6

0

(d)

2
Time (ms)
(e)

4

0

Figure 2. fml characterization. (a) plot of z FM against i FM from experimental data (red curves 'Expt.'), from the calculation of a
restoring function (green curve 'Expt. avg.'), and the comparison to an ideal non-linear (-1,0) element (gray curve 'Ideal model').
't1'-'t6' are temporal markers. (b) plot of di FM dt against v FM, exhibiting a pinched hysteresis loop. (c) plot of dz FM dt against
1
) against z FM for the experimentally derived data (raw and smoothed).
z FM, the phase portrait. (d) plot of ^di FM dt h v FM (or L -FM
Ideal model corresponds to Equation 4. (e)-(f) plots of powers and energies vs. time, for the restoring function, or reversible stored
flux ('fml') and the dissipative component ('d') that causes heating during the ferromagnetic switching.

route map) [12] of the system by plotting dz FM dt vs.
z FM (Fig. 2c), wherein the zero-crossings ^ dz FM dt h = 0
represent regions of slowest state speed for the driven
system near the time markers t2 and t5, while the values
of z FM at the maximum and the minimum of dz FM dt
are the regions of maximum state speed near t3 and t6,
where the trajectory of the system sweeps rapidly past
z FM = 0.
How can we construct a compact dynamical nonlinear model for the FML? [13], [14] One approach is to
consider the FML to be a black box, [15] and to use an
'unfolding' process [16] to determine functions characterizing the static and dynamical properties of the FML
as polynomials in one or more variables. This method
can yield a very accurate mathematical representation
of the electrical characteristics of the FML if the polynomials have a large number of terms, but does not provide much physical insight into the functioning of the
FML. We desired the simplest model that provides intuition at the possible expense of accuracy, which meant
that instead of following a well-defined procedure we
used a trial and error approach using our data to deter30

IEEE cIrcuIts and systEms magazInE

mine an approximate functional form. We therefore created a first approximation with the equations defining a
voltage-controlled (-1,0) element:
di = L -1 ^z h v,
FM
FM
dt
dz FM
=v
dt

(1)
(2)

where L -FM1 = ^di dz FM h is the inverse differential (or
incremental) inductance, which is not the same as the
inverse integral inductance L -FM1 = ^i z FM h for a nonlinear system. We used the inverse L -FM1 rather than L FM
because the latter diverges, is undefined for certain
values of z FM and is therefore an inappropriate mathematical description of the element. Equation 2 is Chua's
definition of flux and is also the dynamical equation for
the voltage-controlled ideal non-linear (-1,0) element.
Together the equations imply that a plot of di dt vs.
v should be a pinched hysteresis loop, shown by the
experimental data (Fig. 2b) and confirming that the device is indeed a non-linear (-1,0) element. [1], [17] The
name 'meminductor' is appropriate for a hypothetical
sEcOnd quartEr 2018



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