IEEE Circuits and Systems Magazine - Q2 2018 - 29

iFM
RS
vext

LFM

+

vL

-
RL

(a)
t1

t2

t3

t4

t5

t6

vFM (V)

0.7
0.0
-0.7

iFM (mA)

5
0

diFM /dt (A /s)

-5

70
0
-70

dvFM /dt (V/ms)

10
5
0
-5
-10
0.5

φFM (mV.s)

a mathematical formula that directly connects two characteristics ^a, b h; for a memory element, this relationship
will be represented by a nonlinear function. The usual
passive resistor, capacitor and inductor are identified as
(nearly) linear (0, 0), (0, -1) and (-1, 0) elements, respectively, with the memristor forming the non-linear (-1, -1)
element. [2] However, real electronic devices are usually
more complex than these ideal mathematical models,
so Chua has defined a set of mathematical relations of
increasing complexity to accommodate a broader set of
observable behaviors. [1] In addition, any real device may
contain characteristics of two or more model ^a, b h elements-the physical device is usually called by the name
of the dominant model that explains its behavior, but that
can change with different inputs (e.g. voltage amplitudes
or frequencies) or operating environments (e.g. temperature). The recent identification of many previously unclassified electronic devices as memristors has opened up
entire new areas of research, modeling and applications.
[3]-[6] Here we examine a ferromagnetic inductor (FML)
in order to see what new insights might be gained from a
nonlinear circuit analysis of this familiar element.
We constructed the FML using a nanocrystalline ferromagnetic toroidal core of composition Fe73.5 Si13.5B9Nb3Cu1,
described in detail elsewhere [7]-[10], and a wire winding
of known resistance. In our first measurements, we applied a sinusoidal voltage across the FML and a 30 Ω series resistor (R S), as shown in Fig. 1a, and measured the
voltage drop across R S to determine the current i FM (t )
and simultaneously the FML voltage v FM (t ) (corrected for
the voltage drop due to the winding) as shown in Fig. 1b.
We used a sinusoidal driving voltage in order to eliminate
transients in the ac signals, and we averaged over 49 cycles of steady-state operation in order to improve the signal to noise ratio in the data. In this analysis, we followed
Chua and defined the flux in the FML as z FM = 8v FM ^ t h dt,
[1] which was computed numerically from the experimental data and is displayed in Fig. 1b, along with several
other functions derived from the measured data that will
be used below.
A plot of the experimentally derived z FM against i FM
(Fig. 2a, labeled 'Expt.') shows the familiar FML hysteresis curve with relatively abrupt switching characteristics. The area contained within the hysteresis loop
corresponds to the irreversible or instantaneous work
used to switch the magnetization in the ferromagnetic toroid twice. We also plotted di FM dt against v FM (Fig. 2b),
which showed a pinched hysteresis loop, a feature familiar from i-v plots of memristors, [4], [11] that has not
been shown in previous analyses of FMLs. Further, we
produced a phase portrait (also known as a dynamical

0.0
-0.5
0

2
Time (ms)
(b)

4

Figure 1. ferromagnetic Inductor (fml) electrical measurements. (a) the circuit used. dashed rectangle refers to the
fml, while rl is its winding resistance. (b) plots of various
measured and calculated functions from the circuit shown
in (a). all plots are over one complete period of a 200 hz
sinusoidal v ext, averaged over 49 successive periods after
steady-state operation for better signal-to-noise ratio.

Suhas Kumar and R. Stanley Williams are with the Hewlett Packard Labs, 1501 Page Mill Rd, Palo Alto, CA 94304, USA.
sEcOnd quartEr 2018

IEEE cIrcuIts and systEms magazInE

29



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