IEEE Circuits and Systems Magazine - Q2 2018 - 31

non-linear (-2,-1) element, [1], [17] which has been incorrectly identified as an FML. [18] The ferroelectric
capacitor, which can be classified as a non-linear (0,-1)
element, is symmetric with respect to the FML in the
periodic table of circuit elements [1].
The next step in creating the model is to determine a
functional form for L -FM1 ^z FM h . One approach is to utilize a
polynomial formula L -FM1 ^z FM h = a 0 + a 2 z 2FM + a 4 z 4FM + f
A problem with this approach is the difficulty in modeling the rapid saturation of z FM with respect to i in
Fig. 2a. The initial model we have chosen for the inverse
differential inductance of the ideal non-linear (-1,0) element is the following:
L -FM1 ^z FM h = c e

zs
2

z 2s - z FM

(3)

+ p o,

where z s = 0.63 mVs is the saturation flux, and p = - 3
and c = 10 -3 are constants. Alternately, we can write the
expression for the inverse integral inductance:

L

-1
FM

> atanh
^ z FM h = c

c

z FM
m
zs

z FM

H

+p

(4)

A plot of z vs. i obtained by parametrically sweeping z in Equation 4 and using i = z FM L -FM1 produces a
single-valued function of z (Fig. 2a, gray curve labeled
'Ideal Model') that clearly shows two regions of negative differential inductance (NDL) underlying the experimentally observed hysteretic behavior. We note that
the presence of a negative differential behavior is often
a signature of local activity, or the ability of a system
to store and release energy during its operation. [12]
Hence this ideal model represents a locally active nonlinear (-1,0) element. A plot of L -FM1 ^z FM h = ^1 v h^di dt h
vs. z FM for both the experimentally measured data and
the ideal model (Fig. 2d) shows first that there are two
branches in the experimental data instead of the single
curve of the ideal model, and second that the NDL predicted by the model ^ L -FM1 ^z FM h 1 0 h is not accessed by
the experimental measurements.
The fact that the plot of L -FM1 ^z FM h = ^1 v h^di dt h vs.
z FM from experimental data did not yield a single curve
but rather two branches shows that the system is not
ideal. There is likely a second state variable or parameter that causes L -FM1 to vary, such as the temperature of
the device. Newton's law of cooling, [19], [20]
dT = PD - T - Tamb ,
dt
C th
C th R th

(5)

where T is the temperature of the inductor, Tamb is the
ambient temperature of the surroundings, C th is the
heat capacity, R th is the effective thermal resistance,
and PD is the dissipated power, has been utilized sucsEcOnd quartEr 2018

cessfully to model temperature variations in memristors
during operation. Equation 5 defines a second state variable for the system, or if the thermal time constant is
much smaller than the period of the driving voltage, the
steady state approximation ^dT dt h = 0 can be used to
define a power-dependent parameter T. The problem
in evaluating Equation 5 for a FML is that PD ! i FM v FM ;
there are both reversible power in the inductor PFML and
power dissipated PD in switching the magnetization of
the ferromagnetic torus (and to a lesser extent inducing
eddy currents). Fortunately, Chua has provided an ingenious technique to determine restoring and dissipative
functions for systems with hysteresis. [21]-[23] A nonhysteretic z vs. i plot that represents a nonlinear inductor that only stores and releases energy is determined
by simply averaging the two branches of the hysteresis
plot, as shown by the green curve labeled 'Expt. Avg.' in
Fig. 2a. Using this result, we plot PFML (blue curve) and
PD = i FM v FM - PFML (red curve) in Fig. 2e. We see that PFML
has both positive and negative values, corresponding
to storing and releasing energy to the circuit, respectively, whereas PD is always $0. In the FML, energy
is stored during a narrow and sharp positive peak in
PFML and subsequently released back to the circuit as
a negative PFML gradually over time. The corresponding
energies obtained by integrating the powers over time
(Fig. 2f) show that the magnetization switching energy
dissipated is much larger than the energy stored in the
inductor in each cycle. This results in both long-term
heating that leads to a significant temperature increase
of the inductor before a steady state is achieved with the
surroundings and also a cycle-dependent temperature
oscillation, which is likely the cause of the experimental
differential inductance curve in Fig. 2d breaking into
two branches.
The hysteresis in the experimental FML z vs. i plot
meant that there was an instability that did not allow
the measurement to directly access the NDL behavior,
which is similar to the hysteresis observed for a voltage
sweep of a current-controlled negative differential resistance (CC-NDR). In order to stabilize this behavior and
directly observe CC-NDR, a series resistor is added to
the circuit, wherein the voltage decrease in the CC-NDR
device appears as a voltage increase across the series
resistor. [19], [24] Similarly, in the case of NDL, a parallel linear inductor (LL) in the circuit can allow the NDL
current decrease to appear as increased current in the
LL. We therefore measured the circuit after installing a
27 mH LL in parallel to the FML (Fig. 3a) and repeated
the set of measurements and calculations described
above (Fig. 3b), in addition to measuring the current
through the LL. In this case, the plot of z FM against i FM
(Fig. 4a) showed that the hysteresis loop had shrunk
IEEE cIrcuIts and systEms magazInE

31



Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2018

Contents
IEEE Circuits and Systems Magazine - Q2 2018 - Cover1
IEEE Circuits and Systems Magazine - Q2 2018 - Cover2
IEEE Circuits and Systems Magazine - Q2 2018 - Contents
IEEE Circuits and Systems Magazine - Q2 2018 - 2
IEEE Circuits and Systems Magazine - Q2 2018 - 3
IEEE Circuits and Systems Magazine - Q2 2018 - 4
IEEE Circuits and Systems Magazine - Q2 2018 - 5
IEEE Circuits and Systems Magazine - Q2 2018 - 6
IEEE Circuits and Systems Magazine - Q2 2018 - 7
IEEE Circuits and Systems Magazine - Q2 2018 - 8
IEEE Circuits and Systems Magazine - Q2 2018 - 9
IEEE Circuits and Systems Magazine - Q2 2018 - 10
IEEE Circuits and Systems Magazine - Q2 2018 - 11
IEEE Circuits and Systems Magazine - Q2 2018 - 12
IEEE Circuits and Systems Magazine - Q2 2018 - 13
IEEE Circuits and Systems Magazine - Q2 2018 - 14
IEEE Circuits and Systems Magazine - Q2 2018 - 15
IEEE Circuits and Systems Magazine - Q2 2018 - 16
IEEE Circuits and Systems Magazine - Q2 2018 - 17
IEEE Circuits and Systems Magazine - Q2 2018 - 18
IEEE Circuits and Systems Magazine - Q2 2018 - 19
IEEE Circuits and Systems Magazine - Q2 2018 - 20
IEEE Circuits and Systems Magazine - Q2 2018 - 21
IEEE Circuits and Systems Magazine - Q2 2018 - 22
IEEE Circuits and Systems Magazine - Q2 2018 - 23
IEEE Circuits and Systems Magazine - Q2 2018 - 24
IEEE Circuits and Systems Magazine - Q2 2018 - 25
IEEE Circuits and Systems Magazine - Q2 2018 - 26
IEEE Circuits and Systems Magazine - Q2 2018 - 27
IEEE Circuits and Systems Magazine - Q2 2018 - 28
IEEE Circuits and Systems Magazine - Q2 2018 - 29
IEEE Circuits and Systems Magazine - Q2 2018 - 30
IEEE Circuits and Systems Magazine - Q2 2018 - 31
IEEE Circuits and Systems Magazine - Q2 2018 - 32
IEEE Circuits and Systems Magazine - Q2 2018 - 33
IEEE Circuits and Systems Magazine - Q2 2018 - 34
IEEE Circuits and Systems Magazine - Q2 2018 - 35
IEEE Circuits and Systems Magazine - Q2 2018 - 36
IEEE Circuits and Systems Magazine - Q2 2018 - 37
IEEE Circuits and Systems Magazine - Q2 2018 - 38
IEEE Circuits and Systems Magazine - Q2 2018 - 39
IEEE Circuits and Systems Magazine - Q2 2018 - 40
IEEE Circuits and Systems Magazine - Q2 2018 - 41
IEEE Circuits and Systems Magazine - Q2 2018 - 42
IEEE Circuits and Systems Magazine - Q2 2018 - 43
IEEE Circuits and Systems Magazine - Q2 2018 - 44
IEEE Circuits and Systems Magazine - Q2 2018 - 45
IEEE Circuits and Systems Magazine - Q2 2018 - 46
IEEE Circuits and Systems Magazine - Q2 2018 - 47
IEEE Circuits and Systems Magazine - Q2 2018 - 48
IEEE Circuits and Systems Magazine - Q2 2018 - 49
IEEE Circuits and Systems Magazine - Q2 2018 - 50
IEEE Circuits and Systems Magazine - Q2 2018 - 51
IEEE Circuits and Systems Magazine - Q2 2018 - 52
IEEE Circuits and Systems Magazine - Q2 2018 - 53
IEEE Circuits and Systems Magazine - Q2 2018 - 54
IEEE Circuits and Systems Magazine - Q2 2018 - 55
IEEE Circuits and Systems Magazine - Q2 2018 - 56
IEEE Circuits and Systems Magazine - Q2 2018 - 57
IEEE Circuits and Systems Magazine - Q2 2018 - 58
IEEE Circuits and Systems Magazine - Q2 2018 - 59
IEEE Circuits and Systems Magazine - Q2 2018 - 60
IEEE Circuits and Systems Magazine - Q2 2018 - 61
IEEE Circuits and Systems Magazine - Q2 2018 - 62
IEEE Circuits and Systems Magazine - Q2 2018 - 63
IEEE Circuits and Systems Magazine - Q2 2018 - 64
IEEE Circuits and Systems Magazine - Q2 2018 - 65
IEEE Circuits and Systems Magazine - Q2 2018 - 66
IEEE Circuits and Systems Magazine - Q2 2018 - 67
IEEE Circuits and Systems Magazine - Q2 2018 - 68
IEEE Circuits and Systems Magazine - Q2 2018 - 69
IEEE Circuits and Systems Magazine - Q2 2018 - 70
IEEE Circuits and Systems Magazine - Q2 2018 - 71
IEEE Circuits and Systems Magazine - Q2 2018 - 72
IEEE Circuits and Systems Magazine - Q2 2018 - 73
IEEE Circuits and Systems Magazine - Q2 2018 - 74
IEEE Circuits and Systems Magazine - Q2 2018 - 75
IEEE Circuits and Systems Magazine - Q2 2018 - 76
IEEE Circuits and Systems Magazine - Q2 2018 - 77
IEEE Circuits and Systems Magazine - Q2 2018 - 78
IEEE Circuits and Systems Magazine - Q2 2018 - 79
IEEE Circuits and Systems Magazine - Q2 2018 - 80
IEEE Circuits and Systems Magazine - Q2 2018 - 81
IEEE Circuits and Systems Magazine - Q2 2018 - 82
IEEE Circuits and Systems Magazine - Q2 2018 - 83
IEEE Circuits and Systems Magazine - Q2 2018 - 84
IEEE Circuits and Systems Magazine - Q2 2018 - 85
IEEE Circuits and Systems Magazine - Q2 2018 - 86
IEEE Circuits and Systems Magazine - Q2 2018 - 87
IEEE Circuits and Systems Magazine - Q2 2018 - 88
IEEE Circuits and Systems Magazine - Q2 2018 - 89
IEEE Circuits and Systems Magazine - Q2 2018 - 90
IEEE Circuits and Systems Magazine - Q2 2018 - 91
IEEE Circuits and Systems Magazine - Q2 2018 - 92
IEEE Circuits and Systems Magazine - Q2 2018 - 93
IEEE Circuits and Systems Magazine - Q2 2018 - 94
IEEE Circuits and Systems Magazine - Q2 2018 - 95
IEEE Circuits and Systems Magazine - Q2 2018 - 96
IEEE Circuits and Systems Magazine - Q2 2018 - 97
IEEE Circuits and Systems Magazine - Q2 2018 - 98
IEEE Circuits and Systems Magazine - Q2 2018 - 99
IEEE Circuits and Systems Magazine - Q2 2018 - 100
IEEE Circuits and Systems Magazine - Q2 2018 - 101
IEEE Circuits and Systems Magazine - Q2 2018 - 102
IEEE Circuits and Systems Magazine - Q2 2018 - 103
IEEE Circuits and Systems Magazine - Q2 2018 - 104
IEEE Circuits and Systems Magazine - Q2 2018 - 105
IEEE Circuits and Systems Magazine - Q2 2018 - 106
IEEE Circuits and Systems Magazine - Q2 2018 - 107
IEEE Circuits and Systems Magazine - Q2 2018 - 108
IEEE Circuits and Systems Magazine - Q2 2018 - Cover3
IEEE Circuits and Systems Magazine - Q2 2018 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q1
https://www.nxtbookmedia.com