IEEE Circuits and Systems Magazine - Q2 2018 - 32

iFM

iLP

RS
vext

LFM

-
RLp

(a)
t1

vFM (V)

0.7
0.0
-0.7

i (mA)

iFM

0
iLp

diFM /dt (A /s)

-5

70
0
-70

dvFM /dt (V/ms)

10
5
0
-5
-10
φFM (mV.s)

0.5
0.0
-0.5
0

2
Time (ms)
(b)

Figure 3. fml with a 27 mh parallel linear inductor (ll) electrical measurements. (a) the circuit used. dashed rectangle
refers to the fml, while R L is its winding resistance. L p is the
parallel ll. R Lp 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.

IEEE cIrcuIts and systEms magazInE

significantly in area, essentially breaking into two lobes,
and revealed NDL regions where the current decreased
as the flux increased, denoted by t5➔t6 and t2➔t3. The
plots in Figs. 4b,c are similar to the corresponding plots
in Figs. 2b,c (the case with no LL), confirming the identical nature of the underlying dynamics and the circuit classification of the FML despite the substantial changes in
the z FM -i FM hysteresis behavior between Figs. 2a and 4a. In
the plot of L -FM1 ^z FM h vs. z FM in Fig. 4d, there is significant
NDL in the experimental data that appears as two branches each with a local minimum displaced from z FM = 0.
The hysteresis loop did not collapse into a single-valued
function of z FM , as with the ideal model (gray curve,
Fig. 4a). The area in each lobe of the experimental data
(red curves) represents an upper limit to the minimum
reversible work needed to switch the magnetization of the
ferromagnetic toroid, and is directly related to the double
minima in the experimental L -FM1 ^z FM h curve, which was
missing from the ideal model. We used the technique of
constructing a restoring function to determine PFML and
PD for the FML, which are plotted in Fig. 4e along with the
measured power in the parallel LL. The power returned to
the circuit (negative power) by the parallel LL was sharply
peaked, large in amplitude, and available because of the
large decrease in PD of the FML. The corresponding energies, plotted in Fig. 4f reveal that the total energy per
cycle dissipated by the FML was approximately a factor
of three lower than the case with no LL (Fig. 2f), as was
also evident in the corresponding shrinking of the area
within the hysteresis loop in the z vs. i plots (Figs. 2a
and 4a). Thus, the steady state temperature for the FML in
this circuit is significantly lower than for the stand-alone
FML. However, the cycle-dependent temperature oscillations should be larger because the dissipated power in
the FML is more strongly peaked, as seen by comparing
Fig. 4(e) with Fig. 2(e), which has a significant influence
on the electronic behavior during a voltage cycle. The
high peak power returned to the circuit by the LL is
useful for some applications for which amplifiers or active devices are not suitable or are otherwise unavailable.
While an accurate predictive model will require careful static and dynamic temperature measurements to
correlate with the electronic behavior, it is possible to
determine a heuristic temperature-dependent functional form for the inverse differential inductance that
displays the minima observed in the FML experimental
data, as described by Equation 6.
L -FM1 ^z FM , T h = c =

z FM
2
zS

2
z FM

+ p (T ) e

z FM
2

oG,

(6)

where p ^T h is a temperature-dependent parameter.
Since the dynamics of the system cause significant
sEcOnd quartEr 2018

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