1. Time-Domain Simulation of
Nonlinear Inductors Displaying Hysteresis
Reprint of the Record of the 14th Compumag Conference on the Computation of
Electromagnetic Fields, July 2003, pages 82–83.
John Paul∗
, Christos Christopoulos and David W. P. Thomas
School of Electrical and Electronic Engineering,
University of Nottingham, Nottingham, NG7 2RD, U.K.
∗Current E-mail (2016): john.derek.paul@gmail.com
Abstract—A method for the time-
domain simulation of nonlinear induc-
tors displaying hysteresis is outlined.
The technique is based on the incor-
poration of the Jiles-Atherton magne-
tization model into the Transmission-
Line Modeling (TLM) method for
lumped circuits. The approach was
validated using a simple circuit con-
taining a voltage source, a series resis-
tor and a hysteretic inductor. Close
agreements were obtained for the
power lost to hysteresis and the se-
ries resistor compared with the power
delivered by the source.
Introduction
The major difficulty in the simulation
of inductors having a ferromagnetic core is
that the magnetization curve displays hys-
teresis, that is the relationship between the
magnetic field intensity and the magnetic
flux density is not a one-to-one function,
but instead depends on the magnetic his-
tory of the sample. The approach devel-
oped by Jiles and Atherton (J-A) [1, 2] is
well suited for inclusion in Transmission-
Line Modeling (TLM) for lumped compo-
nents because it is formulated in terms of a
first-order differential equation.
In previous work, [3], the modeling of
nonlinear dielectric materials using TLM
has been presented. As an extension to the
technique, the simulation of inductors by
the inclusion of the J-A hysteresis model in
the TLM method is described here. The
paper is organized as follows: Firstly, the
development of the TLM algorithm is de-
scribed, next, an outline of the J-A tech-
nique is given and it’s incorporation into
TLM is described. To validate the algo-
rithm, a simulation is performed for a non-
linear inductor displaying hysteresis.
Formulation
Consider a nonlinear inductor carrying a
current IL with a voltage VL dropped across

2. it. The branch constraint for an inductor is
VL = dΨ/dt where Ψ is the total flux link-
age. The flux φ is related to the total flux
linkage by Ψ = Nφ, where N is the number
of turns in the winding. The flux density
is given by B = φ/A where A is the area
of the core. The constitutive relation for
the flux density is B = µ0(H + M) where
µ0 is the permeability of free-space, H is
the magnetic field intensity and M is the
magnetization intensity. From Amp`ere’s
law, the magnetic field intensity is given by
H = ILN/l where l is the mean length of
the magnetic path. By combining these re-
lationships and defining L0 = µ0N2
A/l and
M = ImN/l where Im is the normalized
magnetization, gives
VL = L0
dIL
dt
+ L0
dIm
dt
= V0 + Vm (1)
where V0 is the voltage drop due to rate
of change of IL and Vm is the voltage drop
due to the rate of change of Im.
Inductor model The TLM model of (1)
follows from the bilinear approximation of
the time-derivative, i.e.,
d
dt
=
2
∆t
1 − z−1
1 + z−1
(2)
where ∆t is the simulation time-step and
z−1
is the time-shift operator. Using (2) in
(1) to obtain V0 and Vm gives
V0 = 2V i
0 + Z0IL Vm = 2V i
m + Z0Im (3)
where 2V i
0 = −z−1
(V0 + Z0IL), 2V i
m =
−z−1
(Vm + Z0Im) and Z0 = 2L0/∆t. The
voltage across the inductor is
VL = Z0IL + Z0Im + 2V i
0 + 2V i
m (4)
For a circuit consisting of a voltage source
Vs in series with a resistor Rs connected to
the inductor,
Vs − VL = ILRs (5)
Using (5) to eliminate VL from (4) gives
(Z0+Rs)IL+Z0Im−Um = 0 = f(IL, Im, Um)
(6)
where Um = Vs − 2V i
0 − 2V i
m. For
an inductor with a ferromagnetic core,
Im = Im(IL, Im), so an iterative process is
necessary to solve for f = 0 in (6).
Jiles-Atherton model In the J-A model,
[1, 2], the magnetization is split into two
parts, the anhysteretic magnetization and
the irreversible magnetization. In normal-
ized form, this is expressed by
Im = βcIan + (1 − βc)Iirr (7)
where 0 ≤ βc ≤ 1, Ian is the normalized
anhysteretic magnetization and Iirr is the
normalized irreversible magnetization. The
anhysteretic magnetization dependence is
given by a modified Langevin function, i.e.
Ian = Is coth
IL + αIm
Ia
−
Ia
IL + αIm
= IsL(γ) (8)
where Is, α, Ia are positive constants,
L(γ) denotes the modified Langevin func-
tion with argument γ = (IL + αIm)/Ia. In
the J-A model, the change in irreversible
magnetization is
∆Iirr =
δM (Ian − Iirr)
δIc − α(Ian − Iirr)
∆IL (9)
In (9), ∆Iirr = Iirr − z−1
Iirr, ∆IL = IL −
z−1
IL, Ic is a positive coefficient and the

3. migration flag δM is given by [2]:
δM =



1 : if ∆IL > 0 and Ian > Iirr
1 : if ∆IL < 0 and Ian < Iirr
0 : otherwise
(10)
Also in (9), the directional flag δ = 1 for
∆IL ≥ 0, otherwise δ = −1.
Numerical Algorithm In the TLM model,
the J-A equations (8—10) are solved as part
of a Newton-Raphson (N-R) iteration to
yield f = 0 in (6). To implement the N-R
procedure, the derivative of the total mag-
netization curve with respect to the mag-
netic field is required, from (7),
∂Im/∂IL = βc∂Ian/∂IL + (1 − βc)∂Iirr/∂IL
(11)
The derivative ∂Ian/∂IL = [−cosech2
(γ) +
1/γ2
]/Ia and ∂Iirr/∂IL = ∆Iirr/∆IL if
|∆IL| > 0, otherwise ∂Iirr/∂IL = 0. At
iteration k, the N-R process is
k+1IL = kIL −
(Z0 + Rs)kIL + Z0 kIm − Um
Z0 + Rs + Z0 k∂Im/∂IL
k+1Im = (Um − (Z0 + Rs)k+1IL) /Z0 (12)
To aid convergence, the starting values for
(12) are the values of IL and Im at the pre-
vious time-step. The algorithm described
requires the storage of six real numbers,
i.e. {V i
0 , V i
m, IL, ∆IL, Im, Iirr}.
Results
As a simple example, consider an induc-
tor consisting of N = 230 turns wound on
a ferromagnetic core having a mean mag-
netic path length of l = 75.4mm and a
cross-sectional area of A = 45.4mm2
. The
material properties of the core were Ms =
-2
-1
0
1
2
3
0 0.05 0.1 0.15 0.2
Inductorvoltage/current(V/A)
Time (s)
Voltage
100 * Current
Figure 1: Inductor voltage and current
275kA/m, Ha = 14.1A/m, α = 5 × 10−5
,
Hc = 17.8A/m and βc = 0.55 [2]. The volt-
age source was specified as Vs = Vs,pk(1 −
exp(−ζt)) sin ω0t where Vs,pk = 5V, ζ =
20s−1
and ω0 = 2πf0, where f0 = 50Hz.
The resistor in series with the voltage source
had a resistance of Rs = 220Ω. The simu-
lation time-step was ∆t = 100µs and the
number of time-steps was 2000. The con-
vergence parameter of (12) was set at 10−8
A
and an average of about 12 steps were re-
quired for convergence.
Fig. 1 shows the inductor voltage and cur-
rent and Fig. 2 shows the B−H relation of
the core. To provide a check on the time-
domain solution, the power lost to hystere-
sis and the power dissipated in the series re-
sistor is compared with the power supplied
by the source. The energy lost to hysteresis
per cycle T = 1/f0 is
wh =
T
Hz dBz ≃
T
Hz ∆Bz (13)
By evaluating numerically the summation
in (13) over the final source cycle gave

4. -0.3
-0.2
-0.1
0
0.1
0.2
0.3
-80 -60 -40 -20 0 20 40 60 80
Magneticfluxdensity(T)
Magnetic field intensity (A/m)
Figure 2: B−H relation of the inductor
wh = 7.744J/m3
, so the power lost in the
inductor is Ph = wh A l f0 = 1.325mW.
The power supplied by the source was ob-
tained from Ps = (1/T) T VsIL∆t. The
power dissipated in the inductor and re-
sistor were found from similar expressions.
Numerical evaluation of these summations
gave Ps = 50.02mW, Pr = 48.70mW and
Pl = 1.324mW≃ Ph.
Conclusion
A time-domain method for the simulation
of nonlinear inductors displaying hystere-
sis was outlined. The approach was based
on the incorporation of the Jiles-Atherton
hysteresis model into TLM. The technique
was validated by comparing the power dis-
sipated due to hysteresis and the series re-
sistance with the power supplied by the
source. In future work, the method will
be applied to magnetic circuit simulation,
pulse problems and extended to deal with
rate-dependent hysteresis.
References
[1] D. C. Jiles, J. B. Thoelke and
M. K. Devine. Numerical Determina-
tion of Hysteresis Parameters for the
Modeling of Magnetic Properties Using
the Theory of Ferromagnetic Hystere-
sis. IEEE Transactions on Magnetics,
28(1):27–35, January 1992.
[2] J. H. B. Deane. Modelling the Dynamics
of Nonlinear Inductor Circuits. IEEE
Transactions on Magnetics, 30(5):2795–
2801, September 1994.
[3] J. Paul, C. Christopoulos and D. W. P.
Thomas. Generalized Material Mod-
els in TLM—Part 3: Materials with
Nonlinear Properties. IEEE Trans-
actions on Antennas and Propagation,
50(7):997–1004, July 2002.