Original scientific paper
Informacije
^efMIDEM
A Innrnal of
Journal of Microelectronics, Electronic Components and Materials Vol. 45, No. 1 (2015), 29 - 38
The parameter estimation of the electrothermal model of inductors
Krzysztof Gorecki, Kalina Detka
Gdynia Maritime University, Department of Marine Electronics, Gdynia, Poland
Abstract: This paper presents the electrothermal model of inductors dedicated to the analysis of dc-dc converters in SPICE and the proposed method of determining parameters of this model. The parameter estimation algorithm of this model is described in detail. The results of verification of the correctness of the model and the estimation procedure for arbitrarily selected choking - coils are presented. Very good agreement between the calculated and measured characteristics of the considered choking-coils was obtained.
Keywords: Inductors; modelling; parameters estimation; self-heating
Ocena parametrov elektrotermičnega modela tuljav
Izvleček: Članek opisuje elektrotermični model in določevanje parametrov tuljav, ki se uporabljajo v dc-dc konverterjih v SPICE. Natančno je opisan algoritem določevanja parametrov modela. Predstavljeni so rezultati verifikacije modela in postopek ocenitve parametrov na izbranih tuljavah. Rezultati simulacij se dobro ujemajo z meritvami.
Ključne besede: Tuljava; modeliranje; ocean parametrov; samogretje
' Corresponding Author's e-mail: gorecki@am.gdynia.pl
1 Introduction
Inductors are important components of switched-mode power converters [1 - 4]. Properties of such converters depend on the properties of their structural components, i.e. the ferromagnetic core and the winding. Ferromagnetic materials used to build the core of the inductor are characterized by magnetization hysteresis characteristics. The magnetic permeability of the core, which is proportional to the inductance of inductors is a non-linear function of magnetic force and temperature [5 - 11].
In designing electronic circuits the computer programs dedicated to their analysis are used. Currently, one of most popular programs for this analysis is SPICE software [12 - 15]. The credibility of calculation results depends on the accuracy of the models of the used elements [16]. The inductor models typically use a linear model of the coil or non-linear model of the core and the linear model of the winding [2, 13, 17]. Nonlinear models of the core were presented in [3, 10, 13, 18, 19], but various modifications of the Jiles-Atherton model are the most commonly used models [6, 7, 11, 18, 19,
20]. This model does not take into account such an important phenomenon as self-heating.
In papers [3, 18] the electrothermal model of the chocking-coil for SPICE using the electrothermal core model presented in [11] is proposed. The electrothermal model of the choking-coil is devoted to calculate parameters of its model for the inductor used in the analyzed circuit. Therefore, it is important to prepare algorithm parameter estimation of such a model. This paper presents a modified form of the electrothermal model of the inductor, proposes the method for determining parameters of the model and provides an example of the results of calculations and measurements to illustrate the correctness of the elaborated method.
2 The electrothermal model of the inductor
The presented electrothermal model of the choking - coil takes into account electrical phenomena occur-
ring in the winding, magnetic phenomena occurring in the core and thermal phenomena in the core and the winding. Due to the fact that the choking - coil core is made of soft magnetic material the hysteresis of the magnetization curve can be omitted in the model [6]. The considered electrothermal model of the choking -coil has the form of a sub-circuit of SPICE. The network representation of the elaborated model is presented in Figure 1. The model is composed of three blocks. The first block is the main circuit and it includes a series connection of controlled voltage sources ELS, ERS, the voltage source VL_ with the zero value and the coil with inductance L equal to 2 ^ H and the parallelly connected capacitor Cw modeling interturn capacitance of the winding.
I Main circuit
O-l-A
V = 0
y^^Mo
Rso ,,
Auxiliary blocks
B I Bsat I DB | c
H I A

Elo R20
Dio RiOi-l Cio I
1 ATr Thermal model
ATu
i(t) «
JPR YGPU1
RthR
CthR
X
(?)
JPU Y^PRi
Cthu
T
Figure 1: Network representation of the electrothermal model of the inductor
The voltage source VL monitors the value of the current of the choking - coil. The coil L make it possible to calculate the time derivative of the current of the choking - coil. Els represents the voltage drop on non-linear inductance of the choking - coil and is described by the formula [3]
Els = Ws	'L^ =
(f + fb)■ L S S (f + f,)■ L If,( + A] + A-Bat^-l^!^,
(1)
where z denotes the number of turns in the choking-coil winding, VL - voltage on the coil L, S^.^ - effective cross-section area of the core, Ba - saturation magnet-
ic flux density, H - magnetic force in the core, - magnetic path in the core, A - the field parameter, l^^ - air gap length in the core, yt^ - permeability of free air, which amounts to 12.57 • 10-7 H/m, dB/dH - magnetic permeability of the core, f - the frequency of the inductor current, f^^ - reference frequency.
The resistor RSO represents series resistance of the inductor at temperature Tg. The value of this resistance is described by formula:
RSO =P-
L
S,
(2)
where p is resistivity of copper equal to 1.72 • 10-8 Q^m at temperature 20° C, l^^ is the length of winding, and Sd is the cross-section of the coil wire.
In turn, the controlled voltage source ERS is described by: RS
Ers = Vrs ap-(Tu - To)+.jßo- p-(l ^apiTu - T,))-l-(/ -)

d
2-n-f
t
+ bn - sin
2-n-f
(3)
+ P.
In the equation (3) there are three components. The first one models the dependence of series resistance on temperature. The V^S is a component of the voltage across the resistor RSO, aP is the temperature coefficient of resistivity of copper, which amounts to 4.45 • 10-3 K-1 and TU is temperature of the winding. The second component models the additional voltage drop at the chok-ing-coil which is a result of the skin effect. To describe these phenomena one takes into account the fact that the current of the choking-coil operating in the dc-dc converter has a periodic triangular waveform. This waveform is modeled with a Fourier series, wherein the number of components is limited to four. The Fourier series coefficients of the model are described by:
= 2-cos(2- n- n(T1 - o,5)-(-1)") = d1-n' -d1)
k =
sin(2- n- n- (d1 - 0,5) - (-1)" 2-n'-" '•d1-(1 - d1)
(4)
(5)
where I is choking-coil current, d - diameter of the coil wire, Iv - average value of the coil current calculated in the auxiliary block, d^ - duty of the converter control signal, and f - frequency of this signal. The third component in the formula (3) represents the choking-coil voltage drop resulting from energy losses in the core. The PR component describe energy losses in the core, Ik is the RMS value of the choking-coil current.
I
ü" - cos
I
=1
ak
C
20
2' -S^^-BarA
In the auxiliary block the following are determined: magnetic force H, magnetic flux density B, the time derivative of the magnetic flux density DB, field parameter A, maximum and average values of the magnetic flux density and of the current, coefficient c defining the influence of the Curie temperature TC on the value of the magnetic flux density. Inductance of the inductor is proportional to the magnetic permeability of the core corresponding to the characteristics B(H) slope [2, 6, 13]. To determine the value of the magnetic flux density the formula described in [6, 21] is used:
B = Bsa, ■
H
H + A
(6)
where B . is the saturation flux density of the core.
On the other hand, the value of the magnetic force is calculated by the formula [5]:
H =
z l - BJ,
(7)
In the auxiliary block, the field parameter A, which makes it possible to take into account the influence of temperature on the magnetization curve and inductance of the inductor, is also determined. The dependence of the parameter A on temperature is described by the empirical formula:
A = A, ■ exp[(- Tr + Ta)laT ]
(8)
where aT is the temperature coefficient of the parameter A.
It should be noted that the saturation flux density in the core also strongly depends on temperature and the inclusion of this impact has been expressed by the dependence [6, 10, 11]:
TBsat = Bsa, 0 ■[I ^aBs ■ (Tr - T,}
(9)
where is the saturation flux density at temperature Tg and a^^^ - the temperature coefficient of Ba.
The c coefficient was defined by:
On the other hand, to calculate the average and peak-to-peak values of the current, and the magnetic flux density, two detectors are defined: the peak-to-peak value detector and the average value detector, consisting of the two-terminal networks R1C1 , R2C2 and R11C11 , diodes D and D11, the controlled voltage sources
'2^21'
E1 and E11, respectively representing the inductor current and the magnetic flux density of the core.
The thermal model is used to determine the core temperature Tr and the winding temperature TU of the inductor using the compact model proposed in [6, 12, 16, 22 , 23]. This model includes two controlled current sources, representing power losses in the core GpR and in the winding GpU, respectively. The included in this two-terminal circuits RthR, CthR and RthU, CthU represent thermal time constants of the core and the winding, so that it is possible to take into account the phenomena of self-heating. These time constants fulfill equations describing the relation between the controlled current
sources and GpU1 used for modeling the thermal coupling between the core and winding. The currents of these sources are respectively 0.8 GpR and 0.8 GpU. Depending how one defines a power loss in the winding, GPU includes resistive losses and the skin effect . The losses in the winding are described by the formula:
Pu ' ■[l + ap -(Tu - To)]-+ Ijd ■ „Jmo^ P^f^ T ^ap-(Tu - T,))-Z-
/ o^n^f	(11)

2 ■^■f t
+ bn -sin
dmx - Iv )2
where I is the maximum coil current calculated in the
mx
auxiliary block.
In turn, the core losses are described by [10]:
Pr = V,
DB
\ß-a
2
■(1 + D^(T, -T^pTo ■ j dJL
dt
(12)
where V^ denotes the equivalent volume of the core, P^^g are volumial power losses in the core, DB is the magnitude of flux density, D - the square temperature coefficient of power losses Pvg, T - period of a inductor current, a and ß are exponents in the dependence of core losses based on frequency and amplitude of the flux density in the choking-coil, respectively, Tm is the temperature, at which losses are minimal.
c =
1 forT, < Tc
1 -O.l (Tr -Tc) forTR < Tc + lOK 0 forTR > TC + lOK
where T^^ denotes temperature of the core.
(10)
3 Parameter estimation
The presented model is described by 20 parameters that can be divided into 3 groups:
a.	electrical parameters,
b.	magnetic parameters,
c.	thermal parameters.
=1
a
0
The proposed estimation algorithm uses the concept of local estimation described in [22, 24]. According to this concept, the model parameters are estimated in groups on the basis of the measured characteristics of the inductor operating in specific conditions.
The magnetic parameters of the choking-coil corresponding to the ferromagnetic core reactor can be divided into three groups:
-	The parameters of ferromagnetic material, of which the core is made, related to the hysteresis loop, such as the saturation flux density the Curie temperature TC, the field parameter A, the air gap length l, the temperature coefficient of saturation flux density changes a^^, the temperature coefficient aT of the magnetic field parameter,
-	The geometric parameters of the core, such as the magnetic path length in the core the equivalent value of the core volume V^, the effective cross-section area of the core
-	The ferromagnetic material parameters corresponding to core losses such as PV0, D, a, ß .
Some parameters associated with the magnetic material used to construct the ferromagnetic core can be read directly from the catalog data supplied by manufacturers e.g. the saturation flux density and the Curie temperature TC [22].
In order to determine the temperature coefficient of saturation flux density changes the designer needs to:
1.	Read from the catalog characteristics , eg, [25, 26], the value of the saturation flux density at the reference temperature Tg and the value of this parameter B^^^j at a different temperature T^.
2.	Calculate the value of the temperature coefficient of saturation flux density changes according to the formula [22] :
=

- To
(13)
The geometric parameters of the cores should be read from the catalog data or should be determined basis of the dimensions of the core and calculated using the basic geometrical relationships. For example, to determine the geometrical parameters of the ring core one should:
1. determine the dimensions of the core (Fig. 2), i.e. the outer diameter d^, the inner diameter dw and height hR (these data are usually contained in the name of the core, e.g. RTP 26,9 x14, 5x11)
Figure 2: Dimensions of the ring core
2. calculate the magnetic path length in the core using the formula
iFe =n 2 •
„ )
(14)
3. calculate the effective cross-section area of the core S^:^ using the formula:
SFe =
(dz - dw y^R
2
(15)
4. calculate the equivalent value of the core volume Ve by:
^ =
^•(dz ' - dw' ^^-hR
4
(16)
In order to determine the values of the parameters A, w, and /^ it is necessary to measure the dependence of inductance L on the DC current using the measurement system described in [27]. The measurement should be performed at the frequency f << fb. In the measured characteristics of L(i), whose typical course is shown in Figure 3 one should select 3 points: X1(/J, L,), X2(/^, L2) and X3(/3, L3). Then, the following system of equations must be solved for wS, / and A:
Figure 3: Typical course of the dependence of inductance of the inductor on the dc part of its current
Li =-
iFe
L =-
WS ■ z' • SFe • Bsat ' A
z • Ii - Bsat - A • X ^ B^at + z' • I/ + A' • X ^ + 2 • A • X •{Bsa, + z • Ii )-2 • B^a, ■ z • Ii
2 • X
■ + A
+ A • Bst ■ I p
WS • z • SFe • Bsat • A
z • 12 - Bs,, - A • X ^ + z' • 122 + A' • X ^ + 2 • A • x •{B^,, + z • 12 ))• B^,, ■ z • 12

L =■
2 • X
• z • SFe • Bsct • A
+ A
+ A • Bsa> ■ lp/ß0
13 - Bs^, - A • X ^ + z' • 132 + A' • X2 + 2 • A • x •{Bs,, + z • 13)-2 • Bs,, ■ z • 13
'Fe
2 • X
+ A
+ A • Bs.> ■ lp/ßo
(17)
WhereX = • lp •{l.e + lp )■
In order to determine the value of the temperature coefficient aT, it is necessary to measure the dependence of L (i) for the temperature T^ > T^, and then to determine the value of the parameter A^ at temperature T^ using the formula (17). The value of aT is given by:
To - Tr \n{AJ Ao)
(18)
In order to determine the parameters describing losses in the core:
1. one should read from the catalog characteristics of the core material describing the dependence of power losses density Pv on the amplitude of the magnetic flux density (Bm) at constants frequency f, the coordinates of two points X4(Bm1, PV1) and X5 (Bm2, The typical course of such characteristics is shown in Figure 4. Next, one should calculate the value of the ß coefficient by the formula:

\Og{BmJ Em,)
(19)
2.
to determine the coefficient a one should read from the catalog characteristics describing the dependence of the power density of the core loss on frequency (Fig. 4b) at the known amplitude Bm in points X6(f1, PV3) and X7(f.,, PV4) and calculate the value of the factor a from the formula:
a =
\og(Pv 3 / Pv 4) \og(fi/ /2)
(20)
to determine the parameter P^^ it is necessary to read point e.g. X6(f1, P^.^) from the characteristics describing the dependence of power density losses PV on frequency at the constant value of Bm (Fig. 4b) and next, calculate the value of P^^^ from:
P =
v3
/f^ • Bß^ (2 • n)a• [0.6336 - 0.1892• \n(a)]
(21)
4.
in the catalog characteristics P(T) at first one should read the value of temperature Tm at which the characteristics of P(T) reaches the minimum at point X8(Tm, P^,.) (Fig. 4c), then one should select point X9(T6, PV6) of some characteristics and calculate the parameter D by the formula:
D =
P^ - Pvr _6_5
P^ •{{ - T j2
5^6 m'
(22)
In turn, the value of the parameter fb associated with the dependence defining the output voltage of the controlled voltage source ELS can be determined from the dependence describing the characteristics of the magnetic permeability ^ of the core of the frequency ^(f), whose typical course is shown in Figure 5.
The frequency f. is calculated by the formula
fb =
= fi •ßi - f i •ßi
ß2 -ßl
(23)
where in the calculations the coordinates of two points X10 (f,, ß,) and X11 (f-^ß.) lying on the curve ß(f) were used.
To the electrical parameters appearing in the description of the electrothermal model of the inductor belong
2
2
T .
Figure 4: Dependences of power losses density on the amplitude of the flux density (a), frequency (b) and temperature (c)

^10
f
Figure 5: Typical dependence of magnetic permeability of the ferromagnetic core on frequency
also length of the winding ld and cross-section of the coil wire S^. These parameters are used to determine series resistance of the coil. The length of the winding for the ring core is estimated by calculating the product of the number of turns z and the girth of cross-section of the core, assuming that it is rectangular, by the formula:
Id = 2. z + - )2jj
(24)
In turn, the cross-sectional area of the wire is calculated on the basis of simple geometric formulas and the known wire diameter d3.
The capacitor Cw is determined by the formula:
= ifr	(25)
where f^ is resonant frequency of the inductor, L0 is the inductance value for IDC = 0. The resonant frequency of the chocking - coil can be read from the course of the dependence of the impedance module, whose typical course is shown in Figure 6, on frequency.
|Z|
fr
f
Figure 6: Typical dependence of the module of inductor impedance on frequency
To determine the thermal parameters a. Rh, it is necessary to perform measurements of their own transient thermal impedance of the winding Z^hu(t) and of the core Z^^/t), as well as the mutual transient thermal impedance between the core and the winding Zhu((t) using the method described in [28]. Based on the measured waveforms Zhu(t), Zh(t) and Zhu(t) the values of capacitance and thermal resistance are calculated using the method described in [22, 23].
4 Experimental results
In order to verify the correctness of the proposed method of estimating parameters of the inductor, the values of the parameters of two arbitrarily selected inductors with ferromagnetic cores were estimated and the calculated and measured characteristics of these inductors were compared. The investigations were performed for two inductors containing ring cores of the same size (26.9 mm x 14.5 mm x 11 mm). The first one was the core RTF of ferrite material F-867 and the other
was the core RTP of powdered iron from the material T106 -26. On both the cores 20 turns of the enameled copper wire of 0.8 mm diameter were wound. Using the estimation algorithm proposed in the previous section, the values of all the model parameter values were read or calculated and collected in Table 1.
The measured and calculated characteristics of the considered inductors are shown in Figures 7 - 8. In these figures the results of measurements are denoted as points, whereas the results of electrothermal analysis are represented by lines.
Table 1: Values of parameters of the electrothermal model of inductors with the cores RTP T106-26 and RTF F 867.
ture equal to 50 °C caused an increase in its inductance even up to 45 %.
80
70
60
„50
Ji40 hJ
30 20 10 0
(a)
Parameter	Bsat0 [T]	lp [^m]	TC [K]		
RTP T106-26	1.38	14	1023	4024	2.8-10-3
RTF F867	0.5	0.1	488	260	2.8-10-3
Parameter	ws			SFe [m2]	z
RTP T106-26	0.5	64.99	4.43-10-6	68.2-10-6	20
RTF F867	0.5	62.8	3.14-10-6	50-10-6	20
Parameter	Sd [m2]	ld [m]	Pv0 [kW/m3]	D [K-2]	a
RTP T106-26	502-10-9	0.6	2	0	1.59
RTF F867	502-10-9	0.6	100	0.5-10-6	1.02
Parameter	ß	□t [1/K] Tm [K] fb [kHz] d [mm]			
RTP T106-26	2.15	100-103	368	546	0.8
RTF F867	2.82	240	343	850	0.8
Figure 7 shows the dependence of inductance on the DC current of the inductor containing the powder core RTP T106 -26 (Fig. 7a) and the inductor with the ferrite core RTF F867 ( Fig.7b ) The tests were performed at frequency of 100 kHz for two ambient temperatures equal to 23 and 75°C. As you can see, good agreement between the results of measurements and calculations was obtained. For both the considered choking-coil the dependence L(i) is a decreasing function of the current, where the choking-coil with the ferrite core with the same geometrical dimensions achieved a higher value of inductance, moreover a wider range of changes in its value was observed. A decrease in inductance of the ferrite core (even two hundreds times) was much larger than for the core of the powdered iron (about 30 %). The different courses of the dependence L(i) for both the inductors were due to the non-linear magnetization curve of ferromagnetic cores. It is worth noticing that the course of the dependence L(i) for the choking-coil with the ferrite core showed the visible influence of the ambient temperature on its course, while for the inductor with the powder core such influence is not observed. In the characteristics of the choking-coil with the ferrite core an increase in tempera-
a i
	T=230C
	T=750C
	
■ Vw	
	^ ■—--- • ' ---
RTF 867	
(b)
2	3
i [A]
Figure 7: Measured and calculated dependence of inductance of inductors with the powder (a) and ferrite (b) cores on the current
Figure 8 shows the dependences of the module of impedance of the choking-coils with the considered cores on frequency at the constant values of the DC current. As it is visible, good agreement between the measurements and calculations results was obtained. By considering winding capacitance in the electrothermal model of the choking-coil the resonance on these characteristics was obtained, which corresponds to the obtained measurement results. The value of the resonant frequency for the ferrite core increases with an increase of the DC current, whereas for the powder core it oscillates in the range of 1.3 MHz to about 2.3 MHz.
In order to illustrate the influence of the nonlinearity of the inductor and the self-heating phenomena in this element on characteristics of dc-dc converters, the results of calculations (lines) and measurement (points) of the boost converter with the core RTP T106-26 [29] were presented in Figs. 9 and 10. In Fig.9 calculated and measured dependences of the output voltage Vout
1000
100
10
6000 5000 4000
3000
2000 1000 0
(a)
			RTP T106-26
V/			Idc=0 a
			Idc=0,5 a
	■ \ \ \		id^5 a
ß'	■	■	' "0
2000 3000 4000 5000 6000 f [kHz]
(b)
Figure 8: Calculated and measured dependences of the module of impedance of inductors with the powder (a) and ferrite (b) cores on frequency
of the examined converter on the load resistance R0 at the fixed value of the duty factor of the control signal d = 0.5 at two values of the frequency of the control signal equal in turn 50 kHz and 400 kHz, are presented. Results of calculations passed with the use of the electrothermal model of the inductor are marked with solid lines, whereas results of calculations obtained by means of the linear model of the inductor are marked with dashed lines.
As one can notice, the use of the electrothermal model of the inductor makes possible to obtain the considerably better agreement between performance of calculations and measurement than with the use of the linear model of the inductor. It is proper to notice that the regard of losses in the inductor and dependences of the inductances on frequency causes a decreasing in the output voltage of the considered converter. The use of the linear model of the inductor can cause the overestimate of results of calculations even about 50%.
In turn, Fig.10 illustrates the dependence of the core temperature TR (solid lines) and the winding temperature TU (dashed lines) on the load resistance corresponding to characteristics from Fig.9. As it is visible, for both considered frequencies the decreasing dependences Tr(R0) and TU(R0) are obtained, whereas an increase in frequency causes a decrease in value of the temperature of the inductor. From the fact, that the winding temperature is lower than the core temperature results, that a main source of losses is the core of the inductor.
100 Ro [W]
Figure 10: Calculated and measured dependences of the core and winding temperatures of the load resistance
40 -
H:
30-
20 -10 -
Figure 9: Calculated and measured dependences of the output voltage of the boost converter on the load resistance
5 Conclusions
This paper describes the electrothermal model of the choking-coil with ferromagnetic the core dedicated for SPICE software and proposes a method of estimating values of magnetic, electrical and thermal parameters of this model. The proposed algorithm is simple to implement and largely uses the data presented by manufacturers of the ferromagnetic core and winding wire in the catalog data.
The investigations were performed for two arbitrarily chosen inductors with the core made of powdered iron and ferrite material. The presented experimental results show that the proposed method of estimating the
0
25000
20000
15000
10000
5000
0
70
0
parameters is correct, which is proved by good agreement between the measured and calculated characteristics of the considered inductors.
The electrothermal model of the inductor together with the proposed estimation method of its parameters can be useful for designers of switch-mode power supplies and in the analysis of the considered class of electronic circuits.
6 Acknowledgements
This project is financed from the funds of National Science Centre which were awarded on the basis of the decision number DEC-2011/01/B/ST7/06738.
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21.	Wilamowski B.M., Jaeger R.C., "Computerized circuit Analysis Using SPICE Programs", McGraw-Hill, New York 1997.
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mentowmagnetycznych". Elektronika, No. 1, 2013, pp. 18-22
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29.	Detka K., Gorecki K.: „Wpfyw samonagrzewania w dfawiku na charakterystyki przetwornicy typu boost". Przeglad Elektrotechniczny, Vol. 90, No. 9, 2014, pp. 19-21.
Arrived: 06. 06. 2014 Accepted: 28. 10. 2014