1 Introduction

Power electronics is currently evolving towards integration, a consequence of the constraints related to the need for miniaturization but also to reduce manufacturing costs. In many cases, hybrid systems offer opportunities for volume reduction and the passive components, inductance, transformer and capacitor, represent a brake on this miniaturization. Inductive components such as coils or transformers are key elements of power electronics. These are well known and mastered components with regard to their discrete form, but their integration is still at the study stage and is still far from industrialization. The development of micro-technologies now offers new perspectives for the manufacture of micro-coils adapted to this type of application [1,2,3,4]. The main objectives of this work are to study, to show the feasibility of planar spiral inductances “sandwich” between two layers of magnetic materials to limit the problems of EMC (Electromagnetic Compatibility) and to increase the value of the inductance, to highlight the interest of these devices, their advantages and technological difficulties, to check their main characteristics by comparing measurement and simulation. Regarding the topology of our inductance, we retained the square shape because it produces the highest value of inductance. This work is completed by a study of the thermal and electromagnetic behavior of our micro-coil. The work presented in this paper was carried out in the LAPLACE laboratory in Toulouse [5].

2 Presentation of Converter

The study is oriented towards a buck converter, which is a DC/DC step-down voltage converter, see Fig. 1. The choice of the structure of the micro-converter was made quite simply, since it is not a question here of trying to find an innovative structure and to study the advantages and disadvantages, but to use a structure of basis so that it can be integrated without much difficulty. Our aim is to design and dimension a planar spiral inductance in order to insert it into the micro-converter. Sizing is done to avoid or eliminate parasitic effects. The thermal and electromagnetic effects are also treated to ensure the good operation of the coil after its insertion into the micro-converter.

Fig. 1
figure 1

Schematic diagram of a buck converter

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2.1 Condition Specifications

The design of a planar coil is an empirical approach and requires several successive and iterative steps, so it is difficult to predict its performance before manufacturing. The aim is to provide a general method for designing a planar coil. It is possible to calculate the geometrical and electrical quantities of the planar coil as well as the volume of the magnetic core and the surface occupied by the spiral. The initial specification, representing the specific needs of the designer and are established to meet a typical application. It includes: Input voltage: Vin = 5 V, Output voltage: Vout = 2.5 V, Output power: Pout = 1 W, Duty cycle: α = 0.5, Operation frequency: f = 500 kHz. The choice of the frequency is essential because it conditions the whole of the study, that it is with respect to the active parts of the micro-converter, as compared to the passive elements whose study is made in a second time.

3 Micro-coil Dimensioning

3.1 Micro-coil Geometric Parameter Calculations

The micro coil spiral is geometrically described by several parameters: the width w, the thickness of the conductor t, the inter-spacing of winding s, the number of turns n, the inside diameter din, and the outside diameter dout and the total conductor length lt, see Fig. 2 [1, 6]. The din and dout should be chosen to optimize the relationship between the inductor and the area occupied on the circuit. It is composed of a conductive spiral of copper laid on three stacked layers; a layer of Kapton insulation, a ferromagnetic layer and a silicon semiconductor layer, all put on a ground plane. Characteristics of inductance, substrate and insulation are presented in Table 1.

Fig. 2
figure 2

Sectional view of the inductance on several layers of materials

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Table 1 Characteristics of inductance, substrate and insulation
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From a specifications and based on the method of Wheeler modified we will dimension the square planar spiral inductor, with the aim of a minimum losses [1, 2]. The equations below allow to calculate the different geometric parameters mentioned in Sect. 3.1.

$$L_{s} = k_{1} \mu \frac{{n^{2} d_{avg} }}{{1 + k_{2} \rho }}$$
(1)
$$n = \sqrt {\frac{{2L_{s} \left[ {\left( {1 + c} \right) + k_{2} \left( {1 - c} \right)} \right]}}{{\mu_{0} k_{1} d_{out} \left( {1 + c} \right)^{2} }}}$$
(2)
$$c = \frac{{d_{in} }}{{d_{out} }},\quad d_{out} = d_{in} + 2n(sw)$$
(3)
$$\rho = \frac{{d_{out} - d_{in} }}{{d_{out} + d_{in} }}$$
(4)
$$l_{t} = \left[ {d_{out} - \left( {n - 1} \right)s - nw} \right] - s$$
(5)
$$d_{avg} = \frac{{d_{out} + d_{in} }}{2}$$
(6)

with k1 = 2.34 and k2 = 2.75.

3.2 Micro-coil Electrical Parameter Calculations

Figure 3a, b shows the electrical model.

Fig. 3
figure 3

Equivalent scheme in π of a spiral micro coil [6]

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This model includes the series inductance Ls, the series resistor Rs, the coupling capacitance between the turns Cs, the capacitance associated with the insulation layer (Kapton) tk with the substrate Ck, the capacity CSi and the resistance RSi of the silicon, Resistance of ferrite RNiFe, show the equivalent scheme in π of a spiral micro coil [1, 2, 4, 6].

$$R_{s} = \frac{{\rho_{Cu} l_{t} }}{{S_{Cu} }} = \frac{{\rho_{Cu} l_{t} }}{wt}$$
(7)
$$C_{k} 1 = C_{k} 2 = \frac{1}{2}l_{t} w\frac{{\varepsilon_{0} \varepsilon_{k} }}{{t_{k} }}$$
(8)
$$C_{Si} 1 = C_{Si} 2 = wl_{t} \frac{{\varepsilon_{0} \varepsilon_{Si} }}{{t_{Si} }}$$
(9)
$$R_{NiFe} 1 = R_{NiFe} 2 = \rho_{NiFe} \frac{{t_{NiFe} }}{S}$$
(10)

where tNiFe is the thickness of the magnetic core, which also acts as the substrate and tk represents the thickness of the Kapton, ρCu and ρ are respectively the resistivity of copper and of ferrite and finally εo and εk are the permittivity of vacuum and the Kapton.

3.3 Micro-coil Temperature Calculation

3.3.1 Mathematical Model

In the sequel we consider a two- dimensional layered model described by the heat equation in which the bars of copper, the Kapton, the NiFe and the silicon substrate are considered as continuous and solid surface, see Fig. 4.

Fig. 4
figure 4

Two-dimensional model of our component

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The following assumptions are considered [3, 4].

  1. 1.

    Assembled blocks must be rectangular, but may be in the form of a multilayer assembly level.

  2. 2.

    The lateral edges of these blocks must be adiabatic. On the limit of the first block, the temperature is set at room temperature Text.

  3. 3.

    The power dissipation per unit area is placed on the upper face of each block.

  4. 4.

    The thermal conductivity of the different layers constituting each block must be independent of the temperature.

Between each subdomain (layer of each material constituting the component) Ωi, i = 0 … 3, a conventional continuity condition at the interfaces corresponding to a heat exchange is considered. Specifically, only the following boundary conditions are taken into account. On the vertical side faces of Ωi, i = 0…3 the normal flow is taken null:

$$\frac{dT}{dn} = 0$$
(11)

T is the temperature to calculate.

$$- \lambda_{i} \frac{{dT_{i} }}{dy} = \phi_{i}$$
(12)

The continuity conditions at the interfaces, for example between Ω0 and Ω1 are given by:

$$\lambda_{i} \frac{{dT_{i} }}{dy} = \lambda_{i + 1} \frac{{dT_{i + 1} }}{dy}$$
(13)

where λi and λi+1 represent the thermal conductivity in Ωi and Ωi+1, the initial temperature is given by (Where Ta ambient temperature):

$$T(x,y) = T_{a} .$$
(14)

3.3.2 Numerical Solution

For the numerical solution of the heat equation, a spatial discretization by the classical finite difference method is applied. For i = 0…3 the vector \(T_{i}^{n}\) being computed at the previous time step, the vector \(T_{i}^{n + 1}\) is obtained by solving the following linear system [4, 7]:

$$(1 + \alpha B_{i} ) \cdot T_{i}^{n + 1} = dt \cdot F_{i}^{n + 1} + T_{i}^{n}$$
(15)

where Bi is the spatial discretization matrix on Ωi, i = 0…3, i is the identity matrix and n represents the index of time, tn = n dt, dt being the time step. Let ∆x and ∆y the spatial discretization steps along the axis x and y. Then when t = tn+1 = (n + 1) dt the numerical scheme is given by:

$$T_{M}^{n + 1} - T_{M}^{n} - \alpha_{x} dt\left( {T_{w}^{n + 1} - 2T_{M}^{n + 1} + T_{E}^{n + 1} } \right) - \alpha_{y} dt\left( {T_{N}^{n + 1} - 2T_{M}^{n + 1} + T_{S}^{n + 1} } \right) = dtF_{M}^{n + 1} + T_{M}^{n}$$
(16)
$$\alpha_{i} \cdot x = \frac{{\lambda_{i} dt}}{{C_{i} \rho_{i} \Delta x\Delta x}}\quad \alpha_{i} \cdot y = \frac{{\lambda_{i} dt}}{{C_{i} \rho_{i} \Delta y\Delta y}}$$
(17)

where \(F_{M}^{n + 1}\) is the heat source, Ci is the specific heat of the material i, ρi is the volume mass of the material i.

4 Results of Geometric, Electric and Thermal Dimensioning

All geometric, electric parameters that enter the dimensioning of the micro-coil are shown in the following summary Tables 2 and 3. This set of dimensions serves as our basis for the performance of the integrated micro coil. Results presented concern the geometric, electric and thermal dimensioning.

Table 2 Results of geometric dimensioning
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Table 3 Results of electric dimensioning
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4.1 Influence of Geometric Parameters on the Behavior of Inductance

Figure 5 shows the influence of the frequency on the inductor for different widths w of the driver which vary from 10 to 100 μm. The inter-turn space s and the number of turns n are constant. We note that the decrease in the width of the conductor leads to an increase in the resistance of the conductor and therefore the increase of the current which has the effect of increasing the inductance.

Fig. 5
figure 5

Inductance versus frequency

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Figure 6 shows the influence of the frequency on the inductance for different inter turns which vary from 60 to 120 μm. The value of the inductance is proportional to the square of the number of turns n while keeping the internal and external diameters constant. By increasing the number of turns, the length lt increases or the inter-turn distance s decreases. In both cases the mutual inductance increases and consequently the inductance L.

Fig. 6
figure 6

Inductance versus number of turns

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Figure 7 presents the variation of the series capacitance between the turns as a function of the inter-turn distance for different values of the number of turns. We note that the increase of the inter-turn space and the number of turns causes a decrease in the value of the series capacitance between the turns.

Fig. 7
figure 7

Serial capacity versus Inter spires

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4.2 Results of Thermal Dimensioning

Figure 8 shows the distribution of the temperature of the component in the stationary state with the conditions contained in the specification (current flowing in the coil equal to Imax = 0.8 A). Temperature is maximum and equal to 32 °C at the level of the conductors which are the source of heat. It decreases as one moves away to stabilize at 26 °C which corresponds to the ambient temperature.

Fig. 8
figure 8

Thermal mapping of the cross section of an integrated square spiral inductor

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4.3 Quality Factor

When the frequency increases several effects degrade the value of the quality factor of the inductor such as the skin effect, the losses in the substrate and the current flowing in the conductor [7]. All these effects depend on the geometry of the micro coil.

$$Q = \frac{{\omega L_{s} }}{{R_{s} }}\frac{{R_{p} }}{{R_{p} + \left[ {\left( {\frac{{\omega L_{s} }}{{R_{s} }}} \right)^{2} + 1} \right]R_{s} }}\left[ {1 - \frac{{R_{s}^{2} \left( {C_{s} + C_{p} } \right)}}{{L_{s} }} - \omega^{2} L_{s} \left( {C_{s} + C_{p} } \right)} \right]$$
(18)
$$R_{p} = \frac{1}{{\omega^{2} C_{k}^{2} R_{Si} }} + \frac{{R_{Si} \left( {C_{k} + C_{Si} } \right)^{2} }}{{C_{k}^{2} }}\quad C_{p} = C_{k} \frac{{1 + \omega^{2} \left( {C_{k} + C_{Si} } \right)C_{Si} R_{Si}^{2} }}{{1 + \omega^{2} \left( {C_{k} + C_{Si} } \right)^{2} R_{Si}^{2} }}$$
(19)

After calculation, we have a quality factor Q equal to 78.23. From this below curve Fig. 9, can be deduced the value of inductance corresponding to the value of the quality factor corresponding to our working frequency.

Fig. 9
figure 9

Quality factor versus frequency

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4.4 3D Simulation of Magneto Dynamic Phenomena in Conductive Ribbons

4.4.1 Magnetic Flux Density Propagation Without Magnetic Core

Figure 10 shows the distribution of the density of the magnetic flux in the micro coil in the absence of the magnetic core. The absence of the magnetic core causes the field lines to be distributed in all directions, and are therefore not confined to the immediate vicinity of the micro-coil.

Fig. 10
figure 10

Propagation of magnetic field lines in a planar micro-coil without magnetic core

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4.4.2 Magnetic Flux Density Propagation with Magnetic Core

Figure 11 shows the distribution of the density of the magnetic flux in our micro coil with a magnetic core. This latter confines the field lines, hence their optimal distribution in the immediate vicinity of the micro coil.

Fig. 11
figure 11

Propagation of magnetic flux density in a planar micro-coil with a magnetic core

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4.4.3 Current Density Distribution in the Micro-coil

Figure 12 shows the distribution of the current density in our coil. It is minimum at the exit of the coil because of the driver resistance value. On another hand, this density is not uniform in this coil because it is more important at the angles that induce inhomogeneous resistances in the conductors [8]. It can also been observed that the effect of skin and proximity are present and manifest especially between adjacent turns.

Fig. 12
figure 12

Distribution of current density in the micro-coil

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5 Micro-coil Manufacturing Process

The manufacturing process is presented for the implementation of an integrated inductance in a layer of magnetic material which comprises a substrate, a magnetic material and a coil of copper [5, 9, 10], the manufacturing process was made with the equipment available at Laplace laboratory (Fig. 13) outlines the steps of realization. Each step must be studied and optimized to ensure the performance and reproducibility of the process.

Fig. 13
figure 13

Manufacturing process of our micro coil

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The manufactured coil is ultimately made of a copper conductor deposited on a magnetic layer of black color NiFe ferrite on a layer of silicon, all laid on the mass. In order to avoid any short circuit, a layer of white color Kapton insulation is placed between the two, Fig. 14. It has the following characteristics: surface of the magnetic circuit. 256 mm2, w = 0.85 mm, t = 35 µm, din = 12 mm, dout = 16 mm, s = 2 mm, n = 3.

Fig. 14
figure 14

realised inductor Inductance realized

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6 Characterisation of the Realised Coil

Laboratory measure bench LAPLACE Fig. 15 is used, for the characterisation of the reels made. This bench consists of a LCR meter HP 419A, a software interface and acquisition, a current/voltage source [5]. The choice of the measurement model is made according to the impedance values predefined by the device and for each specified range. In the present case, the inductance values to be measured are of the order of a few µm. For this purpose, the equivalent circuit model shown in Fig. 15b has been chosen to take measurements.

Fig. 15
figure 15

Characterization bench (a), equivalent circuit model retained (b) and gap of the micro-coil (c)

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7 Measurements Process

The different measurements made at the LAPLACE Laboratory [5, 10] on our micro-coil are on the module of Z, the angle of phase shift and the operating frequency for different values of gap which is the distance between the micro-coil and the top core Fig. 16c which allowed us to draw the following curves: Ls, Rs and Cs as a function of the frequency Fig. 16b. The frequency is from 100 Hz to 500 kHz. To the measurement of the gap, we used an insulating material (in black) of different thicknesses ranging from 0 to 1200 µm corresponding to the different gaps shown in Fig. 16c.

Fig. 16
figure 16

Variation of the inductance versus gap (results of measure)

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8 Results of measures

Figure 16 shows the evolution of the inductance as a function of the gap whose values vary from 0 to 1200 µm. Note the value of the inductance decreases as the gap increases. Beyond a certain value of the gap, the value of the inductance remains constant. It can be likened to a coil with virtually no hood. This leads us to conclude that to have a higher value of inductance, the best thing would be that it is to say contained in two cores. These results of measure were obtained for a frequency equal to 500 kHz.

8.1 Frequency Influence on the Inductance and Resistance for Different Gap Values

Figure 17 shows the evolution of the inductance as a function of frequency for different values of the gap. The frequency ranges from 100 Hz to 500 kHz. All these curves are grouped in the same graph, the inductance decreases as the gap increases. This is explained by the fact that when the nucleus on the upper face of the inductance moves away from it (increase the gap), there is a greater dispersion of the field lines, hence a decrease of the value of the inductance. Figure 18 shows the evolution of the series resistance of the coil with the frequency. The useful section of the wire decreases with frequency, under the skin’s effect, therefore the resistance increases significantly with the frequency.

Fig. 17
figure 17

Variation of the inductance as a function of frequency for different gap

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Fig. 18
figure 18

Variation of the resistance as a function of frequency for different gap

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8.2 Frequency Influence on the Phase and the Module of the Impedance for Different Gap Values

Figures 19 and 20 respectively show the evolution of the phase shift and the modulus of the impedance as a function of frequency. Regarding the Fig. 19, regardless of the value of the gap, the same behavior is observed of the angle of phase shift, it rapidly increases the first frequency values to stabilize at a certain value, this whatever the value frequency. It shows the variation of the phase with the frequency. For large frequencies, it approaches 90°, this for all the thicknesses of the gap, which is correct. As the frequencies decrease, the phase should go up to 0°. As it is measurement results, the phase gets closer to 0°. From Fig. 20, the impedance modulus increases with frequency regardless the value of the gap.

Fig. 19
figure 19

The phase variation as a function of frequency for different gap

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Fig. 20
figure 20

Variation of the impedance module as a function of frequency for different gap

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9 Result Validation

Figures 21a and 22a respectively show the evolution of induction and resistance of our coil as a function of frequency. To validate our dimensioning, we made a comparison of the results of our coil with a gap of 1200 microns with the coil (b) resulting from the literature we notice that the two gaits are identical. Indeed, in both figures, there are two areas corresponding to the transient and permanent regime. This comparison makes it possible to conclude that the sizing procedure and the implementation protocol are correct. This validates all the current work [5, 11].

Fig. 21
figure 21

Evolution of the inductance versus frequency, a our results, b issue of literature [5, 11]

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Fig. 22
figure 22

Evolution of the resistance versus frequency, a our results, b issue of literature [5, 11]

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10 Conclusion

The work presented in this paper focuses on the design, realization and characterization of a planar coil for square topology. The most important results found in the literature, relating to the techniques allowing the improvement of the characteristics of the planar coils are related either to the optimization of the topology of the inductance as well as its substrate. As an indication, the proximity effects increase with the width of the turns. The starting point of the dimensioning of this coil is the specifications of a micro converter in which we will integrate this coil. The results of magneto dynamic and thermal simulation, having validated the geometrical and electrical dimensioning, we went to the stage of realization of this coil with a magnetic circuit (spiral on the Kapton which plays the role of insulator, and on the ferrite which plays the role of magnetic circuit). The various stages of implementation of this coil as well as its characterization were conducted at the laboratory Laplace–Toulouse. Measurements focused on impedance and phase versus frequency. This allowed to determine the series resistance and series inductance of this coil. The measurements allowed to follow the evolution of the impedance Z, the inductance Ls and the resistance Rs as a function of frequency for different gap values. In order to validate the results found in this paper a comparison with those from the literature is done and shows similarities between the two works.