# Non-Linear Inductor Models Comparison for Switched-Mode Power Supplies Applications

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## Abstract

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## Highlights

- Purpose of the paper: to compare two analytical models of non-linear power inductor to be employed in an SMPS in the same test conditions.
- Methodology: the models are identified by an experimental setup. It exploits the same DC/DC converter board used for simulations. The computation time and accuracy are compared, on the same computer, retrieving the characteristic curve, calculating a current profile and performing a simulation of a boost converter employing the identified models.
- Research limits: (a) the absolute computation time evaluation could be performed by the algorithmic complexity function; in this paper we provided a the relative comparison between the two algorithms; (b) most manufacturers do not give enough information to identify the model valid up to the saturation region; as a consequence, a dedicated measurement system is required.
- Practical implications: the user can choose the model according to a trade-off between computation time and operation outside the rated current interval.
- Originality of the paper: it is the first time that the two models are compared in the same test conditions showing the pros and cons of each.

## Abstract

## 1. Introduction

## 2. Modelling a Non-Linear Power Inductor

^{3}to 20 × 10

^{3}; such a high value is obtained thanks to some innovative materials [31]. Different shapes for the core can be easily obtained depending on the applications. The high bulk resistivity of ferrites is advantageous, since it limits the losses at high frequency due to the eddy currents. It should be noted that the losses are reduced as the resistivity of the core rises while the frequency rises.

_{𝑑𝑖𝑓}, is considered (Figure 1). It should be noted that the differential inductance is the slope of the total flux linkage ψ as a function of the magnetizing current [7].

_{𝑑𝑖𝑓}is halved to its rated value [7]. This definition is different from what manufacturers usually consider as the current limit operating point; indeed, in many cases, a drop of 10% of the inductance due to the current is considered. Recently, some manufacturers described the behavior up to a 30% inductance drop.

_{𝑑𝑖𝑓}overlaps the inductance definition (meaning the ratio between flux and current) inside the unsaturated region. Unlike the linear zone, the inductance drop implies a higher current variation to obtain the same flux variation as when the inductor was operated in the linear region. The higher the current, the higher the losses will be, causing an effect of heating [32]. The increasing temperature generally reduces the current at which saturation arises [33]. The inductance (and permeability) drop is avoided by adding a small air-gap in the magnetic path [33]. Therefore, the permeability is kept fixed for a broad range, leading to a coil that has a lower dependence on the core rated permeability. A strategy to prevent deep saturation is increasing the cross-sectional area; however, it increases the weight and the cost, since a lower inductance requires more turns, increasing copper losses; hence, a trade-off is necessary [33]. In general, soft saturation ferrites are an alternative to materials with an air-gap core; for this reason, the models presented in this paper are discussed regarding this material.

#### 2.1. The Arctan Model

_{nom}, L

_{30%}, L

_{70%}, L

_{deepsat}. L

_{nom}is the nominal inductance of the inductor, L

_{30%}and L

_{70%}correspond to the drop of the nominal inductance of 30% and 70% caused by two currents labeled I

_{30%}and I

_{70%}, respectively. L

_{deepsat}is the inductance value in deep saturation.

_{nom}, and L

_{deepsat}can be given from the manufacturer (datasheet and/or inductance curves); however, L

_{30%}and L

_{70%}have to be measured together with the corresponding current. Alternatively, two couples of values obtained for different currents, I

_{α%}and I

_{β%}, can be adopted, where α% and β% are the percentage reduction in the inductance from L

_{nom}and range in the 10–90% range [9]. This model is an asymptotic model: L

_{nom}is the asymptote of $L\left[{I}_{L}\left(t\right)\right]$ when ${I}_{L}\to -\infty $; that is approximated with the closest value of the inductance measurable at ${I}_{L}\u224c0$, and L

_{deepsat}is reached for ${I}_{L}\to \infty .$

#### 2.2. The Polynomial Model

_{nom}, L

_{30%}, L

_{70%}, L

_{deepsat}; however, better results can be retrieved by an augmented data set obtained by a measurement system. The coefficients of (5) are calculated by a least-squares regression (LSR). It should be remarked that not all the third-order curves can be used to represent the inductance, because it is expected that approaching the deep saturation value, the curve rises contrarily to the real inductance that must slightly decrease and then remains constant. On the other hand, the deep saturation is not interesting for power electronics applications.

#### 2.3. Temperature Dependence

_{L}*.

## 3. Inductor Characterization

#### 3.1. The Test Rig

^{3}[36]; in this test, it has been replaced with the Coilcraft inductor chosen so that the maximum current imposed by the load causes a drop of 50% of the rated inductance. As a consequence, the core is smaller, corresponding to a volume of 206 mm

^{3}, meaning a reduction of about 93% [35]. From this, it can be noted that the exploitation of the inductor with a lower saturation current allows for a size reduction in the component and, therefore, a power density improvement of the converter. Finally, the performance of the converter in terms of the output voltage remains the same; as a second-order effect, a slight increase in the voltage ripple superimposed to the DC value could be appreciated.

#### 3.2. The Inductor under Test

#### 3.3. Inductance Measurements and Characterization

_{L}is the voltage applied to the inductor that, for a boost converter, coincides with the supply voltage V

_{s}, and is constant during T

_{ON}. The current slope is calculated based on an LSR performed on a set of 50 samples at a time, and this differential inductance value is associated with the current mean value calculated on the same samples. Repeating the calculation on the whole set of samples, the characteristic curve of the inductor is obtained. A picture of the test rig for measurement is shown in Figure 5. The current acquired by the oscilloscope flowing through the inductor is shown in Figure 6; it can be noted that the slope of the current starts with a constant value, then it increases (corresponding to a lessening of the inductance) until it reaches the deep saturation where the slope remains constant despite the further increase in the current, as expected. The top trace, labelled CLOCK, is the internal clock reference signal accessed by the pin “SYNC” of the EVM, it has the frequency of the PWM carrier but out of phase with respect to the state of the switch. The current has been sampled with a sampling time of 1 ns obtaining a vector of 1920 samples and a vector of 38 points representing the inductance has been obtained (the time interval in which the current is sampled is one-half of the period corresponding to a frequency of 260 kHz, meaning 1.92 µs). From this dataset, both the arctan model and the polynomial model can be identified.

## 4. Analysis and Comparison

#### 4.1. Identification of the Characteristic Curve

_{nom}, L

_{30%}, L

_{70%}, and L

_{deepsat}are extracted. The following values are obtained: L

_{nom}= 11.3 µH; L

_{30%}= 7.91 µH; L

_{70%}= 3.39 µH; L

_{deepsat}= 1 µH; L

_{30%}and L

_{70%}correspond to a current of 2.24 A and 3.07 A, respectively. Then, the characteristic parameters (3) are calculated; finally, by Equation (4) the characteristic curve can be plotted. The time required to retrieve the parameters (3) has been calculated by the function “tic” and “toc” by MATLAB

^{®}(version R2014b has been used in this paper). A time of 0.142 µs has been required (for the sake of precision, each calculation is repeated 1000 times, and the result is divided by 1000, the final results is calculated as a mean value of several tests; in the computer, only MATLAB software is running during calculation). A vector containing the current is built to obtain the characteristic curve L(i). It contains 6001 samples from 0 to 6 A with a step of 1 mA. The inductance is evaluated by (6) for each current point and then it can be plotted. The time required to calculate the inductance was 474 µs.

_{3}= 0.0069 × 10

^{−4}, L

_{2}= −0.0495 × 10

^{−4}, L

_{1}= 0.0564 × 10

^{−4}, and L

_{0}= 0.1253 × 10

^{−4}. It can be noted that the rated value of the inductance measured at low current and corresponding to the coefficient L

_{0}gives a higher value compared to the rated one.

#### 4.2. Current Profile Evaluation

_{ON}, the voltage applied to the inductor is constant, and the current is given by the constitutive equation in which the inductance depends on the current:

#### 4.3. Simulation of a DC/DC Converter with Non-Linear Inductor

## 5. Discussion

^{®}function or a similar tool that is more time-consuming; nevertheless, this calculation is performed only once for a given temperature.

^{®}library has been used. As a consequence, this calculation time could be reduced for both models by optimizing the algorithm directly into the microprocessor or DSP.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**The EVM board modified to connect the non-linear inductor. A current probe is connected to sense the inductor current.

**Figure 3.**Comparison of the original inductor equipped on the EVM (left side) with the one applied in this paper (right side).

**Figure 6.**Current flowing through the inductor during the test for the characterization (bottom trace). The top trace, labelled as CLOCK, is the internal reference signal accessed by the pin “SYNC” of the EVM.

**Figure 7.**Characteristic curve of the non-linear inductor obtained by the arctan and polynomial model.

**Figure 8.**Experimental curve of the current (bottom). The top trace, labelled as CLOCK, is the internal reference signal accessed by the pin “SYNC” of the EVM.

**Figure 10.**Absolute error between the experimental values and the current calculated by the two models.

**Figure 11.**DC/DC boost converter implementation in Micro-Cap Spice simulator with a non-linear inductor using the arctan model.

**Figure 12.**DC/DC boost converter implementation in Micro-Cap Spice simulator with a non-linear inductor implementing the polynomial model.

**Figure 13.**Simulation results of the DC/DC boost converter in Micro-Cap Spice simulator with a non-linear inductor implementing the arctan model. Top figure depicts the output voltage of the converter and the inductor current. Bottom figure depicts a detail of the inductor current and output voltage profiles.

**Figure 14.**Simulation results of the DC/DC boost converter in Micro-Cap Spice simulator with a non-linear inductor implementing the polynomial model. Top figure depicts the output voltage of the converter and the inductor current. Bottom figure depicts a detail of the inductor current and output voltage profiles.

Model | |||
---|---|---|---|

Arctan | Poly | Experimental | |

DC RMS [A] | 1.6388 | 1.6276 | 1.7 |

AC RMS [A] | 0.6435 | 0.6019 | 0.574 |

MEAN [A] | 1.5071 | 1.5122 | 1.6 |

RMSE | 2.0042 | 1.7525 | - |

Arctan | Poly | Note | |
---|---|---|---|

**(1)****Identification of the characteristic curve**
| |||

Measurements | Both methods need a measurement setup. | ||

Model Parameter calculation | 0.142 µs | 19.42 ms | Required only once. |

Inductance vs. current calculation | 474 µs | 50 µs | The inductance is evaluated based on a current vector of 6000 points. |

**(2)****Current profile evaluation**
| 131 µs | 121 µs | Obtained by solving differential equation in discrete form. |

**(3)****Micro-Cap Spice simulation**
| 5.88 s | 5.10 s | Simulation of a Boost DC/DC converter implementing the non-linear model of the inductor with a simulation time of 1 ms. |

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## Share and Cite

**MDPI and ACS Style**

Scirè, D.; Lullo, G.; Vitale, G.
Non-Linear Inductor Models Comparison for Switched-Mode Power Supplies Applications. *Electronics* **2022**, *11*, 2472.
https://doi.org/10.3390/electronics11152472

**AMA Style**

Scirè D, Lullo G, Vitale G.
Non-Linear Inductor Models Comparison for Switched-Mode Power Supplies Applications. *Electronics*. 2022; 11(15):2472.
https://doi.org/10.3390/electronics11152472

**Chicago/Turabian Style**

Scirè, Daniele, Giuseppe Lullo, and Gianpaolo Vitale.
2022. "Non-Linear Inductor Models Comparison for Switched-Mode Power Supplies Applications" *Electronics* 11, no. 15: 2472.
https://doi.org/10.3390/electronics11152472