# Assessment of the Current for a Non-Linear Power Inductor Including Temperature in DC-DC Converters

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## Abstract

**:**

## 1. Introduction

_{sat}) compared to its rated maximum value [2], and depends on the magnetic core temperature [10]. Therefore, characterizing the thermal behavior is of paramount importance for the designer [10,11,12,13]. The current peak is strictly correlated with the conduction time of the power switch (T

_{ON}) because the current through the inductor usually reaches its maximum value at the end of the T

_{ON}. Hence, the knowledge of the current profile allows for avoiding overcurrent and preserving both the inductor and power switch.

_{ON}in SMPSs with a linear inductor is straightforward; contrarily, when the inductor works outside the linear zone, the analysis is more complicated because of non-linearity and temperature dependence.

_{deepsat}. The value of L

_{sat}is often lacking in the manufacturers’ datasheet, where saturation is often considered to be the point at which the inductance is reduced by 10% only. For these reasons, a suitable characterization by a test rig was performed to retrieve a complete knowledge of the inductor [14]. Furthermore, the knowledge of the inductor-saturated model is particularly important for implementing virtual sensors or control actions that allow the estimation of the inductor current [15].

_{ON}time is completed. This time interval imposes the computation time threshold, and it is crucial, especially for converters operated with very high switching frequency (adopting, as an example, GaN devices) where the switching period is reduced.

_{ON}under these operating conditions would also imply an unacceptable current increase or deep saturation of the inductor. In contrast, diminishing T

_{ON}leads to an inductor in the linear zone, eliminating non-linear exploitation. Therefore, we demonstrated the need to tune the T

_{ON}value and calculate the maximum T

_{ON}, considering the temperature for a given current [29].

## 2. Analytical Model of the Non-Linear Inductor

_{m}are polynomial coefficients and the linear dependence of the inductor temperature T

_{core}(°C) is described by the proportionality factors β

_{m}. The parameter L

_{deepsat}represents the inductance value when hard saturation is reached. The first term in (1) models the thermal behavior when the inductance is linear. Once the coefficients of (1) are known, the model can be easily implemented and requires fewer calculation resources compared, as an example, with a model based on hyperbolic functions. In addition, (1) shows a continuous derivative in the practical operating region. The coefficients L

_{m}, β

_{m}, and L

_{deepsat}must be identified; they are obtained experimentally using the method proposed in [14] and briefly explained in Section 4.

## 3. Current Profile Calculation

_{ON}) of the power switch. This can be generalized to any SMPSs by considering an appropriate voltage value applied to the inductor.

_{L}at the inductor terminals is constant. In addition, the inductance drop in the saturation region is highly dependent on the magnetic core temperature.

_{Lnl,min}decreases. However, the current peak i

_{Lnl,max}increased more than that of the previous decrease. It is worth noting that the current variation in the non-linear case Δi

_{Lnl}, is higher than the value observed in the linear case Δi

_{L}.

_{core}, shows a slow variation (practically, the temperature remains constant during a switching period, as the latter is much shorter than the system thermal constant), (2) gives:

_{ON}, is used as a starting point because this current peak is known because it is imposed by the design (i.e., the current at which the inductance is halved to fully exploit saturation). Therefore, the waveform of i

_{L}during the conduction time was obtained by evaluating (4), starting from t = T

_{ON}and proceeding backward until t = 0.

## 4. Inductor Characterization

^{®}. A complete description of the characterization system can be found in [14]; here, only the fundamentals are given. The inductor under the test was placed in a DC/DC converter with a variable active load. The LabVIEW instrument imposes a switching frequency, duty cycle of the power switch, and DC bias current by varying the load. Finally, it calculates inductance L using the ratio between the voltage applied to the inductor (maintained constant) and the slope of the current:

_{L}is the effective voltage applied to the inductor without parasitic resistive voltage drop. This avoids the fact that the error is crucial, particularly for high currents. The automatic system measures inductance based on (7) by increasing the current to saturation. Each measurement was performed while maintaining a constant temperature of the inductor. The polynomial model described by (1) was then retrieved by interpolation. The output of this procedure consists of the coefficients used to model inductors L

_{m}and β

_{m}, and the deep saturation value L

_{deepsat}. To avoid that noise corrupting the evaluation of the current derivative, the measurement was repeated many times, calculating the average and standard deviation, corresponding to the true value and error, respectively. Based on the propagation error formula applied in (7), it can be noted that the error ΔL on the value of L decreases when the derivative increases, that is, for the values of the inductance outside the linear region:

_{L}indicates the time derivative of the current in the denominator of (8).

_{NOMINAL}220 µH). This component was characterized by the system described in [14] and summarized in Section 4, and the coefficients of (1) were obtained and are summarized in Table 1. The experimental dataset and the corresponding polynomial model are compared in Figure 3.

## 5. The Boost Converter Used as Case Study

_{NOMINAL}220 µH). The converter employed for the validation was a boost supplied by Vs = 24 V, and the switching frequency F

_{S}was equal to 30 kHz. It adopts an FDP12N60NZ MOSFET as the switch and a rectifier diode, STTH806. It has been loaded with a resistance of 20 Ω, and the output capacitance is equal to 33 μF. These values allow us to reach a maximum current of about 5 A, which is beyond the saturation limit (as noticeable from Figure 3), assessing that it is possible to extend the working operating range of a linear inductor with a suitable analysis. In fact, the rated current of the inductor is 2.4 A. Besides, an inductor with a rated current of about 5A operated in linear conditions would require a bigger, and more expensive, core. All measurements have been performed with a digital oscilloscope with a sample rate of 1 GS/s. The test rig used for the measurements is shown in Figure 4a and depicted in a pictorial diagram in Figure 4b. The test rig is controlled by a PC able to deliver the duty cycle based on the required maximum current and magnetic core temperature.

_{ON}on the inductor current peak. Figure 5 shows oscilloscope plots of the measurements performed on the current flowing through the MOSFET. The current was measured using a current probe TEKTRONIK TCP0020. The converter was operated in the CCM. Figure 5a, corresponding to T

_{ON}= 15 μs, shows a linear current, whereas Figure 5b, corresponding to T

_{ON}= 17 μs, shows a non-linear effect owed to the inductor. It is remarkable that a 13% increase in T

_{ON}caused a 57% rise in the AC current peak; it increased from 0.81 A to 1.27 A, shifting the inductor operation from the linear to the non-linear zone.

_{ON}) and on the temperature as described in the following section.

## 6. Results

_{ON}, leads to inductor operation outside the linear region. Moreover, the current profile depends on temperature. Because both the duty cycle and temperature can vary during the converter operation, this confirms the importance of using a tool for calculating the current profile. The results demonstrate that the proposed method can calculate the current profile under different working conditions up to its maximum value by exploiting the non-linearity of the inductor. Figure 6 is arranged for recognizing the effects of the temperature and of the conduction time on the maximum current value at a glance. Indeed, a rise in the maximum value can be appreciated by either increasing temperature or the duty cycle. Besides, the increase of the duty cycle (corresponding to an increase in the conduction time of the power switch) highlights the shape that no longer fits with a triangular waveform emphasizing the nonlinear behavior. The numerical values of the peaks are summarized in Table 2 where it can be noticed that the increase of the duty cycle induces an increase in the current peak that is more relevant for the highest temperature.

^{4}steps. Figure 7, Figure 8, Figure 9 and Figure 10 compare the simulated and experimental data of the current waveform as a function of core temperature T

_{Core}for fixed duty cycles of 46, 48, 50, and 51%, respectively. By combining losses and inductor modeling, the relevant reduction in the peak current for increasing T

_{Core}, experimentally measured for each duty cycle, was correctly reproduced correctly. This effect is mainly due to the increased losses in the inductor due to the rising temperature.

## 7. Conclusions

_{ON}. The results showed that, based on a suitable inductor model, the proposed approach provides a complete assessment of the current, including the profile, its peak, RMS value, and spectrum. The analytical and simulation results agree with the experimental data measured using a boost converter. In addition, our analysis explains how the non-linear behavior of the inductor can be exploited when the duty cycle of an SMPS varies during operation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**An oscilloscope plot of the current flowing through the power switch shows the influence of T

_{ON}on saturation: (

**a**) a linear trend is observed for T

_{ON}= 15 µs, and (

**b**) a non-linear trend for T

_{ON}= 17 µs.

**Figure 6.**Oscilloscope plots of the current flowing through the inductor (blue), output voltage (cyan), and gate voltage (green) for increasing duty cycle ratio and temperature at a switching frequency of 30 kHz.

**Figure 7.**Comparison of the experimental current waveform with the simulation results from SPICE and the differential equation for a duty cycle of 46% at different temperatures.

**Figure 8.**Comparison of the experimental current waveform with the simulation results from SPICE and the differential equation for a duty cycle of 48% at different temperatures.

**Figure 9.**Comparison of the experimental current waveform with the simulation results from SPICE and the differential equation for a duty cycle of 50% at different temperatures.

**Figure 10.**Comparison of the experimental current waveform with the simulation results from SPICE and the differential equation for a duty cycle of 51% at different temperatures.

**Figure 11.**(

**a**) Comparison of the experimental current waveform with the simulation results from SPICE and differential equation for a duty cycle of 51% at 70 °C. (

**b**) Comparison of mean absolute error for SPICE and differential equation simulations.

**Figure 12.**FFT AC spectra of experimental inductor current for a duty cycle of 50% at different temperatures.

Coefficient | Value | Coefficient | Value (1/°C) |
---|---|---|---|

L_{0} | 262 × 10^{−6} | β_{0} | −0.0006 |

L_{1} | −28.8 × 10^{−6} | β_{1} | −0.0240 |

L_{2} | 22.9 × 10^{−6} | β_{2} | −0.0150 |

L_{3} | −3.72 × 10^{−6} | β_{3} | −0.0090 |

L_{deepsat} | 50 × 10^{−6} |

(A) | Temperature | |||
---|---|---|---|---|

Duty Cycle | 40 °C | 50 °C | 60 °C | 70 °C |

46% | 4.26 | 4.31 | 4.37 | 4.37 |

48% | 4.50 | 4.64 | 4.71 | 4.65 |

50% | 4.76 | 4.93 | 5.06 | 5.12 |

51% | 4.98 | 5.00 | 5.31 | 5.39 |

Temperature (°C) | Duty Cycle (%) | Experimental (A) | SPICE Simulation (A) | Differential Equation Simulation (A) |
---|---|---|---|---|

70 | 51 | 3.94 | 4.28 (+8.6%) | 4.06 (+3.0%) |

60 | 51 | 4.05 | 4.22 (+4.2%) | 4.10 (+1.2%) |

50 | 51 | 4.22 | 4.45 (+5.5%) | 4.30 (+1.9%) |

40 | 51 | 4.30 | 4.51 (+4.9%) | 4.37 (+1.6%) |

**Table 4.**Mean absolute error (in mA) of the inductor current for different duty cycles and temperatures for the SPICE simulations.

(mA) | Temperature | |||
---|---|---|---|---|

Duty Cycle | 40 °C | 50 °C | 60 °C | 70 °C |

46% | 45.8 | 30.3 | 44.9 | 40.5 |

48% | 31.2 | 26.1 | 24.4 | 29.8 |

50% | 92 | 38.9 | 53.4 | 27.4 |

51% | 59.7 | 85.6 | 55.5 | 94.6 |

**Table 5.**Mean absolute error (in mA) of the inductor current for different duty cycles and temperatures for the differential equation simulations.

(mA) | Temperature | |||
---|---|---|---|---|

Duty Cycle | 40 °C | 50 °C | 60 °C | 70 °C |

46% | 36.4 | 33.3 | 37.6 | 97.9 |

48% | 25.4 | 33.2 | 34.2 | 21.6 |

50% | 46.6 | 51.2 | 68.2 | 37.4 |

51% | 73.2 | 83.1 | 35.1 | 113.4 |

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**MDPI and ACS Style**

Scirè, D.; Lullo, G.; Vitale, G.
Assessment of the Current for a Non-Linear Power Inductor Including Temperature in DC-DC Converters. *Electronics* **2023**, *12*, 579.
https://doi.org/10.3390/electronics12030579

**AMA Style**

Scirè D, Lullo G, Vitale G.
Assessment of the Current for a Non-Linear Power Inductor Including Temperature in DC-DC Converters. *Electronics*. 2023; 12(3):579.
https://doi.org/10.3390/electronics12030579

**Chicago/Turabian Style**

Scirè, Daniele, Giuseppe Lullo, and Gianpaolo Vitale.
2023. "Assessment of the Current for a Non-Linear Power Inductor Including Temperature in DC-DC Converters" *Electronics* 12, no. 3: 579.
https://doi.org/10.3390/electronics12030579