# Identification of DC Thermal Steady-State Differential Inductance of Ferrite Power Inductors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Impact of Losses in an Inductor for DC–DC Converter

#### 2.2. Experimental Method for the Differential Inductance Measurement

_{SAT}= 150 A) was added in series. For each load current condition, after a reasonable amount of time passed to reach thermal equilibrium, the choke was short-circuited, and the inductor voltage and current were quickly measured, assuming the temperature was unchanged. Measurements were performed with a digital oscilloscope, using a differential probe to measure inductor voltage and a current probe to measure the relative current (Figure 1). The measured voltage was corrected from the winding resistive voltage drop, as described in [9]

_{L}(t), Φ(t) is obtained as

_{L}) characteristic is obtained. Due to the magnetic dissipative effects, it is a loop with a non-zero area [20]. The measurements were performed at duty cycle δ = 0.5, allowing us to assume the loss in the two half-periods was equal. Thus, the loop was symmetrical. Therefore, the average characteristic was adopted as the experimental Φ(i

_{L}) one. Assuming that in an appropriately defined range of current sweep around the average current, the Φ(i

_{L}) characteristic is linear, as shown in Figure 2; the differential inductance at each tested load current value is determined as

- input voltage: 24 V;
- duty cycle: 0.5;
- switching frequency: 500 kHz;
- output current: 1–5.6 A.

_{L}) curve is now parametrized according to [4]

_{H}, L

_{L}, σ, I*] that better describe the experimental curve were obtained with a deterministic pattern search optimization algorithm [21], minimizing an objective function (f

_{min}) defined as the difference between the experimental and the parametrized differential inductance profile [9]:

- input voltage: 24 V;
- duty cycle: 0.5;
- switching frequency: 500 kHz;
- output current: 1–6.2 A.

## 3. Current Waveform Simulation Results

_{L}) characteristics, referring to the thermal behavior of the inductor caused by DC losses only, could be used to simulate current waveforms under different operating conditions tested experimentally. The nonlinear differential equations of a saturable inductor in a DC–DC buck converter circuit were solved through the polarization fixed point method [22,23].

#### 3.1. MSS1260T-273 Inductor

#### 3.2. SER1390-333 Inductor

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

I_{SAT} | DC current at 25 °C that causes a 30% inductance drop from its value without current. Declared by the manufacturer. |

v_{L,exp} | Measured voltage across the inductor. |

v_{L} | Inductive component of the voltage across the inductor. |

R_{s} | Winding resistance. |

i_{L} | Inductive current. |

Φ | Magnetic flux. |

L(i_{L}) | Differential inductance. |

L_{H} | Upper horizontal asymptote of the differential inductance curve. |

L_{L} | Lower horizontal asymptote of the differential inductance curve. |

I* | Abscissa of the inflection point of the differential inductance curve. |

σ | Coefficient proportional to the slope of the curve in I*. |

I_{L,max} | Maximum value of the tested average current. |

I_{L,min} | Minimum value of the tested average current. |

L_{exp} | Experimental differential inductance curve. |

L_{par} | Parametrised differential inductance curve. |

f_{min} | Objective function of the optimization problem. |

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**Figure 1.**Circuit diagram of the experimental setup for DC thermal steady-state differential inductance identification. A high inductance choke allows imposing negligible current ripple on the inductor under test. After reaching the DC thermal steady-state for a given average output current, the choke is short-circuited, and the inductor voltage and current waveforms are measured with a digital oscilloscope.

**Figure 2.**Average Φ(i

_{L}) characteristics obtained for each measured output current value. The magnetic flux of each section is defined minus a constant term.

**Figure 3.**Experimental DC thermal steady-state differential inductance curve for the Coilcraft MSS1260T-273 inductor compared to the curve obtained from ambient temperature measurements (25 °C).

**Figure 4.**Fitting of the parametrized characteristic on the experimental differential inductance curve for the Coilcraft MSS1260T-273 inductor.

**Figure 5.**Fitting of the parametrized characteristic on the experimental differential inductance curve for the Coilcraft SER1390-333 inductor.

**Figure 7.**Coilcraft MSS1260T-273 inductor, 500 kHz. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range; (

**c**) relative error of the estimation of the RMS value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range.

**Figure 8.**Coilcraft MSS1260T-273 inductor, 750 kHz. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current values. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range; (

**c**) relative error of the estimation of the RMS value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range.

**Figure 9.**Coilcraft MSS1260T-273 inductor, 1 MHz. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range; (

**c**) relative error of the estimation of the RMS value of the inductive current, for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range.

**Figure 10.**Coilcraft MSS1260T-273 inductor, 12 V. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current; (

**c**) relative error of the estimation of the RMS value of the inductive current.

**Figure 11.**Coilcraft MSS1260T-273 inductor, 32 V. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current; (

**c**) relative error of the estimation of the RMS value of the inductive current.

**Figure 12.**Coilcraft SER1390-333 inductor, 500 kHz. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range; (

**c**) relative error of the estimation of the RMS value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range.

**Figure 13.**Coilcraft SER1390-333 inductor, 750 kHz. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range; (

**c**) relative error of the estimation of the RMS value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range.

**Figure 14.**Coilcraft SER1390-333 inductor, 1 MHz. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range; (

**c**) relative error of the estimation of the RMS value of the inductive current for each tested output current value. The orange bars represent the test cases reported in (

**a**), the blue bars are related to the other measurements of the evaluated current range.

**Figure 15.**Coilcraft SER1390-333 inductor, 12 V. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current value; (

**c**) relative error of the estimation of the RMS value of the inductive current for each tested output current value.

**Figure 16.**Coilcraft SER1390-333 inductor, 32 V. (

**a**) Comparison of experimental and simulated current waveforms for different output current values; (

**b**) relative error of the estimation of the peak value of the inductive current for each tested output current value; (

**c**) relative error of the estimation of the RMS value of the inductive current for each tested output current value.

**Figure 17.**Coilcraft MSS1260T-273 inductor. Comparison of experimental and simulated current waveforms for different output current values and different switching frequencies.

MSS1260T-273 (I_{SAT} = 4.7 A) | |||||

Input Voltage (V) | Duty Cycle | Switching Frequency (kHz) | Output Current (A) | DC Losses (mW) | AC Losses (Core + AC Winding) (mW) |

24 | 0.5 | 500 | 4 | 768 | 74 |

32 | 0.5 | 500 | 4 | 768 | 132 |

24 | 0.5 | 1000 | 4 | 768 | 31 |

SER1390-333 (I_{SAT} = 4.8 A) | |||||

Input Voltage (V) | Duty Cycle | Switching Frequency (kHz) | Output Current (A) | DC Losses (mW) | AC Losses (Core + AC Winding) (mW) |

24 | 0.5 | 500 | 4 | 336 | 44 |

32 | 0.5 | 500 | 4 | 336 | 79 |

24 | 0.5 | 1000 | 4 | 336 | 34 |

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**MDPI and ACS Style**

Musumeci, S.; Solimene, L.; Ragusa, C.S.
Identification of DC Thermal Steady-State Differential Inductance of Ferrite Power Inductors. *Energies* **2021**, *14*, 3854.
https://doi.org/10.3390/en14133854

**AMA Style**

Musumeci S, Solimene L, Ragusa CS.
Identification of DC Thermal Steady-State Differential Inductance of Ferrite Power Inductors. *Energies*. 2021; 14(13):3854.
https://doi.org/10.3390/en14133854

**Chicago/Turabian Style**

Musumeci, Salvatore, Luigi Solimene, and Carlo Stefano Ragusa.
2021. "Identification of DC Thermal Steady-State Differential Inductance of Ferrite Power Inductors" *Energies* 14, no. 13: 3854.
https://doi.org/10.3390/en14133854