# Modeling and Measurements of Properties of Coupled Inductors

^{1}

^{2}

^{*}

## Abstract

**:**

_{DC}, of currents in the windings on the parameters of the considered element. A description of the model and methods of measuring parameters of the inductor using an impedance analyzer and a chamber for thermal measurements is given. The obtained results of measurements are compared with the results of calculations proving a satisfactory match.

## 1. Introduction

## 2. Coupled Inductor

_{i}, B

_{i}(i = 1, 2, …, n) corresponding to the winding terminals of the modeled inductor are used for the analysis of electronic systems.

_{i}, and a branch where there are 4 elements connected in series: a voltage source, V

_{Li}, of a zero value; an inductor, L

_{i}; the controlled voltage source, E

_{LSi}; a resistor, R

_{Si}. The voltage source (V

_{Li}) monitors the value of the inductor’s (L

_{i}) current. Additionally, only one branch has the third branch that includes the controlled current sources, G

_{PR1}, modeling power losses in the inductor core. The L

_{i}inductor enables for the determination of a time derivative of the inductor’s current, and its inductance is arbitrarily assumed to be 10 µH. The voltage source, E

_{LS}

_{i}, corresponds to the inductance of the i-th inductor winding and is expressed by the equation, which is a modified version of the dependence describing inductance of an inductor containing one winding, given in [26]. This equation has the form:

^{2}, z is the number of turns in the i-th winding, S

_{Fe}is the effective cross-section area of the core, α

_{s}is the coefficient that depends on the magnetic saturation of the core, l

_{Fe}is the magnetic flux path length in the core, V(L

_{i}) is the voltage on the linear inductor L

_{i}, and µ is the magnetic permeability of the inductor core expressed by the empirical dependence in the form:

_{H}is the coefficient depending on the magnetic field.

_{i}, represents the capacity of the winding, and the resistor, R

_{Si}, represents the series resistance of a single-given winding for the direct current at the winding temperature equal to the reference temperature T

_{0}. The resistance of the resistor, R

_{Si}, is described by the formula given in the papers [16,42]:

_{d}is the winding length, and S

_{d}is the cross-sectional area of the winding wire.

_{PR1}, takes into account the losses in the inductor core resulting from the magnetic field excited in the windings. The efficiency of this source is expressed by the equation

_{v}

_{0}, α are parameters that depend on the type of the material used to build the inductor core.

_{0}denotes a model parameter.

_{LMi}, described by following formula:

_{i}are the coupling coefficients between the individual coils, and V(E

_{LSj}) is the output voltage on the controlled voltage source, E

_{LSj}.

## 3. Investigated Component

_{Fe}, l

_{Fe}, l

_{d}, and S

_{d}—determined on the basis of the catalog data available on the manufacturer’s website [43], and magnetic parameters—α

_{H}, α

_{S}, k

_{1}and k

_{2}—obtained with the use of the idea of local estimation described in [44]. The input data for this procedure were the measured L

_{i}(i) and M(i) characteristics of the considered inductor and the manufacturer’s data [43].

## 4. Measurement Setup

_{x}. The resistor R = 10 Ω was used to limit the value of the direct winding current of the investigated inductor from the V

_{DC}power source. The value of this current in the range from 0 to 10 A was measured with a UNIT-T 803 multimeter (A). However, a current value above 10 A was measured using a PINTEK PA-677 (CP) current probe and recorded by a RIGOL DS 1052E (OSC) oscilloscope.

_{1}and C

_{2}, with a of capacity 6.7 µF were used to protect the measuring bridge against the direct current, and L inductor of inductance 1.45 mH was used to suppress alternating current flowing through the V

_{DC}power supply. The measurements of inductance were performed with the use of an automatic bridge of the type RLC Motech MT4090 (RLC) [45]. The RLC bridge operated in the inductance measurement mode in a series equivalent circuit. The amplitude of the voltage generated by the RLC bridge was 1 V.

_{x}had to be met. The blocking inductor L consisted of 5 identical inductors connected in series. Each of them contained a powdered iron core (material type −102) with 20 turns of copper wire in enamel with a diameter of 2 mm. The geometric dimensions of the inductor: outer diameter, 10.3 cm; inner diameter, 5.6 cm; height, 6.5 cm. Figure 5 shows the measured dependence L(I

_{DC}) of the blocking inductor. The measurements were performed in the setup shown in Figure 3.

_{DC}in the range from 0 to 5 A was almost constant. For I

_{DC}> 5 A, inductance of the inductor decreased to a value of approximately 800 µH at the current I

_{DC}of approximately 19 A.

## 5. Results

#### 5.1. Validation of the Coupled Inductor Model

_{DC}on the parameters of the inductor were performed using the measuring setup presented in Section 4. The obtained measurement results were compared with the calculation results obtained with the use of the model discussed in Section 2. Except Figure 6 for other figures presented in this section, the principle was adopted that the points denote the measurement results and the lines are the calculations results obtained using the model described in Section 2.

_{DC}) in the setup shown in Figure 3 and using the dependence [42]:

_{1}of a single winding, inductances of the series connected two windings L

_{12}and the series connected three windings L

_{123}of the considered inductor on direct current I

_{DC}at a frequency of the exciting signal f = 200 kHz from the RLC Bridge.

_{DC}) presented in Figure 8 show that the series connection of the successive windings on a common core causes an almost two-fold increase in inductance L

_{12}of the inductor when two coils are connected in a series, and more than a three-fold increase in inductance of the inductor in the case of a series connection of three coils. The results obtained from the calculations with the use of the proposed model are in good agreement with the measurement results. The mean square deviation of inductance between the results of measurements and calculations was equal to 1.87 µH in the range of I

_{DC}changes from 0 to 10 A. For I

_{DC}≥ 10 A the mentioned deviation of inductance was equal to 0.88 µH. The measurements of inductance of the coupled inductors containing windings wound in different directions were also performed and the obtained value of this parameter was equal to about 1 nH.

_{1}, in the investigated inductor on frequency. The calculations and measurements were conducted in the setup shown in Figure 3. The measurements were made with the value of the current I

_{DC}equal to zero. The measurements were made using the setup presented in Figure 3, but instead of the Motech RLC bridge, the LCR-8000G GWINTEK bridge [46] was used.

_{1}of the series connected two windings L

_{12}, and the series connected three windings L

_{123}in the investigated inductor on frequency. The measurements were made with the value of current I

_{DC}equal to zero.

_{123}and 1 MHz for L

_{1}.

#### 5.2. Results of the Computations Characteristics of a Flyback Converter

_{o}, and the duty cycle, d. The computations were performed for frequency, f, of the control signal equal to 200 kHz. The influence of the mentioned parameters on the value of the output voltage, V

_{out}, of this flyback converter was investigated.

_{1}of the type IRF840, the diode D

_{1}of the type D1N5822, the capacitor C

_{1}= 47 µH. The number of windings of the coupled inductors L

_{1}and L

_{2}was the same z

_{1}= z

_{2}= 6. The voltage source V

_{2}models a square wave generator with voltage levels equal to 0 and 10 V, respectively. It was assumed that the value of the input voltage V

_{in}is equal to 12 V.

_{DC}= 0, A was equal to approximately 338 µH. Thus, in the computations using the linear model, it was assumed that the value of L

_{1}= L

_{2}= 338 µH. In order to calculate the characteristics of the considered converter, the simulation circuit presented in Figure 12 was used. For clarity the auxiliary block was omitted in this figure.

_{out}of a flyback converter on load resistance, R

_{0}, (Figure 13a) and on the duty cycle, d, of the control signal (Figure 13b).

_{o}, of the considered converter caused an increase in the converter output voltage value in the whole considered range of R

_{o}from 1 to 10 kΏ. For a load resistance higher than 800 Ώ, it was observed that the converter changed the operating mode from the continuous current mode (CCM) to the discontinuous current mode (DCM). Additionally, the values of the output voltage obtained using the nonlinear model of the coupled inductor in the load resistance range from 1 to 10 Ω, and R

_{o}> 8 kΩ, respectively, were approximately 45% and 15% lower than the value of the output voltage obtained using the linear model of the coupled inductor.

_{o}= 1 Ω, the local maximum of V

_{out}voltage was observed for the duty cycle d = 0.7. However, for the mentioned value of d, the value of V

_{out}voltage computed using the linear model was equal to about 15.8 V, whereas using the nonlinear model it was equal to approximately 6.92 V only. On the other hand, for load resistance R

_{o}= 100 Ω, the output voltage of a flyback converter increased in function of the duty cycle of the control signal, and the greatest differences between the considered models were observed for d > 0.65. Using the nonlinear model allowed to obtain an approximately 15% lower value of the output voltage than if using the linear model of coupled inductor.

_{D}of the transistor included in the considered converter at the steady state at load resistances equal to 1 Ω (Figure 14a) and 100 Ω (Figure 14b).

_{D}(t) was observed, which is a result of the nonlinear dependence of the inductor’s inductance on the current.

_{in}is the input voltage of the boost converter, d is the duty cycle of the control signal, f is the frequency of this signal, and L is the inductance of the winding.

_{CR}(L) is a decreasing function of inductance. The highest values of the critical current I

_{CR}= 4 A were obtained when the current flowed only through one winding. The series connection of two windings caused an almost two-fold reduction in the value of the critical current and a more than three-fold reduction in its value in the case of a series connection of three windings of the considered inductor. Additionally, it was observed that for inductances above 100 µH, the value of the critical current for all three considered cases was almost 100 times lower than the value of the critical current with inductance of approximately 2 µH. It can also be seen that the value of the critical current when the three windings were connected in series decreased three times compared with the value of the critical current for a single winding of the considered inductor and was equal to 13 mA for L

_{123}. This means that the series connection of n windings reduces the value of the critical current and may change the boost converter operating mode from discontinuous (DCM) to continuous (CCM) at a constant load.

## 6. Discussion

_{DC}from 10 to 20 A, for which this element is dedicated.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Measured dependence of inductance of the coupled inductor on frequency for three values of ambient temperature with I

_{DC}= 0.

**Figure 7.**Dependence of magnetic flux density in the core on the magnetic force of the investigated component.

**Figure 8.**Dependence of inductance L

_{1}, L

_{12}, and L

_{123}on direct current I

_{DC}at f = 200 kHz.

**Figure 10.**Dependence of the module (

**a**) and phase (

**b**) of impedance on frequency for the inductors containing one, two, or three windings connected in a series.

**Figure 13.**Dependences of the output voltage of a flyback converter on load resistance (

**a**) and on the duty cycle (

**b**) of the control signal.

**Figure 14.**Waveforms of the drain current of the transistor operating in the flyback converter at R

_{0}= 1 Ω (

**a**) and R

_{o}= 100 Ω (

**b**).

**Figure 15.**Dependence of the critical current of the boost converter on inductance of the winding of the inductor.

z | S_{Fe} (m^{2}) | l_{Fe} (m) |
---|---|---|

6 | 114 × 10^{−6} | 0.102 |

α_{s} (A/m) | α_{H} | α |

1.79 | 0.034 | 0.01 |

A (A/m) | ρ (Ω · m) | S_{d} (m^{2}) |

6.5 | 17.24 × 10^{−9} | 3.14 × 10^{−6} |

k_{1} | k_{2} | P_{v}_{0} (W/s^{α}) |

0.9 | 0.9 | 3554.4 |

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**MDPI and ACS Style**

Detka, K.; Górecki, K.; Grzejszczak, P.; Barlik, R.
Modeling and Measurements of Properties of Coupled Inductors. *Energies* **2021**, *14*, 4088.
https://doi.org/10.3390/en14144088

**AMA Style**

Detka K, Górecki K, Grzejszczak P, Barlik R.
Modeling and Measurements of Properties of Coupled Inductors. *Energies*. 2021; 14(14):4088.
https://doi.org/10.3390/en14144088

**Chicago/Turabian Style**

Detka, Kalina, Krzysztof Górecki, Piotr Grzejszczak, and Roman Barlik.
2021. "Modeling and Measurements of Properties of Coupled Inductors" *Energies* 14, no. 14: 4088.
https://doi.org/10.3390/en14144088