Electromagnetic field analysis on resonance phenomenon of inductor driven by inverter considering stray capacitance

Abstract

In inductors excited by the inverter using gallium nitride power devices, the ringing phenomenon, which occurs due to the multi-resonances during the short rise and fall time, increases noise and loss. To clarify the mechanism of the multi-resonances, the high frequency (HF) behaviors of a ring core inductor in frequency domain are measured and simulated by using the 2D electromagnetic field finite element analysis taking account of the stray capacitance. It is shown that the multi-resonances can be simulated, and the simulated results are in good agreement with the measured ones. To explain this phenomenon, a distributed-element circuit model including the stray capacitance of the ring core inductor is proposed based on the simulated results, it is shown that the proposed circuit model can represent the HF behaviors of the ring core inductor well.

Keywords

distributed-element circuit electromagnetic field analysis inductor multiple resonances stray capacitance

Introduction

Recently to make power electronics equipment more compact and higher efficiency, the carrier frequency is increasing by applying the gallium nitride (GaN) – Field Effect Transistor, etc.¹ In inductors excited by the inverter using GaN power devices, the ringing phenomenon,^2,3 which is the oscillation in the waveform of current occurs due to the multi-resonances during the short rise and fall time of input voltage, and it increases the noise and iron loss. The resonances are simulated by the equivalent circuits^4,5 and the electromagnetic field finite element analysis (FEA)⁵ taking account of the stray capacitance. However, the mechanism of multi-resonances seems not to be clear well.

In this paper, to clarify the physical mechanism of the multi-resonances, first, the high frequency (HF) characteristics of the input admittance $\dot{Y}$ of a ring core inductor are measured. Then, the 2D electromagnetic field FEA in frequency domain taking account of the stray capacitance is carried out and verified by comparing with the measured results. In FEA, to model the wire of winding completely within smaller computer cost, a 2D simple model, in which a numerical connection of the wires is introduced and the skin effect of the core is represented by the effective permeability, is proposed. Then, the physical mechanism of the multi-resonance is explained from the viewpoint of the wave propagation based on the results obtained from the FEA. Moreover, for further clarification, the distributed-element circuit of the ring core inductor is proposed and verified.

Measurement

Measurement model

The measurement model of the ring core inductor, in which a winding is wound around a plastic casing outside the ring core, is shown in Figure 1. The core is a magnetic solid one (JIS: SUS430) and the height h_c of 35.35 mm is set to be much longer than the width w_c of 2.46 mm so that 2D FEA can be applied. The winding has 254 turns of copper wire with the diameter ϕ_w of 1 mm and the total length l_w of the wire is 24.26 m.

Figure 1.

Measurement model, (a) whole view, (b) sectional view.

Measured result

The frequency characteristics of the input admittance $\dot{Y}$ of the ring core inductor are measured by using the impedance spectrum analyzer (NF Corporation: ZA57630), and the results are shown in Figure 2. In Figure 2(a), the multi-resonances occur at the frequencies of 8.28 MHz, 18.9 MHz, and 27 MHz, which lead to the ringing phenomenon in the inductor. From the phase characteristics of the admittance $\dot{Y}$ in Figure 2(b), we found that the inductor behavior is switching between the inductive one and the capacitive one due to the effect of the stray capacitance at HF.

Figure 2.

Input admittance $\dot{Y}$ of the ring core inductor (measurement), (a) amplitude, (b) phase.

Electromagnetic field analysis

Analysis method

The electromagnetic field is analyzed by using the A -ϕ method ( A : magnetic vector potential, ϕ: electric scalar potential) in frequency domain taking account of the displacement current which leads to the stray capacitance. The fundamental equations are as follows:

rot (\frac{1}{\dot{μ}} rot \dot{A}) = - (σ + j ω ε) (j ω \dot{A} + grad \dot{ϕ})

(1)

div {- (σ + j ω ε) (j ω \dot{A} + grad \dot{ϕ})} = 0

(2)

where μ, σ, and ε are the permeability, the conductivity, and the permittivity, respectively, and

ω

is the angular frequency. The superscript (

\dot{}

) denotes the complex number.

Analysis model and conditions

In the analysis, the measurement model of the ring core inductor shown in Figure 1 is simplified into the 2D finite element (FE) model shown in Figure 3.

Figure 3.

Simplified 2D FE model of ring core inductor, (a) full model, (b) enlarged view of region R.

The winding is simplified as the straight conductive bars, which are connected into a winding by using the numerical connection of nodes on the boundary. The voltage ${\dot{V}}_{i n}$ between the terminals A and B is given by ${\dot{ϕ}}_{A}$ = 5 V and ${\dot{ϕ}}_{B}$ = −5 V, and the input current ${\dot{I}}_{i n}$ is induced in the conductor. The height h of the conductive bars is enlarged from 35.35 mm to 47.76 mm to keep the sum of the lengths of all conductive bars same as the total length l_w of the wire in the measurement model. Moreover, to keep the sectional area of the core the same as the measurement model, the width w_c of the core is reduced from 2.46 mm to 1.82 mm. And the distances between the winding and the core are kept the same as the measurement model.

The magnetic characteristics of iron core are assumed to be linear and the eddy current in the iron core is considered by the relative effective permeability ${\dot{μ}}_{r e}$ , which is a complex variable, obtained by the eddy current analysis of core only.⁶

Results and discussion

Admittance $\dot{Y}$

First, the admittance $\dot{Y}$ of the ring core inductor is calculated by dividing the input current ${\dot{I}}_{i n}$ of the winding by the applied voltage ${\dot{V}}_{i n}$ (= ${\dot{ϕ}}_{A} - {\dot{ϕ}}_{B}$ ). In Figure 4, The input admittance $\dot{Y}$ obtained from the FEA is compared with the measured one shown in Figure 2, and they are in good agreement with considering the stray capacitance, although there are some differences in the region above 16 MHz, which is thought to be due to the 2D approximation and the differences between the material constants given in the analysis and the actual ones.

Figure 4.

Input admittance Y of the ring core inductor (measurement and FEA), (a) amplitude, (b) phase.

Distributions of displacement current and electric scalar potential $\dot{ϕ}$

Next, to investigate the mechanism of the resonance phenomenon in the ring core inductor, the distributions of displacement current and voltage are shown in Figure 5. In Figure 5(a), the displacement currents between the wires and core and between each turn of the wires can be observed, which lead to the stray capacitance of the winding. In Figure 5(b), from the distribution of the absolute value of the electric scalar potential ϕ at the wire and core versus azimuth θ at the resonant frequencies, the standing waves can be observed and ϕ becomes lager than the input source at terminal A of the winding.

Figure 5.

Distribution of displacement current density and voltage, (a) displacement current outside the wire and core at region R (f = 8 MHz), (b) voltage of the wire and core.

The wavelength of the electromagnetic wave (f = 8 MHz) is from 21.65 m to 37.50 m with different velocity in air (ε_r= 1, μ_r= 1, c_air= 3.00 × 10⁸m/s) and plastic case (ε_r= 3, μ_r= 1, c_air= 1.73 × 10⁸m/s), and the length of the wire is 24.26 m, which is in this range. Therefore, the multi-resonance phenomenon can be explained by the wave propagation process and self-resonance explained by the distributed-element circuit of the transmission line, besides, the distributed-element circuit analysis is performed to verify this explanation in the Section 4.

Distributed-element circuit analysis

Circuit model

To explain the multi-resonances of the ring core inductor from the perspective of circuit analysis, the distributed-element circuit shown in Figure 6 is proposed based on the electromagnetic field analysis and the transmission line theory. ${\dot{L}}_{s}$ is the self-inductance of the winding in each turn and ${\dot{M}}_{n m}$ is the mutual inductance between the n^th and the m^th turns of the winding, which are complex variables due to the eddy current effect. C_w is the stray capacitance between each turn of the winding and C_c is the stray capacitance between the winding and the core. $\dot{ϕ} (n)$ and $\dot{I} (n)$ are the electric scalar potential and current of the n^th turn of the winding, respectively, and ${\dot{ϕ}}_{c o r e}$ is the electric scalar potential of the core, which is set to be uniform according to the results in the Figure 5(b). N is the total turn number of the winding.

Figure 6.

Distributed-element circuit of ring core inductor.

Circuit analysis method

The basic circuit equations of the distributed-element circuit shown in Figure 6 are as follows:

\dot{ϕ} (n) - \dot{ϕ} (n + 1) - j ω {\dot{L}}_{s} \sum_{m = 1}^{N} {\dot{k}}_{n m} {\dot{I} (m) - j ω C_{w} [\dot{ϕ} (m) - \dot{ϕ} (m + 1)]} = 0

(3)

\dot{I} (n) - \dot{I} (n + 1) - j ω C_{c} [\dot{ϕ} (n + 1) - {\dot{ϕ}}_{c o r e}] = 0

(4)

where

{\dot{k}}_{n m}

is the coupling coefficient between the n^th turn and the m^th turn of the winding.

Results and discussion

The circuit parameters can be obtained using the analysis model shown in Figure 3. The stray capacitances C_w, C_c are calculated by using the electrostatic field analysis. The calculated values are C_w= 3.15 pF/turn, C_c= 1.1 pF/turn. The self-inductance ${\dot{L}}_{s}$ and the coupling coefficient ${\dot{k}}_{n m}$ can be calculated by using the linear magnetic field analysis using the relative effective permeability ${\dot{μ}}_{r e}$ that varies with frequency due to the skin effect.

The frequency characteristics of $\dot{Y}$ obtained from the distributed-element circuit are compared with the results obtained from FEA in Figure 7. The results obtained from the proposed circuit model are in good agreement with the results obtained from FEA, and it can successfully explain the HF behaviors of the ring core inductor.

Figure 7.

Input admittance Y of the ring core inductor (FEA and circuit analysis), (a) amplitude, (b) phase.

Conclusion

The frequency characteristics of the ring core inductor at HF are analyzed by using the 2D electromagnetic field analysis taking account of the displacement current. The simulated input admittance $\dot{Y}$ is in good agreement with the measured one. Moreover, the distributed-element circuit of the ring core inductor is proposed based on the FEA. The HF behaviors of the ring core inductor obtained from the proposed circuit model are in good agreement with those obtained from the FEA. Therefore, it can be concluded that the multi-resonances are explained by the wave propagation process in the transmission line theory.

In the future research, the FEA taking account of the nonlinearities or frequency characteristics of the material constants of the iron core and plastic case should be carried out. Moreover, the method to remove the ringing phenomenon in inductors will be investigated using the electromagnetic field analysis and the circuit analysis proposed in this paper.

Footnotes

Acknowledgements

This work was partly supported by JSPS KAKENHI Grant Number JP23K03820.

ORCID iD

Xuanda Hou

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Electromagnetic field analysis on resonance phenomenon of inductor driven by inverter considering stray capacitance

Abstract

Keywords

Introduction

Measurement

Measurement model

Measured result

Electromagnetic field analysis

Analysis method

Analysis model and conditions

Results and discussion

Admittance Y ˙

Distributions of displacement current and electric scalar potential ϕ ˙

Distributed-element circuit analysis

Circuit model

Circuit analysis method

Results and discussion

Conclusion

Footnotes

Acknowledgements

ORCID iD

Funding

Declaration of conflicting interests

References

Admittance $\dot{Y}$

Distributions of displacement current and electric scalar potential $\dot{ϕ}$