2019 International Conference on Artificial Intelligence and Computing Science (ICAICS 2019) ISBN: 978-1-60595-615-2

Study on the Inductance Calculation Methods of Spiral Coils at High Frequency Conditions

A. Almenweer REEM, Yi-xin SU and Xi-xiu WU* School of Automation, Wuhan University of Technology, Luoshi Road No.205,

Wuhan 430070, Hubei, China

*Corresponding author

Keywords: Spiral coil, Inductance, High-frequency transient current, ANSYS Maxwell, MATLAB.

Abstract. In this paper, a deep analysis of modelling and simulation is conducted to calculate a fairly precise value of inductance of a coil at high-frequency using industry standard ANSYS Maxwell simulation tool and MATLAB. Two different approaches to the simulation of inductance calculation are presented. The calculation of self-inductance in the first approach is computed by assigning the high-frequency components as a variable, which is used for sweeping analysis, which makes the analysis simpler and achieves the purpose of this research. Results proved that the self-inductance of a coil falls off with the frequency. Furthermore, an optimization algorithm is carried out in the second approach to design a coil, based on a specific inductance value at high-frequency when transient current flows through it.

Introduction

The distribution of low-frequency current through the wires of a coil is commensurate over the cross sections of the wires. However, with increasing the frequency, the current density becomes major at the surface of the wire which means that this uniform density current does not dominate any longer. The theory of this distribution in straight wires has been thoroughly worked out by Lord Rayleigh and by Stefan and showed that the effect of increasing frequency is to diminish self-inductance [1]. However, some references have shown a set of inductance calculation results for different shaped conductors [2-4]. Reference [5] gives correction factors to account for high-frequency operation and calculates inductance of different types of wire loops. Furthermore, reference [6] calculates high-frequency effects in inductors and covers skin and proximity effect.

Since many references calculate inductance of different types of shaped conductors, this study attempts not only to calculate the inductance, but also to find methods which could firstly calculate the inductance of any model especially when it is complicated, and secondly find an approach which could be used to design a coil based on a specific inductance value. This paper discusses the methods whose purpose is to calculate self-inductance at high frequency during the transient occurrence. In this paper, different two methods to calculate a fairly precise value of inductance of a coil are presented: First method takes into account the effects of high-frequency fields with all the geometrical parameters of a coil by ANSYS Maxwell software based on reference [1] which explains clearly the influence of frequency upon the self-inductance of coils. The second method depends on a mathematical calculation to find the best design of a coil by determining a specific value of inductance, a number of turns, high-frequency value, and current peak value during the transient occurrence as input parameters for GUI of MATLAB software, thus the dimensions of the designed coil are calculated.

This method gives the designed coil of a specific inductance value, but in the first method, a specific model is imported to ANSYS software for inductance calculation when a high-frequency current flows through it. This paper is organized as follows, Section 2 presents the calculation theory, Section 3 presents the methods used to perform the simulations and analysis, and Section 4 presents the conclusions of the study.

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Theoretical Consideration

A clear explanation of the theoretical consideration is presented in reference [1]. The resistance and self-inductance for any frequency for a straight conductor traversed by sinusoidal currents are given by the following expressions in infinite series as Lord Rayleigh illustrated:

2 41 1[1 ]

12 180R R B B

2 41 1 13[ ( )]

2 48 8640L l A B B

(1)

where: /B l R , R is the resistance for steady currents, µ is the magnetic permeability of the conductor, l is the length of the conductor, ω is 2×frequency of the current. A is determined by the following formula for the self-inductance for currents of low frequency:

1( )

2L l A

(2) 2

/R l is the resistance for steady currents, where ρ is the radius of the wire and σ is the specific conductivity of the wire. When the frequency is great, or rather when µ, by the frequency has a great value, these expressions become:

1/21( )2

R l R

1/2[ ( ) ]2

RL l A

l

(3)

For any frequency, the apparent resistance of the wire and also the new value of the self-inductance could be calculated. Now for coils, the mutual inductance between two parallel circles, over a circular cross-section is obtained by integration of Maxwell's series [1]:

2 2 3 2

2 3

2 2 3 2

2 3

8 3 34 . lg (1 )

2 16 32

3 64 ( 2 )

2 16 48

a x x v x xyM a

r a a a

x x y x xya

a a a

(4)

where the radiuses of the two circles are a and a x , where y is the distance between their planes and r the shortest distance between them, then it is obtained the following formula for the self-inductance of a circular coil of the circular cross section:

2 2

2 2

84 [(1 )lg 1.75 0.0083 ]

8

aL a

a a

(5)

where ρ is the radius of the cross-section of the wires and a is the mean radius of the coil. By considering inside 1r radius, 2r outside radius, and d thickness for large values of ω.

2 2 21 1 / 2

1

5 14 [1 ( ) ]

4 ( 2 )

dL N r

r d

(6)

In the formula, 2 2 214 N r is the self-inductance per unit length of a coil with a mean radius 1r , and

L is the self-inductance for large values of ω. The theory of our research study is supported by this

theoretical framework with some modifications and additions. However, inductance calculation methods are illustrated in detail in the next section

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Inductance Calculation Methods

In this section, the inductance calculation methods are shown for a finite-length designed coil that its application is founded in high-frequency equipment:

ANSYS Maxwell

To implement the inductance calculation at high-frequency by the Full-Maxwell finite element method, so some assumptions are proposed for the ease of simulation [7]. At first, the simple geometry model is imported to ANSYS Maxwell with the frequency of operation and transient current peak value by choosing the Eddy Current solution type. This model depends on many variables according to ANSYS analysis: material type, the geometry of the design, frequency of operation and transient current. Then, mesh operation can be added from the menu item Maxwell 2D/3D>Mesh Operations > Assign > On Selection > Length Based. Moreover, select Maxwell > Excitations > Assign > Current to assign a transient current peak value as current excitation. After that, in the Project Manager window of ANSYS Maxwell, right click on the Setup, and go to the tab sweep definitions and add its sweep setup as shown in Fig. 1(a).

Finally, and after specifying the sweep parameters, the file is saved in Maxwell, and the Run button on the toolbar starts the simulation. The self-inductance of the coil is calculated for every iteration for the sweep parameters specified for the frequency value. Once the simulation is carried out the results are viewed by right-clicking on Data Table of the Results tab of the Project Manager window of Maxwell (see Fig. 1(a)). Fig. 1(b) shows the result of inductance calculation of the coil which clarifies the relation between the inductance and high frequency. Consequently, self-inductance of a coil falls off with the frequency which has been proved in [1] by some mathematical calculations as shown clearly in Fig. 1(b) by ANSYS Maxwell method.

(a) (b) Figure 1. (a) ANSYS project procedure and frequency sweep, (b) Inductance calculation by ANSYS Maxwell.

MATLAB Optimization Algorithm

As we referred above that the main purpose of this research work is to design a coil which its application is in high-frequency field. Thus, another concept is presented in this paper which depends on finding the best design of a coil with an aluminum core which could resist the power losses when high-frequency currents flow through it. Firstly, the main idea is to find a method to calculate the inductance for a designed coil with an air gap at high frequency after choosing a suitable value of AWG (Gage of the wire) determined by a specific value of frequency up to 2.8 MHz (this value is chosen in our case study). Secondly, determine the desired inductance value and the number of turns. After that, the magnetic flux density is calculated. Finally, power losses are calculated to find the final design according to a mathematical calculation. By MATLAB optimization algorithm, the diameter of the coil is calculated, and the final design is determined.

Depending on the concept being introduced in this section, Litz wire is a suitable choice because of its ability to enable low-resistance high-current conductors at frequencies up to hundreds of kHz, so

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Litz has become an essential tool for power electronics. However, the main function is to calculate the inductance for the stranded litz coil which can be applied at high frequencies and can reduce A.C. losses in high-frequency windings due to the operating frequency. Furthermore, a simple method for litz design and configuration, which considers the winding as a whole, was presented by Charles Sullivan & Richard Zhang to select a suitable Litz Wire design as explained deeply in reference [8]. Fig. 2 illustrates the flowchart of the second approach.

Figure 2. Flowchart of the procedure to determine the coil design.

The main point in our design is the optimal selection of a shape and dimensions of air-gap, which has an impact on magnetic devices by increasing of saturation current, linearizes B-H curve of a magnetic circuit and causes reducing of the inductance. Furthermore, the air-gap calculation is carried out to get the Magnetic Flux Density value in order to calculate the power loss in Litz wire for the optimal design of the coil. Thus for a typical design:

Determine cA of the core which is in our state is Aluminum cylinder. Then, determine the number

of turns which commensurate with the design. Finally, determine the desired value of the inductance which achieves an effective design to avoid the saturation.

Calculate the air-gap length as a beginning to calculate the magnetic flux density for using it to calculate the power loss in litz wire for the optimal design of the coil. Check Eq.7 for air-gap length calculation:

Length of the air-gap

20

( )c

g

N Al

L

(7)

However, all the parameters of the designed litz coil (number of strands, strand diameter, and the diameter of litz wire) are calculated as shown in Fig. 3. It is worth noting that any anomalous value of the inductance or turns will not lead to the logical output or even error message by MATLAB in its command window. Fig. 3(a) shows the coil geometrical parameters. The obtained coil is a litz-wire coil with 60 turns, 64 strands to shape the litz wire, the diameter of each strand is 0.359 mm, and the litz wire diameter is 23 mm. The air-gap length and the magnetic flux density are clear in Fig. 3(b) for one case study determined by transient current characteristics. It is worth to be noted this coil designed to resist the high-frequency current during the transient occurrence which in turns lasts few Nano seconds as a critical case; it could be suitable at normal conditions essentially with add some types of thermoplastic insulation and care about the over-load during the normal operating. However, litz wire has been classified as a cost-effective choice to reduce eddy current loss high-frequency transformer and inductor windings. Some solutions could be successful to increase the dielectric strength for high voltage or for safety requirements like add thermoplastic insulation as PTFE which

Output the litz coil configuration Diameter of the litz wire and number of strands

Use MATLAB optimization algorithm to find stranded litz wire configuration

Inter all the input parameters: Inductance value, transient current parameters, number of turns

Calculate Power losses in the coil as a critical factor

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could be used for bundle insulation on litz wire in some applications [9-10]. Other thermoplastic insulation can be extruded over a stranded-litz wire winding, such as PTFE (Teflon), PVC, polyurethane, polyester, or polypropylene. Relative cost and operating temperature of several bundle insulation materials are explained clearly in [10].

(a) (b)

Figure 3. (a) GUI output by MATLAB for coil parameters, (b) design and characteristics of the coil in Command Window.

Summary

This paper presents two approaches for inductance calculation at high-frequency. Hence from the simulation of the designed coil carried out in ANSYS Maxwell software, it is clear that as the frequency increases the inductance value decreases and vice versa. Another inductance calculation method, which could resist the transient current during transient phenomena, is also presented in order to design a spiral coil.

Specifically, this study aims to calculate the more considerable value of the equivalent inductance which in turn leads to an obvious suppression effect on the Very Fast Transient Overvoltage (VFTO). Consequently, this research contributes to finding methods which could assist in damping of VFTO generated by operation of disconnect-switch (DS) in Gas Insulated Station (GIS).

References

[1] J.G. Coffin, The Influence of Frequency upon the Self-Inductance of Coils, in Proceedings of the American Academy of Arts and Sciences, vol. 41, no. 34, pp. 275–296 (1906).

[2] M.T. Thompson, Inductance Calculation Techniques - Part II: Approximations and Handbook Methods, in Power Control and Intelligent Motion, (1999), pp. 1–11.

[3] E.B. Rosa, The self and mutual-inductances of linear conductors, Bulletin of the Bureau of Standards, vol. 4, no. 2. pp. 301–344 (1908).

[4] J.C. Maxwell, Electricity and magnetism, vol. 2 (1954).

[5] F.W. Grover, Inductance calculations: working formulas and tables (2004).

[6] H.B. Dwight, Electrical Coils and Conductors: Their Electrical Characteristics (1945).

[7] R. Bargallo, Finite Elements for Electrical Engineering, Electrical Engineering Department, Polytechnic University of Catalonia, Catalonia (2006).

[8] C.R. Sullivan and R. Y. Zhang, Simplified Design Method for Litz Wire, In IEEE Applied Power Electronics Conference (APEC), Fort Worth, TX, USA, pp. 2667–2674 (2014).

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[9] P.N. Murgatroyd, Calculation of proximity losses in multistranded conductor bunches, IEE Proc. Sci. Meas. Tech., vol. 136, no. 3, pp. 115–120 (1989).

[10] X. Tang and C.R. Sullivan, Optimization of Stranded-Wire Windings and Comparison with Litz Wire on the Basis of Cost and Loss, 2004 IEEE 35th Annual Power Electronics Specialists Conference, Aachen, Germany, pp. 854–860 (2004).

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