Calculation Inductance of Toroidal Inductor Wound by Rectangular Cross-Sectional Wire

In this article, we present two methods for calculation of the inductance of toroidal core power inductors wound by rectangular cross-sectional wire, considering that the current density is inversely proportional to the circular coil radius. The first method is to simplify the helical toroidal coil into a thick-walled toroidal, and based on Grover’s toroidal inductor formula, the inductance is obtained by calculation the magnetic flux and the calculation method is simple, but the applicability is poor. The second method is to simplify the helical toroidal coil into a collection of self-closing circular coils, the calculation method is complex but has high accuracy, and the mutual inductance between the circular coils is calculated by the filament method based on the adjusted Grover’s mutual inductance of circular coils with inclined axes. We verify the adjusted Grover’s mutual inductance of filamentary circular coils with inclined axes and the mutual inductance between inclined circular coils with a rectangular cross section. Finally, we compared and analyzed the results calculated by the two methods proposed in this article and the results calculated by the finite element method.

The various advantages of toroidal inductors, which are cooler, smaller and more EMI-resistant are discussed. With toroidal inductors, there is an advantage to maintaining a single layer of windings due to which the inductor behaves closer to an ideal component of lower levels of parasitic capacitance. Multi-layer toroidal inductors involve both turn-to-turn capacitance and layer-to-layer capacitance and a very significat start/finish gap capacitance since there is no start/finish gap. This increases the total amount of parasitic capacitance by orders of magnitude.

This paper presents a micromachined implementation of embedded toroidal solenoids for high-performance on-chip inductors and transformers, which is highly demanded in radio-frequency integrated circuits (RFICs). Microfabricated on CMOS compatible silicon wafers with post-CMOS micromachining techniques, the RF toroidal components can constrain the magnetic flux into a well-defined path and away from other on-chip RF devices, thereby, being in favor of decrease in RF loss, increase in Q-factor and elimination of electromagnetic interference. By using a technical combination of an anisotropic wet etch and an isotropic dry etc., the micromachined toroidal structure can be used for the formation of metal solenoid by copper electroplating. Processed under low temperature (Max 120 °C for photoresist hard-baking), the three mask microfabrication can be compatible with CMOS IC fabrication in a post-process way. The formed toroidal inductors with 4.92 nH and 8.48 nH inductance are tested, and we obtain maximum Q-factors of 25.7 and 17.8 at 3.6 GHz and 3.1 GHz, while the self-resonant frequencies are 17.3 GHz and 7.4 GHz, respectively. On the other hand, two types of toroidal transformers are also formed and tested, resulting in satisfactory RF-performance. Therefore, the novel techniques for close-loop solenoid inductors are promising for high-performance RF ICs.

In electrical engineering a toroidal inductor is used to measure or monitor the electric currents of an AC power circuit as a function of the harmonic distortion [1,2]. A galvanically isolated current measurement is required, such that the advantages of lower losses nd measurement signals processed directly must be attained [3]. The ferrite core toroid inductors produces a reduced current accurately proporonal to the measured current. The toroidal inductor can be also commonly used for feedback control, and other applications [4].The design method of toroidal inductors have been developed by a non iterative method, which introduce an equation for estimation of the core size required as function of the wanted inductance and the maximum values specified for induction and current [5]. Another method solution consists in modeling inductors along with the equivalent circuits, calculation of the leakage inductance, core material characteristics, and geometrical configuretion for the minimization of volume inductors in order to simplify the design procedure [3]. Nevertheless, the trend for the current monitoring is driven by cost reduction, an increased functionality, and limited weight/space in some applications [3,4].This finally results in constantly increasing frequencies, which comes along with and increased bandwidth and poor stability.Based on electric and magnetic properties, like saturation magnetization, and toroidal-core losses, here is proposed the possibility of application of the grain-oriented silicon-iron cores for current monitoring, because these can reduce phase error and improve its accuracy in measurements of AC current at low frequencies (50 - 60 Hz) [6,7]. A simple method for toroidal-inductor design at minimum losses is suggested to calculate several inductors accepting a broad tolerance of the core material features.In general, the inductor design procedure described in literature makes use of numerous monograms, and the final result is achieved through several iterations. In special, toroidal inductors have been designed by several engineers with tedious methods [8-10]. For that reason, the lack of deeper understanding of the fundamental electromagnetic laws, it makes many engineers to consider the design of inductive components a difficult task.The purpose here is to explain a design method based on well-known tools by engineers [11]; presenting in a simple and easy way the relationships that exists between equivalent circuit and transfer function of a toroidal inductor. The proposed work is developed to meet the following objectives:1) To explain the relationship between equivalent circuit and magnetic parameters of a toroidal inductor;2) To develop the method based on normalized parameters;3) To demonstrate the method validation with a current-signal sensor and evaluate the EN-50160-2-2 standard as a function of single harmonic distortion (SHD) in home use loads [1].

The design method for an anti interference toroidal inductor is proposed as an alternative to power-quality evaluation. The method is based on well-known tools by the engineers in which is presented the relationships that exist between equivalent circuit and transfer function of a toroidal inductor. The proposed design method has been explained with normalized functions based on physical parameters of a toroidal inductor. This work presents the main arguments of the suggested methodology and as demonstration of the design method as function of normalized parameters, is developed a current-signal sensor which has been validated in the laboratory by the EN-50160-2-2 standard to evaluate the power quality in home use loads.

In this work a method of design based on normalized parameters for a high flux toroidal inductor were proposed. Based on proposed method a current-signal sensor was designed to monitoring of the AC current waveforms.Two normalized functions have been found. One is the magnetizing inductance, Lm(α), another is magnetizing impedance, Zm(α). These parameter leads to obtain in general an optimal design of any toroidal inductor as a function of α parameter.A toroid was built with recycled grain-oriented silicon-iron foils. From the results was observed that the home use loads do not satisfy the EN-50160-2-2 standard which should be corrected in the future. Also, with some suggestions, the proposed method can be expanded to special design of toroidal inductors for other applications.

Ferrites are of great interest for power electronics due to their low power losses [1,2] and they form an essential part of inductors and transformers used in their main applications areas [3,4,5]. Therefore, it is necessary to investigate and model the magnetic properties and the nonlinear behavior of ferrites, which exhibit saturation, hysteresis and power losses. These effects and the great variety of core geometries (e.g., E, RM, POT, toroidal) and other parameters, such as the number of turns, make it difficult to obtain a single model that is both simple and precise. Of all the geometries, the toroidal core (Figure 1a) is the most studied in current literature [6,7,8,9,10,11,12]. Nevertheless, despite these publications there are not enough studies that calculate parameters to be used in circuit simulators, such as inductances in all working regions of the ferrite (linear, intermediate and saturation), geometries and number of turns. In this context, our previous publications have focused on 2D Finite Element Analysis procedures for RM and POT ferrite cores [13,14]. In addition, in the case of the RM core, we have shown an application of our procedures in a commercial circuit simulator [15].

In this paper we focus on the modeling of ferrite inductors with toroidal cores and their nonlinear behavior. We present a specific procedure to compute the inductance of an inductor with a toroidal ferrite core.

We present the 2D model (cross-section of the real inductor) and then we study, by numerical simulations and experimental measurements, if this model is suitable for the simulation of ferrite cores in the linear, intermediate and saturation regions. To do so, we compare 3D and 2D results with experimental measurements. The validation is carried out based on convergence and computational cost, spatial distribution of the magnetic fields and flux and inductance curves. At the same time we present preliminary studies of convergence and computational cost in 2D and 3D, showing the reduction of the computational cost and the similarity of the results.

The outline of the paper is as follows. In Section 2, we present and describe the specific Finite Element procedure to calculate the inductance. We also describe in detail the measuring procedure we use to validate the procedure and to calculate the input parameters. In Section 3 we provide results obtained in the preliminary study of the convergence and computational cost, and we show the results of the magnetic flux, inductance and magnetic fields. Finally, Section 4 briefly summarizes our main conclusions.

In this article, we present two methods for calculation of the inductance of toroidal core power inductors wound by rectangular cross-sectional wire, considering that the current density is inversely proportional to the circular coil radius. The first method is to simplify the helical toroidal coil into a thick-walled toroidal, and based on Grover’s toroidal inductor formula, the inductance is obtained by calculation the magnetic flux and the calculation method is simple, but the applicability is poor. The second method is to simplify the helical toroidal coil into a collection of self-closing circular coils, the calculation method is complex but has high accuracy, and the mutual inductance between the circular coils is calculated by the filament method based on the adjusted Grover’s mutual inductance of circular coils with inclined axes. We verify the adjusted Grover’s mutual inductance of filamentary circular coils with inclined axes and the mutual inductance between inclined circular coils with a rectangular cross section. Finally, we compared and analyzed the results calculated by the two methods proposed in this article and the results calculated by the finite element method.

The various advantages of toroidal inductors, which are cooler, smaller and more EMI-resistant are discussed. With toroidal inductors, there is an advantage to maintaining a single layer of windings due to which the inductor behaves closer to an ideal component of lower levels of parasitic capacitance. Multi-layer toroidal inductors involve both turn-to-turn capacitance and layer-to-layer capacitance and a very significat start/finish gap capacitance since there is no start/finish gap. This increases the total amount of parasitic capacitance by orders of magnitude.

This paper presents a micromachined implementation of embedded toroidal solenoids for high-performance on-chip inductors and transformers, which is highly demanded in radio-frequency integrated circuits (RFICs). Microfabricated on CMOS compatible silicon wafers with post-CMOS micromachining techniques, the RF toroidal components can constrain the magnetic flux into a well-defined path and away from other on-chip RF devices, thereby, being in favor of decrease in RF loss, increase in Q-factor and elimination of electromagnetic interference. By using a technical combination of an anisotropic wet etch and an isotropic dry etc., the micromachined toroidal structure can be used for the formation of metal solenoid by copper electroplating. Processed under low temperature (Max 120 °C for photoresist hard-baking), the three mask microfabrication can be compatible with CMOS IC fabrication in a post-process way. The formed toroidal inductors with 4.92 nH and 8.48 nH inductance are tested, and we obtain maximum Q-factors of 25.7 and 17.8 at 3.6 GHz and 3.1 GHz, while the self-resonant frequencies are 17.3 GHz and 7.4 GHz, respectively. On the other hand, two types of toroidal transformers are also formed and tested, resulting in satisfactory RF-performance. Therefore, the novel techniques for close-loop solenoid inductors are promising for high-performance RF ICs.

In electrical engineering a toroidal inductor is used to measure or monitor the electric currents of an AC power circuit as a function of the harmonic distortion [1,2]. A galvanically isolated current measurement is required, such that the advantages of lower losses nd measurement signals processed directly must be attained [3]. The ferrite core toroid inductors produces a reduced current accurately proporonal to the measured current. The toroidal inductor can be also commonly used for feedback control, and other applications [4].The design method of toroidal inductors have been developed by a non iterative method, which introduce an equation for estimation of the core size required as function of the wanted inductance and the maximum values specified for induction and current [5]. Another method solution consists in modeling inductors along with the equivalent circuits, calculation of the leakage inductance, core material characteristics, and geometrical configuretion for the minimization of volume inductors in order to simplify the design procedure [3]. Nevertheless, the trend for the current monitoring is driven by cost reduction, an increased functionality, and limited weight/space in some applications [3,4].This finally results in constantly increasing frequencies, which comes along with and increased bandwidth and poor stability.Based on electric and magnetic properties, like saturation magnetization, and toroidal-core losses, here is proposed the possibility of application of the grain-oriented silicon-iron cores for current monitoring, because these can reduce phase error and improve its accuracy in measurements of AC current at low frequencies (50 - 60 Hz) [6,7]. A simple method for toroidal-inductor design at minimum losses is suggested to calculate several inductors accepting a broad tolerance of the core material features.In general, the inductor design procedure described in literature makes use of numerous monograms, and the final result is achieved through several iterations. In special, toroidal inductors have been designed by several engineers with tedious methods [8-10]. For that reason, the lack of deeper understanding of the fundamental electromagnetic laws, it makes many engineers to consider the design of inductive components a difficult task.The purpose here is to explain a design method based on well-known tools by engineers [11]; presenting in a simple and easy way the relationships that exists between equivalent circuit and transfer function of a toroidal inductor. The proposed work is developed to meet the following objectives:1) To explain the relationship between equivalent circuit and magnetic parameters of a toroidal inductor;2) To develop the method based on normalized parameters;3) To demonstrate the method validation with a current-signal sensor and evaluate the EN-50160-2-2 standard as a function of single harmonic distortion (SHD) in home use loads [1].

The design method for an anti interference toroidal inductor is proposed as an alternative to power-quality evaluation. The method is based on well-known tools by the engineers in which is presented the relationships that exist between equivalent circuit and transfer function of a toroidal inductor. The proposed design method has been explained with normalized functions based on physical parameters of a toroidal inductor. This work presents the main arguments of the suggested methodology and as demonstration of the design method as function of normalized parameters, is developed a current-signal sensor which has been validated in the laboratory by the EN-50160-2-2 standard to evaluate the power quality in home use loads.

In this work a method of design based on normalized parameters for a high flux toroidal inductor were proposed. Based on proposed method a current-signal sensor was designed to monitoring of the AC current waveforms.Two normalized functions have been found. One is the magnetizing inductance, Lm(α), another is magnetizing impedance, Zm(α). These parameter leads to obtain in general an optimal design of any toroidal inductor as a function of α parameter.A toroid was built with recycled grain-oriented silicon-iron foils. From the results was observed that the home use loads do not satisfy the EN-50160-2-2 standard which should be corrected in the future. Also, with some suggestions, the proposed method can be expanded to special design of toroidal inductors for other applications.

Ferrites are of great interest for power electronics due to their low power losses [1,2] and they form an essential part of inductors and transformers used in their main applications areas [3,4,5]. Therefore, it is necessary to investigate and model the magnetic properties and the nonlinear behavior of ferrites, which exhibit saturation, hysteresis and power losses. These effects and the great variety of core geometries (e.g., E, RM, POT, toroidal) and other parameters, such as the number of turns, make it difficult to obtain a single model that is both simple and precise. Of all the geometries, the toroidal core (Figure 1a) is the most studied in current literature [6,7,8,9,10,11,12]. Nevertheless, despite these publications there are not enough studies that calculate parameters to be used in circuit simulators, such as inductances in all working regions of the ferrite (linear, intermediate and saturation), geometries and number of turns. In this context, our previous publications have focused on 2D Finite Element Analysis procedures for RM and POT ferrite cores [13,14]. In addition, in the case of the RM core, we have shown an application of our procedures in a commercial circuit simulator [15].

In this paper we focus on the modeling of ferrite inductors with toroidal cores and their nonlinear behavior. We present a specific procedure to compute the inductance of an inductor with a toroidal ferrite core.

We present the 2D model (cross-section of the real inductor) and then we study, by numerical simulations and experimental measurements, if this model is suitable for the simulation of ferrite cores in the linear, intermediate and saturation regions. To do so, we compare 3D and 2D results with experimental measurements. The validation is carried out based on convergence and computational cost, spatial distribution of the magnetic fields and flux and inductance curves. At the same time we present preliminary studies of convergence and computational cost in 2D and 3D, showing the reduction of the computational cost and the similarity of the results.

The outline of the paper is as follows. In Section 2, we present and describe the specific Finite Element procedure to calculate the inductance. We also describe in detail the measuring procedure we use to validate the procedure and to calculate the input parameters. In Section 3 we provide results obtained in the preliminary study of the convergence and computational cost, and we show the results of the magnetic flux, inductance and magnetic fields. Finally, Section 4 briefly summarizes our main conclusions.