A new AC motor driven by travelling magnetic field excited on LC ladder circuit without its nonlinearity

Abstract

In three-phase AC motors, three-phase AC voltage system is essential to generate a rotating magnetic field. Three- phase AC motors also require a three-phase inverter for variable speed and torque operation. In general, the number of switching devices used in three-phase inverter is not less than six and decreasing its number is difficult. This paper focuses on an LC ladder circuit consisting of linear inductors and capacitors. This paper presents a new operating method of the AC motor based on a propagating magnetic field. It is successfully excited by connecting one single-phase AC power supply to the end of the LC ladder circuit. The induction motor operated by the presented method produces torque as in general induction motors.

Keywords

AC motors distributed-element circuit electrical machines electromagnetic field analysis LC ladder circuit rotating magnetic field

1. Introduction

AC motors driven by three-phase alternating current (AC) are one of the most important industrial machines indispensable to modern society. The principle of rotation of a three-phase motor is explained by a rotating magnetic field. The rotating magnetic field requires the three-phase AC to maintain its phase difference in time (U,V, and W) and in space [1]. If any of the phase differences are eliminated, the rotating magnetic field becomes simply a non-rotating alternating magnetic field. Three-phase inverters are used for variable speed and variable torque operation of three-phase motors. The three-phase inverter consists of six power transistors to convert direct current (DC) to three-phase AC. Although some research uses more than six switching devices to ensure inverter reliability and redundancy [2], any reports cannot be found in terms of reducing the number of switching devices. This is because six transistors have been generally considered to be the minimum configuration, which is a natural consequence of the operating principle of three-phase inverters.

Based on the above background, the authors focused on a nonlinear LC ladder circuit, which is a type of distributed-element circuit. This circuit connects linear capacitors with nonlinear inductors in a ladder-like configuration. The nonlinearity of the inductor is characterized by the magnetic saturation of an iron core used. The LC ladder circuit is able to excite special nonlinear wave phenomena such as intrinsic localized modes [3–5] or magnetic solitons [6]. In these phenomena, waves of magnetic energy propagate along the LC ladder circuit, which can be regarded as a kind of rotating magnetic field. Since the wave nature of the distributed-element circuit propagates the wave of the magnetic energy intrinsically, the propagation succeeds by only one single-phase AC power supply connected to beginning of the LC ladder circuit. Therefore, we can reduce the number of switching elements from six (three-phase half-bridge inverter) to four (single-phase half-bridge inverter).

However, linear laws such as phasor representation cannot be applied due to the nonlinearity of the LC ladder circuit. We have faced with the difficulty of solving the nonlinear LC ladder circuit (nonlinear differential equation) analytically, based on our knowledge of mathematics. In this paper, we linearize the inductor by avoiding the magnetic saturation, and treat it as a linear LC ladder circuit (hereinafter referred to as “LC ladder circuit”). First, three LC ladder circuits with different coupling schemes are described and solved analytically by means of phasor representation. Next, the current propagation characteristics of a gapped transformer with an annular structure (hereinafter referred to as “annular transformer”), which is realistic model of the LC ladder circuit, are computed through finite-element magnetic field analysis. Finally, a new induction motor (IM) is proposed using a rotating magnetic field excited by applying a single-phase alternating current at the beginning of the LC ladder circuit. Its torque characteristic is compared with that of a conventional three-phase IM.

2. Analytical model of LC ladder circuit by phasor representation

This paper defines an LC ladder circuit as a circuit in which unit cells consisting of an inductor and a capacitor are continuously connected like a ladder. The circuit equations are derived for the three types of coupling schemes described below. It is shown that all of them finally become the well-known wave equation by the continuum approximation.

Fig. 1.

Magnetically coupled LC ladder circuit diagram.

2.1. Magnetically coupled (circuit A)

The circuit diagram is shown in Fig. 1. As shown in Fig. 1(a), each unit cell is independent. However, due to the mutual induction between the adjacent inductors, the equivalent circuit (Fig. 1(b)) becomes an LC ladder circuit. Applying Kirchhoff’s voltage law to the nth unit cell yields the following circuit equation with current I as the unknown. $\begin{eqnarray}L\frac{d}{dt}I_{n}+\frac{1}{C}\int I_{n}dt+kL\frac{d}{dt}(I_{n-1}+I_{n+1})=0\end{eqnarray}$ (1) where L is the self-inductance, R is the equivalent series resistance of the inductor, C is the capacitance of the capacitor, and k is the coupling factor between the nearest inductors. Mutual induction is assumed to occur only from the most proximate inductors. The equivalent series and parallel resistances of the capacitor are assumed to be negligible compared to R. A continuous variable x = nh are introduced and then I_n = I (x, t) is considered. Let the Taylor series for I (x, t) be $\begin{eqnarray}I_{n\pm 1}=I\pm h\frac{\partial I}{\partial x}+\frac{h^{2}}{2}\frac{\partial ^{2}I}{\partial x^{2}}+\cdots\end{eqnarray}$ (2) Substituting (2) for the time derivative of (1) give the following differential equation $\begin{eqnarray}L\frac{d^{2}I}{dt^{2}}+R\frac{d}{dt}I+\frac{I}{C}+kL\frac{d^{2}}{dt^{2}}\left(2I+h^{2}\frac{\partial ^{2}I}{\partial x^{2}}\right)=0.\end{eqnarray}$ (3) The time derivative operation $\frac{d}{dt}$ in (3) is replaced with jω by phasor representation, where j is the imaginary number and ω is the angular frequency of the solution. Finally, the wave equation is derived as $\begin{eqnarray}\frac{d^{2}I}{dx^{2}}=\frac{{\gamma}^{2}}{h^{2}}I\end{eqnarray}$ (4) where, $\begin{eqnarray}{\gamma}_{\text{A}}=\sqrt{\frac{(1-(1+2k)LC{\omega}^{2})+RC{\omega}j}{kLC{\omega}^{2}}}\end{eqnarray}$ (5) 𝛾_A is the propagation constant well-known in the field of distributed-element circuit [7]. Its real and imaginary parts are called the attenuation and phase constants, respectively. General solution for (4) is expressed as follows. $\begin{eqnarray}\begin{array}{@{}rcl@{}}\displaystyle I(x,t) & = & \displaystyle I_{a}e^{(-\frac{{\gamma}}{h}x)}\\ \displaystyle \Leftrightarrow I_{n} & = & \displaystyle I_{a}e^{(-{\gamma}n)}.\end{array}\end{eqnarray}$ (6) This equation suggests that the amplitude of the progating current decays exponentially with respect to the number of the unit cells n.

Fig. 2.

Electrically connected LC ladder circuit diagram.

2.2. Electrically coupled (circuit B)

The circuit diagram is shown in Fig. 2. The LC ladder circuit is configured by connecting the unit cells electrically. Applying Kirchhoff’s voltage and current laws to the nth unit cell, the following two equations are obtained. $\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle V_{n+1}-V_{n}=\displaystyle RI_{n}+L\frac{dI_{n}}{dt}\\[5.0pt] \displaystyle I_{n}-I_{n-1}=\displaystyle C\frac{dV_{n}}{dt}.\end{array}\end{eqnarray}$ (7) Eliminating the unknown variable V from ((7)) gives the following second-order differential equation on the current I $\begin{eqnarray}I_{n+1}-2I_{n}+I_{n-1}=LC\frac{d^{2}I_{n}}{dt^{2}}+RC\frac{dI_{n}}{dt}.\end{eqnarray}$ (8) As in Section 2.1, by introducing continuous variables x = nh and a phasor representation $\frac{d}{dt}=j{\omega}$ , ((8)) is transformed into ((4)), leading to a wave-like solution. However, the propagation constant 𝛾_B is different as follows, $\begin{eqnarray}{\gamma}_{\text{B}}=\sqrt{(-{\omega}^{2}LC+jRC{\omega})}.\end{eqnarray}$ (9)

2.3. Magnetically and electrically coupled (circuit C)

Figure 3 shows the circuit diagram after equivalent conversion. The magnetic coupling due to mutual induction and the electrical coupling due to wiring are superimposed. From ((1)) and ((7)), $\begin{eqnarray}I_{n+1}-2I_{n}+I_{n-1}=LC\frac{d^{2}I_{n}}{dt^{2}}+RC\frac{dI_{n}}{dt}+kLC\frac{d^{2}}{dt^{2}}(I_{n+1}+I_{n-1})\end{eqnarray}$ (10) is obtained. As in Section 2.1, by introducing continuous variables x = nh and a phasor representation $\frac{d}{dt}=j{\omega}$ , ((10)) is transformed into ((4)), leading to a wave-like solution. However, the propagation constant 𝛾_C is different as follows, $\begin{eqnarray}{\gamma}_{\text{C}}=\sqrt{\frac{(R+j{\omega}(1+2k)L)j{\omega}C}{1+{\omega}^{2}kLC}}.\end{eqnarray}$ (11)

2.4. Comparison of current propagation characteristics

When the LC ladder circuits are utilized to drive AC motors, it is desirable for the current to propagate along the neighboring unit cells without attenuation. Therefore, this section compares the attenuation constants of three types of LC ladder circuits. The propagation constant includes five variables: the circuit constants (R, L, C, k) and the angular frequency of the power supply, ω. It is difficult to analyze the attenuation constant by changing all these parameters. The inductance L and capacitance C are normalized, and quality factor Q $\begin{eqnarray}Q=\frac{2}{R}\sqrt{\frac{L}{C}}\end{eqnarray}$ (12) is introduced for simplified evaluation. Moreover, the angular frequency of the AC power supply ω is also normalized. As a result, five parameters are reduced into two (quality factor Q and coupling factor k). Figure 4 shows the variation of the attenuation constant for three LC ladder circuits. Let 𝛼_th be a threshold for the specific attenuation constant. The first, second, and third rows in Fig. 4 correspond to 𝛼th 0.01, 0.05, and 0.10, respectively. The area filled with black in Fig. 4 satisfies $\begin{eqnarray}\text{Re}({\gamma}_{i}(Q,k))<{\alpha}_{\text{th}}\quad (i=\text{A},\text{B},\text{C}).\end{eqnarray}$ (13)

When 𝛼_th is 0.01, in the circuit B, there is no combination of quality factor Q and coupling factor k that satisfies (13). However, in circuit C, the range of the solution are enhanced to the direction of the higher Q, thanks to the mutual induction between the neighbouring inductors. The similar tendency are also observed When 𝛼_th is 0.05 or 0.10. Lower Q factor allows the equivalent series resistance R to increase. The mutual induction enhances the selectivity of the circuit constants in LC ladder circuits.

Fig. 3.

Magnetically coupled and electrically connected LC ladder circuit diagram.

Fig. 4.

Variation of attenuation constant.

3. New induction motor

3.1. Design methodology

From the theoretical analysis of the LC ladder circuit in Chapter 2, it was found that a multi-phase traveling magnetic field (rotating magnetic field) is successfully generated with only a single-phase power supply at the beginning of the circuit. In this chapter, a new AC motor is considered and its torque characteristics are evaluated from the magnetic field analysis. This section describes the design methodology.

Permanent magnet synchronous motors (PMSMs) and induction motors (IMs) are representative AC motors. They differ in terms of synchronous/asynchronous operation. Especially, IMs can be introduced by deforming a transformer as shown in Fig. 5. Furthermore, the propagation of the rotating magnetic field generated on the LC ladder circuit is passive and asynchronous to the rotational angle of the rotor in the present circuit configuration. For these two reasons, this paper focuses on designing a new IM.

Fig. 5.

Similarity between transformer and induction motor.

Fig. 6.

Configuration of induction motors.

3.2. Results and discussion

The basic structure of the proposed induction motor is shown in Fig. 6. Its structure is based on the 4-pole, 24-slot three-phase IM [9]. Parameters are compared in Table 1. In the proposed motor, a single-phase AC current source with frequency of 160 Hz is connected at the beginning of circuit B, whereas a three-phase AC current source with the same frequency is connected for the conventional motor. Figure 7 shows the current waveform of the proposed motor when the rotor is restrained and the slip is 1. The sinusoidal current propagates along the adjacent phases with decay. Because phase G has a phase difference of 180 degrees with respect to phase A, 12-phase AC is successfully constructed. Figure 8 compares the motor torque of conventional and proposed IMs. The average torque of the proposed motor is 0.06 Nm, which is about 45% lower than the average torque of the conventional motor of 0.109 Nm. This is due to the attenuation of the current during propagation. Despite the reduction in torque, the proposed motor successfully generates continuous torque using the rotating magnetic field generated by the combination of the LC ladder circuit and the single-phase current source. It validates the effectiveness of the proposed IM.

Table 1
Parameters

Conventional motor Proposed motor

Resistance (Ω/phase) 1.4 0.35 = 1.4/4

Number of series 4 1

Coil pitch 5 6

Number of coil turns (Turn/slot) 66 66

Capacitance (μF) – 100

	Conventional motor	Proposed motor
Resistance (Ω/phase)	1.4	0.35 = 1.4/4
Number of series	4	1
Coil pitch	5	6
Number of coil turns (Turn/slot)	66	66
Capacitance (μF)	–	100

Fig. 7.

Propagation current of proposed IM.

Fig. 8.

Comparison of motor torque.

3.3. Relationship between frequency and current phase

In Section 3.2, the frequency of AC power supply was selected so that a 12-phase AC is formed. However, the phase constant corresponding to the imaginary part of equation ((9)) depends on the frequency. Figure 9 shows the current vector diagram for the conventional and proposed motor. The current vectors of U, V, and W phases (conventional motor) have the same magnitude, and the angle between the vectors is independent on the frequency of AC power supply. On the other hand, the current vectors of the proposed motor vary in magnitude, and the phase difference depends on the frequency. The apparent number of phases calculated from the phase difference of the current vectors is 22.7 and 7.2 phases at 80 Hz and 320 Hz, respectively, which are non-integer values. The current and flux density distributions at each frequency are shown in Fig. 10. Comparing the current density contours of the coils, a rotating magnetic field with four poles is generated at 160 Hz, while the desired number of poles is not achieved at 320 Hz and 80 Hz. The non-integer phase is a special property inherent to the proposed motor. Further study will be implemented on investigating the usefulness of the apparent non-integer phase.

Fig. 9.

Current vector diagram.

Fig. 10.

Current and flux density distributions.

4. Conclusion

In this paper, an induction motor using a rotating magnetic field generated on an LC ladder circuit is considered. Propagation constants were obtained from the circuit equations of three LC ladder circuits with different coupling schemes, and it was found that the mutual inductive action enhances the selectivity of the circuit constants. Current propagation characteristics of a gapped transformer with an annular structure were obtained from magnetic field analysis. The LC ladder circuit, in which the elements are electrically connected to each other, showed the best current propagation characteristics. We proposed a new induction motor that generates a rotating magnetic field using the circuit scheme described above, and found that it can generate torque in the same way as an ordinary induction motor. It was found that the phase difference of the current flowing in each phase depends on the power supply frequency, and thus the motor is apparently non-integer phase, which is a property unique to the proposed motor. The length of the LC ladder circuit was set to 12 in this paper because of the focus on distributed constant circuits. However, because of exponential current decay, effective current flow was only observed in a few coils from the beginning of the circuit. In the future, we plan to verify the effectiveness of our proposal by setting the length of the LC ladder circuit to 3 and using a circuit topology closer to normal 3-phase AC. In addition, we will also study the feasibility of other types of AC motors, such as synchronous motors and reluctance motors.

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https://www.jmag-international.com/catalog/161_3phaseinductionmachine_linestart/.

A new AC motor driven by travelling magnetic field excited on LC ladder circuit without its nonlinearity

Abstract

Keywords

1. Introduction

2. Analytical model of LC ladder circuit by phasor representation

3.1. Design methodology

Table 1 Parameters Conventional motor Proposed motor Resistance (Ω/phase) 1.4 0.35 = 1.4/4 Number of series 4 1 Coil pitch 5 6 Number of coil turns (Turn/slot) 66 66 Capacitance (μF) – 100

References

Table 1
Parameters

Conventional motor Proposed motor

Resistance (Ω/phase) 1.4 0.35 = 1.4/4

Number of series 4 1

Coil pitch 5 6

Number of coil turns (Turn/slot) 66 66

Capacitance (μF) – 100