Starting Pulse Vibration Torque Analysis of Aviation Variable Frequency Asynchronous Motor Based on Low-Frequency Step-Down Starting Methods

Abstract

The more electric aircraft provides 115 V/360~800 Hz variable frequency power supply for the variable frequency asynchronous motor, and the motor operation characteristics change with the power frequency variation. It causes the starting pulse vibration torque with the low-frequency power supply (360 Hz) to increase greatly and the mechanical shock and fatigue damage. Therefore, this paper proposes step-down starting method with the low-frequency power supply. Based on the electromagnetic principles generated by the starting pulse vibration torque, this paper uses a novel method of simulation data fitting to establish an approximate model of the starting pulse vibration torque. The parameter design formulas of the low-frequency step-down starting methods, including the reducing voltage, the series resistance, and the series inductance are proposed. The effectiveness of the different methods are verified, and the performance is compared through the simulation. The experimental verification of a small power asynchronous motor is completed. The simulation and experiment results show that the low-frequency step-down starting methods effectively reduce the starting peak torque, and also suppress the shock impact of starting current on the power supply.

1. Introduction

Regarding the asynchronous motor starting characteristic, the research and analysis contents pay more attention to issues such as the large starting current and the small starting torque [1,2]. In addition, when the asynchronous motor starts, there is the starting pulse vibration torque. It is transmitted to the mechanical system through the coupling of multiple physical domains and brings adverse effects for the mechanical system. The ref [3,4] mainly indicate that the starting pulse vibration torque effect of 50 Hz constant frequency asynchronous motor is transmitted to the mechanical system through the couplings of multiple physical domains, which leads to the bearing resonance and fracture. The ref [5] similarly describes that the impact of starting pulse vibration torque is transmitted to the mechanical system, which eventually damages and breaks the gearbox. Therefore, the starting pulse vibration torque brings the large damage to the mechanical system, which is the problem that cannot be ignored.

For the starting pulse vibration torque impact of asynchronous motors on the mechanical system, the mechanical system design is generally improved, and the mechanical system strength is increased to prevent the damage. The ref [6,7] reduce the starting pulse vibration torque by changing the motor structure, such as groove shape or the mechanical bearing stiffness. The ref [8] proposes a method by reducing the starting voltage to decrease the starting pulse vibration pulse. However, this method further reduces the starting torque as the starting pulse vibration torque decreases and can only be applied to drive systems with no-load and light-load starting.

The variable frequency power supply system of 115 V/360~800 Hz is applied by the more electric aircraft [9,10,11,12]. Compared with constant frequency power supply, the performance of equipment and system should adapt to this change [13,14,15,16]. For the asynchronous motor powered by the variable frequency power supply, the operation characteristics change with the power frequency variation. The ref [17,18] shows that the starting torque, the critical torque, the starting current, and the starting pulse vibration torque will decrease as the power frequency increases. Therefore, the starting pulse vibration of aviation variable frequency asynchronous motors is very large at a low-frequency (360 Hz) power supply, and the impact on the mechanical system is more serious [19]. It directly affects the reliability, fatigue, and life of mechanical system [20,21]. The higher requirement is proposed for the mechanical system design.

In order to solve the problem of mechanical shock, it is necessary to analyze the formation principle of starting pulse vibration torque. The ref [22] analyses the electromagnetic principle of the starting pulse vibration torque, and it is a complex coupling process formed by the interaction of different magnetic fields between the stator and rotor. However, it is only qualitative analysis and the related parameter derivation is not given. For the complex coupling problem, the ref [23] simplifies the complex mechanical transmission models, such as the gears and bearings from high-order to low-order, which is convenient for analyzing the coupling and resonance form the electrical system of the more electric aircraft to the mechanical system. Therefore, based on the simplified analysis method, a novel approximate modeling method by the data fitting for the peak torque generated by the pulse vibration torque is proposed and provides the basic evidence for the step-down starting design method.

For the asynchronous motor powered by the variable frequency power supply, it has a large margin of the starting torque and the critical torque with the low-frequency power supply. This paper proposes a control method of low-frequency step-down starting to reduce the starting pulse vibration torque with the low-frequency power. The power supply frequency range can be divided into two parts. In the low-frequency band, the voltage reduction, the series resistance, or the series inductance are used to perform the power supply step-down starting. In the high-frequency range, the method of rated power supply voltage starting is used. Based on the electromagnetic principle of starting pulse vibration torque, the paper establishes a mathematical model of the starting pulse vibration torque through the method of simulation data fitting, and proposes that the minimum peak torque caused by the starting pulse vibration torque is applied as the design goal. It also derives the parameter calculation formulas of the low-frequency step-down starting, including the voltage reduction, the series resistance, and the series inductance. Finally, the method feasibility is verified by simulation and experiments.

2. Torque Characteristics of Aviation Variable Frequency Asynchronous Motor

2.1. Mechanical Characteristics of Variable Frequency Asynchronous Motor

When the asynchronous motor is supplied by the aviation variable frequency power supply, the synchronous speed changes by the influence of power supply frequency, which causes the mechanical characteristic change. Figure 1 shows the mechanical characteristics of four-pole asynchronous motor with the aviation variable frequency power supply.

As seen from the mechanical characteristics of Figure 1, both the critical torque and the starting torque decrease with the increase of power supply frequency. It can be approximated as:

$$T_{st}=\frac{3p{U}_1^2{R^{\prime }}_2}{2\pi f_1\left[{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_1^2{(L_{l1}+{L^{\prime }}_{l2})}^2\right]}\approx \frac{3p{U}_1^2{R^{\prime }}_2}{8{\pi }^3{f}_1^3{(L_{l1}+{L^{\prime }}_{l2})}^2}$$

(1)

$$T_{em}=\frac{3p{U}_1^2}{4\pi f_1[R_1+\sqrt{R_1^2+4{\pi }^2{f}_1^2{(L_{l1}+{L^{\prime }}_{l2})}^2]}}\approx \frac{3p{U}_1^2}{8{\pi }^2{f}_1{}^2(L_{l1}+{L^{\prime }}_{l2})}$$

(2)

where, T_st is the starting torque, T_em is the critical torque, p is the number of pole pairs, U₁ is the power supply voltage, f₁ is the power supply frequency, R₁ is the stator resistance, L_l1 is the stator leakage inductance, R’₂ is the rotor resistance, and L’_l2 is the rotor leakage inductance. The critical torque is the approximately square descent as the power supply frequency rises, and the starting torque is the approximately cubic descent as the power supply frequency rises. Therefore, in the design of variable frequency asynchronous motor, it is necessary to provide an index of starting torque T_st and the critical torque T_em with the high-frequency power supply (f₁ = f_1H = 800 Hz). Generally expressed by the starting torque coefficient k_st and the critical torque coefficient k_m, they are respectively defined as:

$$k_{st}=\frac{T_{stH}}{T_{nH}},k_m=\frac{T_{emH}}{T_{nH}}$$

(3)

where, T_nH is the working torque with f_1H, T_stH is the starting torque with f_1H, and T_emH is the critical torque with f_1H. According to the characteristics of Equations (1) and (2), it can be known that the realization conditions of k_st in the Equation (3) are more rigorous than k_m. In the motor design, the motor starting with the loads needs to meet k_st, and the motor starting with the light loads only needs to meet k_m.

2.2. Starting Pulse Vibration Torque of Variable Frequency Asynchronous Motor

In the mechanical design of asynchronous motors, it is necessary to analyze the maximum torque T_max of shaft during the operation, and the T_max generally appears as the asynchronous motor starts. It may be the critical torque T_em or the peak torque caused by the starting pulse vibration torque, which is defined by the T_stm.

For the asynchronous motor with the traditional constant frequency power supply, the T_stm is usually less than the T_em. The starting process of 400 Hz aviation constant frequency asynchronous motor is shown in Figure 2a. Even if T_stm is larger than the T_em, it is only necessary to appropriately increase the design safety factor, and the starting pulse vibration torque influence may not be considered. However, for meeting the mechanical characteristic requirement with the power supply frequency change, such as the Formula (3), the T_max caused by the motor design parameters during the starting is no longer the T_em, for the asynchronous motor with the variable frequency asynchronous motor. The T_max is the T_stm caused by the starting pulse vibration torque, as shown in Figure 2b.

The relationship between the T_stm caused by the starting pulse vibration torque and the power supply frequency f₁ during the variable frequency asynchronous motor starting is shown in Figure 2c. It can be seen that the T_stm is already larger than the T_em with the f_1H = 800 Hz, and the T_stm is significantly larger than the T_em with the f_1L = 360 Hz. Obviously, the T_max of mechanical shaft formed by the T_stm with the low-frequency power supply is very large, which has a great effect on the fatigue strength of mechanical shaft.

3. Starting Pulse Vibration Torque Model

The principle of starting pulse vibration torque of asynchronous motor is that the stator and rotor magnetic fields cannot be synchronized immediately, forming electromagnetic pulse vibration torque, due to the transient process existence in the stator and rotor circuits during the starting.

3.1. Transient Characteristics of Starting Current and Torque

The voltage balance formula of asynchronous motor in the α-β two-phase stationary coordinate system is:

$$\left[\begin{array}{c}{u}_{\alpha 1}\\ u_{\beta 1}\\ 0\\ 0\end{array}\right]=\left[\begin{array}{cccc}{R}_1+L_{s}p& 0& L_{m}p& 0\\ 0& R_1+L_{s}p& 0& L_{m}p\\ L_{m}p& \omega L_m& R_2+L_{r}p& \omega L_r\\ -\omega L_m& L_{m}p& -\omega L_r& R_2+L_{r}p\end{array}\right]\left[\begin{array}{c}{i}_{\alpha 1}\\ i_{\beta 1}\\ i_{\alpha 2}\\ i_{\beta 2}\end{array}\right]$$

(4)

where, R₁ and R₂ are the stator and rotor resistances, respectively, L_s and L_r are the stator and rotor winding self-inductances, respectively, and L_m is the mutual inductance between the stator and rotor windings.

The stator and rotor current expressions can be obtained as:

$$\left[\begin{array}{c}{i}_{\alpha 1}\\ i_{\beta 1}\end{array}\right]=\frac{1/R_1}{\left(1+T_{1}p\right)}\left\{\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]-L_{m}p\left[\begin{array}{c}{i}_{\alpha 2}\\ i_{\beta 2}\end{array}\right]\right\}$$

(5)

$$\left[\begin{array}{c}{i}_{\alpha 2}\\ i_{\beta 2}\end{array}\right]=-\frac{L_m}{L_r}\left\{\left[\begin{array}{c}1\\ 1\end{array}\right]-\left[\begin{array}{cc}{W}_{11}(p)& W_{12}(p)\\ W_{21}(p)& W_{22}(p)\end{array}\right]\right\}\left[\begin{array}{c}{i}_{\alpha 1}\\ i_{\beta 1}\end{array}\right]$$

(6)

The transfer function in the Equation (6) is:

$$\begin{array}{ll}{W}_{11}(s)=W_{22}(s)=\frac{T_{2}p+1}{T_2^2{p}^2+2T_{2}p+1+T_2^2{\omega }^2}\hfill & \hfill \\ W_{12}(s)=-W_{21}(s)=\frac{T_2\omega }{T_2^2{p}^2+2T_{2}p+1+T_2^2{\omega }^2}\hfill & \hfill \end{array}$$

(7)

In the Formulas (5) and (7), T₁ and T₂ can be expressed as T₁ = L_s/R₁ and T₂ = L_r/R₂.

Figure 3a is the current waveform caused by the transient process of Equations (5)–(7). The stator currents $i_{\alpha 1}$, $i_{\beta 1}$ are the AC current with the frequency f₁ superimposed on the DC current transient process, and the rotor currents $i_{\alpha 2}$,$i_{\beta 2}$ are AC current with the slip frequency f_s superimposed on the DC current transient process.

Since the stator and rotor currents exist in the DC transient process, the starting torque includes the pulse vibration torque shown in Figure 3b, which can be expressed as:

$$T_{st}=p{L}_m\left(i_{\alpha 2}{i}_{\beta 1}-i_{\alpha 1}{i}_{\beta 2}\right)={\overline{T}}_{st}+{\tilde{T}}_{st}$$

(8)

It consists of average torque ${\overline{T}}_{st}$ and the pulse vibration torque ${\tilde{T}}_{st}$ during the starting operation.

The peak torque caused by pulse vibration torque is:

$$T_{stm}=\max \left[p{L}_m\left(i_{\alpha 2}{i}_{\beta 1}-i_{\alpha 1}{i}_{\beta 2}\right)\right]=\max \left({\overline{T}}_{st}+{\tilde{T}}_{st}\right)$$

(9)

3.2. Basic Principle of Starting Torque

Obviously, the cause of pulse vibration torque is that the power is suddenly applied to the stator and rotor windings generates the transient currents containing the AC and DC components [22]. If it is expressed as:

$$\begin{array}{ll}{i}_{\alpha 1}={\tilde{i}}_{\alpha 1}+{\overline{i}}_{\alpha 10}\hfill & \hfill \\ i_{\beta 1}={\tilde{i}}_{\beta 1}+{\overline{i}}_{\beta 10}\hfill & \hfill \\ i_{\alpha 2}={\tilde{i}}_{\alpha 2}+{\overline{i}}_{\alpha 20}\hfill & \hfill \\ i_{\beta 2}={\tilde{i}}_{\beta 2}+{\overline{i}}_{\beta 20}\hfill & \hfill \end{array}$$

(10)

These currents generate the magnetic potential in the motor air gap, as shown in Figure 4.

(1)

F_ss: the rotating magnetic potential on the stator with the synchronous speed (ω₁) generated by the stator currents ${\tilde{i}}_{\alpha 1}$,${\tilde{i}}_{\beta 1}$ of frequency f₁.

(2)

F_0s: the static magnetic potential on the stator with the speed of zero caused by the DC component of stator currents ${\overline{i}}_{\alpha 10}$,${\overline{i}}_{\beta 10}$.

(3)

F_0r: the rotating magnetic potential on the rotor with the synchronous speed (ω_m) generated by the DC component of rotor currents ${\overline{i}}_{\alpha 20}$,${\overline{i}}_{\beta 20}$.

(4)

F_sr: the rotating magnetic potential on the rotor with the synchronous speed (ω₁ = ω_s + ω_m) generated by the rotor current ${\overline{i}}_{\alpha 2}$,${\overline{i}}_{\beta 2}$ of frequency f_s.

The four kinds of magnetic potentials interact to generate the electromagnetic torque shown in Figure 3b. The F_ss and F_sr interact to produce the average electromagnetic torque ${\overline{T}}_{st}$ in Equation (8), the F_0r and F_sr with the f₁ interact to produce the pulse vibration torque ${\tilde{T}}_{st}$, the F_ss and F_0r with the f_s interact to produce the pulse vibration torque ${\tilde{T}}_{st2}$, and the F_0s and F_0r with the fm interact to produce the pulse vibration torque ${\tilde{T}}_{st3}$, which is the relatively complicated transient process.

Obviously, if the analytical formula of pulse vibration torque is derived according to Formulas (5)–(8), it will be very complicated and inconvenient for the engineering design and application.

3.3. Approximate Model of Starting Peak Torque

For the research on the starting pulse vibration torque, it analyzes the T_max of motor design, that is, the torque shock of mechanical device caused by the starting pulse vibration torque. Therefore, the starting peak torque T_stm is taken as the research target, which requires a model that can express the law of T_stm and is convenient for the engineering design and application.

For the torque characteristic of Equation (8) from t = 0 to Δt, the ${\tilde{T}}_{st}$ generated by the interaction between F_ss and F_sr is usually described by the Formula (1) at the t = 0, and the starting current I_st can be expressed as:

$$I_{st}=\frac{U_1^{}}{\sqrt{{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_1^2{(L_{l1}+{L^{\prime }}_{l2})}^2}}$$

(11)

$${\overline{T}}_{st}=\frac{3p_n{U}_1^2{R^{\prime }}_2}{2\pi f_1\left[{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_1^2{(L_{l1}+{L^{\prime }}_{l2})}^2\right]}=\frac{3p_n{R^{\prime }}_2}{2\pi f_1}{I}_{st}^2={\overline{C}}_{st}{I}_{st}^2$$

(12)

where, the stator current and rotor current are approximately equal during the starting. They are proportional to the power supply voltage U₁, and the starting torque ${\overline{T}}_{st}$ is proportional to the starting current $I_{st}^2$.

For the ${\tilde{T}}_{st}$, the same method is applied. Since T_stm only appears at the motor starting moment, the magnitude of DC component ${\overline{i}}_{\alpha 10}$,${\overline{i}}_{\beta 10}$,${\overline{i}}_{\alpha 20}$,${\overline{i}}_{\beta 20}$ should be determined by the U₁ suddenly applied to the windings. It is close to the initial value of transition process, which is approximately proportional to U₁. It can be seen that the F_0s and F_0r magnitudes are approximately proportional to U₁. According to the Formula (11), it can also be approximately proportional to I_st. If the magnetic potential phase effect is ignored, the ${\tilde{T}}_{st}$ can also be assumed to be proportional to $I_{st}^2$, the T_stm with the different f₁ is approximately assumed to be:

$$T_{stm}\approx C_{stm}{I}_{st}^2=\frac{C_{stm}{U}_1^2}{{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_1^2{(L_{l1}+{L^{\prime }}_{l2})}^2}$$

(13)

According to Formula (12), when 360~800 Hz aviation power supply is applied, the coefficient ${\overline{C}}_{st}$ is inversely proportional to the power frequency f₁. However, the motor operates in magnetic saturation state with the low frequency, and in weak magnetic state with the high frequency. The relationship between ${\overline{C}}_{st}$ and f₁ is complex and nonlinear. The nonlinear relationship between the coefficient C_tms of Formula (13) and is more complicated.

The 7.5 kW variable frequency asynchronous motor is taken as an example, the approximate law of T_stm is analyzed by the simulation to verify the feasibility of Formula (13). The parameters of aviation variable frequency asynchronous motor are shown in

Table 1. Parameters of variable frequency asynchronous motor.

Parameters	Value
PN (kW)	7.5
p	4
R1 (Ω)	0.12
Ll1 (mH)	0.06035
R’2 (Ω)	0.147
L’l2 (mH)	0.08715
Lm (mH)	1.986

.

Firstly, the power supply of asynchronous motor is set to different frequencies, then it changes the U₁ at each frequency f₁. The basic motor parameters are obtained by the Ansoft simulation. The motor parameters are set in Simulink simulation for the motor model. The voltage and frequency of supply power are also changed, the I_st and T_stm are obtained through the dynamic characteristic simulation. The data relation between the I_st and T_stm is fitted according to the Formula (13). The fitting results are shown in

Table 2. Starting peak torque coefficient.

Frequency	Cstm	kstm
360	0.00166	1
400	0.00153	0.92
500	0.00129	0.78
600	0.00114	0.69
700	0.00103	0.62
800	0.00094	0.57

and the characteristics obtained are shown in Figure 5.

The similarity of fitted results is analyzed by using the total sum square (TSS) and the residual sum square (RSS) [24]. The calculated goodness of fit is above 0.98. Therefore, the T_stm description of motor is feasible.

4. Low-Frequency Step-Down Starting Method

As can be seen from Figure 2c, the variable frequency asynchronous motor has the maximum of T_stm with the low-frequency (360 Hz) power supply, and the mechanical damage is the greatest. Therefore, the T_stm with the low frequency is the research target.

4.1. Principle of Low-Frequency Step-Down Starting

From the Equation (3), it can be seen that the k_st and k_m of aviation variable frequency asynchronous motor are proposed for the f_1H power supply. The k_st and k_m have a large margin with the low-frequency f_1L power supply, that is, the T_st and T_em are greater than the design requirements. Therefore, if the power supply voltage is appropriately reduced with the low frequency, T_stm can be effectively reduced on the premise of meeting the T_st and T_em requirements.

The frequency of variable frequency power supply is divided into the high-frequency band and the low-frequency band. The high-frequency band is started with the rated voltage U_N, and the low-frequency band is started with the low voltage U_A. The f_A is the boundary between the two frequency bands. The starting voltage U_st is selected as follows: when f₁ ≥ f_A, U_st = U_N; when f₁ < f_A, U_st = U_A. The T_stm characteristic is shown in Figure 6.

The selection principle of frequency boundary f_A and the step-down voltage U_A is given according to Figure 6. The design principle of step-down starting:

(1)

Principle I: The power supply voltage U_A must ensure that k_st and k_m meet the design requirements with the power frequency f_A. If the working torque is T_nA, the starting torque T_stA ≥ k_stT_nA, and the critical torque T_emA ≥ k_mT_nA;

(2)

Principle II: The T_stmL with the power supply frequency f_1L and the voltage U_A is equal to the T_stmA with the power supply frequency f_1A and the voltage U_N, that is, the maximum T_stm is the lowest in the entire frequency range.

For the low-frequency step-down starting shown in Figure 6, the voltage reduction, the series resistance and the series inductance can be adopted.

4.2. Design of Low-Frequency Voltage Reduction

The design method of voltage reduction with the low-frequency power supply is applied. Firstly, according to the design principle I, the reduced voltage U_A should meet k_st and k_m at the frequency boundary point f_A. If the design is applied based on the starting torque T_stA = k_stT_nA, it is:

$$k_{st}{T}_{nA}=\frac{3p{U}_A^2{R^{\prime }}_2}{2\pi f_A\left[{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_A^2{(L_{l1}+{L^{\prime }}_{l2})}^2\right]}$$

(14)

It is arranged to obtain:

$$U_A^2=\frac{2\pi f_A{T}_{st}}{3p{R^{\prime }}_2}\left[{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_A^2{(L_{l1}+{L^{\prime }}_{l2})}^2\right]$$

(15)

Designed with the critical torque T_emA = k_mT_nA in the Equation (3), U_A should meet at the frequency point f_A:

$$k_m{T}_{nA}=\frac{3p{U}_A^2}{4\pi f_A[R_1+\sqrt{R_1^2+4{\pi }^2{f}_A^2{(L_{l1}+{L^{\prime }}_{l2})}^2]}}$$

(16)

It is arranged to obtain:

$$U_A^2=\frac{k_m{T}_{nA}4\pi f_A}{3p}[R_1+\sqrt{R_1^2+4{\pi }^2{f}_A^2{(L_{l1}+{L^{\prime }}_{l2})}^2]}$$

(17)

According to the design principle II, the formula (13) is expressed as:

$$\frac{k_{stm}{U}_N{}^2}{{\left(R_1+{R^{\prime }}_2\right)}^2+4{\pi }^2{f}_A^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2}=\frac{U_A{}^2}{{\left(R_1+{R^{\prime }}_2\right)}^2+4{\pi }^2{f}_{1L}^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2}$$

(18)

In order to simplify the calculation, it is approximately ${\left(R_1+{R^{\prime }}_2\right)}^2<<4{\pi }^2{f}_A^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2$ and written as:

$$U_A{}^2\approx k_{stm}\frac{{\left(R_1+{R^{\prime }}_2\right)}^2+4{\pi }^2{f}_{1L}^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2}{4{\pi }^2{f}_A^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2}{U}_{1N}{}^2$$

(19)

Coupled, the (15) and (19) or (17) and (19), the f_A and U_A can be estimated.

4.3. Design of Low-Frequency Series Resistance

If the stator winding with the series resistance is used to reduce the power supply voltage for the low-frequency power supply, it is necessary to estimate the frequency boundary point f_A and the series resistance R_A.

Similarly, according to the design principle I, it is calculated by the starting torque T_stA = k_stT_nA at the point f_A, and the starting torque should satisfy:

$$k_{st}{T}_{nA}=\frac{3p{U}_n^2{R^{\prime }}_2}{2\pi f_A\left[{(R_1+R_A+{R^{\prime }}_2)}^2+4{\pi }^2{f}_A^2{(L_{l1}+{L^{\prime }}_{l2})}^2\right]}$$

(20)

The total resistance value R_∑ = R₁ + R_A + R’₂ should meet:

$$R_{\Sigma }^2=\frac{3U_N^2{R^{\prime }}_2}{k_{st}{P}_N}-4{\pi }^2{f}_A^2{(L_{l1}+{L^{\prime }}_{l2})}^2$$

(21)

However, it is calculated by the T_em = k_mT_nA at the point f_A, defines R_1∑ = R₁ + R_A, and meets:

$$k_m{T}_{nA}=\frac{3p{U}_n^2}{4\pi f_A[R_{1\Sigma }+\sqrt{R_{1\Sigma }^2+4{\pi }^2{f}_A^2{(L_{l1}+{L^{\prime }}_{l2})}^2]}}$$

(22)

The R_1∑ is:

$$R_{1\Sigma }^{}=\frac{3p{U}_n^2}{8\pi f_A{k}_m{T}_{nA}}-\frac{8{\pi }^3{k}_m{T}_{nA}}{3p{U}_n^2}{f}_A^3{(L_{l1}+{L^{\prime }}_{l2})}^2$$

(23)

According to the design Principle II, the Equation (13) is:

$$\frac{k_{stm}{U}_N{}^2}{{\left(R_1+{R^{\prime }}_2\right)}^2+4{\pi }^2{f}_A^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2}=\frac{U_N{}^2}{{\left(R_1+{R^{\prime }}_2+R_A\right)}^2+4{\pi }^2{f}_{1L}^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2}$$

(24)

Due to the f_A > f_1L, it is approximately ${\left(R_1+{R^{\prime }}_2\right)}^2<<4{\pi }^2{f}_A^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2$, and written as:

$$R_{\Sigma }^2\approx 4{\pi }^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2\left[\frac{1}{k_{stm}}{f}_A^2-f_{1L}^2\right]$$

(25)

Coupled the (21) and (25) or (23) and (25), the f_A and R_A can be estimated.

4.4. Design of Low-Frequency Series Inductance

It reduces the power supply voltage of asynchronous motor by the series inductance. However, the voltage drop across the inductor is not only related to the starting current, but also to the frequency. Its characteristics are different from the series resistance. The frequency boundary point f_A and the series resistance L_A are estimated.

Similarly, according to the design principle I, it is calculated by the starting torque T_stA = k_stT_nA at the point f_A, and it should satisfy:

$$k_{st}{T}_{nA}=\frac{3p{U}_n^2{R^{\prime }}_2}{2\pi f_A\left[{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_A^2{(L_{l1}+L_A+{L^{\prime }}_{l2})}^2\right]}$$

(26)

The total inductance value L_∑ = L_l1 + L_A + L’_l2 should meet:

$$L_{\Sigma }^2=\frac{1}{4{\pi }^2{f}_A^2}\left[\frac{3p{U}_n^2{R^{\prime }}_2}{2\pi f_A{k}_{st}{T}_{nA}}-{(R_1+{R^{\prime }}_2)}^2\right]$$

(27)

However, it is calculated by the T_em = k_mT_nA at the point f_A, and meets:

$$k_m{T}_{nA}=\frac{3p{U}_n^2}{4\pi f_A[R_1+\sqrt{R_1^2+4{\pi }^2{f}_A^2{(L_{l1}+L_A+{L^{\prime }}_{l2})}^2]}}$$

(28)

It is expressed as:

$$L_{\Sigma }^2=\frac{3p{U}_n^2}{16{\pi }^3{f}_A^4{k}_m{T}_{nA}}\left(\frac{3p{U}_n^2}{4\pi k_m{T}_{nA}}-2R_1{f}_A\right)$$

(29)

According to the design Principle II, the Equation (13) is:

$$\frac{k_{stm}{U}_N{}^2}{{\left(R_1+{R^{\prime }}_2\right)}^2+4{\pi }^2{f}_A^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2}=\frac{U_N{}^2}{{\left(R_1+{R^{\prime }}_2\right)}^2+4{\pi }^2{f}_{1L}^2{\left(L_{l1}+L_A+{L^{\prime }}_{l2}\right)}^2}$$

(30)

It is approximately $\left(1/k_{stm}-1\right){\left(R_1+{R^{\prime }}_2\right)}^2<<4{\pi }^2{f}_A^2{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2$, and written as:

$$L_{\Sigma }^2={\left(L_{l1}+L_A+{L^{\prime }}_{l2}\right)}^2\approx \frac{f_A^2}{k_{stm}{f}_{1L}^2}{\left(L_{l1}+{L^{\prime }}_{l2}\right)}^2$$

(31)

Coupled the (27) and (31) or (29) and (31), the f_A and L_A can be estimated.

4.5. Design Example of Low-Frequency Step-Down Starting

The variable frequency asynchronous motor shown in Table 1 is taken as an example, and the design methods of voltage reduction, the series resistance, and the series inductance are respectively applied. It is assumed that the constant power load is driven, and the motor starts with load at the same time for meeting k_st the design requires k_st = 1.2~1.4. Coupled with the (15) and (19), (21) and (25), (27) and (31), respectively. The simulation results are shown in

Table 3. Design data of low-frequency step-down starting.

Control Parameters	Voltage Reduction	Series Resistance	Series Inductance
fA (Hz)	510	590	480
Step-down Data	UA = 84 V	RA = 0.242 Ω	LA = 0.091 mH
TnA (N·m)	9.75	8.45	10.38
TstA (N·m)	13.23	11.15	12.68
kstA	1.36	1.32	1.22
TstmA (N·m)	70.2	52	77.71
TstmL (N·m)	73.63	55.5	83.13

.

The starting peak torque characteristics of voltage reduction methods are shown in Figure 7.

The motor starts with the rated voltage, and the peak torque T_stm of power supply with f_1L (360 Hz) is 135.49 N·m. As can be seen from the data in the Table 3, it respectively reduces to 54%, 41% and 61% of the the peak torque T_stm with 360 Hz power supply. The effect of series resistance is the best.

The above mentioned design methods may not achieve the goal of principle II, since the approximate relationship is used and it is affected by magnetic saturation. If the gap between T_stmL and T_stmA is large, it needs to adjust appropriately. When T_stmL is larger than T_stmA, the voltage U_A can be appropriately reduced, or the step-down resistor R_A or inductance L_A can be increased. At the same time, in order to ensure the requirement of k_st, the frequency boundary point f_A can be appropriately reduced.

5. Characteristic Analysis of Low-Frequency Step-Down Starting

5.1. Characteristics of Starting Peak Torque

The aviation variable frequency asynchronous motors apply three kinds of low-frequency step-down starting methods by the voltage reduction, the series resistance, and the series inductance. According to the characteristics of starting peak torque T_stm shown in Figure 7, it has the following characteristics:

(1)

The series resistance method has the better suppression effect on the starting peak torque T_stm, as shown in Figure 8a. When the power supply frequency f₁ decreases in the low-frequency stage, the voltage drop across the resistor R_A increases due to the increase of starting current I_st. The voltage on the stator winding reduces to make T_stm decrease effectively.

(2)

The series inductance method has the poor suppression effect on the starting peak torque T_stm, as shown in Figure 8a. When the power supply frequency f₁ decreases in the low-frequency stage, the impedance on the inductor L_A decreases linearly. However, the starting current I_st does not increase linearly. The voltage drop on the inductor L_A reduces and the voltage on the stator winding increases, causing the larger low-frequency T_stm.

(3)

The disadvantage of series resistance method is that it consumes the parts of power and causes the resistor heat. Therefore, it needs to be analyzed for the asynchronous motor that requires frequent starting. For the long-time work motor, the power consumed can be ignored since the starting time is short.

5.2. Characteristic Analysis of Starting Current

The surge current during the traditional asynchronous motor starting is always considered as an important factor affecting the power supply stability. The starting current of variable frequency asynchronous motor can be expressed as:

$$I_{st}=\frac{U_1}{\sqrt{{(R_1+{R^{\prime }}_2)}^2+4{\pi }^2{f}_1^2{(L_{l1}+{L^{\prime }}_{l2})}^2}}\approx \frac{3p_n{U}_1^2{R^{\prime }}_2}{2\pi f_1(L_{l1}+{L^{\prime }}_{l2})}$$

(32)

As seen from Equation (32), the starting current I_st is approximately inverse proportional to the power frequency f₁. For the variable frequency asynchronous motor with the rated voltage, the starting current I_st with the high-frequency (800 Hz) is 167.2A, and the starting current I_st with low frequency (360 Hz) is 286.3A, as shown in Figure 8b.

The low-frequency step-down starting can also suppress the surge current I_st during the starting. The starting current I_st of step-down methods is shown in Figure 8b. The maximum is at the frequency boundary point with the rated voltage.

(1)

The frequency boundary point of series resistance method is the highest, and the starting current is the smallest with the rated voltage, which is 214A. The frequency boundary point of series inductance method is the lowest, and the starting current is the largest, which is 243A. The voltage reduction method is center, and the current is 236A.

(2)

For the starting current at the frequency f_1L, the series resistance method has the lowest voltage, and the starting current is also the lowest, which is 195A. The starting currents of voltage reduction and the series inductance are 212A and 216A, respectively. The effect of series resistance is the best.

5.3. Simulation Test and Analysis of Starting Process

The power supply frequency of variable frequency asynchronous motor is set with 400 Hz, and the load torque is 12.46 N·m, which is the rated torque. Three kinds of low-frequency step-down starting methods are used for the simulation test, and the normal power supply is restored after the starting at the 0.28 s. The characteristics of torque and phase current are shown in Figure 9.

The

Table 4. Simulation data of starting process by different step-down methods.

Control Methods	Starting Characteristics			Switching Characteristics
Parameters	Tstm (N·m)	Tst (N·m)	Ist (A)		∆Tem (N·m)	∆IAm (A)
Voltage reduction	60.73	24.07	219.85		20.25	71.06
Series resistance	47.87	23.23	193.58		5.61	7.37
Series inductance	66.43	21.19	231.58		8.82	14.08

shows the simulation data of starting process with the step-down voltage methods. The steady-state torque shown in Figure 9 is 12.46 N·m, the effective value of steady-state phase current is 35.15A, and the peak value is 49.7A.

The ∆I_Am of switching characteristics is the fluctuation value based on the steady-state peak current (49.7A), which can be seen from the data in Table 4:

(1)

The starting torque T_stm of three methods is consistent with the characteristics of Figure 7. The series resistance method is the smallest, and the starting torque T_st is not a huge different.

(2)

The starting current I_st of three methods is consistent with the characteristics of Figure 8b, and series resistance method is the smallest.

(3)

After the step-down starting is completed, the normal power supply switching is restored, and the ΔT_em, ΔI_Am of series resistor method are the smallest.

5.4. Experiment of Pump-Type Variable Frequency Asynchronous Motor

The 850 W pump-type variable frequency asynchronous motor with the narrow frequency conversion (360~600 Hz) is used to test the starting characteristics. The parameters of prototype are shown in

Table 5. Parameters of pump-type motor.

Parameters	Value
R1 (Ω)	1.381
Ll1 (mH)	0.612
R’2 (Ω)	1.762
L’l2 (mH)	1.328
Lm (mH)	10.226

.

For the experiments of the variable frequency asynchronous motor, two experiments are performed, namely the starting (stall) experiment and the starting torque dynamic experiment. The AC variable frequency power supply uses a capacity of 30 kVA. The output frequency of power supply can be adjusted in a wide range from 300~800 Hz, and the output voltage range is from 92.0~150.0 V. For the strating (stall) test, the prototype is placed on the torque bench and connected to the 30kVA power electronic variable frequency power supply and carried out the stall test with diffrernt frequncies, shown in Figure 10. It is approximately taken as the starting torque analysis. The straight line is the simulation characteristic with the motor design data, and the “*” is the experimental data. It can be seen that the experimental torque is significantly lower than the simulation value with the low frequency, and the reason is the low-frequency saturation factor.

For the starting torque dynamic experiment, the prototype was placed on a loadable speed regulation test bench. The experimental equipments mainly include the 30KVA AC variable frequency power supply, the pump-type variable frequency asynchronous motor, the adjustable resistance box, and the digital oscilloscope. According to the series resistance design method of low-frequency step-down starting, the series resistance is 2 Ω. The purpose of experiment is to verify the current and torque starting characteristics of variable frequency asynchronous motor with the direct starting and the series resistance starting. For the direct starting experiment, the AC variable frequency power supply is adjusted to the 115 V/400 Hz output, and directly connected to the variable frequency motor at a certain time, recording the current data through the oscilloscope. For the series resistance starting, the AC variable frequency power supply is also adjusted to the 115 V/400 Hz output, and connected the resistor of 2 Ω. The variable frequency motor is connected at a certain time, recording the current data through the oscilloscope.

The dynamic experiment of starting process can measure the three-phase voltage and current. The A-phase voltage and current waveforms are shown in Figure 11. Figure 11a–c shows the voltage waveforms with the direct starting and the series resistance starting. When the motor starts, the voltage drops the 3.5% and 5.2%, respectively. Figure 11b–d shows the current waveforms with the direct starting and the series resistance starting, and the peak current reaches 28.12A and 20.04A, respectively.

In order to obtain the approximate trend of stating pulse vibration torque, the rotor current is calculated by using Formula (6) and the dynamic torque is calculated by using Formula (9). Figure 12a–b shows the torque waveforms with the the direct starting and the series resistor starting.

The peak torque reaches 7.78 N·m and 4.21 N·m, respectively. The peak torque is 45.89% less than the peak torque with the direct starting.

Obviously, the data of A point and B point shown in Figure 10 can be expressed separately the change trend with the direct starting and the series resistance starting. The peak torque of experimental data is smaller than that of simulation, since the power supply exhibits the voltage transient drop effect during the starting.

6. Conclusions

The strating pulse vibration torque of aviation variable frequency asynchronous motor is taken as the research object. The design methods of low-frequency step-down starting is proposed, which are the voltage reduction, the series resistance, and the series inductance according to a novel approximate model by the data fitting for peak torque generated by the starting pulse vibration torque. The paper also proposes the estimation algorithm of parameters and analyzes and compares the starting pulse vibration torque characteristics. The research and experiment results show that the low-frequency step-down starting can effectively reduce the pulse vibration torque and the surge current during the low-frequency starting. The series resistance method has the better effect on the peak torque caused by the starting pulse vibration torque and the starting current suppression, which is suitable for the variable frequency asynchronous motor drive system with the infrequent starting. The actual motor model establishment is worth considering in future research.