A steady-state and dynamic response model for working characteristics prediction of the electromechanical transmission system including eddy-current magnetic coupler

Abstract

In this paper, a steady-state and dynamic response model is proposed to predict the working characteristics of the electromechanical transmission system including eddy-current magnetic couplers. Considering the influence of mechanical characteristics of motor and load device on the performance of eddy-current magnetic couplers, the system is divided into input and output for kinematic analysis. The expressions of electromagnetic torque versus speed are obtained through layer analysis of the eddy-current magnetic coupler during steady-state and then, the dynamic response model of the system is established by combining the kinematic analysis with the motor, the eddy-current magnetic coupler and three typical loads (constant torque load, quadratic rate load and constant power load). The dynamic characteristics during starting and speed-regulation of the system under constant torque loads are analyzed using the proposed model. The variations of the torque and speed with time of input and output are calculated by numerical integration. Finally, the speed and torque during starting and speed-regulation computed with the dynamic response model are compared with those obtained from the experimental results. The result shows a good agreement.

Keywords

Electromechanical transmission system eddy-current magnetic couplers a steady-state and dynamic response working characteristics mechanical characteristic kinematic analysis

1 Introduction

As a new type of transmission device, eddy-current magnetic couplers have the advantages of reliable structure, convenient installation, low cost, energy-saving, and environment friendliness. They are usually applied in cases where flexible transmission or non-contact transmission is required, such as ship, mining, and electric power generation [1–3].

Eddy-current magnetic couplers have been studied in considerable literatures. Theoretical methods are popular, which can analyze the magnetic field and the performance of eddy-current magnetic couplers by simplifying the 3D model into a 2D layer model [4,5]. However, theoretical methods usually need intensive computation, and they cannot handle complex geometries. Recently, the theoretical models using magnetic equivalent circuit [6–11] and magnetic vector potential [12–15] have been proposed, which can overcome the drawbacks of conventional analytical methods in accuracy and consider material properties. In [6], a new equivalent magnetic circuit model is proposed, in which the eddy current effects are considered through treating the reaction magnetic flux as a leakage flux. In [15], based on the theoretical field calculations, a 2-D model is presented to obtain the torque characteristics, and the saturation effects and PM shapes are considered. Most of these literatures focus on how to make the calculation of torque more accurate, and lack of analysis of torque characteristics and speed regulation performance after adding load.

Moreover, most of the aforementioned research focuses on the performance of eddy-current magnetic couplers, and only the steady-state characteristics are investigated, rarely on both the steady-state and dynamic characteristics. In [16], an approach for evaluation performances of eddy-current magnetic couplers is proposed, which is available for steady-state and transient performance, and the variation of the output torque with time has been studied. In addition, some scholars have studied the dynamic characteristics of the eddy-current magnetic couplers [17–19]. However, the mechanical characteristics of motor and different loads have an effect on the dynamic performance of eddy-current magnetic couplers, which is not included in previous research [20–23]. Therefore, studying the dynamic characteristics of the electromechanical transmission system including eddy-current magnetic couplers is important.

In addressing the above-mentioned problems, a steady-state and dynamic response model is developed to predict the working characteristics of the electromechanical transmission system including eddy-current magnetic couplers. The system is divided into input and output to consider the influence of mechanical characteristics of motor and load on eddy-current magnetic couplers. The electromagnetic torque under steady-state, the variations of the torque and speed during starting and speed-regulation under dynamic are obtained using the proposed model. Finally, an experimental platform under constant torque loads is built to verify the accuracy of the model.

2 Steady-state theoretical model of electromechanical transmission system

2.1 Structure of eddy-current magnetic couplers

The structure of electromechanical transmission system is showed in Figure 1, which is composed of a motor, an eddy-current coupler (including an input rotor and an output rotor), and a load device. The motor is connected to the input rotor, and the load is connected to the output rotor. The input rotor is equipped with a set of permanent magnets (PMs) arranged alternately in the circumferential direction. The PMs are axially magnetized. The output rotor is equipped with a solid conductor. The output speed is adjustable by regulating the axial distance of the two rotors. The structure of eddy coupler is shown in Figure 2 and the geometric parameters are listed in Table 1.

Figure 1.

Structure of electromechanical transmission system.

Figure 2.

Structure of eddy-current magnetic couplers.

Table 1.

Specifications of the eddy-current coupler.

Symbol	Quantity	Values
p	Numbers of pole pairs	9
δ_c	Thickness of conductor back iron	10 mm
h _c	Thickness of conductor	5 mm
g	Thickness of air gap	2–14 mm
h _m	Thickness of PMs	10 mm
δ_m	Thickness of PM back iron	10 mm
r ₁	Inner radius of PMs	80 mm
r ₂	Outer radius of PMs	125 mm
r ₃	Inner radius of conductor layer	65 mm
r ₄	Outer radius of conductor layer	140 mm
μ _2r	Relative recoil permeability	1.099
B _r	Remanence of the PMs (NdFeB)	1.21 T
σ	Conductivity of conducting plate	57 MS/m

Considering that the eddy-current magnetic coupler transmits torque without contact, the input rotor and output rotor of the eddy-current magnetic coupler are in different states during the actual operation. The kinematic analysis of the system needs to be divided into two parts: input and output. The former works under the effect of the torque (T_M) provided by the motor and the electromagnetic torque (T_EC) of the eddy-current coupler. Similarly, the latter works under the effect of the electromagnetic torque (T_EC) and the torque of load (T_L).

2.2 Electromagnetic torque of the eddy-current magnetic couplers

In order to simplify the 3-D problem in Figure 2 into a 2-D problem, the five-layer model as illustrated in Figure 3 can be created by imagining that the coupler is cut at the average radius of the magnets r = (r₁ + r₂)∕2 and spread flat in the circumferential direction.

Figure 3.

Two-dimensional model of the eddy-current magnetic couplers at the mean radius of the magnets r = (r₁ + r₂)∕2.

The main assumptions underlying the theoretical model areas follows.

(1) The material properties of each layer of the magnetic couplers are evenly distributed to ensure that the permeability and conductivity of the material are certain, while ignoring the influence of external factors such as temperature on the material performance parameters;

(2) The strengthening and weakening effect of external magnetic field on the permanent magnet magnetic field is ignored;

(3) The conductor layer and the back iron are stationary, and the permanent magnet and the back iron move along the x-axis, and the thickness of the air gap remains unchanged;

(4) Wind resistance and other losses in the process of motion are ignored.

(5) The thicknesses of two back iron layers are limited and the flux-parallel boundary condition is satisfied both on Γ₁ and on Γ₅.

A magnetic vector potential formulation ( $A^{i} = A_{z n}^{i} (x, y)$ $e_{z}, i = 1, 2, 3, 4, 5$ ) is used to solve the layer problem. From the Maxwell equations and considering the adopted assumptions, we have to solve the following equations in each region, and the field equations associated with the n-th harmonic can be written in scalar form as:

{\begin{matrix} \frac{\partial A_{z n}^{I} (x, y)}{\partial y^{2}} - a_{n}^{2} A_{z n}^{I} (x, y) = 0 \\ \frac{\partial A_{z n}^{II} (x, y)}{\partial y^{2}} - a_{n}^{2} A_{z n}^{II} (x, y) = 0 \\ \frac{\partial A_{z n}^{III} (x, y)}{\partial y^{2}} - a_{n}^{2} A_{z n}^{III} (x, y) = 0 \\ \frac{\partial A_{z n}^{IV} (x, y)}{\partial y^{2}} - (a_{n}^{2} - \frac{j n p μ_{4} σ_{4} s ω_{m}}{r}) A_{z n}^{IV} (x, y) = 0 \\ \frac{\partial A_{z n}^{V} (x, y)}{\partial y^{2}} - (a_{n}^{2} - \frac{j n p μ_{5} σ_{5} s ω_{m}}{r}) A_{z n}^{V} (x, y) = 0 \end{matrix}

(1)

with

{\begin{matrix} α_{n} = \frac{n p}{2 r} \\ s = \frac{n_{i n} - n_{o u t}}{n_{i n}} \\ ω_{m} = \frac{π n_{i n} p}{30} \end{matrix}

(2)

where μ₀ is the vacuum permeability, n_in is the rotation speeds of the primary, n_out is the rotation speeds of the secondary and M_yn(x) is the axial magnetization of the magnets. The distribution of the axial magnetization M_yn(x) is plotted in Figure 4. The axial magnetization can be expressed in Fourier’s series.

\vec{M} = \sum_{n = 1, 3, 5 \dots} Re (M_{y n} (x) e^{j n s ω_{m} t}) \vec{e_{y}}

(3)

with

M_{y n} (x) = \frac{4 B_{r}}{μ_{0} n π} \sin (\frac{n π τ_{m}}{2 τ_{p}}) e^{- j α_{n} x}

(4)

where τ_m is the length between adjacent PMs, τ_p is the length between centers of adjacent PMs, t is time, and j is imaginary number.

Figure 4.

Magnetization distribution along the x-direction (Region II).

According to the symmetry of the magnetic field distribution a for such problem, the solutions of (1) are derived separately as

{\begin{matrix} A_{z n}^{I} = (C^{I} e^{α_{n} y} + D^{I} e^{- α_{n} y}) e^{- j α_{n} x} \\ A_{z n}^{II} = (C^{II} e^{α_{n} y} + D^{II} e^{- α_{n} y}) e^{- j α_{n} x} - j μ_{0} M_{y n} / α_{n} \\ A_{z n}^{III} = (C^{III} e^{α_{n} y} + D^{III} e^{- α_{n} y}) e^{- j α_{n} x} \\ A_{z n}^{IV} = (C^{IV} e^{β_{c n} y} + D^{IV} e^{- β_{c n} y}) e^{- j α_{n} x} \\ A_{z n}^{V} = (C^{V} e^{β_{b n} y} + D^{V} e^{- β_{b n} y}) e^{- j α_{n} x} \end{matrix}

(5)

with

{\begin{matrix} β_{c n} = \sqrt{{α_{n}}^{2} - \frac{j n p μ_{4} σ_{4} s ω_{m}}{r}} \\ β_{b n} = \sqrt{{α_{n}}^{2} - \frac{j n p μ_{5} σ_{5} s ω_{m}}{r}} . \end{matrix}

(6)

The boundary conditions to be satisfied are

\frac{\partial A^{I}}{\partial x} = 0 on Γ_{1}

(7)

{\begin{matrix} \frac{\partial A^{I}}{\partial x} = \frac{\partial A^{II}}{\partial x} \\ \frac{1}{μ_{1}} \frac{\partial A^{I}}{\partial y} = \frac{1}{μ_{2 x}} \frac{\partial A^{II}}{\partial y} \end{matrix} on Γ_{12}

(8)

{\begin{matrix} \frac{\partial A^{II}}{\partial x} = \frac{\partial A^{III}}{\partial x} \\ \frac{1}{μ_{2 x}} \frac{\partial A^{II}}{\partial y} = \frac{1}{μ_{3}} \frac{\partial A^{III}}{\partial y} \end{matrix} on Γ_{23}

(9)

{\begin{matrix} \frac{\partial A^{III}}{\partial x} = \frac{\partial A^{IV}}{\partial x} \\ \frac{1}{μ_{3}} \frac{\partial A^{III}}{\partial y} = \frac{1}{μ_{4}} \frac{\partial A^{IV}}{\partial y} \end{matrix} on Γ_{34}

(10)

{\begin{matrix} \frac{\partial A^{IV}}{\partial x} = \frac{\partial A^{V}}{\partial x} \\ \frac{1}{μ_{4}} \frac{\partial A^{IV}}{\partial y} = \frac{1}{μ_{5}} \frac{\partial A^{V}}{\partial y} \end{matrix} on Γ_{45}

(11)

\frac{\partial A^{V}}{\partial x} = 0 on Γ_{5}

(12)

with

{\begin{matrix} μ_{2 x} = \frac{μ_{0} μ_{2 r}}{τ_{m} / τ_{p} + μ_{2 r} (1 - τ_{m} / τ_{p})} \\ μ_{3} = μ_{0} \\ μ_{4} = μ_{0} \\ μ_{5} = μ_{1} = 2000 μ_{0} . \end{matrix}

(13)

The C^I ∼ C^V and D^I ∼ D^V can be easily found by using scientific computing environments.

The induced current density in the conductor can be deduced from Ampere’s law

\vec{J} = - σ \frac{\partial \vec{A}}{\partial t} = - σ \frac{\partial \vec{A}}{\partial t} \cdot \frac{\partial t}{\partial x} \cdot \frac{\partial x}{\partial t} = - σ s ω_{m} r \frac{\partial \vec{A}}{\partial x}

(14)

and

\vec{B} = \nabla \times \vec{A} = \vec{B_{xn}} + \vec{B_{yn}} = \frac{\partial A_{zn}}{\partial y} \vec{e_{x}} - \frac{\partial A_{zn}}{\partial x} \vec{e_{y}} .

(15)

Supposing that the magnetic field travels with the velocity ν = sω_m along x-axis, the traction force in Region IV and Region V is then calculated by

F_{2 D} = p \sum_{n = 1, 3, 5 \dots} (\int_{y_{5}}^{y_{34}} \int_{- τ_{p}}^{τ_{p}} \frac{1}{2} Re (\vec{J} \times \vec{B_{y n}}) d x d y)

(16)

where F_2D is the traction force in Region IV and Region V, y₃₄ and y₅ are the ordinates respectively corresponding to Γ₃₄ and Γ₅.

To take the 3-D end effects into account, the correction factor k_s is adopted

k_{s} = 1 - \frac{\tanh (\frac{π w_{m}}{2 τ_{p}})}{\frac{π w_{m}}{2 τ_{p}} [1 + 2 \tan h (\frac{π w_{m}}{2 τ_{p}}) \tanh (\frac{π (w_{c} - w_{m})}{2 τ_{p}})]}

(17)

with

w_{m} = r_{2} - r_{1}

(18)

w_{c} = r_{4} - r_{3} .

(19)

Finally, the electromagnetic torque T_EC during steady-state is further calculated by

T_{E C} = | r \cdot k_{s} w_{m} F_{^{2 D}} | .

(20)

2.3 3D FEM of the eddy-current magnetic couplers

The parameters of the finite element model (FEM) are consistent with Table 1. Figure 5 shows the structure of the 3D model. In addition, the meshing of the PMs, air gap, and conductor is refined to ensure the accuracy of simulation results. The FE model meshed is shown in Figure 6.

Figure 5.

3D model of the studied couplers.

Figure 6.

The meshing model of the studied couplers.

Static-state magnetic induction intensity is shown in Figure 7. The magnetic field is periodically distributed, and the number of cycles is the same as the polar pairs.

Figure 7.

Static-state magnetic induction intensity.

Collating the data in Figure 7, the one pole-pair air gap magnetic density distribution at different positions is shown in Figure 8, which distance is the space between the sampling point and the permanent magnet. With the distance increasing from 0 mm to 5 mm, both axial and circumferential flux density decrease.

Figure 8.

Air gap flux density distribution (g = 6 mm) (a) axial; (b) circumferential.

The one pole-pair air gap magnetic density amplitudes at different positions is shown in Figure 9. From Figure 9, it is shown that the two-dimensional waveform distribution of the axial air gap flux density is saddle shaped, and waveform of the circumferential is distributed in a triangular wave pattern. With the decrease of distance, the amplitude of axial and circumferential flux density decreases, and the waveform is more stable.

Figure 9.

Air gap flux density amplitudes (a) axial; (b) circumferential.

The mechanical characteristic curves of the eddy-current magnetic couplers at different air gap thicknesses are shown in Figure 10. It can be observed that there is a critical value of the speed difference (n_c), and the critical values at different air gap thicknesses are almost the same. When the slip speed is less than the critical value, the torque increases rapidly, however, when the slip speed exceeds the critical value, the torque decreases gradually.

Figure 10.

Mechanical characteristic curves of the magnetic couplers at different air gap thicknesses (n_in = 1500 rpm).

2.4 Speed-regulation analysis of the electromechanical transmission system

The speed-regulation analysis of the electromechanical transmission system requires the expressions of the motor torque T_M, the electromagnetic torque T_EC and the load torque T_L. Then the relationship between motor torque T_M versus input speed n_in can be expressed by polynomial as

T_{M} = \frac{2 λ_{max} T_{N}}{\frac{n_{0} - n_{i n}}{(n_{0} - n_{N}) (λ_{max} + \sqrt{λ_{max}^{2} - 1})} + \frac{(n_{0} - n_{N}) (λ_{max} + \sqrt{λ_{max}^{2} - 1})}{n_{0} - n_{i n}}}

(21)

where n₀ is synchronization speed of motor, n_N is rated speed of motor, n_in is input speed of motor, λ_max is maximum torque coefficient of motor, and T_N is rated torque of motor.

For load devices, there are varieties of production equipment, which have different load characteristics. Among them, there are three typical types: constant torque load, quadratic rate torque load and constant power load. Thus, the load torque expressions corresponding to the three type load devices can be given by

T_{L} = {\begin{matrix} c_{1} (constant \,torque\, load) \\ k_{2} {n_{o u t}}^{2} (quadratic\, rate\, load) \\ 9550 P_{3} / n_{o u t} (constant\, power\, load) \end{matrix}

(22)

where c₁ is the load torque of the constant torque load, k₂ is the load coefficient of quadratic rate torque, and P₃ is the power of the constant power load.

The YE3-225S-4s three-phase asynchronous motor is selected to provide the power, which its rated power is 37 kW, rated speed is 1500 rpm, and λ_max is 1.8. For the load devices, c₁, k₂ and P₃ are taken as 50 N⋅m, 2.05 × 10⁻⁵ and 9.6 kW, respectively. Speed-regulation curves of the electromechanical transmission system under different loads are given in Figure 11.

As shown in Figure 11, the stable operation of electromechanical transmission system needs to meet: (1) there is an intersection between the load characteristic curve and the mechanical characteristic curve in the stable operation stage of magnetic couplers; (2) There is a point on the mechanical characteristic curve in the stable operation stage of the motor, which is equal to the intersection torque in (1). Therefore, a, b and c are stable working points of magnetic couplers under three different loads, while a’, b’ and c’ are the corresponding working points of the motor.

Figure 11.

Speed-regulation curves of the electromechanical coupler system under different loads.

Figure 12.

Speed-regulation curves of the electromechanical transmission system under constant torque load.

Figure 12 shows the speed regulation curves of the electromechanical transmission system under the constant torque load. In Figure 12, there are two speed regulation modes: (1) when the required torque is T₁ and the working point is point d, the torque may change from T₁ to T₀ and the working point may change from point d to point b due to coupler loss. At this time, the air gap thickness can be increased to 6 mm to change the torque back to T₁ and the working point to point a; (2) when the required speed is n₂ and the working point is point b, due to the sudden change of external load, the speed may be reduced from n₂ to n₁ and the working point may be changed from the original point b to point d. At this time, the air gap thickness can be increased to 6 mm, so that the speed changes back to n₂ and the working point changes to point a.

Figure 13 shows the speed regulation curves of the electromechanical transmission system under the quadratic rate load. Taking the g = 4 mm as an example, the load curve with variable load coefficients of k₁, k₂ and k₃ and the mechanical characteristic curve intersect at points a, b and c respectively, while corresponding motor operating points are a’, b’ and c’. When k₁ = 4.1 × 10⁻⁵, the speed corresponding to point a is n₁, and the speed regulation interval is 0 ∼ n₁. Similarly, when k₂ = 7 × 10⁻⁵, the speed regulation interval is 0 ∼ n₂, and when k₃ = 13.5 × 10⁻⁵, the speed regulation interval is small is 0 ∼ n₃. It can be seen from the figure that the closer the k is to 0, the larger the speed regulation range is.

Figure 13.

Speed-regulation curves of the electromechanical transmission system under quadratic rate load.

2.5 Prototype and experimental platform

The test platforms are established to verify the accuracy of the proposed model. Figure 14 shows the electromechanical transmission system under the constant torque load. Component 1 is YE3-225S-4s three-phase asynchronous motor with capacity of 37 kW, which is adopted as power source. Component 2 and 4 are speed torque sensors which are connected to the input and output of the eddy-current coupler respectively, which send data to computer for real-time monitoring. Component 3 is an eddy-current coupler, the parameters of which are described in Table 1 detailedly. Component 5 is GZ-20 magnetic powder brake with torque capacity of 200 Nm, which is adjusted to provide constant torque load for the test system.

Figure 14.

Experimental platform of different load conditions: (1 is motor; 2 is sensor; 3 is eddy-current coupler; 4 is sensor; 5 is magnetic powder brake for constant torque load and fan for quadratic rate load).

2.6 Comparison with experimental results

The results obtained by the 3-D FEM and experimental results are compared with those obtained by the theoretical calculation.

Figure 15 shows the comparison of the torque-slip characteristics predicted by the method proposed and 3D FEM in the case of g = 6 mm. It is observed that the torque values of the two method are in good agreement at low speed difference. With the increase of slip, the torque values of method proposed deviates from the one of 3D FEM gradually, the error is 2.2% when the slip is 0.073. Since magnetic couplers usually work at low speed difference to reduce eddy current loss, the method proposed can be used for torque calculation.

Figure 15.

Torque versus slip.

The mechanical characteristic curves of magnetic couplers with different speed are shown in Figure 16. The output speed decreases with the increase of torque, and the higher the torque, the faster the output speed decreases. Comparing the theoretica values with the experimental one, it is obviously observed that the theoretica values is well agree with experimental values under the condition of low torque. Besides, beyond 80 Nm, the higher the torque is, the greater the error of the theoretical formula is.

Figure 16.

The mechanical characteristic curves of magnetic couplers with different speed (g = 6 mm).

3 Dynamic theoretical model of electromechanical transmission system

3.1 Dynamic response model

Considering the non-contact characteristics of the eddy-current magnetic couplers, converting the rotational inertia of the output in the kinematic analysis of the input is unnecessary. The kinematic equation of the input can be expressed as:

T_{M} - T_{E C} = \frac{J_{i n} π}{30} \frac{d n_{i n}}{d t}

(23)

where J_in is the rotational inertia of the input (J_in = 5.27 × 10⁻² kg⋅m²), and n_in is the input speed.

Similarly, the kinematic equation of the output can be expressed as

T_{E C} - T_{L} = \frac{J_{o u t} π}{30} \frac{d n_{o u t}}{d t}

(24)

where J_out is the rotational inertia of the output (J_out = 7.16 × 10⁻² kg⋅m²), and n_out is the output speed.

The accelerations of the input rotor and output rotor of the magnetic coupler are calculated as:

{\begin{matrix} a_{i n} = \frac{d n_{i n}}{d t} = \frac{30 (T_{M} - T_{E C})}{J_{i n} π} \\ a_{o u t} = \frac{d n_{o u t}}{d t} = \frac{30 (T_{E C} - T_{L})}{J_{o u t} π} . \end{matrix}

(25)

Assuming that the moment when the coupler starts or begins to adjust the speed is time 0, then the variations of the input speed and the output speed from the time 0 to the current time t can be given by

{\begin{matrix} Δ n_{i n} = \int_{0}^{t} a_{i n} d t \\ Δ n_{o u t} = \int_{0}^{t} a_{o u t} d t . \end{matrix}

(26)

By substituting Eqs (23) to (25) into Eq. (26), the Eq. (26) can be re-expressed as

{\begin{matrix} Δ n_{i n} = \frac{30}{J_{i n} π} \int_{0}^{t} (T_{M} - T_{E C}) d t \\ Δ n_{o u t} = \frac{30}{J_{o u t} π} \int_{0}^{t} (T_{E C} - T_{L}) d t . \end{matrix}

(27)

The initial values of the input speed and output speed are 0 when the coupler starts, thus, the dynamic input speed n_in and output speed n_out during starting are given by

{\begin{matrix} n_{i n} = Δ n_{i n} \\ n_{o u t} = Δ n_{o u t} . \end{matrix}

(28)

If the input speed and output speed in the steady-state are $n_{i n}^{^{'}}$ and $n_{o u t}^{^{'}}$ respectively, then the input speed and output speed during speed-regulation are calculated as follows:

{\begin{matrix} n_{i n} = n_{i n}^{'} + Δ n_{i n} \\ n_{o u t} = n_{o u t}^{'} + Δ n_{o u t} . \end{matrix}

(29)

Based on the established model, the dynamic characteristics during starting and speed-regulation can be obtained. The variations of torque and speed with time are calculated by the numerical analysis software.

3.2 Dynamic characteristics during starting

Assuming that the initial values of n_in and n_out during starting are 0.1 and 0 rpm respectively, the variations of the torque and speed with the starting-response time under the constant torque load are analyzed.

The variations of speed with time at the air gap thickness of 6 and 14 mm under the constant torque load (T_L = 10 Nm) are shown in Figure 17. When the air gap thickness is 6 mm, the trends of the input speed and output speed are similar, and they reach the stable operation stage almost simultaneously. However, when the air gap thickness is 14 mm, the input speed quickly reaches the stable operation stage, which shortens the existence time of the starting current of the motor, whereas the output speed slowly reaches the stable operation stage, with smaller acceleration and less impact force on load devices. Therefore, in order to prolong the life of the motor and load, it is preferable to start the eddy-current coupler at a large air gap.

Figure 17.

Variation curves of speed during starting under the constant torque load (a) g = 6 mm; (b) g = 14 mm.

The variations of torque with time at the air gap thickness of 6 and 14 mm under the constant torque load (T_L = 10 Nm) are shown in Figure 18. When the air gap thickness is 6 mm, the torque reaches the peak value and then stabilizes quickly. However, when the air gap thickness is 14 mm, the torque gradually reaches a peak to become stable. This phenomenon is caused by the variation of the slip speed. As shown in Figure 17, when the air gap thickness is 14 mm, the increase rate of the input speed exceeds that of the input speed, causing the slip speed to reach the critical value rapidly. Thus, the torque reaches the peak value quickly and begins to decrease gradually. Afterward, the input speed tends to be stable, and the output speed increases gradually. Consequently, the slip speed will decrease and reach the critical value, thus, the torque reaches the peak value again.

Figure 18.

Variation curves of torque during starting under the constant torque load.

3.3 Dynamic characteristics during speed-regulation

Assuming that the initial values of n_in and n_out during speed-regulation are steady-state values before speed regulation, combining the dynamic response model with numerical analysis software, the variation curves of the input speed and output speed under the constant torque load (T_L = 10 Nm) are obtained, as shown in Figure 19. When the air gap thickness changes, the slip speed remains constant because of the effect of inertia, whereas the torque changes immediately. Thus, the system becomes unstable. When the air gap thickness is adjusted from 2 mm to 14 mm, the input speed increases slightly in a short time and then returns to the original steady-state value, and the output speed decreases gradually. When the air gap thickness is adjusted from 14 mm to 2 mm, the input speed decreases obviously in a short time and then returns to the original steady-state speed, and the output speed increases gradually. Notably, in the latter speed-regulation mode, it takes less time for the system to re-enter the stable state, so the stability is better.

Figure 19.

Variation curves of speed with time during speed-regulation under the constant torque load.

The dynamic response model is used to solve the input speed when the system is stable at different air gap thicknesses. The input speeds under the constant torque load, quadratic rate load and constant power load are calculated using numerical analysis software, as shown in Figure 20. It can be seen that as the air gap thickness increases, the variations of the input speed under three operating conditions are different. For the constant torque load, the input speed remains constant. For the quadratic rate load, the input speed increases slightly by 14.5 rpm. This finding indicates that a decrease in the output speed under the quadratic rate load results in a decrease in the torque of load, thus, the resistance torque of the motor decreases, and the input speed increases. For the constant power load, the input speed decreases slightly by 2.47 rpm.

Figure 20.

Variation curves steady-state input speed with air gap thickness under different loads.

Figure 21.

Variation curves of torque during starting (g = 6 mm).

3.4 Experiments of speed and torque during starting

The experiment of dynamic characteristics during starting under the constant torque load is performed. The steady current of the magnetic powder brake is adjusted to maintain the torque of the load at 10 Nm. The frequency of the motor is 50 Hz. The sampling interval is set to 1 s. The air gap thickness is adjusted to 6 mm, and the speeds and torques of the input and output during starting are recorded by sensors 2 and 4. The analytical results are compared with the measured results, as shown in Figures 21 and 22. Then, the air gap thickness is adjusted to 14 mm for measurement, and the comparisons are shown in Figures 23 and 24. The analytical results and measured results are in good agreement. However, the input and output speeds obtained by the proposed dynamic response model are slightly smaller than those obtained by the experiment, which are caused by the factors such as friction and wind-resistance in the measurement.

Figure 22.

Variation curves of speed during starting (g = 6 mm).

Figure 23.

Variation curves of torque during starting (g = 14 mm).

Figure 24.

Variation curves of speed during starting (g = 14 mm).

3.5 Experiments of input speed during speed-regulation

Experiments of dynamic characteristics during speed-regulation under two types of load are performed. The initial air gap thickness is 2 mm, which is then adjusted to 4, 6, 8, 10, and 12 mm gradually, and the corresponding input speeds are recorded when the system is stable.

The comparisons of the results obtained by the proposed model and experiment are shown in Figure 25. The results are highly consistent, which indicate that the proposed model is accurate. In addition, with the increase of air gap thickness, the input speed under constant torque load remains, whereas that of quadratic rate load tends to increase.

Figure 25.

Variation curves steady-state input speed with air gap thickness.

4 Conclusion

In this paper, a steady-state and dynamic response model for working characteristics of electromechanical transmission system is proposed, which considers the influence of motor’s and load’s characteristics on the performance of eddy-current magnetic couplers. The proposed model can predict electromagnetic torque at low speed difference under steady-state operation, the variations of the torque and speed with time during starting and speed-regulation under dynamic operation. The analytical results are in good agreement with the experimental results, which indicate that the proposed model is effective to predict the working characteristics of eddy-current magnetic couplers.

A prototype of test has been built, and the comparison between analytical results and tests has shown the precision of the model. The analytical results on electromagnetic torque during steady-state are in good agreement with the experimental under the condition of low torque. Through detailed dynamic analysis, we have shown that when the eddy-current coupler starts at a large air gap thickness, the input reaches the steady-state earlier than the output, and the torque fluctuation is smaller, which is beneficial to prolong the life of the motor and load.

Footnotes

Acknowledgement

This work is supported by National Natural Science Foundation of China (Grant No. 51875254), and the state key laboratory mechanical transmission for advanced equipment open fund (Grant No. SKLMT-MSKFKT-202331).

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