# RTC (reaction torque compensated) induction motor and its open-loop control

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## Abstract

Rapid change of the rotating speed of an induction motor causes large reaction torque or excessive vibration of the system base. This paper proposes a passive reaction torque compensated (RTC) induction motor considering the dynamic characteristic of open-loop speed control with a variable frequency drive. RTC mechanism converts the reaction torque into the dissipative inertial energy of the rotary stator and the potential energy of the torsion spring torque so that a part of the reaction torque or the spring torque is transmitted to the system base. First, dynamic equation and simulation model for the RTC induction motor are introduced and verified with experiments. The acceleration response of an induction motor with variable speed drive is approximated as first-order ordinary differential equation. Then, the RTC mechanism such as spring and additional inertia are designed considering derived acceleration response or torque profile. Then, an experimental set-up and its control system for the RTC induction motor are built. Finally, effectiveness of the RTC induction motor is verified comparing open-loop speed control of both RTC and conventional induction motor.

## Keywords

Induction motor Variable speed control Reaction torque compensation (RTC)## Abbreviations

*c*_{m,}*c*_{s,}*c*_{b}Damping of rotor, stator and system base

*d*_{0}Initial deflection of tension spring at balance position

*F*Force caused by tension spring

*i*_{rq,}*i*_{sq}*q*-component of rotor and stator current*J*_{m,}*J*_{s,}*J*_{b}Inertia of rotor, stator and system base

*K*Stiffness of tension spring

*k*_{s,}*k*_{b}Stiffness of equivalent stator and system base springs

*L*_{0}Undeformed length of tension spring

*L*_{m,}*L*_{r}Magnetizing and rotor inductance

*p*Number of poles

*R*Distance from center of motor to pin of attached spring

*R*_{r}Rotor resistance

*r*Radius of the pins

*T*_{e,}*T*_{L}Motor and load torque

*T*_{ks}Torque generated by spring between stator and base

*T*_{Trans}Transmitted torque to the system base

*γ*_{m}Rotor acceleration

*η*_{m,}*τ*_{m}Motor constants

*λ*_{rd}*d*-component of rotor flux*θ*_{m,}*θ*_{s,}*θ*_{b}Rotary angle of rotor, stator, system base

*ω*Frequency

*ω*_{d,}*ω*_{m}Rotating flux and rotor speed

## 1 Introduction

AC induction motors are the most common motors in the world and used in industrial fields as well as home appliances [1, 2, 3, 4] due to their unique advantages such as simple and rugged design, low-cost, low maintenance and direction connection to an AC-power source [5].

Variable torque and speed load are most commonly found in the industry and variable speed operation of an induction motor allows to optimize the process under varying load conditions [6]. However, rapid change of the rotating speed of an induction motor causes large reaction torque. The reaction torque of the induction motor may generate excessive vibration of the system base, which results in reduction of its life span or damage of the entire system [7].

A passive, Active and semi-active RFCs (reaction force compensation) to reduce the system vibration have been studied only for a linear motor motion stage [8, 9, 10, 11]. Although the passive RFC mechanism is compact and cost-effective, the passive RFC does not allow in situ modification of the dynamic characteristic of the RFC system. An active RFC mechanism using an additional coil can tune its dynamic characteristic and minimize the transmitted force against the motion profile variation [9]. In addition, a semi-active RFC can modify the damping of the RFC by adjusting the external resistor or open–close time ratio of an additional coil [9, 10, 11].

Reaction torque compensations of an electric motor or driving system with additional motor or structure were proposed for mechanical stabilization [12, 13]. In addition, control algorithms were studied to cancel the reaction torques of the transmission system such gear or wire [14, 15].

This paper proposes an RTC induction motor considering dynamic characteristic of open-loop speed control with a variable frequency drive. The RTC induction motor has a rotary stator and torsion spring as well as rotor, bearing and housing. Through the torsion spring, only part of reaction torque is transmitted to the system base, while some reaction energy is dissipated as oscillating stator motion. First, dynamic equation of the RTC induction motor is introduced. A simulation model for the RTC induction motor is built and verified with experiments. Then, the acceleration response of an induction motor with variable frequency drive is approximated as first-order ordinary differential equation. The RTC mechanism such as spring and additional inertia are optimized under given performance indices considering derived acceleration or torque profile. Experimental set-up for the RTC induction motor is built and its control system is constructed using a DSP. Finally, effectiveness of the RTC induction motor is verified comparing step responses of both RTC and conventional induction motor.

## 2 RTC induction motor

### 2.1 Principle

### 2.2 Mathematical modeling

*J*

_{m},

*J*

_{s},

*J*

_{b}are the inertias of the rotor, the stator and the base, respectively;

*c*

_{m},

*c*

_{s},

*c*

_{b}are damping coefficients of the rotor, the stator and the base; \(\theta_{\text{m}} , \theta_{\text{s}} , \theta_{\text{b}}\) are the rotary angles of the rotor, the stator and the base; \(\dot{\theta }_{\text{m}} , \dot{\theta }_{\text{s}} , \dot{\theta }_{\text{b}}\) are the speed of the rotor, the stator and the base; \(\ddot{\theta }_{\text{m}} , \ddot{\theta }_{\text{s}} , \ddot{\theta }_{\text{b}}\) are the acceleration of the rotor, the stator and the motor;

*T*

_{L}is the load torque at the output shaft of the rotor,

*T*

_{e}is the motor torque. Finally,

*T*

_{ks}is the torque generated by spring between the stator and the base and is generally a function of the angle displacement of the stator.

In this study, dynamics of the system base is neglected in designing the RTC induction motor. During the conceptual design of the RTC induction motor, the dynamics of the system base is not known and an iterative design between the RTC and the system base is necessary [19]. In addition, the first step of the iterative design is to preliminarily determine the parameters of the RTC mechanism ignoring the base dynamics.

*k*

_{s}, the stator rotation and transmitted torque can be expressed in Eqs. (5) and (6).

## 3 Design of RTC mechanism

### 3.1 Simulation model and motor validation

Parameters of the induction motor

Rotor inertia | 0.065 × 10 |

Rotor damping coef. | 0.033 × 10 |

Rotor resistance | 1.45 |

Stator resistance | 2.35 |

Mutual inductance | 0.03 |

Leakage inductance of stator winding | 0.005 |

Leakage inductance of stator winding | 0.004 |

^{2}) by adjusting frequency and amplitude of the applied voltage on the three phase stator windings, as shown in Fig. 5. The blue line presents the profile for the motor speed while the red line illustrates the applied voltage on one phase coil among the three-phase stator windings. In addition, the corresponding phase currents of the induction motor by simulation and experiment are compared in Fig. 6. Furthermore, resulting motor speed and torque by simulation and experiment are compared in Fig. 7. Simulation and experiment agree well with each other, which confirm that the simulation model has enough accuracy to design the RTC mechanism.

### 3.2 RTC mechanism

*K*, undeformed length

*L*

_{0}and initial deflection

*d*

_{0}produces torque via spring force

*F*as the stator rotates. The equivalent torsion stiffness of the RTC mechanism is derived but not a linear function of stator rotation angle

*θ*

_{s}, as shown in Eq. (7). Torque of the RTC mechanism according to the stator rotation is calculated and shown in Fig. 8b. Furthermore, by changing initial deflection of spring, equivalent torsion stiffness is also justified according to results from Fig. 8c. Although the equivalent torsion is not linear, it can be approximated as a linear spring

*k*

_{s}during the preliminary design of the RTC mechanism.

*R*,

*r*are the radii of the stator and the pin, respectively, as shown in Fig. 8a,

*θ*

_{s}is the rotation angle of the stator, and

*A*=

*L*

_{0}+

*d*

_{0}+

*R*−

*2r*.

### 3.3 Approximate acceleration during open-loop speed control

^{2}(dash red line). The RTC mechanism should be designed considering the speed profile of the induction motor.

Since the speed response of the induction motor would not track accurately the speed command under open-loop control of the V/F method, it is necessary to find an approximated profile of the motor acceleration for proper design of the RTC mechanism.

*q*-axis after eliminating rotor flux component on that axis 0. Then, the rotor speed can be calculated with Eq. (9).

*d*-axis is assumed to be aligned with the orientation of the rotor flux,

*q*-components of the rotor and stator currents have the relationship shown in Eq. (10). From Eq. (9) and (10), the rotor speed can be calculated with Eq. (11).

*q*-component of stator current can be determined using the rotor torque (Eq. 12) and dynamic equation of the motor speed (Eq. 13). Thus,

*q*-component of the stator current can finally be derived as Eq. (14).

*η*

_{m}and

*τ*

_{m}are constants in Eqs. (16) and (17).

*d*-axis component of the rotor flux is not constant under open-loop control, its value does not change much during the operating. Taking derivative of Eq. (15) under the assumption of constant rotor flux, dynamic equation of the rotor acceleration γ

_{m}can be expressed with Eq. (18).

### 3.4 Design of RTC mechanism

*T*

_{L}and

*c*

_{m}

*ω*

_{m}are becoming essentially smaller than in comparison with the other component in Eq. (13). Thus, as the motor torque is calculated by multiplying the approximated acceleration and the rotor inertia (\(T_{e} \approx J_{m} \ddot{\theta }_{m}\)) and used as input for Eq. (1), (2), (3), (4), maximum rotation angle of the stator and transmitted torque with various stator inertia and torsion stiffness are calculated and shown in Fig. 11a and b, respectively. The maximum stator rotation and the transmitted torque to the base can be adjusted by changing design parameters of the RTC mechanism such as stator inertia and torsion stiffness. Bigger stator inertia results in smaller stator rotation and transmitted torque, while stronger torsion spring leads to smaller stator rotation but higher transmitted torque. The upper limit of the stator rotation is 30 degrees or the maximum stator angle under 40 mm initial deflection from Fig. 8b, while the desired maximum transmitted torque is chosen to be smaller than 70% with respect to the motor torque. In particular, the maximum transmitted ration need to be tuned considering the geometric constraints (maximum stator rotation and initial deflection) or spring stiffness. Those two performance indices are shown by dashed red line in Fig. 11a and b, respectively. Due to these conditions, the RTC region can be determined as shown in Fig. 11c. Moreover, boundary lines to distinguish between RTC region (white area) and undemand region (gray area) is defined based on two conditions such as: limit of stator rotation (≤ 30

^{o}) and maximum transmitted torque (≤ 70%) and all points in RTC region is satisfied for RTC system. Finally, a design point is optimal one for the lowest stator inertia of motor (0.00228 kg m

^{2}) which do not require more additional mass in the original motor stator. At design point, the spring stiffness is known as value of 0.12Nm/rad, and initial defection is 7.36 mm.

## 4 Results

### 4.1 Experimental set-up

Parameters of the RTC mechanism

Stator inertia | 2.28 × 10 |

Stator damping coef. | 5.8 × 10 |

Tension spring stiffness | 20 |

Initial deflection of the stator spring | 7.36 |

### 4.2 Performance of RTC induction motor

Speed responses of the induction motor with and without RTC mechanism are compared in Fig. 13. The RTC induction motor is investigated during acceleration or speed change from 200 to 2000 rpm in 0.17 s (about 1100 rad/s^{2}). Although the rotor speed could not follow the speed profile perfectly, the rotor speed rapidly reaches its stable status in both cases of with and without RTC. Although the rotor speed of the RTC induction motor has very slow oscillation due to the stator rotation, the oscillation is so small to be ignored.

## 5 Conclusions

This paper proposed an RTC induction motor considering the dynamic characteristic of open-loop speed control with a variable frequency drive. The RTC induction motor has a rotary stator and torsion spring as well as rotor, bearing and housing. First, dynamic equation of RTC induction motor is introduced and a simulation model for the RTC induction motor is built. Then, the acceleration response of the induction motor with variable speed drive is approximated as a first-order ordinary differential equation. The RTC mechanism such as spring and additional inertia are determined considering the derived acceleration or torque profile. Experimental set-up for the RTC induction motor is built and its control system is constructed using a DSP. Finally, effectiveness of the RTC induction motor is verified with experiments.

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