Linear ultrasonic motor loss separation and experimental testing method
By measuring the resonant frequency of the linear ultrasonic motor and applying voltage excitation at different frequencies, an equivalent stator circuit was established, and the loss resistance was calculated. This solved the problem of loss separation in linear ultrasonic motors, provided an accurate loss assessment tool, simplified the technical problems that were not solved in the prior art, and achieved effective loss separation and assessment.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- HEFEI UNIV OF TECH
- Filing Date
- 2023-09-18
- Publication Date
- 2026-06-02
AI Technical Summary
Existing technologies struggle to accurately assess the various types of losses in linear ultrasonic motors, and there is a lack of effective experimental methods for loss separation and measurement, leading to difficulties in design and optimization.
By measuring the resonant frequency of the linear ultrasonic motor in both free and contact states, applying voltage excitation signals of different frequencies, establishing the stator equivalent circuit, calculating the resistance of mechanical damping loss, hysteresis damping loss, and contact loss, and using correction coefficients to optimize the loss assessment.
It achieves effective separation of linear ultrasonic motor losses, provides an accurate loss assessment tool, simplifies the calculation process, and is applicable to loss testing of traveling wave type rotary ultrasonic motors.
Smart Images
Figure CN117269621B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of ultrasonic motor modeling technology, specifically to a method for loss separation and experimental testing of linear ultrasonic motors. Background Technology
[0002] Linear ultrasonic motors have many advantages that electromagnetic motors cannot match in high-precision linear drive and positioning platforms. Accurate assessment of various losses in linear ultrasonic motors is one of the key factors that must be considered in the modeling, analysis and optimization design of high-quality linear ultrasonic motor systems.
[0003] Currently, the assessment of losses in linear ultrasonic motors primarily relies on analytical modeling methods that incorporate numerous assumptions. These methods simplify the contact boundary conditions between the stator and actuator of the ultrasonic motor, the actual characteristics of the piezoelectric ceramic material, and the electromechanical coupling relationship, making it difficult to separate the various types of losses in linear ultrasonic motors. Furthermore, they fail to consider the mechanisms by which different types of losses change and interact with operating conditions such as temperature rise and vibration. Consequently, the losses in the analytical results deviate from the actual values. Moreover, this method involves cumbersome calculations and is only suitable for specific linear ultrasonic motors, making it difficult to widely promote.
[0004] Furthermore, there is currently no operational experimental method for separating and testing the losses of linear ultrasonic motors. Existing experimental measurement methods for linear ultrasonic motor losses lack completeness and systematicity, neglecting the interrelationship of material losses such as piezoelectric ceramics and metal components. They either only measure the overall loss of the motor at a single frequency or only roughly obtain the mechanical damping loss, and do not consider the impact friction loss caused by impedance changes under the contact state of the stator and mover. There is an urgent need for a clear experimental method combined with basic model calculations to achieve the separate measurement of multiple types of losses in linear ultrasonic motors. This makes it difficult to compare and optimize the theoretical analytical values of various losses with the actual measured values. Summary of the Invention
[0005] Given the difficulty of using existing experimental methods to separate and measure various types of losses in linear ultrasonic motors to guide motor design and optimization, this invention provides a method for separating and experimentally testing losses in linear ultrasonic motors.
[0006] The technical solution of the present invention includes the following steps:
[0007] S1. Resonant frequency measurement steps: Measure the resonant frequency of the stator of the linear ultrasonic motor in both the free and contact states. The resonant frequency of the stator in the free state is denoted as the first resonant frequency f. r1 The resonant frequency of the motor stator in the contact state is denoted as the second resonant frequency f. r2 ;
[0008] S2. Power loss measurement procedure: In the free state, measure the power loss p of the motor stator under excitation at the first resonant frequency. m The power loss p of the motor stator under excitation at twice the first resonant frequency was measured. pzt Under contact conditions, the power loss p of the motor stator under excitation at the second resonant frequency was measured. mc ;
[0009] S3. Loss resistance calculation step: Estimate the mechanical damping loss p of the linear ultrasonic motor based on the power loss under three different states obtained in the S2 loss power measurement step. m Hysteresis damping loss p pzt and the total power loss p under contact conditions mc ; and calculate the corresponding mechanical damping loss resistance R. m Hysteresis damping loss resistor R pzt and contact damping loss resistance R c .
[0010] Preferably, the S3 loss resistance calculation step further includes an S31 measurement correction step, which corrects the power loss measured in the S2 loss power measurement step under different conditions. The correction formula is as follows:
[0011]
[0012] in, This is the correction value for mechanical damping loss. This is a correction value for the hysteresis damping loss of the piezoelectric ceramic. This is a correction value for the total power loss under contact conditions. β m β is the correction factor for mechanical damping loss. pzt β is the correction factor for the hysteresis damping loss of piezoelectric ceramics. mc This is a correction factor for the total power loss under contact conditions.
[0013] Preferably, the correction factor β for mechanical damping loss m ∈[0.8, 0.9], the correction factor β for the hysteresis damping loss of piezoelectric ceramics pzt ∈[0.5, 0.9], correction factor β for total power loss in contact state mc ∈[0.8, 0.9].
[0014] Preferably, the mechanical damping loss resistance R of the linear ultrasonic motor m Hysteresis damping loss resistor R pzt and contact damping loss resistance R c Calculate using the following formula:
[0015]
[0016] Among them, R mc For the total power loss resistance, I f1 I is the current value of the motor stator under free state excitation at the first resonant frequency. 2f1 I is the current value of the motor stator in its free state under excitation at twice the first resonant frequency. f2 This represents the current value of the motor stator under the second resonant frequency excitation in the contact state.
[0017] Preferably, the mechanical damping loss resistance R of the linear ultrasonic motor m Hysteresis damping loss resistor R pzt and contact damping loss resistance R c Calculate using the following formula:
[0018]
[0019] Among them, R mc For the total power loss resistance, U f1 U is the voltage value of the motor stator under free state excitation at the first resonant frequency. 2f1 U is the voltage value of the motor stator in its free state under excitation at twice the first resonant frequency. f2 This represents the voltage value of the motor stator under the second resonant frequency excitation in the contact state.
[0020] Preferably, in the S2 power loss measurement step,
[0021] Mechanical damping loss is measured by applying a sinusoidal voltage excitation of u = U sin w1t to the stator of the motor in a free state, where w1 = 2πf. r1 U is the voltage amplitude;
[0022] Hysteresis damping loss is measured under free conditions when a sinusoidal voltage excitation of u = U sin2w1t is applied to the motor stator, where w1 = 2πf. r1 U is the voltage amplitude;
[0023] Total power loss is measured by applying a sinusoidal voltage excitation of u = Usin2w2t to the motor stator under contact conditions, where w2 = 2πf r2 U is the voltage amplitude.
[0024] Compared with the prior art, the beneficial effects of the present invention are as follows:
[0025] 1. The stator equivalent circuit established in this invention can simultaneously simulate both the free state and the stator / mover contact state by inputting three different frequencies f. r1 f r2 and 2f r1The voltage excitation signal can effectively separate the piezoelectric ceramic hysteresis loss, stator mechanical damping loss, and stator / mover contact loss of the linear ultrasonic motor. The calculation formula is simple and reliable, and a corrected calculation formula for evaluating the three losses under the rated operating conditions of the motor is given. This provides an effective theoretical tool for accurately evaluating various losses during the operation of the linear ultrasonic motor.
[0026] 2. The experimental testing method for various losses of linear ultrasonic motors based on stator equivalent circuits proposed in this invention is reliable in principle, simple to operate, and can be extended to the loss testing of traveling wave type rotating ultrasonic motors, providing a practical and feasible solution for the experimental testing method of ultrasonic motor losses.
[0027] 3. The experimental test scheme for loss of linear ultrasonic motor proposed in this invention can be used to identify various loss resistance values in the equivalent circuit, and the physical meaning is clear, providing necessary support for constructing an accurate equivalent circuit model of linear ultrasonic motor. Attached Figure Description
[0028] Figure 1 This is a schematic diagram of the structure of the V-type linear ultrasonic motor of the present invention;
[0029] Figure 2 This is a schematic diagram of the equivalent circuit of the V-type linear ultrasonic motor of the present invention;
[0030] Figure 3 This is a schematic diagram of the measurement circuit of the V-type linear ultrasonic motor of the present invention;
[0031] Figure 4 This is a schematic diagram of the V-type linear ultrasonic motor of the present invention;
[0032] Figure 5 This is a schematic diagram of the equivalent circuit of the V-type linear ultrasonic motor of the present invention in a free state;
[0033] Figure 6 This is a schematic diagram of the equivalent circuit of the V-type linear ultrasonic motor of the present invention in contact state;
[0034] Figure 7 This is a schematic diagram showing the data flow for calculating the loss resistance of the V-type linear ultrasonic motor of the present invention.
[0035] In the picture:
[0036] 1. Motor stator; 2. Piezoelectric ceramic; 3. Preload adjustment screw; 4. Motor mover. Detailed Implementation
[0037] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. In this specification, the drawings are limited to illustrating the corresponding structures or method principles and should be considered as one embodiment of the technical solution, not as the entire scope of protection of this application. The reference numerals used in the drawings or process descriptions of this application do not indicate the order of steps. For steps with specific order requirements, the order is implicit in the inherent relationship between the steps. For steps with a specific order but unclear relationship, this application will explicitly describe them. Steps that do not meet the above two conditions should generally be considered to have no necessary order between them.
[0038] This method for separating and testing losses in linear ultrasonic motors is mainly applied to linear ultrasonic motors with V-type stators, aiming to separate and identify motor parameters and provide a basis for motor modeling and optimization. Figure 1 The schematic diagram of the linear ultrasonic motor is used in the loss separation and experimental testing of a representative V-type stator linear ultrasonic motor; for example... Figure 1 As shown, the V-type stator linear ultrasonic motor is a symmetrical structure composed of two typical Langevin stators. The two stators 1 are arranged in a V-shape, with their contacting ends flexibly fixed together. This flexibility means that although the connection is relatively fixed or integral, it can deform or undergo micro-displacement during ultrasonic vibration depending on the modal changes of the stators. Piezoelectric ceramics 2 are mounted on the stators 1 to drive them to vibrate at a specified frequency. The connection point of the stators 1 forms a driving foot that is linearly connected to the mover 2. When the stators 1 vibrate, the driving foot undergoes micro-displacement, thereby driving the mover 2 to move and achieve the predetermined function of the linear ultrasonic motor.
[0039] For modeling and analysis of V-type linear ultrasonic motors, the equivalent circuit method can be used. For example... Figure 2 As shown, an equivalent stator circuit is established that can simultaneously simulate both the free state and the stator / mover contact state. Based on the electromechanical analogy principle, the stator equivalent circuit is modeled as two parts: an electrical branch and a mechanical branch. The stator electrical branch in both the free state and the stator / mover contact state is represented by a resistor R representing the piezoelectric ceramic hysteresis loss. p The branch formed by connecting in parallel with the static capacitor C0 constitutes the stator mechanical branch in the free state, which is composed of R. m L m C m The three components form a series RLC resonant branch. Finally, the stator mechanical branch in the stator / mover contact state is based on the stator mechanical branch in the free state, with a variable capacitor C connected in series. c and variable resistor R cThe series branch, consisting of both components, represents the boundary conditions when the stator and mover are in contact in the mechanical branch. The switch S in the mechanical branch controls the switching between the free state (switch to "1") and the contact state (switch to "2"). Correspondingly, the measurement circuit of this invention, corresponding to the above equivalent circuit, is as follows: Figure 3 As shown.
[0040] The V-type linear ultrasonic motor operates in different modes under different frequency conditions, thus enabling the separate measurement of its losses. Simply put, when the system operates at its resonant frequency, power is primarily applied to the mechanical branches; when the system operates in a non-resonant state, power is distributed between the mechanical and electrical branches. Therefore, the measurement method of this invention generally follows the following flow: Figure 4 As shown, the resonant frequency of the system under different states is first determined. Then, under different states, different operating frequencies are set to measure the system power loss under different states. Based on this, the actual power loss of different types is evaluated. Finally, the loss resistance corresponding to different power losses is obtained based on the different power losses, thereby guiding the modeling, design and optimization of the ultrasonic motor.
[0041] S1. Resonant frequency measurement steps.
[0042] In order to separate the power loss and then calculate the loss resistance of different types of loss, it is necessary to measure the power of the V-type linear ultrasonic motor in different states. The different states refer to the power relative to the resonant frequency of the V-type linear ultrasonic motor.
[0043] When the stator of a V-type linear ultrasonic motor is in a free state or in a contact state, the structure of the system to which the stator belongs changes; the former does not include the motor mover, while the latter does. When the preload is removed and the stator is in a free state, that is, when the stator and motor mover are no longer in contact, it is equivalent to... Figure 2 In the equivalent circuit, the switch is set to "1". The actual equivalent circuit is as follows: Figure 5 As shown. When preload is applied, the motor stator and motor mover are in contact, equivalent to the switch being in position "2". The actual equivalent circuit at this time is as follows. Figure 6 As shown.
[0044] In this case, the resonant frequencies of the V-type linear ultrasonic motor are obviously different in the two states, and both resonant frequencies need to be measured. Therefore, the S1 resonant frequency measurement step actually includes two relatively independent steps: S11 measuring the resonant frequency in the stator free state and S12 measuring the first resonant frequency in the stator contact state. There is no specific order between the two steps, and the order can be set arbitrarily. In step 11, measuring the resonant frequency in the stator free state, the preload of the motor stator is removed, causing the motor stator to disengage from the motor rotor. An impedance analyzer is used to measure the impedance curve of the motor stator at this time; by reading the frequency corresponding to the minimum value of the impedance curve, the first resonant frequency f of the motor stator in the free state is determined. r1 .
[0045] In step S12, which measures the resonant frequency under stator contact conditions, a predetermined preload is applied to the motor stator to bring it into contact with the motor rotor. An impedance analyzer is then used to measure the impedance curve of the motor stator at this point. By reading the frequency corresponding to the minimum value of the impedance curve, the second resonant frequency f of the motor stator under contact conditions is determined. r2 .
[0046] S2, Power Loss Measurement.
[0047] according to Figure 3 The schematic diagram shows the connection of measuring instruments. Connect the voltmeter, wattmeter, and ammeter to the stator in sequence, and observe the voltmeter reading. Adjust the effective voltage value to the rated operating voltage of the motor.
[0048] In a free state, there are two rather special vibration states: one is the resonance state, and the other is the harmonic vibration state.
[0049] S21, Mechanical damping loss p m Measurement.
[0050] In the resonant state, the frequency of the voltage excitation signal is equal to the resonant frequency of the stator in the free state, causing the mechanical branch to exhibit a very small impedance value, much lower than that of the electrical branch. Therefore, most of the input electrical power is supplied to the mechanical branch. At the same time, the mechanical branch reaches resonance, and the inductive reactance and capacitive reactance cancel each other out. At this time, the active component of the input electrical power is approximately equal to the power loss consumed by the resistance in the mechanical branch. Thus, the mechanical damping loss p generated by the high-frequency vibration of the stator can be separated. m Therefore, a sinusoidal voltage excitation of u = U sin w2t is applied to the stator, where w2 = 2πf r1 Then, the power reading of the power meter at this time is the mechanical damping loss p of the motor stator. m Record the reading I of the ammeter connected to the stator at this time. f1 Or the voltmeter reading U f1 .
[0051] S22, Mechanical damping loss p pzt Measurement.
[0052] On the other hand, in a free state, if the stator is in a double-frequency vibration state, the excitation signal frequency is far from the stator's resonant frequency and is twice the resonant frequency. According to the impedance relationship of the equivalent circuit, the impedance that represents a minimum value at the resonant frequency has a maximum magnitude under this excitation signal. Therefore, almost all the stator current flows through the electrical branch, that is, almost all the input electrical power is supplied to the electrical branch. At this time, the active component of the input electrical power is equal to the power loss consumed by the resistor in the electrical branch, which is the hysteresis loss p of the piezoelectric ceramic. pzt Therefore, a sinusoidal voltage excitation of u = U sin 2w1t is applied to the stator, where w1 = 2πf r1 Then, the power reading of the power meter at this time is the hysteresis loss p of the piezoelectric ceramic of the motor stator. pzt Record the reading I of the ammeter connected to the stator at this time. 2f1 Or the voltmeter reading U 2f1 .
[0053] S23, Total power loss p mc Measurement.
[0054] When the motor stator is in contact, it's equivalent to setting the switch to position "2" in the equivalent circuit. In this case, if it's in resonance and the motor mover remains stationary, the excitation signal frequency is still equal to the stator's resonant frequency in the stator / motor contact state. This causes the mechanical branch, including the contact branch, to exhibit a very small impedance, far lower than the electrical branch. Therefore, most of the input power is supplied to the mechanical branch, and since the mechanical branch reaches resonance, the inductive reactance and capacitive reactance cancel each other out. At this point, the active component of the input power is approximately equal to the resistance R in the mechanical branch. m and R c Total power loss p mc Therefore, a sinusoidal voltage excitation of u = U sin 2w²t is applied to the stator, where w² = 2πf. r2 Then, reading the power meter at this time will give the total power loss p of the motor stator. mc Record the reading I of the ammeter connected to the stator at this time. f2 Or the voltmeter reading U f2 .
[0055] S3, Calculation of loss resistance.
[0056] S31, Measurement Correction
[0057] As mentioned earlier, when the power meter reading is used as the active power consumption of the corresponding branch, in reality, the measured power also includes the power consumed in other branches, albeit at a very small percentage. To improve the accuracy of the results, the measured value can be corrected by setting an appropriate correction factor. This correction factor can be determined based on historical experience or a priori by combining other simulation and experimental results. The specific determination method is not within the scope of this invention and is omitted here. Therefore:
[0058]
[0059] in, This is the correction value for mechanical damping loss. This is a correction value for the hysteresis damping loss of the piezoelectric ceramic. This is a correction value for the total power loss under contact conditions. β m β is the correction factor for mechanical damping loss. pzt β is the correction factor for the hysteresis damping loss of piezoelectric ceramics. mc This is a correction factor for the total power loss under contact conditions. Generally, β can be taken as... m ∈[0.8, 0.9], β pzt ∈[0.8, 0.9], β mc ∈[0.8, 0.9], the correction process is optional.
[0060] S32, Calculation of loss resistance.
[0061] The equivalent circuit resistance of the V-type linear ultrasonic motor can be calculated based on the corrected power losses, thus determining the parameters in the modeling and optimization process of the V-type linear ultrasonic motor. For specific procedures, please refer to... Figure 7 A diagram illustrating the data flow.
[0062] S321, Calculation of mechanical damping loss resistance.
[0063] exist Figure 5 In the equivalent circuit, since the excitation signal frequency is equal to the stator resonant frequency f in the free state... r1 This results in the mechanical branch exhibiting an extremely low impedance, far lower than that of the electrical branch. Therefore, the vast majority of the input power is supplied to the mechanical branch. Simultaneously, the mechanical branch reaches resonance, and the inductive reactance and capacitive reactance cancel each other out. At this point, the active component of the input power is approximately equal to the resistance R in the mechanical branch. m The power loss consumed is calculated using the following formula: or
[0064] S322, Calculation of hysteresis damping loss resistance.
[0065] exist Figure 5In the equivalent circuit, since the excitation signal frequency is equal to twice the stator resonant frequency f in the free state... r1 This causes the impedance, which has a minimum value at the resonant frequency, to have a maximum magnitude under this excitation signal. Therefore, almost all the stator current flows through the electrical branches, meaning almost all the input electrical power is supplied to the electrical branches, which is almost equivalent to the electrical power acting entirely on the piezoelectric ceramic. Therefore, the hysteresis damping loss R of the piezoelectric ceramic... pzt The calculation formula is or
[0066] S333, Calculation of contact loss resistance.
[0067] exist Figure 6 In the equivalent circuit, since the frequency of the excitation signal is equal to the resonant frequency of the stator in the stator / mover contact state during contact, the mechanical branch exhibits a very small impedance value, much lower than that of the electrical branch. Therefore, most of the input power is supplied to the mechanical branch, and the mechanical branch reaches resonance, where the inductive reactance and capacitive reactance cancel each other out. At this time, the active component of the input power is approximately equal to the mechanical damping loss resistance R in the mechanical branch. m and contact damping loss resistance R c Total power loss P consumed mc Therefore, the total power loss resistance is... or Therefore R c =R mc -R m Obviously, in calculating R... mc R m R pzt Whether a current value or a voltage value is used is completely equivalent to this application, and therefore either one should be considered within the scope of protection of the corresponding claim.
[0068] In summary, the calculation expressions for each loss resistor are as follows:
[0069]
[0070] It should be understood that for calculating R mc R m R pzt There is no predetermined or better order of calculation, therefore the order is not limited in this application.
[0071] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.
Claims
1. A method for loss separation and experimental testing of a linear ultrasonic motor, characterized in that, Includes the following steps: S1. Resonant frequency measurement steps: Measure the resonant frequency of the stator of the linear ultrasonic motor in both the free and contact states. The resonant frequency of the stator in the free state is denoted as the first resonant frequency f. r1 The resonant frequency of the motor stator in the contact state is denoted as the second resonant frequency f. r2 ; S2. Power loss measurement procedure: In the free state, measure the power loss p of the motor stator under excitation at the first resonant frequency. m The power loss p of the motor stator under excitation at twice the first resonant frequency was measured. pzt Under contact conditions, the power loss p of the motor stator under excitation at the second resonant frequency was measured. mc ; S3. Loss resistance calculation step: Estimate the mechanical damping loss p of the linear ultrasonic motor based on the power loss under three different states obtained in the S2 loss power measurement step. m Hysteresis damping loss p pzt and the total power loss p under contact conditions mc ; and calculate the corresponding mechanical damping loss resistance R. m Hysteresis damping loss resistor R pzt and contact damping loss resistance R c .
2. The method for loss separation and experimental testing of a linear ultrasonic motor as described in claim 1, characterized in that, The S3 loss resistance calculation step also includes an S31 measurement correction step, which corrects the power loss measured in the S2 loss power measurement step under different conditions. The correction formula is: in, This is the correction value for mechanical damping loss. This is a correction value for the hysteresis damping loss of the piezoelectric ceramic. This is a correction value for the total power loss under contact conditions; β m β is the correction factor for mechanical damping loss. pzt β is the correction factor for the hysteresis damping loss of piezoelectric ceramics. mc This is a correction factor for the total power loss under contact conditions.
3. The method for loss separation and experimental testing of a linear ultrasonic motor as described in claim 2, characterized in that, Correction factor β for mechanical damping loss m ∈[0.8, 0.9], the correction factor β for the hysteresis damping loss of piezoelectric ceramics pzt ∈[0.5, 0.9], correction factor β for total power loss in contact state mc ∈[0.8, 0.9].
4. The method for loss separation and experimental testing of a linear ultrasonic motor as described in claim 2, characterized in that, The mechanical damping loss resistance R of the linear ultrasonic motor m Hysteresis damping loss resistor R pzt and contact damping loss resistance R c Calculate using the following formula: Among them, R mc For the total power loss resistance, I f1 I is the current value of the motor stator under free state excitation at the first resonant frequency. 2f1 I is the current value of the motor stator in its free state under excitation at twice the first resonant frequency. f2 This represents the current value of the motor stator under the second resonant frequency excitation in the contact state.
5. The method for loss separation and experimental testing of a linear ultrasonic motor as described in claim 4, characterized in that, The mechanical damping loss resistance R of the linear ultrasonic motor m Hysteresis damping loss resistor R pzt and contact damping loss resistance R c Calculate using the following formula: Among them, R mc For the total power loss resistance, U f1 U is the voltage value of the motor stator under free state excitation at the first resonant frequency. 2f1 U is the voltage value of the motor stator in its free state under excitation at twice the first resonant frequency. f2 This represents the voltage value of the motor stator under the second resonant frequency excitation in the contact state.
6. The method for loss separation and experimental testing of a linear ultrasonic motor as described in claim 1, characterized in that, In the S2 power loss measurement step Mechanical damping loss is measured by applying a sinusoidal voltage excitation of u = U sin w1t to the stator of the motor in a free state, where w1 = 2πf. r1 U is the voltage amplitude; Hysteresis damping loss is measured under free conditions when a sinusoidal voltage excitation of u = U sin 2w1t is applied to the motor stator, where w1 = 2πf. r1 U is the voltage amplitude; Total power loss is measured by applying a sinusoidal voltage excitation of u = Usin2w2t to the motor stator under contact conditions, where w2 = 2πf r2 U is the voltage amplitude.