Motor simulator considering voltage-current-torque harmonic simulation of motor

Abstract

The invention discloses a motor simulator considering motor voltage-current-torque harmonic simulation. The motor simulator comprises a simulation motor electromagnetic harmonic model, a simulation motor motion harmonic model, a voltage modulation optimization model, a connection inductor and an inverter. Wherein the simulation motor electromagnetic harmonic wave model takes a three-phase current and a rotor position as input, and voltage and torque ripple of a simulation port considering harmonic waves are obtained; the simulation motor motion harmonic model takes torque pulsation and simulation load parameters as input to obtain a rotor position considering the rigidity and damping of the dual-mass spring-damping system; and the voltage modulation optimization model converts the voltage of the analog port into a switching signal of the power device. The method aims at solving the problem that in the prior art, due to the fact that inherent harmonic characteristics of the motor are ignored, deviation exists between a simulation result and an actual test, and higher-precision motor simulation and controller test are achieved by establishing the harmonic model accurately reflecting the inherent characteristics of the motor and integrating power level simulation.

Description

Technical Field

[0001] This invention relates to the field of motor simulation control, specifically a motor simulator that considers the simulation of motor voltage-current-torque harmonics. Background Technology

[0002] As a core component of new energy electric vehicles, the development of motor controllers requires extensive functional verification, performance testing, and durability testing. Traditional testing methods mainly rely on building mechanical load benches based on real motors, but this approach suffers from a series of problems, including complex system structure, high mechanical losses, poor reusability, and high construction costs.

[0003] To overcome the aforementioned limitations, scholars both domestically and internationally have proposed a motor simulator solution based on Power Hardware-in-the-Loop (PHIL) technology. This solution utilizes high-speed hardware computing modules to achieve real-time simulation of the motor model and reproduces the simulation results at the power level through power electronic control technology, thereby replacing the mechanical rotating components in traditional test benches. Compared to traditional methods, the PHIL motor simulator has significant advantages such as low energy loss, high testing flexibility, and system safety and reliability.

[0004] However, due to factors such as cogging effect, non-sinusoidal back EMF, and magnetic saturation, actual motors inevitably contain a large number of harmonic components that are integer multiples of the rotor frequency. Most existing motor simulators focus on simulating the motor's fundamental characteristics, primarily targeting the fundamental current response under inverter fundamental voltage excitation, often neglecting the motor's inherent harmonic characteristics. This results in significant discrepancies between simulation results and actual bench test results, making it difficult to meet the ever-increasing demand for high-precision testing. Summary of the Invention

[0005] The present invention addresses the shortcomings of the prior art by proposing a motor simulator that considers voltage-current-torque harmonic simulation. The aim is to establish a harmonic model that accurately reflects the inherent characteristics of the motor and is related to the rotor position, and to integrate it into the power level simulation to achieve higher precision motor simulation and controller testing. This solves the problem of deviation between simulation results and actual tests caused by ignoring the inherent harmonic characteristics of the motor in the prior art.

[0006] To achieve the above-mentioned objectives, the present invention adopts the following technical solution: The present invention provides a motor simulator that considers the simulation of motor voltage-current-torque harmonics, characterized in that it includes: a simulated motor electromagnetic harmonic model, a simulated motor motion harmonic model, a voltage modulation optimization model, and a connecting inductor and an inverter; wherein, the simulated motor electromagnetic harmonic model includes: a voltage-current harmonic model and a current-torque harmonic model; Collect the three-phase current between the inverter and the connecting inductor , , Rotor position output by the analog motor motion harmonic model The parameters of the connected inductor are then input into the electromagnetic harmonic model of the simulated motor for processing, thereby obtaining the d-axis reference voltage of the analog port output by the voltage-current harmonic model. and q-axis reference voltage Torque pulsation output by the current-torque harmonic model ; Torque pulsation The simulated load parameters input from the external source are processed in the harmonic model of the simulated motor motion to obtain the rotor position considering the stiffness and damping effect of the dual-mass spring-damped system. ; d-axis reference voltage q-axis reference voltage and rotor position The input voltage modulation optimization model is used to obtain the α-axis reference voltage. and β-axis reference voltage Thus, through voltage modulation and This is converted into switching signals for the power devices in the inverter.

[0007] The motor simulator described in this invention is also characterized in that the voltage-current harmonic model obtains the d-axis reference voltage according to the following steps. and q-axis reference voltage : S1a. Obtain the d-axis flux linkage sequence through finite element parameter simulation. and q-axis flux linkage sequence ,in, This represents the stator current at the m-th current operating point. Indicates fixed stator current The d-axis flux linkage sequence at the m-th current operating point. Indicates fixed The q-axis flux linkage sequence at the m-th current operating point. Indicates the total number of current operating points; S1b. d-axis flux linkage sequence and q-axis flux linkage sequence Two-dimensional Fourier series expansions were performed separately, and the Fourier series coefficients of the same order and all different stator currents were combined. Then, polynomial fitting was performed to obtain the result based on the stator current. The coefficients of the d-axis flux linkage Fourier series of the independent variable and q-axis flux Fourier series coefficients Thus, by using equation (1), we can obtain the following: Torque angle and rotor position The expression for the d-axis flux linkage matrix parameter of the independent variable. and q-axis flux linkage matrix parameter expression : (1) In equation (1), express The highest Fourier series order in the corresponding dimension. express The highest Fourier series order in the corresponding dimension. The dimension is The d-axis flux linkage Fourier series coefficient matrix, The dimension is The coefficient matrix of the Fourier series of magnetic flux linkage on the q-axis. Indicates and The relevant dimensions are The matrix, Indicates and The relevant dimensions are The matrix has: (2) In equation (2), express The fundamental angular frequency in the corresponding dimension, Represents the Euler number. represents an imaginary number, Indicates matrix transpose; S1c. Obtain the d-axis reference voltage using equation (3). and q-axis reference voltage : (3) In equation (3), This represents the d-axis current between the inverter and the connecting inductor. This represents the q-axis current between the inverter and the connecting inductor. This represents the stator resistance of the motor simulator. This represents the angular velocity of the motor simulator. This represents the equivalent resistance value of the connected inductor. This indicates the inductance value of the connected inductor.

[0008] Furthermore, the current-torque harmonic model is obtained by following these steps to obtain the torque ripple. : S2a. Obtaining the torque pulsation sequence through finite element parameter simulation. ,in, Indicates fixed The torque ripple sequence at the m-th current operating point; S2b. Torque pulsation sequence Two-dimensional Fourier series expansions were performed separately. The Fourier series coefficients of the same order and all different stator currents were combined, and then polynomial fitting was performed to obtain the result. Torque ripple Fourier series coefficients of the independent variable Thus, the torque pulsation can be obtained using equation (4). : (4) In equation (4), The dimension is The torque pulsation Fourier series coefficient matrix.

[0009] Furthermore, the simulated motor motion harmonic model includes a dual-mass spring-damped system, and the rotor position considering the stiffness and damping effect of the dual-mass spring-damped system is obtained according to the following steps. : S3a. Let the state variables of the two-mass spring-damped system be... The input vector is ,in, This indicates the relative angular displacement between the motor and the load. This represents the angular velocity of the entire vehicle. This represents the load torque, i.e., the simulated load parameter input from the outside; S3b. Using equation (5), the state-space expression of the two-mass spring-damped system is obtained: (5) In equation (5), and Represents two coefficient matrices; (6) In equation (6), This represents the moment of inertia of the motor rotor. This represents the vehicle's rotational inertia equivalent to the input of the transmission. This represents the torsional stiffness of the half-shaft and tires equivalent to the input of the gearbox. This represents the equivalent damping of a two-mass spring-damped system. S3c. Using equation (7), the transfer function expression between the input and output of the two-mass spring-damped system is obtained: (7) In equation (7), This represents a 3×3 identity matrix. Represents complex frequency. Represents the inverse of a matrix; S3d. Using equation (8), the rotor position considering the stiffness and damping effect of the dual-mass spring-damped system is obtained. : (8) In equation (8), express The element located in the first row and first column is the angular velocity of the motor simulator. With torque pulsation The transfer function between them express Laplace transform, This represents the inverse Laplace transform.

[0010] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. In constructing the electromagnetic harmonic model of the simulated motor, the motor simulator of this invention constructs a voltage-current harmonic model and a current-torque harmonic model. The voltage-current harmonic model is constructed by obtaining the stator flux linkage values ​​of the dq axis under different current operating points through finite element parameter simulation, and then performing two-dimensional Fourier series expansion and polynomial fitting to obtain the reference voltage of the simulation port. The current-torque harmonic model is also obtained by obtaining the torque ripple value through finite element parameter simulation, and then performing Fourier series expansion and polynomial fitting to obtain the torque ripple. Thus, this invention can accurately reflect the inherent voltage-current harmonic and current-torque harmonic characteristics of the motor, achieving higher precision motor simulation.

[0011] 2. In constructing the harmonic model of the simulated motor motion, the motor simulator of this invention employs a dual-mass spring-damping system, considering the influence of the stiffness and damping effect of the dual-mass spring-damping system on the rotor position. Therefore, this invention can more accurately simulate the motion relationship between the motor and the load, thereby improving the overall accuracy of the simulation and making the simulation results closer to reality.

[0012] 3. This invention utilizes power hardware-in-the-loop technology to reproduce power levels and overcomes the problems of complex system structure, high mechanical loss, and poor reusability of traditional testing methods by combining electromagnetic harmonic models and motion harmonic models of simulated motors. Compared with traditional methods, the motor simulator designed in this invention has significant advantages such as low energy loss, high testing flexibility, and system safety and reliability. Attached Figure Description

[0013] Figure 1 This is a block diagram of the motor simulator application of the present invention; Figure 2 This is a schematic diagram illustrating the principle of zero-sequence current suppression in this invention. Figure 3 This is a graph showing the speed / torque response results of the present invention; Figure 4 The figure shows the dq axis current response results of the tested motor controller under simulated operating conditions. Figure 5 The figure shows the three-phase current response of the tested motor controller under simulated operating conditions. Figure 6 The figure shows the dq axis voltage response results of the tested motor controller under simulated operating conditions. Figure 7 The figure shows the dq-axis flux linkage response results of the controller under test in the present invention under simulated conditions. Detailed Implementation

[0014] In this embodiment, a motor simulator considering the simulation of motor voltage-current-torque harmonics is provided, such as... Figure 1 As shown, this is applied to a controller for a motor under test powered by a high-voltage DC voltage, and includes: a connecting inductor, an inverter, an electromagnetic harmonic model of the simulated motor, a motion harmonic model of the simulated motor, and a voltage modulation optimization model.

[0015] One end of the inductor is connected to the three-phase AC terminal of the motor simulator, and the other end serves as the electrical port of the simulated motor, used to connect to the three-phase AC terminal of the controller of the motor under test.

[0016] An inverter converts direct current (DC) to alternating current (AC). By controlling the switching signals in the inverter, three-phase AC power is generated, thereby simulating the operating state of a motor.

[0017] Collect the three-phase current between the inverter and the connecting inductor , , Rotor position output by the analog motor motion harmonic model The parameters of the connected inductor are then input into the electromagnetic harmonic model of the simulated motor for processing, thereby obtaining the d-axis reference voltage of the analog port output by the voltage-current harmonic model. and q-axis reference voltage Torque pulsation output by the current-torque harmonic model .

[0018] In this embodiment, the electromagnetic harmonic model of the simulated motor includes: a voltage-current harmonic model and a current-torque harmonic model; The voltage-current harmonic model is processed according to the following steps: S1a. Obtain the d-axis flux linkage sequence through finite element parameter simulation. and q-axis flux linkage sequence ,in, This represents the stator current at the m-th current operating point. Indicates fixed stator current The d-axis flux linkage sequence at the m-th current operating point. Indicates fixed The q-axis flux linkage sequence at the m-th current operating point. This indicates the total number of current operating points.

[0019] S1b. d-axis flux linkage sequence and q-axis flux linkage sequence Two-dimensional Fourier series expansions were performed separately, and the Fourier series coefficients of the same order and all different stator currents were combined. Then, polynomial fitting was performed to obtain the result based on the stator current. The coefficients of the d-axis flux linkage Fourier series of the independent variable and q-axis flux Fourier series coefficients Thus, by using equation (1), we can obtain the following: Torque angle and rotor position The expression for the d-axis flux linkage matrix parameter of the independent variable. and q-axis flux linkage matrix parameter expression : (1) In equation (1), express The highest Fourier series order in the corresponding dimension. express The highest Fourier series order in the corresponding dimension. The dimension is The d-axis flux linkage Fourier series coefficient matrix, The dimension is The coefficient matrix of the Fourier series of magnetic flux linkage on the q-axis. Indicates and The relevant dimensions are The matrix, Indicates and The relevant dimensions are The matrix; and we have: (2) In equation (2), express The fundamental angular frequency in the corresponding dimension, Represents the Euler number. represents an imaginary number, This indicates the matrix transpose.

[0020] S1c. Obtain the d-axis reference voltage using equation (3). and q-axis reference voltage : (3) In equation (3), This represents the d-axis current between the inverter and the connecting inductor. This represents the q-axis current between the inverter and the connecting inductor. This represents the stator resistance of the motor simulator. This represents the angular velocity of the motor simulator. This represents the equivalent resistance value of the connected inductor. This indicates the inductance value of the connected inductor.

[0021] The current-torque harmonic model is constructed according to the following steps: S2a. Obtaining the torque pulsation sequence through finite element parameter simulation. ,in, Indicates fixed The torque pulsation sequence at the m-th current operating point.

[0022] S2b. Torque pulsation sequence Two-dimensional Fourier series expansions were performed separately. The Fourier series coefficients of the same order and all different stator currents were combined, and then polynomial fitting was performed to obtain the result. Torque ripple Fourier series coefficients of the independent variable Thus, the torque pulsation can be obtained using equation (4). : (4) In equation (4), The dimension is The torque pulsation Fourier series coefficient matrix; Torque pulsation The simulated load parameters input from the external source are processed in the harmonic model of the simulated motor motion to obtain the rotor position considering the stiffness and damping effect of the dual-mass spring-damped system. .

[0023] The simulated motor motion harmonic model includes a dual-mass spring-damped system, and is processed according to the following steps: S3a. Let the state variables of the two-mass spring-damped system be... The input vector is ,in, This indicates the relative angular displacement between the motor and the load. This represents the angular velocity of the entire vehicle. This represents the load torque, i.e., the simulated load parameter input from the outside; S3b. Using equation (5), the state-space expression of the two-mass spring-damped system is obtained: (5) In equation (5), and Represents two coefficient matrices; (6) In equation (6), This represents the moment of inertia of the motor rotor. This represents the vehicle's rotational inertia equivalent to the input of the transmission. This represents the torsional stiffness of the half-shaft and tires equivalent to the input of the gearbox. This represents the equivalent damping of a two-mass spring-damped system.

[0024] S3c. Using equation (7), the transfer function expression between the input and output of the two-mass spring-damped system is obtained: (7) In equation (7), This represents a 3×3 identity matrix. Represents complex frequency. This represents the inverse of a matrix.

[0025] S3d. Using equation (8), the rotor position considering the stiffness and damping effect of the dual-mass spring-damped system is obtained. : (8) In equation (8), express The element located in the first row and first column is the angular velocity of the motor simulator. With torque pulsation The transfer function between them express Laplace transform, This represents the inverse Laplace transform.

[0026] d-axis reference voltage q-axis reference voltage and rotor position The input voltage modulation optimization model is used to obtain the α-axis reference voltage. and β-axis reference voltage Thus, through voltage modulation and Converted into switching signals for the power devices in the inverter; The voltage modulation optimization model is processed according to the following steps: S4a. Using formula (9) and Voltage converted to stationary coordinate system and : (9) In equation (9), This represents the rotor equivalent electrical angle considering the switching cycle delay. , This indicates the voltage modulation period.

[0027] S4b. Utilizing SVPWM voltage vector modulation method based on and Calculate the initial conduction time of the three-phase bridge arm , , And serve as the initial switching signal.

[0028] S4c. Active compensation is performed to address the nonlinear characteristics of the inverter, resulting in the compensation times for the three-phase bridge arms as follows: , , Thus, the conduction time of the three-phase bridge arm after nonlinear compensation can be obtained using equation (10). , , : (10) S4d. Construct a proportional resonant controller to suppress zero-sequence harmonic currents. For example... Figure 2 As shown, the controller references the target. The input to the controller is zero; the feedback input is the sum of the three-phase currents; the controller output is the zero-sequence voltage controller. ,right After normalization, the compensation time for suppressing zero-sequence current is obtained using equation (11): (11) In equation (11), This indicates the maximum value of the modulation voltage.

[0029] S4e. Calculate the conduction time of the three-phase bridge arm to suppress the zero-sequence current component. , , , as the optimized switching signal.

[0030] Example: To verify the motor simulator design method of the present invention, a motor simulator model was built, and a finite element simulation model was introduced for comparison.

[0031] The parameters of the simulated motor are shown in Table 1: Table 1 Simulated Motor Parameters The simulation conditions set include 1) constant speed with different torques; 2) constant torque with different speeds, as shown in Table 2: Table 2 Simulation Conditions See Figure 3 Speed ​​represents the desired speed of the simulated motor, Te_Set represents the desired torque of the simulated motor, Te_FEA represents the actual torque of the simulated motor in the finite element simulation model, and Te_FB represents the actual torque of the simulated motor in the motor simulator model. It can be seen that under both operating conditions, the motor simulator can track the torque during the motor simulation process, and the results are almost completely consistent with the output of the finite element simulation model. It not only accurately describes the average torque during the dynamic torque response process, but more importantly, it precisely describes the torque ripple of the motor in each state.

[0032] See Figure 4 Id_Set represents the expected d-axis current of the simulated motor, Iq_Set represents the expected q-axis current of the simulated motor, Id_FEA represents the actual d-axis current of the simulated motor in the finite element simulation model, Iq_FEA represents the actual q-axis current of the simulated motor in the finite element simulation model, and Id_FB represents the actual d-axis current of the simulated motor in the motor simulator model. It can be seen that under both operating conditions, the motor simulator can track the d- and q-axis currents during motor simulation, and the results are basically consistent with the output of the finite element simulation model.

[0033] See Figure 5In the finite element simulation model, Ia_FEA represents the A-phase current, Ib_FEA represents the B-phase current, and Ic_FEA represents the C-phase current. Similarly, Ia_FB represents the A-phase current, Ib_FB represents the B-phase current, and Ic_FB represents the C-phase current in the motor simulator model. It can be seen that under both operating conditions, the motor simulator can track the three-phase currents during motor simulation, and the results are almost identical to those output by the finite element simulation model.

[0034] See Figure 6 Ud_FEA represents the d-axis voltage of the simulated motor in the finite element simulation model, Uq_FEA represents the q-axis voltage of the simulated motor in the finite element simulation model, Ud_FB represents the d-axis voltage of the simulated motor in the motor simulator model, and Uq_FB represents the q-axis voltage of the simulated motor in the motor simulator model. It can be seen that under both operating conditions, the motor simulator can track the d- and q-axis voltages during motor simulation, and the results are basically consistent with the output of the finite element simulation model.

[0035] See Figure 7 Fluxd_FEA represents the d-axis flux linkage of the simulated motor in the finite element simulation model, Fluxq_FEA represents the q-axis flux linkage of the simulated motor in the finite element simulation model, Fluxd_FB represents the d-axis flux linkage of the simulated motor in the motor simulator model, and Fluxq_FB represents the q-axis flux linkage of the simulated motor in the motor simulator model. It can be seen that in the initial stage, due to the stator current... The flux linkage results in the motor simulator model are relatively small, and oscillations occur. With... As the value increases, the motor simulator model quickly stabilizes and completes the tracking of the dq axis flux under two working conditions, which is basically consistent with the output results of the finite element simulation model.

Claims

1. A motor simulator considering motor voltage-current-torque harmonic simulation, characterized by, Comprise: Analog motor electromagnetic harmonic model, analog motor motion harmonic model, voltage modulation optimization model, connection inductance and inverter; wherein, the analog motor electromagnetic harmonic model comprises: voltage-current harmonic model and current-torque harmonic model; Collecting three-phase currents between the inverter and the connecting inductance , , and the rotor position output by the analog motor motion harmonic model , together with the parameters of the connecting inductance, are input into the analog motor electromagnetic harmonic model for processing, so as to obtain the d-axis reference voltage and the q-axis reference voltage of the analog port output by the voltage-current harmonic model torque ripple output by the current-torque harmonic model ;​ torque pulsations and external inputted analog load parameter into analog motor motion harmonic model to process, get rotor position considering double mass spring-damping system stiffness and damping effect ; The d-axis reference voltage , q-axis reference voltage and rotor position are processed in a voltage modulation optimization model to obtain the α-axis reference voltage and β-axis reference voltage , so that and are converted into switching signals of power devices in the inverter by voltage modulation.

2. The motor simulator of claim 1, wherein, The voltage-current harmonic model is obtained for the d-axis reference voltage and the q-axis reference voltage as follows: S1a. Obtain the d-axis flux linkage sequence by finite element parameter simulation and the q-axis flux linkage sequence wherein, denotes the stator current of the mth current operating point, denotes the fixed stator current the d-axis flux linkage sequence of the mth current operating point, denotes the fixed the q-axis flux linkage sequence of the mth current operating point, denotes the total number of current operating points; S1b. The d-axis flux linkage sequence and the q-axis flux linkage sequence are respectively subjected to two-dimensional Fourier series expansion, and the Fourier series coefficients of all different stator currents of the same Fourier series order are combined and then subjected to polynomial fitting, to obtain the d-axis flux linkage Fourier series coefficients and the q-axis flux linkage Fourier series coefficients with the stator currents as the independent variables, so that the d-axis flux linkage matrix parameter expression , the torque angle and the rotor position are obtained with the stator currents as the independent variables, and the q-axis flux linkage matrix parameter expression is obtained. (1) in formula (1), denotes the highest Fourier series order in the dimension concerned, denotes the highest Fourier series order in the dimension concerned, denotes the d-axis flux linkage Fourier series coefficient matrix of dimension denotes the q-axis flux linkage Fourier series coefficient matrix of dimension denotes the d-axis flux linkage Fourier series coefficient matrix of dimension denotes the q-axis flux linkage Fourier series coefficient matrix of dimension denotes the matrix of dimension related to denotes the matrix of dimension related to denotes the matrix of dimension related to (2) In formula (2), denotes denotes the fundamental angular frequency in the dimension corresponding to denotes Euler's number, denotes the imaginary unit, denotes the matrix transpose; S1c. Obtain a d-axis reference voltage using equation (3) and a q-axis reference voltage : (3) in formula (3), represents the d-axis current between the inverter and the connecting inductance, represents the q-axis current between the inverter and the connecting inductance, represents the stator resistance of the motor simulator, represents the angular velocity of the motor simulator, represents the equivalent resistance value of the connecting inductance, represents the inductance value of the connecting inductance.

3. The motor simulator of claim 1, wherein, The current-torque harmonic model is obtained from the torque pulsation : S2a. Obtain the torque ripple sequence by finite element parameter simulation wherein, denotes the fixed the torque ripple sequence of the mth current operating point in the lower row S2b. Torque ripple sequence The two-dimensional Fourier series expansion is performed respectively, the Fourier series coefficients of the same Fourier series order and all different stator currents are combined, and then polynomial fitting is performed to obtain the torque ripple Fourier series coefficients with as the independent variable , so that the torque ripple is obtained by formula (4) : (4) In formula (4), denotes a torque ripple Fourier series coefficient matrix of dimension denotes a torque ripple Fourier series coefficient matrix of dimension 4. The motor simulator of claim 1, wherein, The analog motor motion harmonic model comprises a double-mass spring-damper system, and the rotor position considering the stiffness and damping effect of the double-mass spring-damper system is obtained according to the following steps : S3a. Let the state variable of the double mass spring-damper system be , and the input vector be , where denotes the relative angular displacement between the motor and the load, denotes the angular velocity of the whole vehicle, denotes the load torque, i.e. the analog load parameter input from outside. S3b. Obtain the state space expression of the double-mass spring-damper system using formula (5): (5) In formula (5), and denote two coefficient matrices; (6) in formula (6), denotes the moment of inertia of the motor rotor, denotes the overall vehicle moment of inertia equivalent to the gearbox input, denotes the torsional stiffness of the half shaft and tire equivalent to the gearbox input, denotes the equivalent damping of the dual mass spring-damper system; S3c. Obtain the transfer function expression between the input and output of the double-mass spring-damper system using formula (7): (7) In formula (7), denotes a unit matrix of dimension 3x3, denotes the complex frequency, denotes the inverse of a matrix; S3d. Obtain the rotor position taking into account the stiffness and damping action of the double-mass spring-damper system with formula (8) : (8) In formula (8), denotes the element located in the first row and the first column in the matrix, i.e. the angular speed of the motor simulator and the torque ripple, the transfer function between them, denotes the Laplace transform of denotes the inverse Laplace transform.

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