Abstract: This paper analyzes the characteristics of a variable speed system in the low-frequency region, describes some problems existing in the system in the low-frequency region, and proposes corresponding improvement measures. Keywords: low frequency characteristics, system analysis, improvement measures 1 Overview A common problem with AC speed control systems composed of frequency converters is that their performance is not ideal when operating in the low-frequency region. This is mainly manifested in the small starting torque during low-frequency startup, causing difficulty in starting the system or even failure to start. The high-order harmonics generated by the nonlinearity of the frequency converter cause torque pulsation and motor heating in the motor, and also increase the motor's operating noise. During low-frequency steady-state operation, fluctuations in grid voltage or changes in system load, as well as unusual changes in the frequency converter's output voltage waveform, will cause motor jitter. When the distance between the frequency converter and the motor is large, and the interference of high-order harmonics on the control circuit, motor creep is easily caused. Due to the aforementioned phenomena, the speed regulation characteristics and dynamic quality indicators of the speed control system composed of frequency converters are severely reduced. This paper analyzes the low-frequency mechanical characteristics of the system and the low-frequency characteristics of the frequency converter, and proposes corresponding measures to improve the low-frequency operating performance of the system. 2. Low-Frequency Mechanical Characteristics of Frequency Converters 2.1 Low-Frequency Starting Characteristics Changing the stator frequency F1 of an asynchronous motor can smoothly adjust the synchronous speed of the motor. However, as F1 changes, the mechanical characteristics of the motor will also change, especially in the low-frequency region. According to the maximum torque formula of an asynchronous motor: Temax = 3/2{np(U1/W1)²}/{R1/W1 + (R2/W1)² + (LL1 + LL2)²} where np—number of pole pairs of the motor; R1—resistance per phase of the stator; R2—resistance per phase of the rotor referred to the stator side; LL1—leakage inductance per phase of the stator; LL2—leakage inductance per phase of the rotor referred to the stator side; U1—voltage per phase of the stator; W1—power supply angular frequency. It can be seen that Temax decreases as W1 decreases. At low frequencies, R1 is no longer negligible. Temax will decrease as W1 decreases, and the starting torque will also decrease, possibly even failing to drive the load. 2.2 Low-Frequency Steady-State Characteristics The torque formula for a motor during steady-state operation is as follows: TL = 3np(U1/W1)2SW1R2/{(SR1+R2)2+S2W2(LL1+LL2)2} When the angular frequency W1 is at its rated value, R1 can be ignored. However, at low frequencies, R1 cannot be ignored. Therefore, in the low-frequency region, due to the increased proportion of voltage drop across R1, it will be impossible to maintain a constant M, especially when the grid voltage and load change, the system will exhibit jitter and creep. 3 Low-Frequency Characteristics of Inverter Speed ​​Control System 3.1 Harmonic Analysis In a speed control system composed of inverters, due to the nonlinearity of the inverter, in addition to the fundamental current, there are also harmonic currents in the motor stator. Due to the presence of higher harmonics, the motor loss and inductive reactance increase, reducing cosφ, thereby affecting the output torque and generating a pulsating torque 6 times the fundamental frequency. Analyzing the 5th and 7th harmonics in the current waveform, the 5th harmonic frequency in the stator current of a three-phase motor is F5 = 5F1 (F1 is the fundamental current frequency). It generates a negative-sequence magnetomotive force and magnetic field in the motor's air gap. The rotational speed n51 of this magnetic field is 5 times the rotational speed n11 of the magnetic field generated by the fundamental current, and it rotates in the opposite direction to the fundamental magnetic field. Since the motor speed is constant and assumed to be close to n11, the 5th harmonic magnetomotive force induces a rotor current 6 times the fundamental frequency in the rotor. The combined effect of this current and the fundamental magnetomotive force in the air gap produces a pulsating torque 6 times the fundamental frequency. The magnetic field generated by the 7th harmonic is in phase with the fundamental, but its rotating magnetic field rotates at 7 times the rotational speed of the fundamental rotating magnetic field. Therefore, the relative rotational speed between the corresponding rotor current harmonic and the main magnetic field in the air gap is also 6 times the fundamental frequency, also producing a pulsating torque 6 times the fundamental frequency. The two pulsating torques, each six times the fundamental frequency, together cause the electromagnetic torque of the motor to pulsate. Although their average value is zero, the pulsating torque makes the motor speed uneven, with the greatest impact at low frequencies. 3.2 Generation of Pulsating Torque in Quasi-Square Wave Mode Let ψ1 and ψ2 be the spatial vectors of the stator flux linkage and rotor flux linkage, respectively. In steady-state quasi-square wave (QSW) operation mode (the thyristors in the bridge are triggered by 180° electrical angle pulses), ψ1 moves along the periphery of a regular hexagon within the output cycle. ψ2 moves along a circle concentric with the hexagon. In quasi-square wave operation mode, the motion of ψ1 and ψ2 is continuous, but they have significant differences. When vector ψ2 rotates at a constant stator voltage angular velocity W1, vector ψ1 moves along the periphery of the regular hexagon at a constant linear velocity. The constant linear velocity of vector ψ1 causes a change in its angular velocity, which in turn causes a change in the angle δ between ψ1 and ψ2. In addition, when ψ1 moves along the hexagonal trajectory, its amplitude also changes to some extent. When the motor is unloaded, the angle between ψ1 and ψ2 and the torque T are zero when W1t = 0, π/6, and π/3, but when W1T ≠ 0, π/6, and π/3, δ is not zero. This, along with the amplitude change of ψ1 mentioned above, causes low-frequency torque pulsation, the frequency of which is 6 times the fundamental frequency of the stator voltage. When the motor is under load, it corresponds to a constant average value of δ, and the low-frequency torque pulsation will be superimposed on the constant average value of torque. 4. Measures to improve the low-frequency characteristics of the system 4.1 Increase the starting torque Due to the voltage drop across R1 at low frequencies, the starting torque of the system decreases as W1 decreases. Therefore, the frequency converter has a torque increase function, which can adjust the torque of the motor in the low-frequency range to match the load and increase the starting torque. Automatic and manual torque boost modes are available. The principle is that increasing the stator voltage correspondingly increases the starting torque. However, setting the boost voltage too high will lead to excessive current, causing motor saturation, overheating, or overcurrent tripping. For example, the torque boost function of the 1336PLUS series inverter can automatically adjust the boost voltage to generate the required voltage. The boost voltage can be selected based on the current required for the predetermined torque. The torque boost controls the current while ensuring the motor operates at its optimal state. When selecting manual torque boost, the torque boost value should be set according to the actual situation. 4.2 Improving Low-Frequency Torque Pulsation Low-frequency torque pulsation in AC speed control systems composed of frequency converters directly affects the system's dynamic characteristics. Both frequency converter manufacturers and system integration engineers are striving to improve this technical problem. For example, flux control and sinusoidal PWM control methods are employed. Instead of controlling the GTR's conduction and cutoff based on the intersection of the modulated sine wave and the carrier wave, these methods consistently ensure that the asynchronous motor's flux is close to a sine wave, with the rotating magnetic field trajectory being circular to determine the GTR's conduction pattern. This ensures uniform rotation of the asynchronous motor at very low frequencies, thereby expanding the frequency conversion speed range and suppressing the asynchronous motor's vibration and noise. The circular rotating magnetic field is achieved by detecting the flux, allowing the control loop to continuously determine whether the actual flux exceeds the error range, thus changing the GTR's operating mode to ensure the rotating magnetic field trajectory is circular, reducing torque pulsation. 4.3 Circular PWM Method for Reducing Torque Ripple The term "circular" refers to the stator flux linkage ψ1 space vector following a polygon in the Gaussian plane that is very close to a circle. The width and position of the voltage pulse are determined with the aim of reducing motor ripple torque. The three-phase inverter is a full-wave bridge structure. When it operates in such a mode, if one of the AC output terminals (a, b, c) is connected to the DC bus at any time (it should be connected to the other DC bus simultaneously), this principle is clearly shown in Figure 1(a). Obviously, there are six ways for the AC output terminal to be connected to the DC bus, resulting in six positions for the stator voltage U1 space vector. These six positions are shown in Figure 1(b). The six on/off states in Figure 1(b) correspond to the six positions of U1. The thick lines in the figure indicate that switches 1, 3, and 6 are in the on position. The instantaneous phase voltages generated by the projection are as follows: Va = Vb = 1/3Vdc Vc = -2/3Vdc Similarly, Va, Vb, and Vc represent the instantaneous phase voltage values ​​of the three-phase output voltage. If Ia + Ib + Ic = 0, Va, Vb, and Vc can be obtained by the vertical projection of the space vector on the A, B, and C axes. In addition to the above six on/off states, there are also two states where switches 1, 3, and 5 or 2, 4, and 6 are turned off simultaneously. In this case, the AC output terminals a, b, and c are connected to the same potential, and U1 and Ua, Ub, and Uc become zero in sequence. Applying this operating mode to a three-level PWM inverter can achieve a lower harmonic component compared to two-level PWM. PWM is a chopper-quasi-square wave modulation. The phase voltage on the load consists of a rectangular segment and a zero-voltage segment (when U1=0). At each voltage pulse moment, vector ψ1 moves at a constant linear velocity and remains stationary in the zero-voltage segment. However, since vector ψ2 rotates at a constant angular velocity W1, the angle δ between ψ1 and ψ2 appears. Therefore, voltage chopping is the main cause of high-frequency torque pulsation, and the frequency is the same as the output voltage torque pulse frequency. This is due to the inherent nature of PWM itself. In fact, high-frequency torque pulsation is difficult to eliminate and is superimposed on low-frequency torque pulsation. To eliminate the low-frequency torque pulsation of the system, the following two methods can be used. (1) At the moment of the midpoint of the voltage pulse, the angle δ between vectors ψ1 and ψ2 should remain constant for all pulses during steady-state operation to eliminate the influence of δ change on low-frequency torque (frequency is 6F1). Under no-load conditions, δ=0. Although the amplitude of ψ1 changes, the low-frequency torque pulsation will still be completely eliminated. (2) Under constant load (δ-cost≠0), only the change in amplitude of ψ1 causes low-frequency torque ripple, while the change in amplitude of ψ2 caused by the load is negligible. Therefore, it is necessary to obtain a ψ1 vector trajectory that is close to a circle. Circular PWM uses the spatial position of the no-load vector ψ1 to determine the midpoint of the voltage pulse, that is, a reasonable combination of the thyristor conduction segment and the zero-voltage segment, which can generate ψ1 with negligible amplitude change. This principle is shown in Figure 1. The stopping time of ψ1 (i.e., the zero-voltage segment) is marked with a dot. The voltage pulse position is determined to make them symmetrical, as shown by the midpoint of each horizontal axis in the figure. The pulse width (i.e., duration) corresponds to the length of the horizontal axis, and the required output voltage is determined. The period of the natural voltage waveform is determined by the time required for the ψ1 vector to rotate one revolution along the polygon. Using this method, while keeping the output voltage variable from zero to the maximum value, low-frequency torque ripple can be effectively eliminated.