1. Introduction Do you want to use brushless servo motors in your new product? We often encounter keywords such as "trapezoidal wave," "sine wave," and "vector control." Only by understanding their true meaning can you choose the right product for your new design. Over the past decade or even two decades, the servo motor market has shifted from brushed servos to brushless servos, primarily due to the low maintenance rate and high stability of brushless servos. In these past ten years, the technology of drive circuits and systems has become very sophisticated. Control methods can now fully realize the functions described by those keywords. Most high-performance servo systems use an internal control loop to control torque. This internal torque loop, in conjunction with external speed and position loops, achieves different control effects. The design of the external control loop is independent of the matched motor, while the design of the internal torque loop is closely related to the performance of the matched motor. Torque control of brushed motors is very simple because brushed motors can perform commutation themselves. The output torque is proportional to the DC voltage input to the two poles of the brushed motor. The torque can also be easily controlled via a PI control loop. The main function of the PI control loop is to adjust the motor's input voltage in real time by detecting the deviation between the actual motor current and the control current. [align=center]Figure 1[/align] Since brushless motors do not have commutation capabilities, the corresponding control method is more complex. Brushless motors have three sets of coils, unlike the two sets of coils in brushed motors. To obtain effective torque, the three sets of coils in a brushless motor must be controlled independently according to the actual position of the rotor. This driving method fully illustrates the complexity of controlling a brushless motor. 2. Basics of Brushless Motors Simply put, a brushless motor mainly consists of a rotating permanent magnet (rotor) and three sets of evenly distributed coils (stator), with the coils surrounding the stator and fixed externally. Current flowing through the coils generates a magnetic field, and the three magnetic fields superimpose to form a vector magnetic field. By controlling the current in each of the three sets of coils, we can make the stator generate a magnetic field of arbitrary direction and magnitude. Simultaneously, the torque can be freely controlled through the mutual attraction and repulsion between the stator and rotor magnetic fields. [align=center]Figure 2[/align] For any angle of rotor rotation, the stator has an optimal magnetic field direction that generates the maximum torque; similarly, the stator can also generate a magnetic field direction with no torque output. Simply put, if the magnetic field generated by the stator is aligned with the magnetic field of the rotor's permanent magnets, the motor will not output any torque. In this case, the two magnetic fields still interact, but since the direction of this force is aligned with the rotor's rotation axis, the two magnetic fields only generate pressure on the bearings and do not generate any rotational force. On the other hand, if the direction of the magnetic field generated by the stator is orthogonal to the direction of the rotor's magnetic field, this will generate a force that causes the rotor to rotate, and this is the location where the maximum torque is generated. Magnetic fields of any direction and magnitude generated by the stator can be decomposed into two components: one parallel and one perpendicular to the direction of the rotor's magnetic field. Thus, mutually orthogonal magnetic fields generate rotational force, while mutually parallel magnetic fields generate pressure on the bearings. For this reason, the function of an efficient brushless motor drive is to reduce mutually parallel magnetic fields and maximize mutually orthogonal magnetic fields. [align=center]Figure 3[/align] For ease of modeling and analysis of the control system, we conventionally control the coil current rather than the stator magnetic field. This is because we can easily detect the motor current, while the magnetic field (actual magnetic flux) is difficult to obtain. In a brushless motor, the current flowing through the three sets of coils directly generates the stator magnetic field. Since these three sets of coils are artificially installed with a 120-degree angle difference between them, the magnetic fields generated by the three sets of coils also have a 120-degree angle difference. The superposition of these three magnetic fields generates the stator magnetic field. To model the magnetic field generated by the current flowing through the stator coils, we introduce the concept of a "space current vector." The space current vector of the fixed coil has a fixed magnetic field direction, which is entirely determined by the interaction between the magnitude of the magnetic flux through the coil and the current flowing through the coil. Thus, we can use the space current vector to characterize the stator magnetic field; this space current vector is the spatial superposition of the current vectors generated by the three sets of coils. An intuitive way to explain the space current vector is to assume that the stator consists of only one set of coils, and the magnetic field generated by the current flowing through this set of coils is the same as the superimposed magnetic field generated by the previous three sets of coils. [align=center]Figure 4[/align] Like the stator magnetic field, the stator space current vector can also be decomposed into two components: one perpendicular to the rotor magnet axis and the other parallel to it. The magnetic field generated by the perpendicular current component is orthogonal to the rotor's magnetic field, thus producing rotational torque. The current component parallel to the rotor's magnetic axis produces the same magnetic field as the rotor's magnetic field and does not produce any torque. Therefore, a good control algorithm needs to minimize this current component parallel to the rotor's magnetic axis, as it only generates excess heat in the motor and exacerbates bearing wear. We need to control the coil current to maximize the current component perpendicular to the rotor's magnetic axis. The resulting motor torque is proportional to the magnitude of this current component. To effectively obtain a continuous and stable torque, we need an ideal, continuously stable magnetic field to generate a stable stator space current vector, and this magnetic field needs to follow the rotor's rotation in real time and remain permanently perpendicular to its magnetic field. From the perspective of rotor rotation, the stator's spatial current vector should be a stable value. Therefore, during motor rotation, the stator's spatial current vector should represent a circular loop. Since the stator's current vector is formed by the superposition of current components generated by three sets of coils, and these three sets of coils are physically spaced 120 degrees apart, the motor's current vector should be formed by the superposition of three ideal sinusoidal signals, with a 120-degree phase angle between these three sets of sinusoidal signals. [align=center]Figure 5[/align] To minimize (make zero) the stator current vector in the same direction as the rotor's magnetic field and maximize the perpendicular magnetic field, the sinusoidal current in the stator coils needs to be phase-adjusted in real time with the rotor's rotation angle. To achieve this ideal state, we have achieved varying degrees of success in controlling brushless motors through various control methods. 3. Trapezoidal Wave Commutation The simplest way to control a brushless DC motor is the so-called "trapezoidal wave" commutation. In this scheme, we control the current in only one pair of motor coils at a time, while the third coil remains unconnected to the power supply in the circuit. Hall effect sensors installed inside the motor detect current every 60 degrees and feed the results back to the motor controller via digital signals. Because only two sets of coils in the trapezoidal wave commutation mode carry the same current, while the third set has zero current, this detection method can only detect current vectors in six directions during one motor revolution. Since the motor current changes every 60 degrees during rotation, each current vector can only calibrate the current within a 30-degree range to the left and right. The current waveform jumps abruptly from zero to the maximum positive current, then back to zero, and then back to the maximum negative current. In this scenario, the motor current jumps regularly within six regions, allowing the motor to operate approximately smoothly. [align=center]Figure 6[/align] Please see Figure 7, which is a framework diagram of the trapezoidal wave control method for brushless motor drives. A PI control loop is used to control the current. We compare the actual measured current with the required current to obtain a deviation signal. This deviation signal is then integrated and amplified to generate an output correction value, which is used to reduce errors. The correction value generated by the PI control loop is then tuned by PWM and provided to the output bridge. The purpose of this process is to ensure that the current in any coil remains stable. Commutation and current control are not related. The position signal generated by the Hall sensor in the motor is only used to select which pair of coils corresponds to the output bridge that needs to be energized, while other bridges remain without current. The current sensing loop is mainly used to detect the current of the energized coil in real time and feed the signal back to the current control loop. [align=center] Figure 7 Framework diagram of trapezoidal wave control method for brushless motor drive[/align] Although the trapezoidal wave commutation control method can meet many different application control needs, it still has some defects. Because in this commutation method, the current vector can only represent six discontinuous directions, it cannot characterize the current change within any 30-degree angle. This causes the motor torque to fluctuate by 15% (1-cos(30)) at six times the motor rotation frequency. This inaccuracy of the current vector also brings about efficiency loss, because part of the current in the coil cannot generate torque for the motor. More importantly, the six current channel switching cycles per revolution of the motor generate harsh noise and make it very difficult to control the motor's precision at low speeds. Trapezoidal commutation cannot achieve smooth and precise control for brushless motors, especially at low speeds. Sine wave commutation, however, solves these problems. The sine wave control method for brushless motors primarily involves simultaneously controlling the current in three sets of coils, allowing them to change smoothly in a sine wave form during motor rotation. The current in the three sets of coils is controlled in real time to achieve a constant magnitude vector perpendicular to the rotor's magnetic field direction. Compared to trapezoidal commutation, this commutation method eliminates torque fluctuations and current jumps during commutation. During rotation, to make the motor current closer to a smooth sine wave form, a high-precision sensor is needed to accurately measure the rotor's rotational position. Hall effect signals can only provide coarse measurements and cannot meet this high-precision requirement; therefore, an encoder or similar device is needed. Figure 8 is a framework diagram of sine wave commutation in a brushless motor drive. This method features two independent current control loops to control the two coils of the motor in real time. Because the motor is WYE-connected, the current in the third coil is equal in magnitude to the sum of the currents in the other two coils, but in the opposite direction (Newton's current law). Therefore, we cannot control the current in the third coil independently. [align=center]Figure 8: Sine Wave Commutation Frame Diagram of Brushless Motor Drive[/align] Because the currents of the three coils must be combined into a stable motor rotation vector current, and these three coils maintain a 120-degree angle with each other, the three currents must be in sine wave form and maintain a 120-degree phase difference. The position encoder mainly provides two sine wave signals, spaced 120 degrees apart. These two signals are superimposed with the torque control signal to form an amplified sine wave signal to obtain the control torque for the motor. The two current signals, after phase superposition, form the current vector that rotates the motor. The sine wave signal obtained by tuning the current signals of the two motor coils is input into a pair of PI controllers. Since the current in the third coil is the negative superposition of the other two, we do not need to control it. The output signal of each PI controller will be modulated by PWM and input to the motor coil through a bridge circuit. The control voltage of the third coil is the negative superposition of the voltages of the other two coils, and the control voltages of these three coils still maintain a 120-degree phase angle. In order to make the actual output current waveform accurately match the current control signal, the tuned current control vector must rotate smoothly, be stable in magnitude, and always remain perpendicular to the direction of the rotor magnetic field as required. Sine wave commutation can achieve smooth motor control that trapezoidal wave commutation cannot achieve. However, this ideal method can only play a very good role in smoothing the motor at low speeds, but has no effect on the motor at high speeds. This is because when the speed increases, the current loop controller must track the sine wave signal with an increasing frequency and overcome the back electromotive force of the motor with an increasing amplitude and frequency. [align=center]Figure 9[/align] Because the gain and response frequency of the PI controller are limited, the instability of this current loop control can easily cause phase lag of the current and control errors. The higher the speed, the greater the error. This also causes the direction of the stator current vector to deviate from the effective perpendicular direction, failing to stably follow the rotor's rotating magnetic field. This results in a smaller output torque, requiring more current to maintain it, thus reducing motor efficiency. The higher the motor speed, the worse this situation becomes. In some states, the direction of the motor current can deviate by as much as 90 degrees, at which point the motor torque will decrease to zero. With sine wave commutation, if the speed exceeds this state, the motor would output negative torque, but this is impossible. The fundamental problem with sine wave control is that it controls the motor current as a variable. As the motor speed increases and the PI controller reaches its bandwidth limit, this control method becomes ineffective. Vector control solves this problem by directly controlling the vector current components corresponding to the rotor magnetic field in the parallel and perpendicular directions to achieve precise control of the stator coil current. Theoretically, the vector current can be decomposed into two current components parallel and perpendicular to the rotor magnetic field. Because the current in these two directions is static, the PI controller can control the current with DC signals, rather than sine wave signals. Therefore, the coil current and voltage output by the controller are constants, not the time-varying variables of the past. This eliminates the limitations of the controller in terms of frequency response and phase drift. If vector control is used to control a brushless motor, the quality of current control is unrelated to the motor speed. [align=center]Figure 10[/align] In vector control, we mainly control the motor current and voltage corresponding to the directions parallel and perpendicular to the rotor magnetic field. This means that the measured motor current must be mathematically calculated by the PI controller and then transformed from the three-phase static structure of the stator to the dynamic structure of the rotor dq (parallel and perpendicular to the rotor magnetic field direction). Similarly, the control voltage at the motor end also needs to be mathematically calculated to transform from the rotor dq structure to the three-phase static structure of the stator before being input to the PWM section for modulation. These transformations require high-speed mathematical processing capabilities, and DSPs and high-performance processors are adopted and become the core of vector control. Although this structural transformation can be completed in one step, it is more convenient to describe it in two steps. The motor current first transforms from a physical 120-degree phase difference three-phase structure in the stator into a stable, dynamic, right-angled orthogonal dq structure, and then from this dynamic stator structure into a three-phase static rotor structure. To ensure effective results, these calculations must be completed within one sampling cycle of the PI controller. This transformation is the exact opposite of the operation required by the PI controller to transform the voltage signal from the dq structure into the three-phase structure of the stator coils. Once the motor current is transformed into the dq structure, control becomes very simple. We need two PI controllers: one to control the current parallel to the rotor magnetic field, and one to control the vertical current. Because the control signal for the parallel current is zero, this also makes the parallel current component of the motor zero, thus driving the entire current vector of the motor to be transformed into the vertical current. Since only the vertical current can generate effective torque, the motor efficiency is maximized. The other PI controller is mainly used to control the vertical current to obtain the required torque corresponding to the input signal. This ensures that the vertical current is controlled as required to obtain the desired torque. [align=center]Figure 11[/align] The output signals of the two PI controllers represent the voltage vectors corresponding to the rotor. Corresponding to the conversion of the motor current signal, these static voltage vectors also undergo a series of reference coordinate transformations to obtain the voltage control signals required by the output bridge. They are first converted from the rotor's dynamic dq reference structure to the stator's static xy structure. Then, the voltage signals are converted from this rectangular coordinate structure into physical structures spaced 120 degrees apart, and then input into the motor's three-phase coils (U, V, and W). These three voltage signals need to be modulated by PWM before being input to the motor coils. The work of converting the time-varying current and voltage sine wave signals in the motor coils into DC signals of the dq structure is the reference coordinate transformation. The essential difference between sine wave commutation and vector control lies in a series of coordinate transformations and the processing of current control. In sine wave commutation, we need to perform commutation first, and then obtain the required sine wave current through PI control. Therefore, the PI control of the system mainly processes the time-varying motor current and voltage sine wave signals, and the motor performance is limited by the controller bandwidth and phase drift. In vector control, the current signal first passes through a PI controller and then undergoes high-speed commutation. Therefore, the PI controller does not need to process time-varying current and voltage signals; the system is also unaffected by the bandwidth and phase drift of the PI controller. So why is vector control superior? Vector signals allow the motor to operate smoothly at both low and high speeds. Sine wave commutation allows the motor to run smoothly at low speeds, but its efficiency drops significantly at high speeds. Trapezoidal wave commutation works relatively well at high speeds, but produces torque fluctuations at low speeds. Therefore, appropriate control is the best control method for brushless motors.