1. Introduction
The performance of the current loop in a motor driver or servo (see Figure 1) directly affects the motor's torque output (crucial for smooth response) as well as precise positioning and speed profiles. A key metric for smooth torque output is torque ripple. This is particularly important for contour cutting and slicing applications, where torque ripple directly translates into achievable end-application accuracy. For automation applications where production efficiency is directly affected by available control bandwidth, parameters such as response time and settling time, which are dynamically related to the current loop, are critical. In addition to the motor design itself, several factors within the driver also directly influence these performance parameters.
A motor driver contains multiple sources of torque ripple. Some originate from the motor itself, such as cogging torque caused by stator windings and stator slot arrangement, as well as rotor EMF harmonics.1 Other sources of torque ripple are related to offset and gain errors in the phase current feedback system 2 (see Figure 1).
Inverter dead time also directly affects torque ripple because it adds low-frequency (mainly 5th and 7th) third harmonic components of the stator frequency to the PWM output voltage. In this case, the impact on the current loop is related to the current loop's immunity to harmonic frequencies.
This paper will focus on torque ripple caused by phase current measurement. We will analyze each type of error and discuss methods to minimize the impact of measurement errors.
Figure 1. Current loop in a motor driver with non-ideal components in the feedback path.
2. Torque ripple caused by current measurement error
The electromagnetic torque formula for a 3-phase permanent magnet motor is:
2.1 Two-phase measurement
2.2 Three-phase measurement
3. Error sampling time
When a three-phase motor is powered by a switching voltage source inverter, the phase current can be considered to consist of two components: the fundamental component and the switching component (see Figure 2A).
Figure 2. (A) Phase current of a 3-phase motor driven by a switching voltage source inverter. (B) Amplified phase current illustrating how current ripple is attenuated by sampling.
For control purposes, switching components must be eliminated, otherwise the performance of the current control loop will be affected. A common method for extracting the average component is to sample the current synchronized with the PWM cycle. The current is averaged at the beginning and middle of the PWM cycle. If the sampling is closely synchronized with these instances, the switching component can be effectively suppressed, as shown in Figure 2B. However, if timing errors exist when sampling the current, aliasing will occur, leading to a degraded performance of the current loop. This section discusses the causes of timing errors, their impact on the current loop, and how to ensure the system's stability can cope with sampling timing errors.
3.1 Sampling Timing Error in Motor Driver
The fundamental component of the phase current is typically in the tens of Hz range, and the bandwidth of the current loop is typically in the several kHz range. It seems counterintuitive that even a small timing error can affect control performance. However, since only the phase inductance limits di/dt, even a small timing error can lead to significant current distortion. For example, a 250V voltage applied across a 5mH inductor for 1μs will result in a 50mA current change. Furthermore, assuming the system uses a 12-bit ADC with a full-scale range of 10A, timing errors will cause the loss of the lower 4.3 bits of the ADC. As shown later, bit loss is the optimal scenario. Aliasing can also cause torque ripple and gain errors in the feedback system.
The most common cause of incorrect sampling timing is:
The link between the lPWM and the ADC is insufficient, preventing sampling at the correct time.
There is a lack of sufficient independent synchronous sample-and-hold circuits (two or three depending on the number of phases being measured).
The gate drive signal propagation delay causes the motor voltage to be out of phase with the PWM timer.
Generally, the severity of timing errors is determined by factors that may affect di/dt. Of course, the magnitude of the timing error is also important, but motor speed, load, motor impedance, and DC bus voltage also have a direct impact on the error.
3.2 Impact of Sampling Error on System Performance
The effect of sampling error can be determined using the derived formula. For two-phase current measurement, assuming ia is sampled at an ideal moment (iae=0), and ib is sampled with a delay, resulting in ibe≠0. In this case, the error term defined by Equation 9 is:
For the three sensors shown in Figures 3A and 3B, note that the ib measurement delay will result in a current (torque ripple) that is twice the fundamental frequency. Also note that the DC components of id and iq are unaffected.
For the two sensors shown in Figures 3C and 3C, note that the ib measurement delay will cause the AC component to be 1.73 times larger than when there are three sensors. Furthermore, the DC components of id and iq will also be affected.
3.3 Minimize the impact of sampling timing errors
As the performance requirements of the control loop increase, it is essential to minimize the impact of sampling timing errors, especially as ADC resolutions tend to increase. A few years ago, 10- to 12-bit ADCs were common, but now 16-bit resolution is the norm. These extra bits should be utilized effectively; otherwise, the values of high-performance ADCs will be affected by low-order bit loss due to system latency.
The most effective way to minimize sampling timing errors is to sample as close as possible to the ideal sampling times for all phases. This may lead to the selection of a controller optimized for a digitally controlled switching power supply converter. Furthermore, optimizing propagation delay/skew in the gate drive circuitry will have a positive impact.
If minimizing timing errors is still insufficient, a significant performance improvement can be achieved by using three current sensors and an ADC with three independent sample-and-hold circuits.
4. Misalignment error
The derived formula can also describe how the system responds to misalignment in the measured current. First, by observing the situation with two sensors and using ide from Equation 9 as an example, the error component can be expressed as:
4.1 Minimize the impact of misalignment error
Current feedback misalignment is one of the main causes of torque ripple in motor drives and should be minimized. Generally, there are two types of misalignment errors in current feedback. First, static misalignment exists at any point in time and at any temperature. Second, misalignment drift is a function of parameters such as temperature and time. A common method to minimize the effects of static misalignment is to perform misalignment calibration, which can be done during manufacturing or every time the motor current is zero (usually when the motor is stopped). If this method is used, static misalignment is generally not a problem.
Misalignment drift is more complex to handle. Because it is a slow drift that typically occurs during motor operation, it is difficult to calibrate online, and the motor usually cannot be stopped. Some observer-based online calibration methods are suggested, but the observer depends on the model of the motor's electrical and mechanical systems. For online estimation to be effective, accurate knowledge of the motor parameters is required, but this is usually not the case.
As discussed earlier, the most effective way to minimize offset drift is to use three-phase current measurement. Assuming the channels use the same type of components, their drift is likely to be similar. If this is the case, the offset will be canceled out, and no torque ripple will be generated. Even if the channels do not drift at the same rate, as long as they drift in the same direction, the three-channel method will have a offset-canceling effect.
For two-phase current measurements, torque ripple persists even if the channels drift at the same rate. In other words, two-sensor systems are highly sensitive to offset drift. In this case, the only way to avoid torque ripple is to ensure the drift remains small, which can increase cost and feedback system complexity. For a given set of performance requirements, a 3-channel feedback system may be a cost-effective solution, a fact proven by experience.
5. Gain error
6. Experimental Verification
The effects of offset error and gain error on the measured current and output torque were verified in the experimental setup described in Figure 4.
Motor drive board | Analog Devices' AC input, 350 V DC , 3-phase, closed-loop, field-oriented controller drive platform |
PM Motor | M-2311S-LN-02D Teknic, 4-pole, 0.42 nm, 6000 rpm, 3-phase PM synchronous motor |
Torque sensor | RWT421-DA, sensor technology, ±2 nm, 0.25% accuracy |
Braking load | Magnetic particle brake, maximum 1.7 nm |
Inertial load | Disconnected 1 kW ABB induction motor |
Figure 4. Test equipment setup
The current feedback circuit in the drive board utilizes Hall effect sensors in all three phases of the motor. Two-phase or three-phase current measurement can be selected in the software. Offset calibration is performed when the motor is not running, so offset and gain errors are quite small during normal operation (when there is no time for drift effects to occur). Such errors typically occur due to temperature drift (despite the calibration procedure), and to describe the impact of these errors, artificial offsets and gain errors are introduced into the control software after the calibration procedure. The measured quantities derived from the control algorithm will differ from the actual quantities, which will include the effects of these errors, as discussed in the previous sections. Figure 5 illustrates the case where the set speed reference is 520 rpm, at which point the motor electrical frequency is 35 Hz.
Clearly, when the driver maintains relatively constant d-axis and q-axis currents to keep the set speed, the actual current contains a significant amount of harmonic components, especially in the case of offset errors. These harmonic components directly affect the output torque ripple, as shown in Figure 6. It must be noted that significant mechanical torque pulsations exist due to slight shaft misalignment in the test equipment. This occurs at the mechanical frequency and some lower harmonics. However, the changes in harmonic components related to offset and gain error sources are still clearly visible. For offset errors, the harmonic components at the electrical frequency (35Hz) increase proportionally to the percentage of offset error, as shown in the figure, while the harmonic components at twice the electrical frequency increase asymmetrically with gain error, just as theoretically predicted.
Furthermore, the effects of the three-phase measurement can be clearly seen in Figure 7. The offset error induced torque ripple is completely eliminated, and the gain error induced torque ripple is reduced by 1.73 times—once again confirming the results of the theoretical calculation.
in conclusion
Through analysis and measurement, this paper describes how non-ideal effects in current feedback systems affect system performance. As previously explained, systems using three-phase current measurement are significantly more tolerant of measurement errors than systems using two-phase current measurement.
Figure 5. Actual values (red) and measured values (blue) (from top to bottom); iq and id with 1% offset error; iq and id with asymmetric gain error (1.05/0.95).
Figure 6. The nominal percentage of torque ripple measured during two-phase current measurement, and the increasing offset error (left) and increasing gain error (right).
Figure 7. The nominal percentage of torque ripple measured during three-phase current measurement, and the increasing offset error (left) and increasing gain error (right).
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