Optimizing Motion Control: Getting the Most Out of Resolvers

设计在艰苦环境中运行的动作控制系统时，对于工程师们来说，最头痛的事情是寻找到能够提供各个运动的机器元件精确的位置数据的硬件。许多新型传感设备不能在艰苦条件下保持功能的可靠性。在这种情况下，工程师们最好的选择是一个从二十世纪四十年代就开始使用的反馈 传感器 — 旋转变压器，这种传感器经过实践证明是非常可靠的。

旋转变压器监视旋转单元（例如电机转轴和齿轮）的轴间角，并将位置数据发送回运动控制系统。该器件的设计使它能够显著减少电噪声和振动的影响。例如，旋转变压器工作频率相对较低，可以对其分量进行通带限制，从而减少了对噪声的敏感度。由于器件不含电子元件（只有磁性元件），它更适应振动和极端温度环境。

工业级旋转变压器的精度达到5-10弧度分，大概相当于11-12比特（图1）。为使旋转变压器达到最佳精度，必须对误差源进行补偿，并需要理解旋转变压器和信号处理单元的原理。

Figure 1. Angular accuracy and digital resolution

How a Resolver Tracks Position

A resolver can be thought of as a rotary transformer with one primary and two secondaries. The coupling between the primary and each secondary varies as the shaft rotates. The induced voltage varies between zero and a maximum value, either in phase with the primary or 180° out of phase. Figure 2 shows one rotation of a resolver. A constant amplitude voltage is used as the reference or input. The output sine and cosine signals can be thought of as position feedback signals.

Figure 2. Resolver output as a function of angle

Observe the phase relationship of the three signals and determine in which quadrant the rotating axis is located (Figure 3).

Figure 3. Quadrant relationship

If the resolver is excited with a voltage of the form:

VE = Sin (ωτ) (1)

then the position of the shaft is determined by the following trigonometric equations:

Vsine = V Sin (θ) Sin (ωτ) (2)
Vcos = V Cos (θ) Sin (ωτ) (3)

where:
θ = shaft angle
ωτ = carrier
V = peak voltage amplitude induced into the resolver feedback coil

The carrier does not contribute to determining position except in phase with the reference voltage. The peak voltage induced into each secondary is not equal to the reference voltage amplitude. Resolvers typically have a transformation factor between input and output.

Modern R/D Converters

R/D converters simplify the process of converting two analog signals into digital signals in modern control systems (Figure 4). A set of analog switches multiplexes the feedback sine and cosine signals, scales them, and compares the result to a set of analog signals controlled by a D/A converter. The result represents an error voltage equivalent to (θ- j), which is the shaft angle minus the estimated shaft angle. The demodulator removes the carrier, leaving the error signal, which is measured by the integrator. The output of the integrator controls a voltage-controlled oscillator, which causes a counter to count up or down, depending on the polarity of the error. The D/A converter controlled by the counter completes the loop. This structure forms a type II servo loop, with the VCO used for secondary integration.

Figure 4. Simplified R/D structure diagram

If the D/R converter presents a step position change to the R/D converter, the response will have a slight overshoot with settling time and bandwidth limitations similar to those of a typical servo system. This characteristic must be considered when implementing resolver conversion in a motion control system. Resolver

Model

Developing simulation models using tools such as SPICE can help predict the performance of a resolver/cable environment (Figure 5). Simulation can also be used during the design process when developing a resolver interface.

Figure 5. Detailed model and equivalent T network model

Remember that the resolver has transformer characteristics. The most obvious one is that it has an air gap, which can create a large leakage inductance that has a significant impact on the cable and accuracy.

The distributed capacitance is small compared to typical frequencies in the audio frequency range, so this model ignores these capacitances. The DC resistance (Rs and Rr) is the resistance of the wire in the coil. The leakage (Ls and Lr) inductance is large due to the air gap. Remember that these are moving mechanical parts and there is some air space between the rotor and stator. Rc and Lm are eddy current and magnetizing inductance losses. The model can be simplified to an equivalent T model by considering the primary and secondary transformation ratios.

Error Sources

There are three main sources of error: cable resonance, reference phase shift, and feedback signal matching. Temperature, source impedance, and load impedance should also be considered.

Cables and Resonance The twisted pair cable can be modeled as a distributed RLC network (Figure 6). The cable has mainly capacitive characteristics, which will cause resonance when coupled with the resolver.

Figure 6. Twisted-pair cable can be modeled as a distributed RLC network

The peak amplitude of the sine and cosine feedback signals can be limited to a small voltage band by using an R/D converter. The inductive voltage source (resolver feedback) and the capacitive load (cable) also create a natural resonant circuit. If the cable is long, it is easy to resonate and produce a large return voltage. This will cause errors because the signal may be clipped at the quadrant boundary.

The solution is to reduce the excitation voltage to bring the feedback signal within the acceptable range. There are several ways to achieve this. You can rotate the shaft to the zero angle position and adjust the input to get the correct output value of the cosine coil. You can also rotate the shaft to the 45° position where all signals have the same amplitude and adjust the input to get the output 0.707 times the desired output. You can also use trigonometric identities, which do not require positioning angles. In the identity, the square of the sine plus the square of the cosine is a constant.

V = V Sin2(θ) + V Cos2(θ) (4)

Adjustments must be made when the motor is stationary.

Reference Phase Shift From the previous model, we can see that there is a delay in the resolver and cable due to the inductance. This means that the returned sine and cosine feedback signals are out of phase with the reference, and the phase shift will result in a position error proportional to the shaft speed.

If this condition already exists, there is no quick fix. The solution is to use two oscillators on the board—one to excite the resolver and one that is in phase with the return sine and cosine signals. If the oscillators are digital and generate analog signals (sine waves) from a common digital clock, the phase relationship will remain constant. The motor should be held still while the adjustment is made.

Feedback Signal Matching If the return cables are long, another source of error will occur; cable imbalance. Most resolvers, especially the feedback coils, have a large inductance, so they can act as a voltage source with internal inductance. Twisted-pair cables are predominantly capacitive. If the pairs are not matched, the impedance presented to the source (sine or cosine feedback) is different, resulting in a differential voltage drop. As a result, the return voltages have different peak voltage amplitudes. Since position is derived from the ratio of the two return signals, the error is further magnified if the cable is unbalanced.

Since standard cables are not perfectly balanced, compensation must be applied. The best approach is to slowly rotate the shaft, observe the peak amplitude as the resolver rotates through the 0°, 90°, 180°, and 270° positions and adjust the variable gain amplifier to compensate. This can be done during commissioning and when the gain values ​​are stored and restored at later startups.

The output and load impedances should keep the voltage source impedance as small as possible. Also be aware of its instability. The amplifier must drive a complex RLC load, so instability or oscillation is likely.

Likewise, the load impedance should be kept large enough to avoid drawing too much current from the resolver. Current in the resolver generates heat, causing performance variations. Note that the feedback coil and source are isolated from each other, so a differential receiver is used to reduce noise.

Developing a resolver T-model using measured resolver parameters

The transformer T-model is developed from measured resolver characteristic parameters from actual applications and SPICE simulations. The measured values ​​are used because they contain the DC, AC, and magnetic field characteristics of the resolver. The primary (rotor) and a secondary (stator) can be considered as a four-terminal two-terminal pair device with inputs and outputs. The four parameters (Zro, Zrs, etc.) should be listed as complex impedances in the resolver data sheet (Figure 7). The goal is to find Z1, Z2, and Z3 based on the complex impedance values ​​provided in the data sheet or measured in the experiment.

The definitions of Z1 and Z2 are shown in Figure 7.

Z1 = Zro - Z3 (5)
Z2 = Zso - Z3 (6)
Z3 =[ Zso ( Zro - Zrs)]1/2 (7)

When substituting the actual values, a quadratic equation is obtained with real and imaginary components. The roots must be determined.

Figure 7. Definition of four impedances

Summary

Resolver inductance is a major concern in precision control. The inductive effect will amplify small differences in voltage matching, but if handled carefully, the maximum precision that the resolver can provide can be obtained.