Resolution explained for magnetic angle sensors

文章作者：Carmine Fiore和Serge Reymond，整体电源系统

由于光学编码器在功能上是相似的，因此“分辨率”通常用作磁编码器的主要特征，该设备在数字形式下提供了附着在机械轴上的磁铁的角度。分辨率是一个关键参数，因为它指示传感器可以解析的最小角度。不幸的是，当比较产品时，用户通常会被误导，因为在商业和技术文档中以不同的方式定义了分辨率。

This article proposes a way to define resolution such that it can be consistently determined across different sensor’s datasheets. We will also show that for magnetic encoders, resolution alone is not sufficient to adequately compare products. Sensor bandwidth, which is often absent in many magnetic position sensor datasheets, is also required to compare magnetic angle sensors.

measurement error

Before defining resolution, it’s important to clarify some points related to measurement error. A measurement error is defined as the difference between the measured value of a quantity and its true value. This error can be divided in two components, described below:

• systematic (or bias) error：系统误差是在相同条件下执行的多个测量中保持恒定的组件。该误差可以估计为大量测量值的平均值与测量的真实值之间的差异。
• 随机错误：随机错误是总误差减去系统错误。它说明了在相同条件下执行的一组测量中的不可预测变化。

图1shows the different combinations of random and systematic error. There are three groups of measurements, with different amounts of random and systematic error. Group A has a larger random error, group B has a larger systematic error, and group C has similar random and systematic errors.

图1Combinations of random and systematic errors are segmented into three groups. Source:monolithic Power Systems

在磁角传感器数据表中，系统和随机误差分别表示为INL和分辨率。为简单起见，本文将假设传感器没有系统的错误，这意味着平均值是真值。

standard deviation and confidence

A metric that can be used to quantify the amount of random error in a measurement is standard deviation (σ). In statistics, σ measures the dispersion of a set of samples around their average. The higher the dispersion, the higher the σ. This parameter is also referred to as the root-mean-square (RMS) noise.

一组测量通常遵循钟形曲线分布，也称为高斯或正常曲线（请参阅图2）。This is the case when the random variation does not depend on the past error. The Gaussian curve peaks at the measurements’ average (µ), and σ characterizes its width. If the total area under the Gaussian curve is normalized to 1, then the area delimited by a range of values [a₁, a₂是measurem的结果的概率ent falls somewhere between a₁and a₂。The larger the range, the greater the confidence that a single measurement falls into that range.

图2Gaussian distribution is shown with µ = 0 and σ = 1. Source: Monolithic Power Systems

表格1列出了将测量包含在[µ –nσ，µ +nσ]范围内的概率或信心。

表格1The list shows the confidence factor for several values of n. Source: Monolithic Power Systems

定义分辨率

美国国家标准技术研究所（NIST）将决议定义为“测量系统检测并忠实地表明测量结果特征的小变化的能力”。

Resolution is then the smallest interval that an instrument can detect. To determine this interval, this article will assume that the distribution of random errors follows the Gaussian distribution. This leads to a question: how far apart should two angles be to distinguish both angles with a reasonably high probability?

When the distance between the two angles is smaller than 6σ, the two noise distributions centered on the angles significantly overlap (denoted as “A” in图3）。If the result of the measurement falls in the overlap region, it’s impossible to know if the true angle is angle a₁or a₂。只有当两个角度之间的距离等于或大于6σ之间，单个测量才能区分这两个点，其置信度等于或高于或高于99.73％（图3中表示为“ B”）。因此，传感器的分辨率为6σ的间隔。

图36σ间隔中包含的样品以µ为中心₁。资料来源：单片电源系统

类似物-to-digital conversion

Generally, the output of position sensors is given in a digital format; for example, it may be provided through an ABZ or SPI interface. In this scenario, the analog signal from the magnetic sensor must be digitized.图4显示数字磁角传感器的简化框图。请注意，该图包括一个过滤器块，将在下一节中进一步讨论。

图4The simplified block diagram of a digital magnetic angle sensor includes a filter block. Source: Monolithic Power Systems

The step size for analog-to-digital conversion (ADC)—namely the range of values in the analog domain divided by the number of steps in the digital domain—is often wrongly interpreted as the sensor’s resolution. This interpretation is only correct when the peak-to-peak noise of the analog signal is smaller than the step size of the ADC.

但是，在大多数情况下，情况并非如此。模拟信号的峰值峰噪声通常超过ADC步骤，因此它出现在传感器的数字输出中，作为输出最小显着位（LSB）的随机闪烁。这就是为什么ADC制造商定义指标，例如“无噪声分辨率”或“峰值峰值分辨率”。

图5shows how noise is carried from the analog domain to the digital domain. In this example, the step size is 1, while the peak-to-peak noise is 6. In addition, the continuous and discrete distributions are shown on the X-axis and Y-axis, respectively. Since the noise exceeds the digital step, decreasing the step size does not improve the resolution.

图5Here is how noise is carried from analog to digital domain. Source: Monolithic Power Systems

When providing a measurement in a digital format, the resolution can also be expressed in bits, calculated with Equation (1):

解析度_bit= log₂FS/6σ (1)

Where FS is the full scale of the quantity to be measured.

In the case of angle measurements, FS = 360°, which means the resolution can be estimated with Equation (2):

解析度_bit= log₂360/6σ (2)

带宽评论

When discussing sensor performance, a key parameter that is often overlooked is the bandwidth, also known as the cutoff frequency. Sensor bandwidth corresponds to a signal’s frequency range, which can be measured by the sensor. Signals with a frequency larger than the sensor bandwidth are attenuated. A detailed characterization of the sensor would require its transfer function under an analytical or graphical form. At the bare minimum, the cutoff frequency should be provided.

图4shows that a low-pass filter stage can be implemented in the sensor. This reduces the noise on the sensor output. In this case, the sensor bandwidth is same as the filter’s bandwidth. If the noise distribution is Gaussian, decreasing the filter bandwidth by a factor of 4 decreases the noise by a factor 2, which increases the resolution by 1 bit. This means that information regarding the noise or resolution should correspond with information regarding the bandwidth.

A bandwidth that is too low for an application can have dramatic effects. If the sensor is used inside a control loop, the system may be unstable, and the motor may exhibit oscillations, noise, and/or loss of efficiency (see图6）。在此图中，r是位置参考，a_mis the motor shaft angle, and A_sis the sensor output. A common design rule is to have filter bandwidth at least 10 times larger than the bandwidth of the control system or control loop.

图6带宽对运动控制回路有重大影响。资料来源：单片电源系统

图7、图8和图9显示效果of a low-pass filter bandwidth (BW) on angle measurement, noise, and control loop performance, respectively.

图7表明在高BW滤波器下的电动机轴角和传感器输出几乎重叠（分别用蓝色和绿线表示）。同时，具有较低BW滤波器的传感器输出无法准确地遵循电动机轴位置（用红线表示）。

图7不同的滤波带宽可以重叠并影响传感器输出。资料来源：单片电源系统

使用BW滤波器以显着降低噪声（请参阅图8）。As the bandwidth gets lower, the noise is more attenuated.

图8Different filter bandwidths can also impact sensor output noise. Source: Monolithic Power Systems

图9shows how different filter bandwidths affect motor control loop performance. If a filter has a lower bandwidth (denoted with the red line), then there is more overshoot and a longer settling time.

图9This is how different filter bandwidths can affect motor control loop performance. Source: Monolithic Power Systems

What to look for in datasheets

To ensure that a sensor is well-suited for your application, it’s crucial to make a distinction between a digital step and the actual sensor resolution.

通常，当使用SPI分辨率，ADC分辨率和ABZ分辨率之类的术语时，它们指定了多少位用于测量的数字表示，而不是实际的传感器分辨率。

如果传感器数据表中包含诸如RMS噪声，峰值噪声，角度噪声或噪声密度之类的规格，它们通常是获得传感器分辨率的最可靠来源。然后，设计师可以使用方程（1）来计算以位表达的分辨率。

Table 2shows an example from a datasheet. In this scenario, the resolution is misleading since it’s actually referring to the digital step. If the filter bandwidth is configurable, multiple noise values may be listed.

Table 2Datasheet example shows how resolution figures can be misleading. Source: Monolithic Power Systems

Using the lowest noise value shown in the table, the resolution can be calculated with Equation (3):

解析度_bit= log₂360/6σ= log₂360/0.06 = 12.55（3）

By comparing both resolution and bandwidth, it’s possible to determine the real performance differences between products. The filter bandwidth can be expressed through several parameters such as time constant, step response, or cutoff frequency. Table 2 shows an example using the filter time constant and filter cutoff frequency.

分辨率和带宽的主要考虑因素

For many Monolithic Power Systems (MPS) angle sensors, the digital representation of the data is 16 bits; meanwhile, the resolution, sensing technique (Hall or TMR), and the filter bandwidth vary between parts.Table 3lists the resolution and bandwidth values for some sensors in the MagAlpha family. Note that some sensors have a configurable filter bandwidth, which allows them to be adapted to different application requirements.

Table 3该列表显示了各种传感器的分辨率和带宽值。资料来源：单片电源系统

The table also illustrates how the tunneling magnetoresistance (TMR) sensing technique used inside themA600与基于霍尔的传感器相比，传感器有助于在更高的带宽下实现强大的分辨率。

The article explained the definition of resolution starting from the description of random error and the statistical concepts of standard deviation and confidence. It also clarified the difference between the digital representation—number of bits provided on the sensor output—and the resolution of the sensor measurement, in case it’s provided in digital form.

By showing the effect of filtering, the article proved that both resolution and bandwidth must be considered to determine a product’s real performance. Finally, it provided an example of a typical magnetic angle sensor datasheet to show how to correctly interpret the information contained in it.

本文最初发表在EDN。

Carmine Fiore is an application engineer atmonolithic Power Systems.

serge Reymond is sensors application manager atmonolithic Power Systems.