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A Low-Cost, Smart Capacitive Position Sensor
Abstract
A new high-performance, low-cost, capacitive position-measuring system is described. By using a highly linear oscillator, shielding and a three-signal approach, most of the errors are eliminated. The accuracy amounts to 1 μm over a 1 mm range. Since the output of the oscillator can directly be connected to a microcontroller, an A/D converter is not needed.
I. INTRODUCTION
This paper describes a novel high-performance, low-cost, capacitive displacement measuring system featuring:
1 mm measuring range,
1 μm accuracy,
0.1 s total measuring time.
Translated to the capacitive domain, the specifications correspond to:
a possible range of 1 pF;
only 50 fF of this range is used for the displacement transducer;
50 aF absolute capacitance-measuring inaccuracy.
Meijer and Schrier [l] and more recently Van Drecht,Meijer, and De Jong [2] have proposed a displacement-measuring system, using a PSD (Position Sensitive Detector) as sensing element. Some disadvantages of using a PSD are the higher costs and the higher power consumption of the PSD and LED (Light-Emitting Diode) as compared to the capacitive sensor elements described in this paper.
The signal processor uses the concepts presented in [2],but is adopted for the use of capacitive elements. By the extensive use of shielding, guarding and smart A/D conversion,the system is able to combine a high accuracy with a very low cost-price. The transducer produces three-period-modulated signals which can be selected and directly read out by a microcontroller. The microcontroller,in return, calculates the displacement and can send this value to a host computer (Fig. 1) or a display or drive an actuator.

Electronic Circuit

Personal Computer
Actuator
Display

Fig. 1. Block diagram of the system

Fig. 2. Perspective and dimensions of the electrode structure
Ⅱ. THE ELECTRODE STRUCTURE
The basic sensing element consists of two simple electrodes with capacitance Cx, (Fig. 2). The smaller one (E2) is surrounded by a guard electrode. Thanks to the use of the guard electrode, the capacitance Cx between the two electrodes is independent of movements (lateral displacements as well as rotations) parallel to the electrode surface.The influence of the parasitic capacitances Cp will be eliminated as will be discussed in Section Ⅲ.
According to Heerens [3], the relative deviation in the capacitance Cx between the two electrodes caused by the finite guard electrode size is smaller than:
δ<e-π(x/d) (1)
where x is the width of the guard and d the distance between the electrodes. This deviation introduces a nonlinearity.Therefore we require that δ is less than 100 ppm.Also the gap between the small electrode and the surrounding guard causes a deviation:
δ<e-π(d/s) (2)
with s the width of the gap. This deviation is negligible compared to (l), when the gap width is less than 1/3 of the distance between the electrodes.
Another cause of errors originates from a possible finite skew angle α between the two electrodes (Fig. 3). Assuming the following conditions:
the potentials on the small electrode and the guard electrode are equal to 0 V,
the potential on the large electrode is equal to V volt,
the guard electrode is large enough,
it can be seen that the electric field will be concentric.

d
l/2
l/2

Fig. 3. Electrodes with angle α.
To keep the calculations simple, we will assume the electrodes to be infinitely large in one direction. Now the problem is a two-dimensional one that can be solved by using polar-coordinates (r, φ). In this case the electrical field can be described by:
(3)
To calculate the charge on the small electrode, we set φ to 0 and integrate over r:
(4)
with Bl the left border of the small electrode:
(5)
and Br the right border:
(6)
Solving (4) results in:
(7)
For small α's this can be approximated by:
(8)
It appears to be desirable to choose l smaller than d, so the error will depend only on the angle α. In our case, a change in the angle of 0.6°will cause an error less than 100 ppm.
With a proper design the parameters εo and l are constant,and then the capacitance between the two electrodes will depend only on the distance d between the electrodes.
Ⅲ.ELIMINATION OF PARASITIC CAPACITANCES
Besides the desired sensor capacitance C, there are also many parasitic capacitances in the actual structure (Fig.2). These capacitances can be modeled as shown in Fig.4. Here Cpl represents the parasitic capacitances from the electrode E1 and Cp2 from the electrode E2 to the guard electrodes and the shielding. Parasitic capacitance Cp3 results from imperfect shielding and forms an offset capacitance. When the transducer capacitance Cx is connected to an AC voltage source and the current through the electrode is measured,Cpl and Cp2 will be eliminated. Cp3 can be eliminated by performing an offset measurement.

Fig. 4. Elimination of parasitic capacitances
The current is measured by the amplifier with shunt feedback, which has a very low input impedance. To obtain the required linearity, the unity-gain bandwidth fT of the amplifier has to satisfy the following condition:
(9)
where T is the period of the input signal.
Since Cp2 consists of cable capacitances and the input capacitance of the op amp, it may indeed be larger than Cf and can not be neglected.
IV. THE CONCEPT OF THE SYSTEM
The system uses the three-signal concept presented in [2], which is based on the following principles. When we measure a capacitor Cx with a linear system, we obtain a value:
(10)
where m is the unknown gain and Moff, the unknown offset.By performing the measurement of a reference quantity Cref, in an identical way and by measuring the offset, Moff,by making m = 0, the parameters m and Moff are eliminated.The final measurement result P is defined as:
(11)
In our case, for the sensor capacitance C, it holds that:
(12)
where Ax is the area of the electrode, do is the initial distance between them, ε is the dielectric constant and △d is the displacement to be measured. For the reference electrodes it holds that:
(13)
with Aref the area and dref the distance. Substitution of (12) and (13) into (10) and then into (11) yields:
(14)
Here, P is a value representing the position while a1 and a0 are unknown, but stable constants. The constant a1 =Aref/Ax is a stable constant provided there is a good mechanical matching between the electrode areas. The constant ao = (Arefd0/(Axdref) will also be a stable constant provided that do and dref are constant. These constants can be determined by a one-time calibration. In many applications this calibration can be omitted; when the displacement sensor is part of a larger system, an overall calibration is required anyway. This overall calibration eliminates the requirement for a separate determination of a1 and a0.
V . THE CAPACITANCE-TO-PERIOD CONVERSION
The signals which are proportional to the capacitor values are converted into a period, using a modified Martin oscillator [4] (Fig. 5j.
When the voltage swing across the capacitor is equal to that across the resistor and the NAND gates are switched off, this oscillator has a period Toff:
Toff = 4RCoff. (15)
Since the value of the resistor is kept constant, the period varies only with the capacitor value. Now, by switching on the right NAND port, the capacitance CX can be connected in parallel to Coff. Then the period becomes:
Tx=4R(Coff+Cx)=4RCx+Toff (16)
The constants R and Toff are eliminated in the way described in Section IV.
In [2] it is shown that the system is immune for most of the nonidealities of the op amp and the comparator, like slewing, limitations of bandwidth and gain, offset voltages,and input bias currents. These nonidealities only cause additive or multiplicative errors which are eliminated by the three-signal approach.
VI. PERIOD MEASUREMENT WITH A MICROCONTROLLER
Performing period measurement with a microcontroller is an easy task. In our case, an INTEL 87C51FA is used,which has 8 kByte ROM, 256 Byte RAM, and UART for serial communication, and the capability to measure periods with a 333 ns resolution. Even though the counters are 16 b wide, they can easily be extended in the software to 24 b or more.
The period measurement takes place mostly in the hardware of the microcontroller. Therefore, it is possible to let the CPU of the microcontroller perform other tasks at the same time (Fig. 6). For instance, simultaneously with the measurement of period Tx, period Tref and period Toff,the relative capacitance with respect to Cref is calculated according to (11), and the result is transferred through the UART to a personal computer.

Fig. 5. Modified Martin oscillator with microcontroller and electrodes.

Fig. 6. Period measurement as background process.

Fig. 7. Position error as function of the position and estimate of the nonlinearity.
VII. EXPERIMENTAL RESULTS
The sensor is not sensitive to fabrication tolerances of the electrodes. Therefore in our experimental setup we used simple printed circuit board technology to fabricate the electrodes, which have an effective area of 12 mm × 12 mm. The guard electrode has a width of 15 mm, while the distance between the electrodes is about 5 mm. When the distance between the electrodes is varied over a 1 mm range, the capacitance changes from 0.25 pF to 0.3 pF.Thanks to the chosen concept, even a simple dual op amp (TLC272AC) and CMOS NAND’s could be used, allowing a single 5 V supply voltage. The total measurement time amounts to only 100 ms, where the oscillator was running at about 10 kHz.
The system was tested in a fully automated setup, using an electrical XY table, the described sensor and a personal computer. To achieve the required measurement accuracy the setup was autozeroed every minute. In this way the nonlinearity, long-term stability and repeatability have been found to better than 1 μm over a range of 1 mm (Fig.7). This is comparable to the accuracy and range of the system based on a PSD as described in [2].
As a result of these experiments, it was found that the resolution amounts to approximately 20 aF. This result was achieved by averaging over 256 oscillator periods. A further increase of the resolution by lengthening the measurement time is not possible due to the l/f noise produced by the first stages in both the integrator and the Comparator.
The absolute accuracy can be derived from the position accuracy. Since a 1 mm displacement corresponds to a change in capacitance of 50 fF, the absolute accuracy of 1 μm in the position amounts to an absolute accuracy of 50 aF.
CONCLUSION
A low-cost, high-performance displacement sensor has been presented. The system is implemented with simple electrodes, an inexpensive microcontroller and a linear
capacitance-to-period converter. When the circuitry is provided with an accurate reference capacitor, the circuit can also be used to replace expensive capacity-measuring systems.
REFERENCES
[1] G. C. M. Meijer and R. Schner, ”A linear high-performance PSD
displacement transducer with a microcontroller interfacing,” Sensors
and Actuators, A21-A23, pp. 538-543, 1990.
[2]J. van Drecht, G. C. M. Meijer, and P. C. de Jong, ”Concepts for the
design of smart sensors and smart signal processors and their application
to PSD displacement transducers,” Digesr of Technical Papers,
Transducers ’91.
[3]W. C. Heerens, ”Application of capacitance techniques in sensor design,”
Phys. E: Sci. Insfrum., vol. 19, pp. 897-906, 1986.
[4]K. Martin, ‘‘A voltage-controlled switched-capacitor relaxation oscillator,”
IEEEJ., vol. SC-16, pp. 412-413, 1981.

一种低成本智能式电容位置传感器
摘要
本文描述了一种新的高性能, 低成本电容位置测量系统。经过使用高线性振荡器, 屏蔽和三信号通道, 大部分误差被消除。其精确度在1毫米范围内达1微米。由于振荡器的输出可直接连接到微控制器, 因此无需用A/D转换器。
Ⅰ.导言
本文介绍了一种新型高性能, 低成本的电容位移测量系统, 特点如下:
l 1毫米测量范围
l 1微米精确度
l 0.1 s总测量时间
对应到电容域, 规格相当于:
l 1皮法的变化范围; 只有这个范围的50fF( fF是法拉乘以10的负15次方。f是femto的缩写) 用于位移传感器。
l 50aF绝对电容测量误差。
梅耶尔和施里尔[1]以及最近的范德雷赫特河, 梅耶尔, 和德容[2]提出了位移测量系统, 采用一个PSD( 位置敏感探测器) 作为传感元件。和本文描述的电容传感器元件相比, 使用PSD的缺点是, PSD和LED( 发光二极管) 有更高的成本和功率消耗。
使用[2]中所提概念的信号处理器, 被采用到电容元件的使用中。经过广泛使用屏蔽, 智能A / D转换, 该系统能够将高精确度和低成本结合。换能器产生能够选择和直接由微控制器读出的三段调制信号。微控制器, 相应的, 计算位移及发送此值到主机电脑( 图1) 或显示或驱动执行器。

电子电路

上位机
执行器
演示

图1 该系统的框图
金属
屏蔽
电极
屏蔽

图2 电极结构的尺寸和透视图
Ⅱ.电极结构
基本传感元件包含电容为Cx的两个简单电极(图2) 。较小的一个( E2) 是由屏蔽电极包围。由于使用屏蔽电极, 两电极间的电容Cx可平行于电极表面独立运动( 横向平移以及旋转) 。寄生电容Cp的影响可被消除, 将在第3节讨论。
据Heerens [3], 由有限屏蔽电极大小造成的两个电极之间电容Cx的相对偏差小于:
δ<e-π(x/d) (1)
其中x是屏蔽的宽度, d是电极之间的距离。这种偏差引入了非线性。因此, 我们规定δ小于100ppm。另外小电极和周围屏蔽之间的间距产生一个偏差:
δ<e-π(d/s) (2)
S是间距的宽度。当间距宽度小于电极之间距离的1/3时, 这偏差和( 1) 相比是微不足道的。
另一个误差的原因可能源自两个电极之间的有限倾斜角α( 图3) 。假设符合下列条件:
l 小电极和屏蔽电极上的电势等于0V
l 大型电极电势等于V伏
l 屏蔽电极足够大
能够看出, 电场将同心。

d
l/2
l/2

图3 倾斜角度α的电极
为了使计算简单, 我们将假设电极在一个方向无限大。问题就成为一个二维问题, 能够用极坐标( Υ, φ) 方法解决。在这种情况下, 电场能够表述为:
(3)
为了计算小电极的损耗, 我们设定φ为0, 整定Υ:
(4)
Bl是小电极的左侧边界:
(5)
Br是右边界:
(6)
求解( 4) 结果:
(7)
对小α的近似:
(8)
选择比d小的l似乎是可行的, 因此该误差将只决定于角度α。在这种情况下, 0.6°的角度变化, 将产生小于100 ppm的误差。
对参数εo和l是常数的设计, 两个电极之间的电容将仅仅取决于电极之间的距离d。
Ⅲ.寄生电容的消除
除了理想传感器电容Cx, 在实际结构中还有许多寄生电容( 图2) 。这些电容能够建模, 如图4所示。这里Cpl代表电极El的寄生电容, Cp2是从电极E2到屏蔽电极和屏蔽层的。寄生电容Cp3造成不完善屏蔽, 形成一个偏移电容。当传感器电容Cx连接到AC电压源, 经过电极的电流可测, Cpl和Cp2, 将被消除。Cp3可经过偏移测量消除。

图4 消除寄生电容
电流经过并联反馈放大器测量, 它具有非常低的输入阻抗。要获取所需的线性度, 放大器的单位增益带宽fT必须符合下列条件:
(9)
T是在此期间的输入信号。
由于Cp2包括电缆电容和运算放大器的输入电容, 它很可能大于Cf而不可忽略。
Ⅳ.本系统的概念
该系统采用了[2]提出的三信号的概念, 它是基于以下原则。当我们用线性系统测量电容Cx, 得到一个值:
(10)
其中m是未知的增益,Moff是未知偏移。以相同的方式, 经过测量参考量Cref, 测量偏移Moff, 使m= 0,参数m和Moff被抵消。最后的测量结果P定义为:
(11)
在我们的例子中, 传感器的电容Cx为:
(12)
其中Ax, 是电极面积, do是它们之间最初的距离, ε是介电常数, △d是要测量的位移。对于参考电极, 它为:
(13)
Aref为面积, dref为距离。将( 12) ( 13) 式代入( 10) 式, 然后代入( 11) 得:
(14)
式中, 当a1和a0未知时, P是一个表示位置的值, 可是稳定的常数。常数a1=Aref /Ax是一个稳定的常数, 表明电极之间的区域有良好的机械匹配。常量a0=( Arefdo /( Axdref) 也是一个稳定的常数, 表明do和dref是常数。这些常量能够由一次性校准确定。在许多应用中校准能够省略; 当位移传感器是一个较大系统的一部分, 全面的校准是必须的。这个整体校准无需单独测定a1和a0。
Ⅴ