We present an inclinometer which incorporates a movable electrode acting as a pendulum inside a parallel plate capacitor. The movable electrode acts as the bottom plate while the top plate is a fixed metal with varying area in the shape of a triangle. When the bottom plate moves under the influence of gravity, the overlapping area of the two plates of the parallel plate capacitor varies corresponding to a change in capacitance. The relation between the angle of tilt, overlapping area, and output capacitance is derived which is used to covert the output capacitance to the tilt angle. The inclinometer has a range of 50° with a resolution of 0.38° and a response time of ∼130 ms. This design has a pronounced advantage over current methods of making inclinometer. Generally, inclinometers incorporate MEMS-based accelerometers which need complex interface circuitry and are expensive to produce while having redundant features that are not required for inclinometers. Other specialized inclinometers use fluids that are prone to environmental changes and complex to manufacture due to the presence of fluids.

Inclinometer, often referred to as a tilt sensor, is used to measure the angle of inclination or the tilt angle of an object. Inclinometers generally determine the angle of inclination based on the angle between a freely hanging object and its component of gravitational force. These sensors are mostly used in areas such as automotive, robotics and construction.1,2 Typically, an inclinometer incorporates an accelerometer to measure the inclination angle in response to the direction of gravity (acceleration).3 Over the past few decades, MEMS-based accelerometers have gained attention as they have high sensitivities and ranges. Some of the common transduction mechanisms in MEMS accelerometers are; capacitive, piezoresistive, resonant, thermal, and optical.4 Capacitive, piezoresistive and resonant accelerometers have a proof mass that accelerates in response to gravitational forces, where the displacement of the proof mass is then related to inclination angle. The proof mass in MEMS-based accelerometers have dimensions in tens of micrometers that is made using microfabrication processes, which increases the cost and complexity of the manufacturing process. Raw output from the capacitive accelerometers has a low signal to noise ratio that requires complex amplification and noise reduction circuits which further adds to the complexity of their large scale manufacturing processes and increases power consumption.5 On the other hand, piezoresistive sensors are sensitive to changes in temperatures and thus require temperature compensation techniques that further increase the complexity of the device.6 Although thermal accelerometers are much easier to fabricate as they do not have a proof mass but their working principle is based on heat transfer and fluid flow, which makes their response very slow.7 Additionally, MEMS-based accelerometers have redundant features that are not needed in the applications where only the inclination angle needs to be found. The gravitational forces are comparatively small to other everyday forces. Movements that take place under the action of gravity do not require large bandwidth and ranges of acceleration that MEMS accelerometers provide. Thus, we need specialized inclinometers that are designed to particularly measure inclination angles with higher performance and lower power consumption compared to MEMS accelerometers.

Specialized Inclinometers use a free mass that actuates under the influence of gravitational forces. Then, a particular mechanism measures the subsequent changes in a certain response. Fluidic tilt sensors measure the change in resistance or capacitance due to the movement of fluid induced by gravitational force. However, the presence of fluid as the sensing element increases the complexity of manufacturing and fluidic properties are affected by the changes in environmental conditions. The simplest and most effective form of inclinometers are pendulum based, but so far only fluid based inclinators have been made that use a pendulum.8–11 Currently, there is no method available to electronically measure the inclination angle of a pendulum to provide information about the inclination angle. Most pendulum based inclinometers are manually read using the naked eye.

Here we present a method to find the inclination angle of a pendulum-like freely moving metal electrode in a parallel plate capacitive structure to eventually form an inclinometer. A parallel plate capacitive structure consists of two metal plates separated by a dielectric material. The effective capacitance of the parallel plate capacitor depends upon the overlapping area of the two metal plates. We designed the parallel plate capacitive structure in such a way the top plate is made up of metal that has a varying size, in the shape of a triangle [Figure 1(a)]. The bottom plate is made up of a movable electrode (pendulum) such that when the structure is tilted, the bottom plate moves under the action of gravity to change the effective overlapping area of the top and bottom electrode, which in turn changes the capacitance of the structure. The two output terminals come from the top and bottom plate of capacitive structure that can be fed directly to an electronic interface that has a capacitance to digital converter. A mathematical relation is derived that converts the value of capacitance into the angle of inclination. Thus, we can directly relate the change in capacitance to the angle of tilt.

The inclinometer has a freely hanging electrode (pendulum) that is attached to a metal hinge such that it can freely rotate around the hinge. The internal axis of the movable electrode forms an angle with the gravitation vector with respect to the direction it is twisted in [Figure 1(b)]. The angle ‘α’ formed between the internal axis and the gravity vector gives information about the angle of tilt. Clockwise and anticlockwise rotations form the angle ‘α’ with the gravitation vector in opposite directions to the internal axis [Figure 1(c) and 1(d)]. This Inclinometer is converted into a parallel plate capacitor by making the freely hanging metallic object (movable electrode) hang on a metallic hinge. Since the metallic hinge is conductive, it acts as the output terminal for the bottom plate. Then the fixed metallic plate on top of it (copper tape) shaped like a triangle is separated by an air gap ‘d’ from the movable plate at the bottom [Figure 1(b)]. The air acts as the dielectric material in this parallel plate capacitive structure. The top plate in the form of a triangle allows the formation of a parallel plate capacitor with a varying overlapping area ‘A’ as the bottom movable electrode rotates around the hinge in response to tilt. This overlapping area ‘A’ is directly related to ’α’ because when the inclinometer is tilted anti-clockwise, the movable electrode tilts to the right and the overlapping area is seen to decrease [Figure 1(c)]. Alternatively, clockwise twist results in an increase in the overlapping area ‘A’ [Figure 1(d)]. As we know from the equation of a parallel plate capacitor given in Equation (1), the overlapping area ‘A’ is directly proportional to capacitance ‘C’. Thus, we are able to relate the tilt angle ‘α’ to capacitance ‘C’ to form an inclinometer.

C=εAd
(1)

Where ‘ε’ is the permittivity of the dielectric and ‘d’ is the thickness of the dielectric. First, we find the mathematical relation of ‘α’ with ‘C’ by plotting ‘A’ for various angles of tilt (α). The overlapping area of the two parallel plate capacitor varies according to the angle ‘α’, where the overlapping area ‘A’ was geometrically calculated for angles between -25° to +25° for every 5° intervals. For each angle, the overlapping area is either in the form of a triangle (from -25° to -10°) or a trapezium (from -10° to 25°) as seen in Figure 1(c) and 1(d) respectively. A video illustration of this concept can be seen in supplementary material Video S1. The calculated overlapping area is plotted in a graph against the tilt angle ‘α’ as seen in Figure 1(e). The relationship between ‘A’ and ‘α’ is a second order polynomial given in Equation (2).

A=0.367α0.008α2+17
(2)

We can see two linear regimes from -25° to 0° and then 0° to 25°. By replacing the value of ‘A’ with ‘C’, we can ultimately find the relation between the output capacitance of the tilt sensor and the angle of tilt, given in Equation 3.

C=ε(0.367α0.008α2+17)d
(3)

This relationship will help us find the capacitance of a parallel plate structure in terms of the tilt angle for an inclinometer.

The assembly process starts with a customized 3D printed enclosure with a hole in the middle of the bottom plate through which a (rigid) metal wire is passed through [Figure 2(a)]. This wire acts as a frictionless pivot point which carries the bottom movable electrode like a pendulum such that the bottom electrode can swing under the force of acceleration (movement) or gravity (tilt). Subsequently, the lid is placed on the enclosure to secure the moving electrode where the height of the enclosure is 2 mm and the thickness of the electrode is 1 mm so that the moving electrode can move freely in the air gap inside the enclosure. Finally, a copper tape shaped like a triangle is attached on top of the lid to act as the top electrode [Figure 2(b)] and further wrapped towards the back side so that both electrodes can be in the same place for easier integration with the electronic interface [Figure 2(c)]. The shape of this top electrode is made in the shape of a triangle such that it has variable area across the width of the sensory platform [Figure 2(b)].

In order to verify the relationship between tilt angle ‘α’, overlapping area ‘A’ and output capacitance ‘C’, we ran FEM simulations of a model of inclinometer made in COMSOL. The electric field patterns between the top varying shaped plate and the movable bottom plate can be seen in Figure 3(a). At -25° the overlapping area is least thus we have a small electric field, which means smaller capacitance, As we move towards +25° we can see an increase in the electric field between the plates which results in increased capacitance. The value of capacitance at each angle is plotted in Figure 3(b). In the same figure, when the mathematical calculated overlapping area from Figure 1(e) is plotted, we can see that the hypothesis is further validated that overlapping area in our inclinometer is indeed directly proportional to the output capacitance. As the overlapping area changes due to the titling of the structure, the output capacitance changes accordingly. The inclinometer showed a repeatable response between -25° to +25° with a linear response between -25° to +15°. This resolution allows detection of small movements of less than one-degree tilt.

The response of the inclinometer output capacitance is plotted against the tilt angle in Figure 4(a). It can be seen that the experimental response coincides with the response we found in the FEM simulation. The sensitivity of 13 fF/° resulted in a resolution of 0.38°. Since the movable electrode in the inclinometer moves under the influence of gravity to detect movement, we get extremely fast response times of ∼130 ms. A video demonstration of the changing output capacitance when the tag is tilted can be seen in Supplementary material Video S2. To demonstrate this effect the inclinometer in the video was made with a transparent enclosure so that the changing overlapping area can be clearly seen. Figure 4(b) shows the response of the inclinometer from a test in which the inclinometer is tilted in different orientation and speed. The tag is started at rest position and sequentially titled left to right at various levels of inclination angles [‘α’ in Figure 1(c)]. With the current width of the 3D printed enclosure, the sensor can record angles from -25° to 25°. The sensor was then moved slowly from -25° to 25° slowly to show the linear response.

The main advantages of our inclinometer design become evident when the design complexity, power consumption, and the overall cost of fabrication are considered. Table I shows a summary of different designs that have been used to from inclinometers, and their properties are compared with our design.12–15 Most of the alternative designs and the accelerometer-based commercial inclinometers have higher resolution but at the cost of increased design complexity and power consumption. With increased design complexity and power consumption, the overall cost of the device goes high. Such inclinometers do find applications in several areas. However, in certain applications, a high level of accuracy is often not necessarily desired. Instead, lower cost and reduced power consumption are preferred. For example, collision detection devices in cars do not require high accuracy as the acceleration produced in an accident is very large, and the device only needs to know if the acceleration produced by the collision is above a certain threshold. Alternatively, a security tag installed on a painting in a museum needs to detect if sufficient movement has occurred which can correspond to a possible attempt of theft. In these scenarios, accuracies below 0.1° become redundant. In such applications, faster response time is preferred. A further advantage of low power consumption is that a small sized flexible solar cell can wrap around the sensor to provide sufficient power to run the whole system in sunlight.16 Either the designs outlined in Table I incorporate objects submerged in liquids or they rely on the movement of liquids due to thermal convection, which significantly reduces the response time, as liquids are more viscous than air.17 In our design, a pendulum hangs in the air inside the enclosure. Additionally, we do not observe hysteresis for the same reason that air has a significantly lower damping factor than liquids. The free electrode displaces in air under the influence of a constant gravitational force. As soon as the inclinometer is tilted, the pendulum electrode comes to the same position every time with respect to the input inclination angle. Consequently, the output capacitance also remains equal for each inclination angle. In order to verify this hypothesis, we conducted an experiment in which the inclinometer was attached to a continuous rotation servo motor. We can precisely control the speed of rotation of the inclinometer using the continuous rotation servomotor in order to observe the output response of inclinometer when it moves from one end to another and then back to the starting position. Figure 4(c) shows the output plot when the inclinometer is moved from +25° to -25°, and then -25° to +25°. We can see that both of the graphs have similar values at each time instant corresponding to the respective inclination angle. Since we are using metal electrodes, heat causes an increase in the capacitance as the resistance of the top electrode copper foil increases. In order to measure the extent of the effect of heat on the performance of the inclinometer, we heated the device using a 100°C heat source for 30 seconds and then let it cool down for 110 seconds. The change in capacitance due to the applied heat is plotted in Figure 4(d) along with the normal response of inclinometer when it moved from +25° to -25°. We can see that the capacitance of the inclinometer goes from 9.08 pF to 9.14 pF for a temperature change of 25 °C to 100 °C. On the other hand, the net change in capacitance of the inclinometer when moved from +25° to 25° is 0.62 pF. Thus even for extreme temperature changes of 75°C, we observe a small increase in output capacitance (0.06 pF) that is 9% of the maximum possible capacitance change (0.62 pF). The total change per degree centigrade comes down to a mere 0.0008 pF/°C. Thus, for small to moderate changes in temperature, the heat does not significantly alter the tilt angle readings. Furthermore, once left to cool down after heating, the output of inclinometer returns to its original position.

We have shown a freely moving mass based inclinometer which can provide digital information about the tilt angle to as low as 0.38° using a parallel plate capacitive structure. The mathematical result area validated using FEM simulations and experiments. The parallel plate capacitor has two output terminals that can be connected directly to an electronic interface to provide accurate information about the angle of inclination. The simplicity of the design and interface allows it to be used as an add-on to enhance the functionality of existing things. The inclinometer attached to a Bluetooth wireless interface can be attached to valuable items such as a painting or a laptop as an add-on to provide notification if the item is displaced from its resting position. The add-on approach has shown its advantages in the past by adding functionality to plants and marine species.18–21 The advantage of using the capacitive structure is that it does not require an active source of power. A capacitance to digital converter only activates the sensor when it needs to extract data, which can vary depending upon the desired logging interval. Thus, our inclinometer design consumes much less power in comparison to resonant and piezoresistive sensors which require an active activation source in order to function.

Supplementary material is available containing videos.

This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under KAUST-KFUPM Special Initiative Award No. OSR-2016-KKI-2880.

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Supplementary Material