Voltage sensor

A voltage sensor includes a vibrator configured to be supported by a mechanical supporting portion and to be given a floating potential, a drive electrode configured to be disposed adjacent to the vibrator and to resonate the vibrator with applied AC voltage, a driver configured to apply an AC voltage that crosses 0 V to the drive electrode, a fixed electrode configured to be disposed adjacent to the vibrator with a gap formed between the fixed electrode and the vibrator, and a calculator configured to detect a magnitude of a measurement target voltage based on a change of a resonant frequency of the vibrator when the measurement target voltage is applied to the fixed electrode.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is based on Japanese Patent Application (No. 2016-199722) filed on Oct. 11, 2016, the contents of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a voltage sensor.

2. Description of the Related Art

A voltage sensor is known which is equipped with a Pockels element, a quarter-wave plate, a polarizer, an analyzer, etc. (refer to JP-A-3-146875, for example). In this voltage sensor, an optical signal that is output from a light source is polarized by the polarizer and enters the Pockels element, where the optical signal is modulated optically according to the magnitude of a voltage. The modulated optical signal is transmitted to the analyzer via the quarter-wave plate. The optical signal that is output from the analyzer is received and detected by a prescribed optical receiver. The voltage applied to the Pockels element can be detected in this manner.

However, the voltage sensor disclosed in JP-A-3-146875 has a problem that it requires a large number of components, that is, the Pockels element, the quarter-wave plate, the polarizer, the analyzer, and other components. Furthermore, it requires complicated assembling work because optical axis alignment etc. need to be carried out.

In view of the above, another type of voltage sensor has been proposed which is equipped with a vibrator that is supported a mechanical suspension and a fixed electrode that is opposed to the vibrator with a certain gap formed between them (refer to JP-A-2013-228367). When a voltage to be measured is applied to the fixed electrode, an electrostatic attractive force acts on the vibrator to change its resonant frequency. The measurement target voltage can thus be calculated.

In this voltage sensor, when a voltage to be measured is applied to the fixed electrode, a resulting electrostatic attractive force changes the spring constant of the suspension substantially, as a result of which the resonant frequency of the vibrator is changed. Since this change has a certain correlation with the magnitude of the measurement target voltage, the measurement target voltage can be measured on the basis of the resonant frequency thus changed. Not requiring the above-mentioned optical components, the voltage sensor disclosed in JP-A-2013-228367 is small in the number of components. Furthermore, since optical axis alignment etc. are not necessary, assembling work is not very complex.

In the voltage sensor disclosed in JP-A-2013-228367, an AC voltage that is applied to a drive electrode to vibrate the vibrator is given a positive DC bias voltage. This measure is employed taking into consideration the vibration efficiency in a case that the vibrator is regarded as a vibration actuator, because the vibrator is given a floating potential.

However, in the voltage sensor disclosed in JP-A-2013-228367, since the AC voltage that is applied to the drive electrode is given the positive DC bias voltage so that the total voltage is kept positive even when the AC voltage takes a bottom value, the vibrator is charged up being placed in an electric field that has a single polarity in one direction all the time, whereby the effective electric field applied to the vibrator is varied. This affects its resonant frequency and thereby increases a fluctuation or a drift of measurement results.

This problem is not limited to the case of an AC voltage that is given a positive DC bias voltage so that the total voltage is kept positive even when the AC voltage takes a bottom value, but arises also in the case of an AC voltage that is given a negative DC bias voltage so that the total voltage is kept negative even when the AC voltage takes a top value.

SUMMARY OF THE INVENTION

The present invention has been made to solve the above problem of the related art, and an object of the present invention is therefore to provide a voltage sensor capable of reducing the fluctuation or the drift of measurement results while preventing increase of the number of components and complication of assembling work.

The invention provides a voltage sensor comprising:

a vibrator configured to be supported by a mechanical supporting portion and to be given a floating potential;

a drive electrode configured to be disposed adjacent to the vibrator and to resonate the vibrator with applied AC voltage;

a driver configured to apply an AC voltage that crosses 0 V to the drive electrode; and

a fixed electrode configured to be disposed adjacent to the vibrator with a gap formed between the fixed electrode and the vibrator;

a calculator configured to detect a magnitude of a measurement target voltage based on a change of a resonant frequency of the vibrator when the measurement target voltage is applied to the fixed electrode.

According to the voltage sensor, since the AC voltage that crosses 0 V is applied to the drive electrode, the vibrator is not placed in an electric field that has a single polarity in one direction all the time. Thus, the phenomenon that the effective field applied to the vibrator is varied due to charging-up of the vibrator can be suppressed, and hence its influence on the resonant frequency can be prevented. Furthermore, since a Pockels element, a quarter-wave plate, a polarizer, an analyzer, and other components are not necessary and optical axis alignment etc. are not required as components of the voltage sensor, increase of the number of components and complication of assembling work can be prevented. In conclusion, it becomes possible to reduce a fluctuation or a drift of measurement results while preventing increase of the number of components and complication of assembling work.

In the above voltage sensor, for example, the driver applies, to the drive electrode, an AC voltage that is kept at 0 V every time the AC voltage reaches 0 V for a prescribed period starting from a time when the AC voltage reaches 0 V.

In this voltage sensor, since the AC voltage is kept at 0 V every time it reaches 0 V for a prescribed period starting from a time when it reaches 0 V, the degree of charging-up of the vibrator is further lowered and hence a fluctuation or a drift of measurement results can be reduced further.

In the above voltage sensor, for example, the driver applies, to the drive electrode, an AC voltage whose time average over one cycle is approximately equal to 0 V.

In this voltage sensor, since the AC voltage whose time average is approximately equal to 0 V is applied to the drive electrode, a variation of the effective electric field applied to the vibrator due to charging-up of the vibrator and hence its influence on the resonant frequency can be suppressed further. As a result, a fluctuation or a drift of measurement results can be reduced further.

The invention can provide a voltage sensor capable of reducing the fluctuation or the drift of measurement results while preventing increase of the number of components and complication of assembling work.

DETAILED DESCRIPTION OF THE EXEMPLARY EMBODIMENTS

Although a preferred embodiment of the present invention will be described below with reference to the drawings, the invention is not limited to the following embodiment.

FIG. 1shows a basic configuration and the principle of operation of a voltage sensor1according to the embodiment of the invention. As shown inFIG. 1, the voltage sensor1according to the embodiment includes a mechanical suspension (supporting portion)10, a vibrator20, a fixed electrode30, and a calculation portion40.

The suspension10supports the vibrator20. The spring coefficient of the suspension10is represented by k. The vibrator20is a flat-plate electrode supported by the suspension10and can be vibrated because of the elasticity force of the suspension10. The mass of the vibrator20is represented by m.

The fixed electrode30is a flat-plate electrode that is opposed to the vibrator20with a prescribed gap formed between them, and the fixed electrode30and the vibrator20constitute parallel-plate electrodes. The area of the opposed surfaces of the vibrator20and the fixed electrode30is represented by S, and their initial gap is represented by g.

When an AC voltage is applied to a drive electrode50(hereinafter described) of the voltage sensor1, the vibrator20, being suspended by the suspension10, is vibrated in the direction in which the distance between the vibrator20and the fixed electrode30is increased and decreased (the left-right direction inFIG. 1). The vibrator20vibrates with the amplified displacement at a resonant frequency frthat is given by the following Equation (1):

A measurement target voltage Vmis further applied to the fixed electrode30. In response, an electrostatic attractive force acts on the vibrator20from the fixed electrode30, whereby the position of the vibrator20is changed by x. The electrostatic attractive force can be expressed as an equivalent spring constant kethat is given by the following Equation (2):

[Formula⁢⁢2]⁢ke=ɛ0⁢SV⁢2(g-x)3(2)
where ε0is the permittivity of the gap g and Vmis the measurement target voltage.

As a result, the vibrator20vibrates at a resonant frequency fr′ that is given by the following Equation (3):

Since the equivalent spring constant kewhich is given by Equation (2) varies depending on the magnitude of the voltage Vmwhich is applied to the fixed electrode30, the resonant frequency fr′ in Equation (3) reflects the magnitude of the voltage Vm.

Thus, the calculation portion40can calculate the magnitude of the measurement target voltage Vmfrom the resonant frequency fr′ of the vibrator20.

FIG. 2is a perspective view of the voltage sensor1according to the embodiment. As shown inFIG. 2, the voltage sensor1is a micro voltage sensor that is manufactured using a MEMS (micro-electro-mechanical systems) processing technique.

In the voltage sensor1shown inFIG. 2, to increase the resonant frequency of the vibrator20by decreasing its mass, the vibrator20has a long and narrow shape. The suspensions10are attached to the two respective ends of the long and narrow vibrator20and thus support the vibrator20from the sides of its two ends. Each suspension10is folded back so as to assume a U shape. Supported by these two suspensions10, the vibrator20is given a floating potential.

As shown inFIG. 2, the voltage sensor1includes not only the components shown inFIG. 1but also the drive electrode50, a first electrode61and a second electrode62which are connected to the ends, opposite to the ends connected to the vibrator20, of the suspensions10, respectively, and stoppers70. The calculation portion40shown inFIG. 1is connected to at least one of the first electrode61and the second electrode62.

The drive electrode50is disposed adjacent to the vibrator20. When an AC voltage is applied to the drive electrode50, the drive electrode50excites and vibrates the vibrator20to cause it to resonate.FIG. 3is an enlarged view of part A ofFIG. 2. As shown inFIG. 3, the drive electrode50has comb tooth electrodes51which extend toward the vibrator20. Likewise, the vibrator20shown inFIG. 2has comb tooth electrodes21which extend toward the drive electrode50. The comb tooth electrodes21and51are arranged alternately so as not to be in contact with each other.

The voltage sensor1shown inFIG. 2can be manufactured by processing an SOI (Silicon-On-Insulator) wafer, for example. More specifically, single-mask patterning and Deep RIE (Reactive Ion Etching) are performed. The vibrator20which is a movable component is released by sacrificial layer etching using vapor HF.

To be able to vibrate, the vibrator20is floated being supported by the suspensions10in the manner described above. In an actual product in which dimensions x, y, and z of a device layer are 1,125 μm, 1,585 μm, and 25 μm, respectively, the vibrator20sinks with respect to the other electrodes30and50by a maximum of 66 nm. However, since the sinking distance is as short as 1/30 of the thickness 2 μm of a sacrificial layer, the vibrator20does not come into contact with a handle layer. Although because of the wayFIG. 2is drown the comb tooth electrodes51and the fixed electrode30may look as if they were also floated, this is because of progress of sacrificial layer etching in partial regions. In actuality, they are fixed to the handle layer.

FIG. 4is an enlarged top view of part of the voltage sensor1shown inFIG. 2. As seen fromFIGS. 2 and 4, the four stoppers70are formed beside the vibrator20, that is, on the side of the fixed electrode30(only two stoppers70are shown inFIG. 4). The stoppers70are floating electrodes; they are not connected electrically to any member. Each stopper70has a main body71, a spring portion72, and a contact portion73.

The main body71is approximately shaped like a square in a top view, and the spring portion72which is long and narrow extends from one apex of the approximately square main body71. A tip portion of the spring portion72projects slightly to the vibrator20side beyond the edge of the fixed electrode30. The spring portion72is formed in such a manner as to project toward the vibrator20side from the apex of the approximately square main body71, then be bent by 90° to extend alongside the edge of the main body71, and be bent again by 90° toward the vibrator20side. That is, the spring portion72is bent by 90° two times and thereby given elasticity.

Because of the presence of the stoppers70having the above structure, even if the vibrator20is pulled excessively toward the fixed electrode30side by an electrostatic attractive force, the vibrator20comes into contact with the contact portions73and hence is prevented from being short-circuited with the fixed electrode30. Even if the vibrator20comes into contact with the stoppers70, resulting impact is reduced by the elasticity of the spring portions72, whereby the degrees of bending of the contact portions73are reduced or they are prevented completely from being bent.

Next, a description will be made of how the voltage sensor1according to the embodiment operates. First, in the voltage sensor1, an AC voltage is applied to the drive electrode50, whereby an electrostatic attractive force is generated. With the AC voltage supplied to the drive electrode50, the vibrator20vibrates at the prescribed resonant frequency fr.

In this state, a measurement target voltage Vm is applied to the fixed electrode30. As a result, an electrostatic attractive force that is given by Equation (2) is generated and the vibrator20comes to vibrate at a resonant frequency fr′ that is given by Equation (3).

The calculation portion40calculates the magnitude of the measurement target voltage Vm from the resonant frequency fr′. To measure the resonant frequency fr′, the calculation portion40needs to measure a displacement of the vibrator20. More specifically, in the voltage sensor1, laser light is shone on the vibrator20and a displacement of the vibrator20is determined from a deflection width of reflection light (optical measurement).

Alternatively, in the voltage sensor1, a displacement of the vibrator20may be measured from a capacitance variation that is caused by a variation of the electrode gap g (electrical measurement). A displacement of the vibrator20may be measured from a capacitance variation by either using the fixed electrode30as it is or disposing another parallel plate electrode dedicated to the displacement measurement.

In the voltage sensor1according to the embodiment, a drive circuit (driver)80is connected to the drive electrode50. The drive circuit80applies, to the drive electrode50, an AC voltage for causing the vibrator20to resonate. In particular, in the embodiment, the drive circuit80applies, to the drive electrode50, an AC voltage that crosses 0 V, more preferably, an AC voltage whose time average (over one cycle) is approximately equal to 0 V.

FIGS. 5A-5Dshow first to fourth examples, respectively, of the AC voltage that the drive circuit80applies to the drive electrode50. As shown inFIG. 5A, the first example AC voltage crosses 0 V and is given a prescribed positive (or negative) DC bias voltage. Thus, the time average of this example AC voltage is not approximately equal to 0 V.

As shown inFIG. 5B, the second example AC voltage is such as to be kept at 0 V every time it reaches 0 V for a prescribed period starting from the time when it reaches 0 V. That is, this example AC voltage has 0-V halt periods.

As shown inFIG. 5C, the third example AC voltage has positive portions and negative portions that are the same in amplitude, instead of being given a positive or negative DC bias voltage.

As shown inFIG. 5D, the fourth example AC voltage has not only positive portions and negative portions that are the same in amplitude but also 0-V halt periods. That is, this example AC voltage is a combination of the second example (FIG. 5B) and the third example (FIG. 5C).

By applying such an AC voltage to the drive electrode50, the variation of the effective electric field applied to the vibrator20due to charging-up of the vibrator20and its resulting influence on the resonant frequency can be suppressed.

FIG. 6is a graph showing how a signal varies with respect to the frequency in the voltage sensor1shown inFIG. 2. For example, an AC voltage as shown inFIG. 5Cwas applied to the drive electrode50and a resulting signal generated from the vibrator20was detected. As the frequency was varied, as shown inFIG. 6a steep signal variation occurred at 22.73 kHz (resonant frequency).

As described later with reference toFIG. 8, the frequency (resonant frequency) at which a signal variation as mentioned above occurred decreased monotonously as the voltage Vm applied to the fixed electrode30was increased. This indicates that the voltage sensor1is sufficiently usable as a voltage sensor.

FIG. 7is a graph showing a correlation between the voltage Vm applied to the fixed electrode30and the resonant frequency in a case that an AC voltage (0 to 60 V) that was given a positive DC bias was applied to the drive electrode50. This AC voltage is called an “AC voltage with an offset voltage 30 V.”

Likewise,FIG. 8is a graph showing a correlation between the voltage Vm applied to the fixed electrode30and the resonant frequency in a case that an AC voltage (−30 to 30 V) having positive portions and negative portions that were the same in amplitude (i.e., the time average was 0 V) was applied to the drive electrode50. This AC voltage is called an “AC voltage with an offset voltage 0 V.”

In the graphs shown inFIGS. 7 and 8, a resonant frequency was measured 10 times for each of voltages Vm applied to the fixed electrode30which were 0 V, 20 V, 40 V, 50 V, 60 V, and 80 V. Resonance frequencies were measured by the above-described electrical measuring method.

As seen fromFIG. 7, in the case where the AC voltage with an offset voltage 30 V was applied to the drive electrode50, a measurement result variation of about 0.005 to 0.013 kHz occurred in each of the cases that the voltage Vm applied to the fixed electrode30was 0 V, 20 V, 40 V, 50 V, 60 V, and 80 V.

In contrast, as seen fromFIG. 8, in the case where the AC voltage with an offset voltage 0 V was applied to the drive electrode50, a measurement result variation of about 0.003 kHz occurred when the voltage Vm applied to the fixed electrode30was 40 V. However, almost no measurement result variation was found in each of the other cases that the voltage Vm was 0 V, 20 V, 50 V, 60 V, and 80 V.

FIG. 9is a graph showing a correlation between the voltage Vm applied to the fixed electrode30and the standard deviation of the resonant frequency in both of the case that the AC voltage with an offset voltage 30 V was applied to the drive electrode50and the case that the AC voltage with an offset voltage 0 V was applied to the drive electrode50.

As seen fromFIG. 9, in the case where the AC voltage with an offset voltage 30 V was applied to the drive electrode50, the standard deviation was larger than or equal to 0.0015 kHz in each of the cases that the voltage Vm applied to the fixed electrode30was 0 V, 20 V, 40V, 50 V, 60 V, and 80 V. In particular, when the voltage Vm was 80 V, the standard deviation was equal to a little lower than 0.0035 kHz. The average of the standard deviations that were obtained in the case where the AC voltage with a positive DC bias voltage was applied to the drive electrode50was equal to 0.00233 kHz.

On the other hand, in the case where the AC voltage with an offset voltage 0 V was applied to the drive electrode50, the standard deviation was a little lower than 0.0010 kHz when the voltage Vm applied to the fixed electrode30was 40 V. However, the standard deviation was lower than 0.0002 kHz in each of the other cases that the voltage Vm was 0 V, 20 V, 50 V, 60 V, and 80 V. The average of the standard deviations that were obtained in the case where the AC voltage having positive and negative portions and a time average 0 V was applied to the drive electrode50was equal to 0.00019 kHz.

As described above, the average of the standard deviations that were obtained in the case where the AC voltage with an offset voltage 0 V was applied to the drive electrode50was smaller than 1/10 of the average of the standard deviations that were obtained in the case where the AC voltage with an offset voltage 30 V was applied to the drive electrode50; that is, the measurement result variations were much smaller in the former case.

It is expected that also in the cases of AC voltages (having amplitude: 30 V) with an offset voltage 10 V or 20 V the standard deviation and hence the measurement result variation would be smaller than in the case of the AC voltage with an offset voltage 30 V.

FIG. 10is a table showing standard deviations in the cases of the AC voltages (having amplitude: 30 V) with an offset voltage 10 V or 20 V as well as the standard deviations in the cases of the AC voltages (having amplitude: 30 V) with an offset voltage 0 V or 30 V.FIGS. 11A-11Dillustrate a method for calculating standard deviations in the cases of the offset voltages 10 V and 20 V, and show waveforms of the AC voltages (having amplitude: 30 V) with the offset voltages 30 V, 20V, 10 V, and 0 V, respectively.

First, as shown inFIG. 11A, an area obtained by subtracting the area of a portion where the voltage is lower than 0 V (there is no such portion in this waveform) from the area of a portion (indicated by symbol “+” inFIG. 11A) where the voltage is higher than 0 V is assumed to be “1.” In this case, as shown inFIG. 10, the standard deviation of the resonant frequency is equal to 0.00233 kHz.

As shown inFIG. 11D, an area obtained by subtracting the area of a portion (indicated by symbol “−” inFIG. 11D) where the voltage is lower than 0 V from the area of a portion (indicated by symbol “+” inFIG. 11D) where the voltage is higher than 0 V is equal to “0.” In this case, as shown inFIG. 10, the standard deviation of the resonant frequency is equal to 0.00019 kHz.

Standard deviations in the cases of the offset voltages 10 V and 20 V are calculated as described below on the basis of the above data according to area ratios. In the case of the offset voltage 20 V, an area obtained by subtracting the area of a portion (indicated by symbol “−” inFIG. 11B) where the voltage is lower than 0 V from the area of a portion (indicated by symbol “+” inFIG. 11B) where the voltage is higher than 0 V is equal to “0.66” because the area subtraction result obtained in the case of the offset voltage 30 V (FIG. 11A) is assumed to be “1.” Thus, it can be said that the standard deviation of the resonant frequency of this case is 0.00162 kHz (this value is written inFIG. 10) according to this area ratio.

In the case of the offset voltage 10 V, an area obtained by subtracting the area of a portion (indicated by symbol “−” inFIG. 11C) where the voltage is lower than 0 V from the area of a portion (indicated by symbol “+” inFIG. 11C) where the voltage is higher than 0 V is equal to “0.33” because the area subtraction result obtained in the case of the offset voltage 30 V (FIG. 11A) is assumed to be “1.” Thus, it can be said that the standard deviation of the resonant frequency of this case is 0.00090 kHz (this value is written inFIG. 10) according to this area ratio.

As described above, it can be said that also in the cases ofFIGS. 11B and 11Cthe resonant frequency variation can be made smaller than in the case ofFIG. 11A. That is, whereas the case is advantageous that an AC voltage whose time average is equal to 0 V is applied to the drive electrode50, it can be said that certain levels of advantages can be obtained even in the case where an AC voltage that crosses 0 V is applied to the drive electrode50.

FIG. 12is a graph showing a correlation between the voltage Vm applied to the fixed electrode30and the resonant frequency in a case that an AC voltage having an offset voltage 0 V and a certain 0-V halt period was applied to the drive electrode50.

As shown inFIG. 12, when an AC voltage that had a certain 0-V halt period and whose time average was 0 V was applied to the drive electrode50, no measurement result variations were found in each of cases that the voltage Vm applied to the fixed electrode30was 0 V, 20 V, 40 V, 50 V, 60 V, and 80 V.

FIG. 13is a graph showing a correlation between the voltage Vm applied to the fixed electrode30and the standard deviation of the resonant frequency in both of the case that the AC voltage with an offset voltage 0 V (no halt period) was applied to the drive electrode50and the case that the AC voltage having an offset voltage 0 V and a certain 0-V halt period was applied to the drive electrode50.

As described above, in the case of the AC voltage having no halt period, the standard deviation of the resonant frequency was a little lower than 0.0010 kHz when the voltage Vm applied to the fixed electrode30was 40 V and was lower than 0.0002 kHz (average: 0.00019 kHz) in each of the other cases that the voltage Vm was 0 V, 20 V, 50 V, 60 V, and 80 V.

On the other hand, in the case of the AC voltage having a halt period, the standard deviation of the resonant frequency was lower than about 0.0001 kHz (average: 0.00007 kHz) in all of the cases that the voltage Vm was 0 V, 20 V, 40 V, 50 V, 60 V, and 80 V.

As described above, the average of the standard deviations of the resonant frequency in the case of the AC voltage having a halt period was smaller than that in the case of the AC voltage having no halt period; that is, the measurement result variations were much smaller in the former case.

As described above, in the voltage sensor1according to the embodiment, since an AC voltage whose time average is approximately equal to 0 V is applied to the drive electrode50, the vibrator20is not placed in an electric field that has a single polarity in one direction all the time. Thus, the phenomenon that the effective electric field applied to the vibrator20is varied due to charging-up of the vibrator20can be suppressed, and hence its influence on the resonant frequency can be prevented. As a result, a fluctuation or a drift of measurement results can be reduced. Furthermore, since a Pockels element, a quarter-wave plate, a polarizer, an analyzer, and other components are not necessary and optical axis alignment etc. need not be carried out, increase of the number of components and complication of assembling work can be prevented. In conclusion, it becomes possible to reduce a fluctuation or a drift of measurement results while preventing increase of the number of components and complication of assembling work.

The AC voltage may be such as to be kept at 0 V every time it reaches 0 V for a prescribed period starting from the time when it reaches 0 V. This further lowers the degree of charging-up of the vibrator20and can thereby reduce a fluctuation or a drift of measurement results further.

Although the invention has been described above in the form of the embodiment, the invention is not limited to the embodiment and various modifications are possible without departing from the spirit and scope of the invention.

For example, although in the embodiment the vibrator20has a long and narrow shape, the invention is not limited to that case; the vibrator20may assume any of other shapes such as a ring shape. That is, the vibrator20may have any shape as long as it enables voltage measurement according to the principle of operation shown inFIG. 1. For example, a ring-shaped vibrator may be employed that is vibrated in a wine glass mode.

Although in the embodiment the vibrator20and the drive electrode50have the comb tooth electrodes21and the comb tooth electrodes51, respectively, the invention is not limited to that case. The vibrator20and the drive electrode50need not have the comb tooth electrodes21and the comb tooth electrodes51if the vibrator20can be vibrated by generating a sufficiently strong electrostatic attractive force.