The force-balance closed-loop configuration overcomes most of the open-loop mass-and- spring inclinometer errors, as well as the speed limitation.

Figure 3 shows a force-balance inclinometer configuration. A force **F=mgsin** is again acting on the mass **m**, causing a deflection **x** which results in a pickoff output **V _{p}** going into an amplifier of gain

**k**. Here the similarity with the open loop ends: the output voltage Vo is now fed to an actuator made of a coil moving inside the magnetic field of a permanent magnet. The coil drives the mass

_{v}**m**with a force

**F**(1), where

_{e}=Bli**B**is the magnetic field density,

**i**is the current through the coil and

**l**is the coil wire length.

**FIGURE 4** *Block diagram of the closed-loop inclinometer*

In this case the spring only function is the suspension and guidance of the mass and it is designed to be as weak as possible, while the spring restoring force is replaced by an electromagnetic balancing force.

As the mass starts moving under the gravitational force, the coil applies an opposing force **F _{e}** proportional to

**V**, which tracks the deflection

_{o}**x**(negative feedback); the mass movement will stop as the forces balance each other.

The electromagnetic spring action results from the restoring force

**F**being proportional to the deflection

_{e}**x**as follows: the current through the coil is

**i=V**(2), where

_{o}/R**R**is the sum of the coil resistance and the sense resistor

**R**, and

_{s}**V**(3).

_{o}= k_{p}k_{v}xThen, from (1), (2) and (3):

**F**(4), where the expression inside the brackets is the electromagnetic stiffness coefficient

_{e}=(Bl k_{p}k_{v}/R)x**k**

_{e}= Bl k_{p}k_{v}/RThe advantages of the force-balance principle are evident from the following:

- As long as the stiffness of the suspension
**k**can be neglected, the inertial force**F**is exactly balanced by the reaction force**F**, since the mass will keep moving until equilibrium is reached. Then_{e}**F =F**and from (1)_{e}**F=(Bl)i**, which shows that the current output is exactly proportional to the inertial force, as long as the coefficient**Bl**is a constant. None of the other components of the sensor affect the output. - Since the current
**i**is the measured output, we do not need a sizeable deflection**x**as in the open loop sensors. Consequently, the stiffness can be increased to obtain a fast response without affecting accuracy.

In most cases, the current **i** is indirectly measured by sensing the voltage **V _{s}** across a sense resistor

**R**in series with the coil.

_{s}Figure 4 describes the closed loop system using servo-systems conventional block diagram methods. The transfer functions used apply to steady state only, and they don’t apply to the dynamics of the system.

**FIGURE 4** *Block diagram of the closed-loop inclinometer*

Using conventional servo loop methods, the transfer function of the forward path is **G=k _{p}k_{v}/(kR)**, while that of the feedback path is

**H=Bl**.

The transfer function of the closed loop is

**i/F=G/(1+GH)**. After some algebraic manipulation:

**i/F=(k**, or

_{p}k_{v}/R)/((k_{p}k_{v}Bl/R)+k)**i/F=k’/(k**(4), where

_{e}+k)**k’= k**.

_{p}k_{v}/RFrom (4), the total stiffness

**K=k**of the closed loop spring is the sum of the electromagnetic and mechanical stiffness coefficients.

_{e}+kThe condition for an effective force balance sensor is for the electrical stiffness coefficient to be much higher than the mechanical one,

**k**. However, the effect of the elastic suspension cannot be neglected in one regard: the stability of the inclinometer bias (residual output for zero input).

_{e}>>kThe elastic suspension of the mass suffers from all the stability problems characteristic to elastic elements related to residual stress and stress relief over time, dimensional changes due to temperature, shock and vibration, etc. Since any force on the mass resulting from those changes will be interpreted as an input and result in a change in the coil current to restore balance, an effect of bias instability will appear.

This effect is the most difficult error source to compensate due to its mostly random or unpredictable character. It is most troubling when the inclinometer is used as a leveling device, or in very low-range tilt sensors.