We present a new technique for 3-D position sensing and active magnetic
levitation of a steel ball for use in a levitating ball viscometer. In order
to achieve a stable levitation, a very sensitive positioning measurement
system is mandatory. For this task the differential transformer principle was
chosen to realize a 3-D position measurement. This leads to a purely magnetic
sensor and actuator system without the need for other transducer types such
as optical readout. The actuation utilizes power efficient switch-mode
electronic circuitry which opens the possibility of upscaling the device, if
demanded, for future applications. It is shown that this switch-mode
actuation can be combined directly with the position measurement when special
switching patterns are applied. A position resolution of

Precise position measurement is a key part of magnetic levitation where
optical position sensing is a common method for demonstrator setups
(

A magnetic levitation system using magnetic position measurement can overcome
these drawbacks as it does not impose any additional constraints onto the
physical properties on the environment of the levitated object as long as
this environment is not magnetizable. In this paper, we illustrate this
approach using the example of a viscometer based on a levitated ball

There are some approaches to building a “self-sensing” active magnetic
bearing based on the measurement of the actuator's self-inductance
(

Using a differential transformer is a more sophisticated approach. The
possibility of using a linear variable differential transformer (LVDT) in
combination with magnetic actuation has been demonstrated in

The remaining content of the paper is structured as follows. In Sect.

Photograph of the setup.

Computer-aided design (CAD) model showing the ball, permanent magnet, actuator coils and sensor coils.

Figure

CAD model showing all functional parts, including the housing.

For this setup, a

Figure

Block diagram of the measurement electronics.

Figure

Figure

The setup is related to our previous work

Quadratic dependency of the magnetic force on the flux density.

In order to create a force on a ferromagnetic ball, a non-uniform magnetic
field, such as that created by a magnetic dipole, is necessary. The resulting
force on the magnetized ball then is oriented in the direction of the
gradient of

The position of the permanent magnet is adjusted such that the force (without
using the solenoids) would be just enough to hold the ball. The upper
actuator coil is then used to stabilize the ball at this equilibrium
position, which otherwise would not be stable according to the Earnshaw
theorem

Each of the six actuator coils is controlled using a bipolar PWM signal generated by H bridge MOSFET switches (Allegro A4950). The PWM duty cycle controls the mean voltage and therefore the mean actuator current. The frequency of the PWM signal is chosen to be sufficiently high such that the movement of the ball is not disturbed by high-frequency vibrations; see also below.

Position measurement principle.

The magnetic field components that arise from PWM control of the actuator
coils are used directly for the excitation of the differential transformer
(see Fig.

The position can be measured for each coordinate separately. For example, to
measure the ball position in the

The actuator coils can be modeled by a serial connection of the coils'
self-inductance

Actuator current and sensor coil voltage due to PWM control.

In order to overcome this issue, both slopes are used for measurement.
Subsequently the difference of both signals

The current in the two actuator coils on the

The position in all three coordinates is measured consecutively using the
method described above. The summary is as follows.

The two actuator coils behind the sensor coils are switched to open circuit (via freewheeling diodes).

After the current in these coils has decayed completely, the measurement can start.

Two subsequent PWM slopes (rising and falling) are used for measurement
of

After measurement, the current in the switched off actuator coils is built up again.

After a pause the whole process is repeated for the next coordinate.

PWM switching pattern for position measurement, showing voltage and current in the six actuator coils.

The sample rate that can be achieved with this method depends on the PWM
frequency and the properties of the actuator coils, as the current in the
actuator coils has to decay completely before measurement can begin. For the
demonstrator setup, the sample rate is

The equilibrium position of the ball is unstable in the vertical direction
(the magnetic force increases when the ball approaches the permanent magnet)
and has to be stabilized by an active controller. For this task a PD
controller that acts on the upper actuator coil is used. The lower actuator
coil is not used for actuation in this setup, but it is necessary for the
position measurement. Note that the mechanical system is much slower than the
PWM switching frequency of

Settling of the vertical controller; top: duty
cycle for the upper actuator coil (0.5

Depending on the targeted vertical position

This means the I controller “searches” for the vertical equilibrium position of the ball. Note that this position also depends on the buoyancy force when the ball is embedded in a fluid. This effect can therefore be utilized to measure the fluid's density.

As soon as the vertical position is stabilized, the ball levitates freely in the chamber. The horizontal position is stable in this setup and does not have to be controlled actively (although this setup would facilitate this). The magnetic flux density has its maximum directly on the axis of the permanent magnet and decreases radially. Therefore the field gradient points in a negative radial direction and thus a restoring force results, which can be interpreted as a nonlinear spring that pulls the ball towards the center.

Once the ball is in stable levitation, there are various methods to set it into motion and obtain information about the fluid surrounding the ball. The method described in the following will serve as an illustrative example.

Block diagram of the vertical position
controller.

Measured orbit of the ball in water at different
excitation frequencies,

To excite the ball to an orbital movement, sine and cosine shaped signals
with the frequency

This system can be described as a 2-D spring-mass system which is damped by the viscous drag. Note that the spring force as well as the viscous drag are nonlinear in general, which leads to a nonlinear dynamic system. Also, the excitation force component is not perfectly sinusoidally varying with time due to the geometry of the actuator coils.

In the following section we assume a circularly shaped orbit and neglect all
nonlinear effects. The horizontal

The function's magnitude represents the orbit's radius with respect to the
excitation amplitude

To measure this transfer function we sequentially excite the ball at various
frequencies

The measurement sequences

Table

List of all liquids used for measurements with
shear viscosity and density at 25

The measurement results are shown in Figs.

Bode plot of the transfer function for the liquids
listed in Table

Corresponding Nyquist plot to Fig.

By the application of a fitting algorithm like

Estimated dissipation factor versus shear
viscosity for the liquids listed in Table

It was demonstrated that magnetic actuation using switch mode circuitry can be combined with 3-D position measurement leading to a stable levitation of a ferromagnetic ball. The ball can then be excited to perform an orbital movement.

The experimentally realized orbits do not have a perfect circular shape.
Instead there is a large variety of shapes, depending on the viscosity,
excitation frequency and amplitude (see Fig.

The ball's orbit in water;

To investigate this phenomenon we calculated the discrete Fourier transform
(DFT) of one period from a typical orbit in Fig.

DFT of one period from an orbit in water,

A spurious oscillation at the second harmonic appears to be present, which is
supposedly excited due to nonlinearities of the excitation forces. The
background spectrum is evenly distributed, which indicates white noise. We
can filter the orbit if we clear out all spectral components with a harmonic
order greater than 2 (see Fig.

Filtered orbit in water,

Looking at the orbit figures, one can see that the noise in the

Spurious effects are supposedly associated with imperfections of the setup
but mainly with the nonlinear character of the excitation mechanism. The
latter could be optimized by varying the shape and the number of actuator
coils or an appropriate predistortion of the driving signals, which, however,
would depend on the current position of the ball. Although some orbits appear
heavily distorted (e.g., Fig.

It turned out that for this setup, viscosities in the range of 1 up to
10 mPa s correspond to quality factors (see Eq.

The presented principle of a magnetically levitated ball viscometer opens the possibility of enclosing the ball in a hermetically sealed measurement chamber which is immune to leakage or contamination even under high pressure or extreme temperatures. The interaction of a moving ball with the fluid closely resembles that of the established falling ball viscometer such that the obtained viscosity values are potentially comparable to those obtained by this classical method. However, the magnetic actuation brings some drawbacks such as the power dissipation of the actuator coils which heats up the fluid under test. Therefore a precise temperature control of the fluid is mandatory for viscosity measurement. Also, the magnetic position measurement is sensitive to fluctuating external magnetic fields and ferromagnetic objects in the vicinity of the system.

The data sets are available at:

This work has been partially supported by the Austrian COMET-K2 program of the Linz Center of Mechatronics (LCM), and was funded by the Austrian federal government and the federal state of Upper Austria. The described method is protected by pending patent A50085/2014.

Financial support was provided by the Austrian research funding association (FFG) within the scope of the COMET program within research project “Industrial Methods for Process Analytical Chemistry – From Measurement Technologies to Information Systems (imPACts)” (contract no. 843546).Edited by: Qingquan Sun Reviewed by: two anonymous referees