| { Richard D. Best, Digital Eqpt. Corp., 85 Swanson Rd., Boxboro, Ma., 01716 }
Design outline for a single coil magnetic position controller
Abstract
The magnetic coil position controller (MCPC) is a device for
levitating and (for objects with a natural axial orientation) arbitrarily
orienting small ferromagnetic objects or larger objects with attached
ferromagnetic 'handles'.
Applications might include laser machining of small metal components
, contact free suspension of experimental samples or measurement devices
in fluid media (e.g. black body measurements) and frictionless support of
rotors for power generation machinery (admittedly speculative).
An analysis of a single coil magnetic positioner is made showing that
it is feasible to simultaneously measure and control the height of
the small ferromagnetic object by appropriate choice of a driving
current signal.
A proposed breakdown into subsystems to implement the controller is
also made, along with some discussion as to appropriate specific
circuit implementations.
Overview
The coil is used as both the actuator and the position sensing element.
The coil and bearing together constitute a coupled electromechanical
system. Two models will be required to explain the operation of the
controller: an electrical model that expresses circuit variables in
terms of mechanical state, and a mechanical model that expresses
mechanical forces in terms of electrical state.
The electrical model will treat the coil & bearing combination as
a variable inductor with the inductance being solely a function of
the height of the bearing off the earth's surface (y). With this reference
frame, it's expected that the inductance, L(y), increases with y.
We deliberately choose not to consider L to be explicitly a function of the
distance of the bearing from the radial axis of the coil. This is not
strictly correct, but it seems plausible that the bearing will be subject to
a radial restoring force for any positive value of current. Variations
in inductance due to increases in r will manifest as changes in y, but for
a first pass, we'll neglect them.
The functional dependency of L on y will be determined
from measurements in which i is a small signal constant amplitude
current source and E is measured for varying y. L can then be
inferred from E=d( L*i )/dt = L(y) * di/dt with driven i and statically
measured y and E.
The mechanical model considers Fy (the net vertical force on the
bearing) to be solely a function of i (the current flowing through the coil),
y, and m (the mass of the bearing). Since m is a constant, the
relation may be written Fy = F(i,y). Y(t) is a given driving function.
Electrical Model
It will now be shown that it is possible to simultaneously control and
measure y by an appropriate choice of the current driving function i(t).
Let the coil be driven with the sum of two signals:
[1] i(t) = Q(t) + I0*cos(w*t)
where Q(t) is a baseband large signal current (A),
I0 is a constant small signal current (A), and
w is a fixed high frequency (Hz).
[2] E(t) = d( L(y)*i(t) )/dt
= i*dL/dt + L*di/dt
We wish to process E(t) (call the result Ep(t)) in such a way as to make
( terms in Ep(t) derived from the L*di/dt term in E(t) ) >>
( terms in Ep(t) resulting from the i*dL/dt term in E(t) ).
If this can be done, then Ep(t) will be a measure of y implicitly through
L(y). Please remember throughout that L(y) is implicitly a function of t
through the driving function y(t).
Substituting [1] into [2] yields:
[3] E(t) = L*dQ/dt - L*I0*w*sin(w*t) + Q*dL/dt + I0*dL/dt*cos(w*t)
^ ^ ^ ^
| | | term_4
| | term_3
| term_2
term_1
Let L(f) = Fourier_transform( L(t), t, f ) and
Q(f) = Fourier_transform( Q(t), t, f ) and
w = 2*PI*f.
We now require both L(f) and Q(f) to be baseband limited.
Let B be some number > max( bandwidth( L(f) ), bandwidth( Q(f) ) ).
Let's make B encompass positive frequencies only.
By convolution, and remembering that differentiation scales the
amplitude spectrum, but does not broaden it (i.e. shift the zeroes)
we have:
[4] bandwidth( L*dQ/dt ) = bandwidth( convolve{ L(f), j*2*PI*f*Q(f) } < 2*B
[5] bandwidth( Q*dL/dt ) = bandwidth( convolve{ Q(f), j*2*PI*f*L(f) } < 2*B
Note that L*dQ/dt and Q*dL/dt are also baseband.
Therefore, if E(t) is ideal bandpass filtered about w with passband width = 2*B,
then the resulting signal, Ef(t) will be deficient in term_1 and term_3:
[6] Ef(t) = - L*I0*w*sin( w*t ) + I0*dL/dt*cos( w*t )
Modulation of Ef(t) by sin( w*t ) yields Em(t):
[7] Em(t) = - L*I0*w*sin^2( w*t ) + I0*dL/dt*cos( w*t )*sin( w*t )
= - L*I0*w*[ (1/2)*(1 - cos( 2*w*t )) ]
+ I0*dL/dt*[ (1/2)*sin( 2*w*t ) ]
( In [7], the double angle formulae have been used to express products
of the form cos( a )*sin( a ) as terms in cos( 2*a ) and sin( 2*a ) ).
Averaging Em(t) over a finite interval will yield a measure of L
(the desired Ep(t)):
[8] Ep(t) = (1/T)*integral( t-T, t, Em(t), t )
[9] = - (1/2*T)*I0*w*integral( t-T, t, L(t), t )
[10] + (1/2*T)*I0*w*integral( t-T, t, L(t)*cos( 2*w*t ), t )
[11] + (1/2*T)*I0*integral( t-T, t, dL(t)/dt*sin( 2*w*t ), t )
Since we earlier assumed that L(f) is baseband limited to B, it can be shown
that the terms on lines [10] and [11] are << the term on line [9] for
sufficiently high w.
[ The above paragraph will require a proof; supply as a lemma. The kernel of
this proof should use L(t) > 0 and therefore L(t) >= L(t)*cos( f*t ) for
all t. A second part will be needed for the dL/dt integrand. ]
[ Rework from here on. ]
Refinement
A preliminary hardware breakdown of the MCPC yields the following subsystems:
1. Controller (MCS96)
2. Digitally Controlled Current Source (DCCS)
3. Voltage Sense Circuitry (VSC)
4. Sense Current Source (SCS)
5. Current Mixer (MIX)
6. Coil (COIL)
Inputs (to the controller)
1. Drive coil voltage estimate.
A digitised measure of the voltage across the coil from the
Voltage Sense Circuitry.
2. Drive coil interrupt input.
A digital control signal from the Voltage Sense Circuitry that
means conversion complete.
3. Reference position control time series.
Some representation of the sequence of positions that the controller
will attempt to drive the bearing to at successive discrete time
instants; initially input by operator at console and stored in
controller memory.
[]
Outputs (from the controller)
1. Drive coil current word.
This is a binary encoded measure of the amplitude of the current
level to be output by the DCCS.
2. Drive coil current strobe.
This is a digital control signal that writes value of drive coil current
word into the DCCS.
[]
States (of the controller)
1. Current time.
2. A vector of recently sampled coil voltage measures and an association
to time of occurrence of each sample (perhaps implicit).
3. A reference position sequence placeholder (presumably the reference
position control series is to be repeated until changes are made,
so a pointer to what position should be output now must be maintained.
This pointer will eventually loop back around to the start of the
reference series).
[]
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| <<< CLT::SCAN$$DISK:[NOTES$LIBRARY]MATH.NOTE;7 >>>
-< Mathematics at DEC >-
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Note 693.4 Dynamics and design of a floater (mostly physics) 4 of 4
EAGLE1::BEST "R D Best, Systems architecture, I/O" 28 lines 21-MAY-1987 18:21
-< could use some help in justifying neglecting the sine integrals >-
--------------------------------------------------------------------------------
I could use some help in justifying the argument that integrals
[10] and [11] in .3 can be safely neglected relative to [9].
Here's one approach:
Consider the integrals over a single period of sin( 2*w*t ) with both
endpoints of the interval chosen as zeroes of sin( 2*w*t ). I may have to
relax this later to make the proof valid. Let the lower endpoint be
t0.
Since L(t) is always > 0, the cosine integral over a single period will be
roughly:
{ sqrt(2)/2 } * { L(t0 + T/2) - L(t0 + 3*T/2) } * T
This is because we can consider L(t) to be relatively constant over
a single lobe of the sine wave. The area under the positive lobe
is the {sqrt(2)/2}*L(t0 + T/2)*T term. We subtract the area under
the negative lobe to get the net contribution to the integral over that
period. The sqrt(2)/2 part arises from the ratio the area under a single
sine wave lobe / the area the circumscribing rectangle
(yes, this is an approximation).
I'd like to show that when you add up all these single period integrals,
(and the corresponding cosine integral), that the integral of
L(t) is much bigger.
Is this a valid approach ? Anybody care to pitch in ?
Another method will be necessary for the sine integral because it has the
dL/dt term.
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