Problem-Solving Example #6

Description of a Voice Coil Motor

A VCM is basically an electromagnetic transducer in which a coil placed in a magnetic pole gap experiences a force proportional to the current passing through the coil. VCM can be generally classified as a dc brushless motor if the coil is wound as the starter and the permanent magnet with the attached payload is allowed to move or rotate. For rotary applications, a number of types of limited travel & brushless motors are available on the market, possessing flat torque output over a region of ±20°. By providing the necessary encoding and electronics, these motors are also used as constant speed motors.

Assuming the force generated is independent of position, the governing equation can be written as

L * di/dt + R i = E - φ v
dv/dt = φ i / J
v = dx/dt

where

L = inductance of the motor coil (henry);

i = current intensity through the motor coil (ampere);

R = resistance of the motor coil (ohm);

E = applied voltage to the coil (volt);

φ = transducer constant; for linear motion, its units are either Newton/ampere or volt/meter/second, and for rotary motion, the units are N-m/A or volt/radians/second;

v = velocity of motion (radians/second or meter/second);

J = total inertia of the motor and access mechanism (kg - m2 or kg);

x = response (radians or meter).

The VCM design objective is to move from point A to B in the shortest time possible.

A physical constraint may exist to limit R / φ**2 to a constant +/- a tolerance. H1 & H2 in attached code show how such a constraint may be handled. R and φ will be varied in order to find a minimal seek time, Total_t, while meeting this physical constraint.

A dc brushless motor is to be used whose coil has an inductance of .015 H in the presence of the permanent magnet. The supply voltage is 24 V, and there is a drop of about 1 V in the electronics before the motor winding. To reduce the thermal gradient, the maximum current should be limited to 0.5 A. Obtain the voltage transition times (Tuai), plus the torque sensitivity (φ) of the dc motor and the resistance (R) of the coil to meet the design objectives.

1. Theoretical Problem/Solution ... Zero time for change in polarity

Problem Seektime	[Voice Coil Motor For An Actuator;
		Objective = Move from Point A to B in desired time (Tf)]
  Allot Tau(3)
  R = 50	Emax = 23	L = .015	[Ohms, Volts, Henry]
  Phi = .5	J = 5e-4	[Volts * Sec./Rad., Kg * M * M * Ohm]
  C = 0	V = 0		X = 0	[Initial Values]
  T = 0	Dt = .001	Tf = .072	[Time In Seconds]
  Tau(1) = .3*Tf	Tau(2) =.4*Tf	Tau(3) = .1*Tf
  Tmech = .12	Tm_tol = .05	[Mechanical Time constant
					& % tolerance]
  Pi = 4*Atan(1.)	Xfinal = Pi / 6	[radians = 30 degrees of travel]
  Call Actuator
End
Model Actuator
  Find Tau   In Bangbang   By Ajax( Cntl2)    To Match Err_pos, Err_vel
End
Controller Cntl2( Ajax)	Damp = .0002	End
Model Bangbang
  T = 0		C = 0	V = 0	X = 0	Tp = 0	[ Initial Values]
  For I = 1 to 3   Do
	Total_t = Tp + Tau( i)
	Dt = Tau( i)/100		Dp = 5 * Dt	Tp = Tp + Dp
	Initiate Isis   For VCMotor
		Equations Cdot/C, Vdot/V, Xdot/X   of T   Step Dt   To Tp
	Polarity = -(-1)**I	E = Emax * Polarity
	Do While (Tp .le. Total_t) then
		Tp = Tp + Dp
		Integrate VCMotor   By Isis
		Print T, V, Vdot
	End Do
  Repeat
  Err_pos = Xfinal - X		Err_vel = V
End
Model VCMotor
  Vdot = Phi * C / J	Xdot = V
  Cdot = (E - Phi * V - R * C) / L
End

2. Theoretical Problem/Solution ... Zero time for change in polarity
New objective: Minimize Seek Time

Problem Seektime	[Voice Coil Motor For An Actuator
			Objective = Minimize Seek Time]
  Allot Tau(3)
  R = 50	Emax = 23	L = .015	[Ohms, Volts, Henry]
		o
		o	(same code as in example 1)
		o
  Pi = 4*Atan(1.)	Xfinal = Pi / 6	[radians = 30 degrees of travel]
  Find Phi, R   In Actuator   By Jupiter( Cntl1)
	Holding H1, H2, Hc	To Minimize Total_t
End
Model Actuator
  H1 = - R * J + Tmech * (1+Tm_tol) * Phi**2	[H1 & H2 must be >= 0]
  H2 = R * J - Tmech * (1-Tm_tol) * Phi**2
  Find Tau   In Bangbang   By Ajax( Cntl2)   To Match Err_pos, Err_vel
End
Controller Cntl1( Hera)	Adjust = 2	End
Controller Cntl2( Ajax)	Summary = 0	End
Model Bangbang
  T = 0		C = 0	V = 0	X = 0	Tp = 0	[ Initial Values]
  For I = 1 to 3   Do
	Total_t = Tp + Tau( i)
	Dt = Tau( i)/100		Dp = 5 * Dt	Tp = Tp + Dp
	Initiate Isis   For VCMotor
		Equations Cdot/C, Vdot/V, Xdot/X   of T   Step Dt   To Tp
	Polarity = -(-1)**I	E = Emax * Polarity
	Do While (Tp .le. Total_t) then
		Tp = Tp + Dp
		Integrate VCMotor   By Isis
		Print T, V, Vdot
	End Do
  Repeat
  Terminate VCMotor
  Err_pos = Xfinal - X		Err_vel = V
End
Model VCMotor
  Vdot = Phi * C / J	Xdot = V
  Cdot = (E - Phi * V - R * C) / L	Hc = .5 - C	[ Hc must be >= 0]
End

3. Practical Problem/Solution ... ArcTan() used to change polarity

Problem Seektime	[Voice Coil Motor For An Actuator
			Objective = Minimize Seek Time with constraint on
			rise/fall times; ie. |Edot| < Emax_slope]
  Allot Tau(3)
		o
		o	(same code as in example 1)
		o
  Pi = 4*Atan(1.)	Xfinal = Pi / 6	[radians = 30 degrees of travel]
  Ypeak = .985 * Pi/2	Xmax = Tan( Ypeak)
  Trise = 5*Tf/100	Pw50 = Trise	Tends = .6*Trise
  Find Phi, R	   In Actuator	By Jupiter( Cntl1)
	Holding H1, H2, Hc	To Minimize Total_t
End
			o
		o	(same code as in example 2)
		o
Model VCMotor
  Vdot = Phi * C / J	Xdot = V
  Call Risetime	[ E = constant except during transition times]
  Cdot = (E - Phi * V - R * C) / L	Hc = .5 - C	[ Hc must be >= 0]
End
Model Risetime
  If( I .Eq. 1) then		[ Calculate E during transition time]
	If( T .lt. Tends ) then	[ ie. E = f(t) during rise/fall times]
		E = (Atan((4*T/Pw50 - 1.)*Xmax) / Ypeak + 1.)/2.
	Endif
  Else If( I .eq. 2) then
	If( T-Xtau(1) .lt. Trise) then  [Switch voltage polarity]
		Tt = T - Xtau(1)
		E = - Atan((2*Tt/Pw50 - 1.)*Xmax) / Ypeak
	Endif
  Else If( I .eq. 3) then
	If( T-Xtau(1)-Xtau(2) .lt. Trise) then	[Switch voltage polarity]
		Tt = T-Xtau(1)-Xtau(2)
		E = Atan((2*Tt/Pw50 - 1.)*Xmax) / Ypeak
	Else If( Time-T .lt. Tends) then	[Drop voltage to zero]
		Tt = T-Time+Tends
		E = - (Atan((4*Tt/Pw50 - 1.)*Xmax) / Ypeak-1.)/2
	Endif
  Endif
End

Applied Voltage waveforms to Coil

Figure 1. Theoretical Problem/Solution ... Zero time for change in polarity
Objective: Specific time (Tf) for movement; i.e. Total_t must equal Tf.

Figure 2. Theoretical Problem/Solution ... Zero time for change in polarity
Objective: Minimize Seek Time (Total_t)

Figure 3. Practical Problem/Solution ... ArcTan( ) used to change polarity
Objective: Minimize Seek Time while constrainting rise/fall time.