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J.C. Compter, Electrical drives for precision engineering designs, 2007

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Page 1: Precision Motors Spec a Motor

J.C. Compter, Electrical drives for precision engineering designs, 2007 1

Page 2: Precision Motors Spec a Motor

J.C. Compter, Electrical drives for precision engineering designs, 2007 2

Page 3: Precision Motors Spec a Motor

J.C. Compter, Electrical drives for precision engineering designs, 2007 3

Preface ....................................................................................................................................................5 1. Introduction....................................................................................................................................6

The iron armature motor ................................................................................................7 The hollow rotor motor..................................................................................................8 The disc armature motor ................................................................................................8 Nominal quantities .........................................................................................................9

2. Torque constant and back-EMF constant ....................................................................................10 3. Power conversion.........................................................................................................................12 4. Servomotor characteristics ..........................................................................................................13

The motor steepness.....................................................................................................13 Thermal limits..............................................................................................................14 Maximum mechanical power.......................................................................................15 Maximum efficiency....................................................................................................16

5. Voltage controlled servomotor.....................................................................................................18 The required motor-voltage .........................................................................................18

6. Thermal aspects ...........................................................................................................................20 6.1 The model ........................................................................................................20 6.2 The ohmic loss and temperature dependent constants.....................................21 6.3 De-rating by the ambient temperature .............................................................21 6.4 Transient analysis.............................................................................................22

7. Electronically commutated motors...............................................................................................25 7.1 DC-brushless....................................................................................................25 7.2 AC-Synchronous Servo Motors.......................................................................28 7.3 Comparison of motors with and without brushes ............................................30 7.4 Attention to ....Losses related to the iron ........................................................31 7.5 Sinusoidal or trapezoidal EMF and the amplifier ............................................33 7.6 Comparison motor types ..................................................................................33 7.6.1 The iron armature motor ..............................................................................33 7.6.2 The hollow rotor motor................................................................................35 7.6.3 The disc armature motor ..............................................................................35 7.7 Load cases........................................................................................................37

8. Voltage Brushless Motors ............................................................................................................40 Introduction..................................................................................................................40 8.1 DC systems ......................................................................................................40 8.2 AC synchronous motor ....................................................................................41 Conclusions..................................................................................................................44

9. Motion profiles .............................................................................................................................45 10. The motor amplifier .................................................................................................................49

10.1 Low cost...........................................................................................................49 10.2 Pulse width and frequency modulation............................................................49 10.3 Four quadrant operation...................................................................................50 10.4 The transfer function of a PWM amplifier ......................................................52 10.5 Gain errors and offset.......................................................................................55 10.6 Cables...............................................................................................................57 10.7 The smaller the better?.....................................................................................57

11 Linear motors and actuators ........................................................................................................58 11.1 Introduction......................................................................................................58 11.2 The application field of linear drives ...............................................................58 11.3 Consequences of a direct drive ........................................................................59

12 Actuators ......................................................................................................................................61

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J.C. Compter, Electrical drives for precision engineering designs, 2007 4

12.1 Electro dynamic actuators................................................................................61 12.2 Force and dissipation .......................................................................................64 12.3 The voltage.......................................................................................................65 12.4 The stroke and the factor K ..............................................................................65 12.5 Heat transfer.....................................................................................................66 12.6 Mechanics ........................................................................................................67 12.7 Electro dynamic actuator types........................................................................72 12.7.1 The loudspeaker ...........................................................................................72 12.7.2 A sledge actuator..........................................................................................74 12.7.3 A flat actuator ..............................................................................................75 12.7.4 Moving magnet ............................................................................................76 12.8 Summary ..........................................................................................................79

13. Linear motors...........................................................................................................................80 13.1 Electronic commutation ...................................................................................81 13.2 DC-brushless....................................................................................................81 13.3 AC-synchronous motors ..................................................................................82 13.4 Comparison ironless and iron core linear motors ............................................84 13.5 Linear motor, moving magnet..........................................................................85

14 The ”constants” ...........................................................................................................................86 14.1 the K-factor ......................................................................................................86 14.2 The resistance R ...............................................................................................87 14.3 The steepness S ................................................................................................88 14.4 The thermal resistance Rth................................................................................88 14.5 Life-time ..........................................................................................................88 14.6 Amplifier choice ..............................................................................................89

15 Literature......................................................................................................................................90

Page 5: Precision Motors Spec a Motor

J.C. Compter, Electrical drives for precision engineering designs, 2007 5

Preface Precision engineering is highly linked with modern production equipment, where accuracy is linked to speed. For example accuracies of nanometers are required in a settling time expressed in milli-seconds in a wafer scanner. Wire-bonders, component mounting equipment, wafer-handlers and dye-handling robots are other examples. This puts high demands on the technologies involved. Machine stability and sufficient high vibration modes have to be obtained to get a good controller performance. Thermal stability is required to allow accurate measurements and to prevent frame or tool deformations. Actuators should produce the exact controller requested force directed to the centre of mass to prevent excitations in other degrees of freedom. The author is active in this field of interest as industrial designer for electrical drive systems for more than 30 years and these lecture notes reflect a part of the knowledge gained with the electrical drives in the central position. Extending the knowledge on electrical drives is mainly driven by the high demanding projects done by Philips Applied Technologies for its customers. The electrical drives are the linking pin in these notes, directed to electrical drives for precision engineering. In preparing these notes the decision is made to give preference to a system approach. Many times a reference was made to thermal aspects, dynamics, control, electronics and the consequences of tolerances. The electrical drives are considered here as a component to be selected, based on its characteristics. The consequence is that typical motor design issues are described, but not analyzed by means of mathematical formulations, as can be found in the literature directed to the design of motors and actuators. J.C. Compter Eindhoven 22-06-2007

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J.C. Compter, Electrical drives for precision engineering designs, 2007 6

1. Introduction In controlled systems the information obtained from sensors or commanded signals is used to influence the state of the system. The signals are used to exert forces or to control other types of “actuators” (fans, valves, heaters, etc.). In electro-mechanical systems it is generally necessary to con-vert electronic signals to mechanical forces or torques. The specific characteristics of the actuator may be of noticeable influence on the performance of controlled systems. Several types of electromechanical actuators exist. Examples are rotating or translating (linear) DC-motors, step motors, AC-synchronous motors and AC-asynchronous motors. All these motors have different characteristics and are specially suited for a typical application. In this section some basic theory on electro-mechanical actuators will be presented. Tools used to select motors for a certain application will be introduced. Finally a first impression of the influences on the control system behaviour will be presented. If we only need a constant speed, for instance for a transport system, we could take an AC-synchronous motor or an AC-asynchronous motor. Looking to the dynamical behaviour one will see however for these motors a considerable time-constant or oscillating response. In a servo system, motors with a well predictable response, as holds for the DC-motor, are preferred. This can also be obtained for the two mentioned AC-motors with the vector control technology, but this requires an advanced and expensive motor controller and the related costs can only be accepted for powers above 1 kW. The majority of servo-systems within Philips are in the range 1 ... 400 watt, so we will concentrate on the DC-motors. Examples of DC-motors are: disc armature (Fig. 1.1), hollow rotor (Fig. 1.2), iron armature (Fig. 1.3) and brushless (Fig. 1.4).

Fig.1.1 Disc armature motor Fig.1.2 Hollow rotor motor

Fig.1.3 Iron armature motor Fig.1.4 Brushless motor

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J.C. Compter, Electrical drives for precision engineering designs, 2007 7

THE IRON ARMATURE MOTOR As the name suggests, it is a feature of such motors that the rotor consists of a wound lamination core. On closer examination we see that the lead-outs of the rotor teeth widen. The function of this widening is to reduce the reluctance of the air gap; in combination with the small air gap (< 0.5 mm) the result is that relatively little magnetic material is needed to achieve field strengths in the rotor plate of 1 Tesla or more.

Fig.1.5 The iron armature motor

The toothed structure of the rotor gives rise to cogging, which are reduced by skewing the rotor slots in the more expensive motors. Strengths

• The power density, defined as Pmech/volume, is by far the highest, because the magnets only have a small air gap to overcome. The rest of the flux path consists of the magnetically highly conductive iron. The end result is less weight, less volume and low price.

• The iron guarantees mechanical robustness; the windings are anchored in it by means of a moulding resin. Considerable attention has generally been given to secure fixing of the winding head and to the connections to the collector. A high resistance to centrifugal forces and accelerations is achieved in this way.

• The thermal capacity is great because of the presence of the rotor iron; short-time peak powers are consequently readily absorbed. The small dimensions resulting from the high power density do generally result in the thermal resistance Rth1 being slightly higher.

• A low mechanical time constant can be achieved through the slim structural shape of the rotor in combination with the high power density; < 5 ms is feasible.

• The large number of producers means that “second sourcing” is possible at good prices.

Weaknesses • The rotating iron produces cogging in addition to eddy current and hysteresis losses. • The fact that the conductors are located in the slots of the iron rotor gives rise to a substantial

self-inductance of the rotor windings. This makes the commutation of the rotor coils difficult. It also gives rise to a time constant in the control loop; with fast torque changes the self-inductance means that the amplifier must have a safety margin on the voltage if it is to overcome L.di/dt.

• The drawbacks of the iron armature motor will not be found in the hollow rotor motor. In this motor the windings are located in the air gap and together form a cage that has been made into a solid entity with resins. This cage has then been fixed to the shaft of the motor.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 8

THE HOLLOW ROTOR MOTOR Inside the cage there is a stationary magnet. The tolerances of the cage force suppliers to select a large air gap. Because of this structure the reluctance for the magnets is factors greater than in the iron armature motor; the magnetic field must cross four air gaps and pass the cage twice. Achieving acceptable field strength requires large magnets in these motors.

Fig.1.6 The hollow rotor motor Strengths

• No iron losses and no cogging. • A small moment of inertia J, because in this case the iron is not turning in the same direction. • The lower self-inductance moves the commutation limit to higher values. The self-inductance

is lower because there is an air gap on both sides of the windings. In addition, the magnets located on one of the two sides also behave like air. (The permeability µr of magnets is very similar to air.)

Weaknesses

• The power density is a factor 3 lower than that of comparable iron armature motors, because appreciably larger magnets must be used.

• Greater vulnerability to peak torques and high speeds. The cage construction has lower limits than the iron armature rotor.

• In the absence of the rotor iron the thermal capacity is low. At the same time the thermal resistance Rth1 is moderate because of the large air gaps and the impeded heat dissipation via the shaft. This motor is therefore better suited to short-time loads. An increase of 3* in the thermal load can be achieved by the introduction of a forced air current via a separate blower.

THE DISC ARMATURE MOTOR The feature of this motor is an axially oriented magnetic field and a disc-shaped rotor. The rotor disc can consist of rotor windings, which have been made into a solid whole with epoxy. Another form of construction is a printed circuit board on which a track pattern has been etched. Magnets are placed on both sides of the disc to obtain a sufficiently strong field.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 9

Fig.1.7 The disc armature motor

Strengths

• No cogging, no iron losses. • A small moment of inertia and a high value of S, possible through the use of a lot of magnetic

material, lead to a low mechanical time constant. • A very low self-inductance and electric time constant, because of the existence of an air gap

and magnetic material respectively on either side of the copper. The commutation forms hardly any limitation in the motor performance.

• Because of the large rotor area the thermal resistance is low. The low thermal capacity of the PCB version is partly offset by the high permissible temperature of 155ºC. The epoxy version has a temperature limit of 110ºC, but does have greater thermal capacity.

• The better controllability of the dimensioning leads to a higher permissible shaft speed, because imbalance and distortion come less into play; 8000 - 10,000 rpm occur. Extremely robust.

• The flat structural shape makes combination with an encoder, brake or tacho for example easier.

Weaknesses

• The low power density and low values for S/volume and S/mass are a consequence of the voluminous stators. Partly because of this prices are high.

• Usually AlNiCo is used as magnetic material. It is a material with a high sensitivity to demagnetisation. Peak currents must therefore be controlled, it being important to remember that these motors have hardly any damping effect by way of a high self-inductance.

• To make re-magnetisation possible by the user, the manufacturer generally has the magnetisation windings located around the magnets.

• Depending on the type of PWM amplifiers, the low self-inductance can prevent the amplifier working properly. The low self-inductance means that there are sharp variations in current. Even with a torque T = 0, this leads to unnecessary ohmic losses; it may be necessary to take the step of using (voluminous) smoothing coils.

• The good control engineering properties mean that the disc motors are frequently used for robots and in servo-controlled machines. But the latest developments in the field of brushless motors (AC synchronous and induction motors) mean that this choice is no longer as self-evident as it was.

NOMINAL QUANTITIES Nominal values of torque and speed (or current and voltage) are generally given for an ambient temperature. The main usefulness this has is that it makes comparison with other motors possible. At the same time a life of the motor can be given under these conditions. For the user of a motor this life

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J.C. Compter, Electrical drives for precision engineering designs, 2007 10

is a reference point. But only very rarely will we come across an application that satisfies the nominal conditions, so a life expectancy must always be given with more than a little caution. 2. Torque constant and back-EMF constant The principle mechanism for electro-motors is the Lorentz-force. In this paragraph this mechanism will be presented.

Figure 2.1, Lorentz-force

In figure 2.1a a charge quantity dq [Coulomb] is given, that moves through a conductor with a length l [metre] with the speed va [metre/sec.]. Perpendicular on the conductor a magnetic field with the strength B [Tesla] is present. Lorentz proved, that for the force acting on the charge holds:

... at vBdqdf = 2.1

It is based on the theory of special relativity. Although very interesting this will not analysed here (see Interpretation of Classical Electromagnetism, W.G.V.Rosser, ISBN 0-7923-4197-2).

The speed of the charge dq is given by:

.dtdlva = 2.2

The current [ampere] in the conductor is, according the definition of Ampere:

.dtdqI = 2.3

The force [newton] on the conductor is obtained by substitution and integration over the length of the wire. The force equals:

... lIBft = 2.4

The field B is assumed to be generated by a permanent magnet. For a motor with N windings on a radius r, the torque [newton.meter] equals:

........ IKrlIBNrfNT tr === 2.5

So, the torque is proportional to the current, when using a permanent magnet. For this reason, motors with a permanent magnet are used in servo systems. This type of motors we will call servomotors.

Now the induced voltage, the back EMF will be analysed. If the conductor is moving in a field B (in the direction given in figure 2.1b), the force on the charge dq is given by: ta vBdqdf ..= . The force

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J.C. Compter, Electrical drives for precision engineering designs, 2007 11

per unit of charge, dfa/dq, is per definition the electrical field strength E [V/m]. The voltage difference over the conductor, known as the back-EMF (electro-magnetic force) is the line integral of the field-strength E:

...0 0∫ ∫ ===l

t

l

lvBdldqaEdlEMF

2.6

For a motor with N windings on a radius r rotating with the speed ω [rad/sec] we have the next relation:

...... ωet K

rvrlBNEMF ==

2.7

So the EMF is proportional to the motor speed if the field B is constant. The back-EMF constant is equal to the torque constant and we will use from now on the symbol K with as units [Nm/A]! Be aware of the units applied; many times one will meet Kt with the units Nm/A and Ke with as units V/1000 rpm. From now on the symbol E will be used for the back EMF, because this is the usual symbol.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 12

3. Power conversion The motor windings have a resistance R. There fore we must supply a voltage U to the motor according:

..RIEU += 3.1

The incoming electrical power Pel is partly converted to mechanical power and dissipation:

RIIEIUPel2.. +== 3.2

Applying the formula for the back-EMF we arrive at:

..... 2 RIIKPel += ω 3.3

Substitution of K.I=T leads to:

,.. 2 RITPel += ω 3.4

what can be recognized as:

.dissmechel PPP += 3.5

The loss in the motor Pdiss results in to a temperature rise of the windings. The maximum temperature of the windings limits the allowable continuous current (and torque) of the motor. An important conclusion is that for the mechanical power holds:

.... IKIEPmech ω== 3.6

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J.C. Compter, Electrical drives for precision engineering designs, 2007 13

4. Servomotor characteristics

THE MOTOR STEEPNESS The relation between torque and the speed of a servomotor with a constant voltage will be analysed now. Based on the voltage equation one obtains:

ω.. KRKTERIU +=+=

4.1

Definition of the no-load speed ω0:

KU

=0ω 4.2

and the steepness S according:

RKS

2

= 4.3

leads with substitution to:

).( 0 ωω −= ST 4.4

With a blocked rotor one obtains the stall-torque Ts:

RUKSTs ==

4.6

T

U=constant

i = U/R , T =K.U/Rs

00,0

T

s

0 = U/KS=T /s 0

Figure 4.1, Torque speed curve

The torque-speed curve is a straight line (figure 4.1). The derivative of this line is the motor steepness, what can be proven by dividing the stall torque and the no-load speed. The higher the steepness S, the better the motor. The background is that at a fixed torque T the loss Pdiss decreases at increasing steepness, because the next formula holds:

dissPT

RIIK

RKS

2

2

222

=== 4.7

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J.C. Compter, Electrical drives for precision engineering designs, 2007 14

It has to be mentioned that the stall torque Ts and no-load speed ω0 depend both linearly on the supply voltage U. Substitution proves:

.o

sTSω

= 4.8

In data sheets one will find the damping factor Km, which is equal to √S.

THERMAL LIMITS The dissipation of the motor is limited by the maximum allowed temperature θmax of the winding isolation. With a thermal resistance Rth between the windings and the ambient, and the temperature θamb one obtains the maximum allowable dissipation Pmax,diss as:

.maxmax,

th

ambdiss R

P θθ −=

4.9

For the dissipation holds:

.2

2

STRIPdis ==

4.10

Combination of the two preceding formulas leads to a maximum allowable continuous torque (T100):

... maxmax,100

th

ambientdiss R

SPST θθ −==

4.11

Servomotors do not operate in continuous duty. The peak torque, necessary during acceleration can be much higher, because the motor accelerates only during a limited period of time. Suppose a motor operates in a cycle of Tcycl; during δ.100% of this time the torque T is produced. The dissipated power is given now as:

.2

STP δ

δ = 4.12

This loss should not exceed Pdiss,max, so:

.22

100max, S

TS

TPdissδδ==

4.13

This leads to:

100.TT δδ = 4.14

This torque level can be drawn in the torque speed curve (figure 3) as a function of δ. One has to be aware that Pdiss,max, and so also Tδ, depends on the ambient temperature!

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J.C. Compter, Electrical drives for precision engineering designs, 2007 15

T

U

00,0

T

s

100 %50 %25 %

10 %

nom

Figure 4.2, Tδ by thermal limits

When the torque changes rapidly in time an other definition is used many times; the effective torque Trms, which is defined as:

∫=ctc

rms dttTt

T 2)(1

4.15

The time interval tc is the cycle time of the process involved. Then holds:

STP rms

2

= 4.16

MAXIMUM MECHANICAL POWER The mechanical power of the motor, produced by the rotating shaft, has a maximum (under the condition of a constant voltage U). The mechanical power Pmech is given by:

=−== ).(. 0 STTTPmech ωω

4.17

Its maximum can be obtained by:

ST

dTdPmech 2

0 −= ω 4.18

The torque level is given consequently as:

220 sTST ==

ω

4.19

The maximum output power equals now:

RUSPmech 44

220

max, ==ω

4.20

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J.C. Compter, Electrical drives for precision engineering designs, 2007 16

This can be found at the middle of the torque speed curve. Indicated in the graph is the line Pmax; on this line all points (ω,T) with maximum output power can be found as a function of the supply voltage U.

0 /2

Pmech

T

T s

T s2 Pmech, max

Speed/power

Torque P max

0,0 Figure 4.3, Mechanical output power

MAXIMUM EFFICIENCY The efficiency of the motor depends also on the operating point (given by Τ fr and T) of the motor:

IUdTT

PP fr

in

out

.)..( ωω

η−−

== 4.21

with Tfr as the friction torque of the motor and d as the viscous damping coefficient (the speed-dependent friction). The torque T for maximum efficiency is determined by:

SdTTST

TdTd sfrs

+

+=⇒=

..0

2η 4.22

torque

speed/power0 /2

Pout

T

ω Figure 4.4, Efficiency

In the next figure all the available curves are drawn in the torque speed plane; the line Pmax represents the points were the maximum mechanical output power can be found at changing voltage; the curve belonging to the maximum efficiency is represented by 0max.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 17

torque

speed

Pmax

η opt

P comm

UT

maxmax

ωmax

S O A

T 100

T 25

Unom0.5 U nom

Figure 4.5, SOA

• Three dashed lines are also added; they are: • maximum torque (brush heating) • maximum speed (brush lift and bearings) • maximum power (commutation limit) • maximum voltage (brush fire)

Figure 4.6, A commutator

The last four lines enclose the so-called “Safe Operation Area”, the SOA. As general advice for motor selection one has to operate always within the Safe Operation Area, preferably operate between the Popt and ηopt line to combine an acceptable efficiency and a high output power. Additionally one has to reconsider the motor chosen when the worst case operation point can be found under the T-ω line belonging to 0.5 Unom (motor under loading). Within Philips one also prevents to exceed momentary the T25-line; this is based on a questionable carefulness; the more one knows concerning the application and motor one can shift this limitation upwards. Finally remains the area with a dashed contour as preferred operation area.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 18

5. Voltage controlled servomotor A servomotor can be controlled in two ways. The most common way in professional systems (factory mechanization, robot systems) is controlling the motor current with a current amplifier, what means that the output current is proportional with the input signal for the amplifier. By controlling the current one control directly the motor torque.

For smaller motors, used in consumer systems, a linear voltage amplifier is often used. By controlling the motor voltage one control more or less the speed. To indicate why is said more or less the inductance of the motor is added to the voltage equation, leading to:

RidtdiLEu .++=

5.1

Let us assume as load an inertia J only. The result of substituting:

0.and,,.,. ωωω KuKTiKE

dtdJT ====

5.2-5.5

gives in Laplace notation:

11

20 ++

=ss mme τττω

ω 5.6

with:

SJ

KRJRL me === 2

.,/ ττ 5.7

The electrical time constant τe is in general much smaller than τm. The conclusion is that one deals with a second order system, what has to be considered as a complication for the design of the controller.

u i T ω

T load+ _K 1

current source

u i T ω

T load+ _K 1

sJ

voltage source 1

R(1+s) τ + _

K

e

sJ

Figure 5.1, Current and voltage control

The solution to prevent that additional time constants are introduced is to apply a current source amplifier, characterized by forcing a current, pre-described by its input, through the load, which is here of course a motor.

THE REQUIRED MOTOR-VOLTAGE The first equation of this paragraph gives the motor-voltage needed. Replacing the current i by T/K leads to:

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J.C. Compter, Electrical drives for precision engineering designs, 2007 19

KRT

dtdT

KLKu .. ++= ω

5.8

The speed and torque as a function of time are in general determined by the application. A complication is that the resistance R rises with α=4 % per 10°K and that the motor constant K falls with some percent per 10°K by the temperature dependence of the magnet strength. Usually symbol for the relative decrease of the motor constant is kt [%/K]. Concerning this last dependency one has to refer to the motor data sheets, where one also will find a usual tolerance on the resistance R and motor-constant K of 5 to 10 %. Taking the effects mentioned above one has to take more affects into account to obtain the required supply voltage for the motor-amplifier. One has to consider also the voltage drops over:

E

+

_

mains

supply amplifier motorcable

connectors

Figure 5.2, Loss of voltage

• the cable and connectors between the amplifier and the motor • the commutation system (brush resistance and voltage drop over the contact layer between

brushes and commutator) • the voltage drop over the end-stage of the amplifier • the voltage drop over the supply at high currents.

These additions also hold for the voltage of a current amplifier. In chapter 14 a more extended list of amplifier choice related items will be given.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 20

6. Thermal aspects

6.1 THE MODEL In the paragraph on the thermal limits of the motor, it was said that the maximum power dissipation is limited. This dissipation will result in a temperature rise of the motor. Because of the thermal capacity of the motor, the temperature will rise according to a first order response, with a time constant. Rotor and stator have their own thermal time constants, which can be found by applying the thermal network model, indicated in the figure.

rotor stator

ambient

PLoss

R

R

R

th1

th0

th2

C Cth2th1

Figure 6.1.1, Thermal model

If one has to consider the time dependent thermal behaviour depends on the ratio between the smallest thermal time constant and the cycle time of the load. The value for the thermal constant is between 10 seconds (<< 10 watt) and 1000 seconds (1 kW motors). The rotor loss consists of the losses i2R, eddy current loss and friction loss in the bearings and brushes. The next figure gives the interactions involved.

Model for a motor with brushes

1pL+R( )θ r

s

-

+U

E

I Tem

d Thyst

sign( )ω

+- -

Tload

+ -pJ

Tfric

ω-

1

I( .R( )+θ ω2.d+θ ) r2

| .T +T ω|

θr θs

θambient

Functions of time: , ,U,i,E,Tω,θ θr s load

Rth1

Rth2Cth Cth2

(ω)

p1

T ( )ΘCog

s -

Θ

s

Figure 6.1.2, Full model for a motor with brushes

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J.C. Compter, Electrical drives for precision engineering designs, 2007 21

6.2 THE OHMIC LOSS AND TEMPERATURE DEPENDENT CONSTANTS A simplified example will be given now, assuming no damping, friction and hystersis in the motor. Target is to link the temperature rise and the rms torque Trms, taking into account the temperature dependency of the resistance (α) and the motor constant (kt).

2th2t

2amb

th2th1amb2

rms2

st2amb

ramb2

rmsr2

s

2rms

.P).Rk.(1K))Rα.P.(R.(1RT

).k.(1K)α..(1RT)R(θ

)K(θTP

+++

=∆+

∆+==

θθ

6.2.1

with P as the copper losses. Be aware of Ramb and Kamb, numbers belonging to the actual ambient temperature, which might differ from the values Rref and Kref at the reference temperature used in data sheets from the motor supplier. Rearrangement leads to:

0).(.2. 221

222

222

223 =−+−++ ambrmsththambrmsambthambtthtamb RTRRRTKPRKkPRkKP α 6.2.2

The roots of this third order equation in the power loss P can be obtained in analytical form either by numerical methods, with as result that a direct link exists between the torque Trms and the losses P. The temperature rise of the rotor and stator follow by the multiplication of the solution P with Rth1+ Rth2 and Rth2 respectively.

6.3 DE-RATING BY THE AMBIENT TEMPERATURE The link between the ambient temperature θamb, the maximum allowed rotor temperature θr,max and the maximum allowed continuous torque T100 can be obtained with the following approach, based on θr,max=100 ˚C.

Figure 6.3.1, De-rating The de-rating by a rising ambient temperature is clearly visible! Substitution of the function K(θamb) and P(θamb) in the function T100 in Fig. 6.3.1 leads to the analytical expression as an alternative for the program given.

0 50 1000

1

2Allowed power loss

Pi

θambi

0 50 1000

0.5

1

1.5De-rating coefficient

Ti

θambi

PiP θambi( )P θref( ):=Ti

T100 θambi( )T100 θref( ):=

Resulting allowed torqueT100 θamb( ) K θamb( ) P θamb( )Rθmax

⋅:=

Resulting motor constantK θamb( ) K 1 kt θs θamb( ) θref−( )⋅+⎡⎣ ⎤⎦⋅:=

Resulting stator temperatureθs θamb( ) θrmax θamb−Rth1 Rth2+( )

Rth2⋅ θamb+:=

Allowed losses nowP θamb( ) θrmax θamb−( )Rth1 Rth2+

:=

Rotor resistance at max rotor temperatureRθmax R 1 α θrmax θref−( )⋅+⎡⎣ ⎤⎦⋅:=

kt 0.002−:=R 6=K 1:=Rth2 0.3=Rth1 0.5=θrmax 100=θref 30=

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J.C. Compter, Electrical drives for precision engineering designs, 2007 22

( )( )( )( )

( )⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−

++−+

−= ref

thth

ambrtht

ththrefr

ambramb RR

RkRRR

KT θθθθθα

θθθ21

max2

21max

max100

.1..1.

.)( 6.3.1

De-rating is not only a question of going to tropical area’s, mounting a motor in a warm housing has the same consequences!

6.4 TRANSIENT ANALYSIS

A transient analysis is given as the next example. Included are now the thermal resistances Rth1, Rth2, Cth1 and C th2, the temperature dependency of the rotor resistance R and the motor constant K, with their initial value based on the reference temperature θref, whereas θamb holds as the actual reference temperature. Let us assume a torque Trms, giving Pref as loss when the rotor and stator temperature equals θref. With Figure 6.1.1 one gets as the set differential equations:

( ) 112

1 .).(1

).(1.

thth

sr

refrt

refr

th

refr

CRkCP

dtd θθ

θθ

θθαθ −−

−+

−+=

6.4.1

2221 .. thth

ambs

thth

srs

CRCRdtd θθθθθ −

−−

= 6.4.2

For the rotor and stator are used θr0 and θs0 respectively as initial values. Let us assume that the motor load, a constant torque, is removed after 450 seconds. The problem is solved with Mathcad as follows in fig. 6.4.1.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 23

Figure 6.4.1, Thermal transient

To be learned from this graph is that the transient behaviour during the interval with torque behaves clearly in another way by the temperature dependency of rotor resistance R and the motor constant K. The thermal time constants have to be measure consequently during cooling down!

The same graph is now made with as only modification Pref=20Watt instead of 18 W, so only a small change of near 5 %in Trms. The rotor temperature rises from 95 to 108.4.

0 200 400 600 800 10000

50

100

Θri

Θsi

Ti

Ti itmaximax

⋅:=Θsi Zi 2,:=Θri Zi 1,:=

i 0 imax 1−..:=Z rkfixed θ 0, tmax, imax, D,( ):=tmax 1000:=imax 1000:=

Call a Runge-Kutta procedurre to solve the set

D t θ,( )P t( )

1 α θ0 θref−( )⋅+

1 kt θ1 θref−( )⋅+⎡⎣ ⎤⎦2

⎡⎢⎢⎣

⎤⎥⎥⎦

⋅1

Cth1⋅

θ0 θ1−( )Cth1 Rth1⋅

θ0 θ1−

Cth2 Rth1⋅

θ1 θamb−( )Cth2 Rth2⋅

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

:=

The right hand terms of the differential equations

P t( ) Pref t 450<if

0 otherwise

:=

Remove after 450 seconden the loss

θθr0

θs0

⎛⎜⎝

⎠≡

Definition of the initial values

α 0.004:=Pref 18:=

kt 0.002−:=θr0 20≡θs0 20≡θref 25:=θamb 20:=

α 0.004:=Rth2 1:=Rth1 2:=Cth2 20:=Cth1 10:=

Definition of the constants

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J.C. Compter, Electrical drives for precision engineering designs, 2007 24

Figure 6.4.2, Exceeding T100 The 5% increase in the torque leads to a 14% temperature rise by the temperature dependencies of R and K. This means, one has to be careful when T>T100 for a time interval approaching the thermal time constants of the motor even in the case that Trms<T100!

The measurement of the rotor temperature of an iron core motor during operation can only be done by putting a probe in a drilled hole in the shaft, filled with a thermally well conducting lubricant. Several experiments indicated, that the copper temperature is close to this temperature. Another method, applicable for all motor types, is to trace the armature resistance (so without brushes) in time immediately after a stop and to extrapolate backwards in time to the moment of switching off, according an exponential curve. With the resistance increase of 4%/10 degrees one gets the initial temperature at the moment the motor has stopped when the armature resistance is known at a well defined temperature.

0 200 400 600 800 10000

50

100

150108.414

20

Θri

Θsi

9990 Ti

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J.C. Compter, Electrical drives for precision engineering designs, 2007 25

7. Electronically commutated motors The mechanical commutation with brushes and a commutator is the main cause of performance limitations for this type of motors. The maximum brush current limits the maximum torque, the maximum speed is determined by the lift of the brushes, the maximum voltage by the limited lamellae voltage and the mechanical output by the commutation power handling capacity of the commutation system. Furthermore it is a fact that the limited reliability of motors with brushes is highly related to mechanical troubles with the brushes. The principal geometry of a servomotor with brushes is given by permanent magnets on the stator and (heat generating) coils on the rotor. So the thermal resistance from the heat source towards the cool ambient is high, because the heat has to be transferred over the air gap. Electronically commutation removes the limitations from the brushes and the other geometry, as given in the next figure, leads to a better heat transfer to the ambient.

Coil a

Coil bCoil c

MagnetStator

Figure 7.1, ECM

On the market ECM’s are offered and sold from mW’s up to MW’s. The higher costs of electronically commutated motors (ECM) are many times accepted as price for an increased reliability. The principal geometry of an ECM is given in the figure. A permanent magnet is mounted on the rotor and the stator is provided with at least 3 coils. The value of the individual coil currents is determined by the (measured) momentary rotor position and by the required torque production. Two types of brushless motors are available. At first there is the DC-brushless type, based on a six-step controller. The second type is AC-synchronous, based on sinusoidally changing currents. In the following paragraphs more details will be given.

7.1 DC-BRUSHLESS The following figure represents the six-step controller in combination with an ECM. Some logical cir-cuits control by means of six power transistors the current entering the coils. In the simplest form the motor is provided with 3 coils, which are mounted in the stator under and spatial angle of 120 degrees. To obtain a trapezoidal induced voltage as function of the rotor position one applies a careful distribution of the winding sides of each coil over the stator slots in combination with a fitted magnet shape and magnetization.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 26

PWM-logic HallDecoder

Iref

+

-

Position

N

Z

S 1

S 2S 3

Current

Figure 7.1.1, DC-brushless

The spatial distribution over 120 degrees leads also to an electrical phase shift of 120 degrees between the 3 induced voltages in the coils (EMF), as indicated by the next figure.

=0 2Coil 1

Coil 2

Coil 3

EMK

0

0

0

Figure 7.1.2, Induced voltage

This figure reveals the 3 induced voltages as a function of the rotor position; at constant speed a constant relation exists between position and the time. The control circuit has as target to realize that each coil carries current during the intervals with a constant induced voltage. The polarity of the current has to be related to the polarity of the induced voltage and the amplitude to the value of the required torque.

This procedure leads to a constant torque production. To prove this we first consider a single coil, assuming a constant speed ω. The K-factor of each coil is constant during each interval with a constant induced voltage, because the following relation holds here:

ωθ )(EK =

7.1

For each coil k holds:

kmechkkkkin PTIKIEMFRIP ,2

, )(.).(... ====− θωθω 7.2

In the next figure the relations between the rotor position and the current, the induced voltage and the transferred power Pmech are indicated. The currents are in phase with the induced voltages, block shaped and have an alternating polarity. Adding the contributions of the 3 coils leads to a constant torque:

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J.C. Compter, Electrical drives for precision engineering designs, 2007 27

IKP

T kkmech

.ˆ.2

3

1,

==∑

=

ω

7.3

=0 2

Coil 1

Coil 2

Coil 3

Current

Pmech

EMK

Figure 7.1.3, Induced voltage

So the value of the torque is determined by the value of the current. Furthermore a characteristic of this type of control is that two coils are carrying current at the same instant of time. The figure also reveals that at 6 positions within one revolution the current distribution over the coils has to be changed. This is the background of the term six-step control.

1

23

1

23

1

23

1

23

1

23

1

23

2 3

5 6

1

4

Figure 7.1.4, Six step

Summarizing we conclude for DC-brushless:

• at six rotor positions commutation has to be done, so only those six positions has to be detected; 3 Hall-sensors positioned under 120 degrees can fulfil this function

• the torque production is determined by the value of the supply-current; a single current sensor is (under idealized assumptions) sufficient

• only two coils are carrying current at the same instant of time. In practice one finds a number of causes that leads to a less ideal behaviour:

• the position of the Hall sensors is not at the correct position. This leads to commutation at a wrong position

• deviations between the three coils • a non-trapezoidal shape of the EMF • the amplitude of the currents are not equal, caused by inequalities in the amplifier end-stages

(offset, gain errors, drift).

The causes mentioned above lead to torque fluctuations near the commutation; this occurs 6 times per revolution. Especially in position servo-systems one has to be aware of this phenomenon, because an

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J.C. Compter, Electrical drives for precision engineering designs, 2007 28

unstable behaviour might be the result near the commutation positions. By adding some hysterisis in the Hall elements instability can be prevented. If the motor consists of more than 1 pole pair; e.g. p pole pairs, the number of fluctuations will be 6.p times per revolution. The figure 7.1.3 of this paragraph suggests that the shape of the current can be as a square wave. However at increasing speed the time constant of the coil (L/R) becomes a limiting factor, also when current control is used. The result is a decreasing value of the mean value of the current with as result a decreasing torque-constant.

7.2 AC-SYNCHRONOUS SERVO MOTORS The operation of the second family of brushless motors is based on a sinusoidally varying EMF in respect to the rotor position. The magnetization of the rotor and the winding geometry in the stator can used to obtain this EMF, as show in the next figure. The current in the three coils have exactly the same phase as the EMF. First the torque production under this condition will be analysed.

Figure 7.2.1, Induced voltage AC-synchronous

Figure 7.2.2, Power AC-synchronous

Suppose the motor is running at a constant speed. The mechanical power produced by each coil j is given by:

)3

2.(sin..ˆ)().( 2,,

πω jtIEtitEP jmech +== 7.2.1

Summarizing the contributions of the coils leads to:

ωωIEP

TIEP mechmech

ˆ.ˆ.5,1ˆ.ˆ.5,1 ==⇒= 7.2.2

Also for this motor holds:

.(:)ˆ ωE 7.2.3

So we may write:

.(:)IT 7.2.4

Fulfilling the condition that the same phase angle holds for the current and EMF of a coil leads to a similar situation as for the DC-brushless motor; the torque production is determined by the current amplitude.

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To obtain the phase-equality of the current and EMF the control circuit has to fed by a signal representing the actual rotor position at each moment of time, whereas a DC-brushless motor is satisfied with 6 pulses per revolution. Another significant difference with the DC-brushless motor is that the current of each phase has to be sensed to obtain the sinusoidal shape, as indicated in the next figure.

PWM-logic

+

-

brake

PI PIPI

+- +

- +-

(Absolute) encoder or resolver

Logicθ

P r o m Generate 3 sinuses

X

XX

I R e f controller

ω

θ

AC synchronous

3

R e f

sin ( θ ) sin ( θ + 1 2 0 ) sin ( θ + 2 4 0 )

Figure 7.2.3, AC-synchronous circuit layout

An advantage of the AC-synchronous brushless motor is the absence of discrete commutation positions, so unstable behaviour in a position servo is prevented by the absence of commutation positions. For the DC-brushless motor the influence of the inductance on the torque production at higher speed is discussed. A limited bandwidth of the amplifier leads also for the AC-synchronous motor to a decreasing torque capability at higher speeds. Suppose the motor is provided with 4 pole-pairs and runs at 6000 rpm. The frequency of the current will be here 400 Hz !!

A realistic value of the amplifier bandwidth is 1000 Hz, so a considerable phase shift φ between the input and output of the amplifier will occur. A similar analysis as given before leads to:

)cos(.ˆ(:) φIT 7.2.5

So the required bandwidth for the amplifier of an AC-synchronous motor is not determined by the specified control loop bandwidth only but also by the required torque capability at high speed.

Also the selection of the supply voltage of the amplifier has to be done carefully; one has to consider the required rise time of the torque, the level of the torque, the speed and the commutation process. It can be proven that one needs as amplifier specification at least:

( )22

....32ˆ TpS

dtdTT

SKU phph τωωτ +⎟

⎠⎞

⎜⎝⎛ ++=−

7.2.6

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J.C. Compter, Electrical drives for precision engineering designs, 2007 30

TdtdT

ST

StTtP ..)()(

2

ωτ++=

7.2.7

With as definition of the symbols used:

phph

phph

phphrms RL

RKS

ITK

=== τ,.5.1

,2

7.2.8-

7.2.10

The symbol p represents the number of pole pairs in the motor.

7.3 COMPARISON OF MOTORS WITH AND WITHOUT BRUSHES The lifetime of electronically commutated motors is determined by the bearings only. The torque performance is given by the thermal limitations of the electrical isolation. In rare cases one also meets a torque limitation to prevent demagnetisation of the magnets.

T

n (speed)

thermal limit brushes ordemagnetisation

sparking; limit output power

jumping brushes

brush fireSafe Operation Area (S.O.A.)

0,0

T

T

T

25

50

100

ECM-limitsTorque

Figure 7.3.1, Limitations for motors with and without brushes

The figure shows the limitation of a motor with brushes; added to this are the limitation for a brushless motor, indicated by the dashed lines. The speed for this last type is limited by the bearings and/or the heat production by eddy-currents in the rotor. The horizontal line is the torque limit related with demagnetisation. The hyperbola in the figure represents the maximum mechanical output of a motor with brushes. (Going over this line holds Pmech=ω.T is constant). For the ECM represents the working point at the maximum speed and torque the point of maximum mechanical output power. In the electronically commutated motors one can find the heat producing windings at the stator. In comparison with the mechanically commutated motors does this mean a considerably reduced thermal resistance from the windings towards the cool environment. The differences given leads to the conclusion that the absence of mechanical commutation leads to a considerable higher mechanical output for the same motor volume. For reasons of costs one finds in motors with brushes the classical magnetic materials as ferrites and AlNiCo, whereas SmCo and NdFeB magnets, with at least four times higher ratio between magnetic field production capability and volume, are applied in ECM’s. The cost of NdFeB drops dramatically and so also motors with brushes and NdFeB magnets are offered nowadays on the market.

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Material Br (Tesla)

Hc (kA/m)

Br/°C specific resistivity ohm.m

Hc /°C Relative costs/kg

Ferrite <0.4 250 -0.2% 106 +0.34% 1

SmCo 1 700 -0.05% 0.5.10-6 -0.3% 75

AlNiCo 1.3 130 -0.02% 0.5 10-6 -0.03% 7.5

NdFeB 1.2 800 -0.13% 1.4.10-6 -0.6% 37

An overview on the strong points of the ECM: 1) Life-time of more than 30000 hour, 2) high torque production at the same position allowed; this is limited for the brush motor by local

commutator heating, 3) increased reliability (one has to consider the bearings only), 4) it is possible to use a supply without mains transformer (select the 300 V motor-type), 5) no pollution by carbon dust and no sparking, 6) very limited sensitivity for air-pollution, 7) robust mechanical construction, 8) the measurement of the temperature of the stator windings is easy; guarding the motor temperature

allows now that one applies the motor at the performance limits, 9) the heat generation in an ECM occurs in the stator; the heat-resistance from the heat source

towards the ambient is significantly lower than for a motor with brushes, with the heat source on the rotor. At the end leads this to a higher ratio between performance and volume,

10) the high heat-capacity of the stator allows a longer overload time, 11) reduced dimension by the absence of the brushes and commutator.

7.4 ATTENTION TO ....LOSSES RELATED TO THE IRON In general high power magnets as NdFeB or SmCo are in applied in the rotor of an ECM. Those magnets generate high field densities in the stator iron. This is very attractive when one prefers a high ratio between performance and volume. However this implies also iron-losses in the stator when the motor rotates. In an experiment the stator temperature of an ECM is considered at 5000 rpm; a second motor realizes this speed. The ECM was disconnected from its supply. Under these conditions only eddy-current losses and hysteresis losses can be found in the stator. These losses resulted into a temperature rise of 50 degrees! This phenomenon can also be approach from another way, namely look to the required current to obtain a certain value of the torque at increasing speed. The following figure reveals an increase of 50 %. The stator copper loss rises to more then 200 % and one has to deal also with the heating by iron-losses. This coupled phenomenon limits the performance of an ECM!

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J.C. Compter, Electrical drives for precision engineering designs, 2007 32

1

2

3

4

5

6

7

0 500 1000 1500 2000 2500 3000 3500

Current vs speed at a fixed torque

Speed (rpm)

Current (A rms)

Figure 7.4.1, The current at a fixed shaft torque at a rising speed

As indication the T10, T30, T50, T80, and the T100 curves are given for an ECM. The background is that the allowable copper loss is reduced at increasing speed by the fact that the iron loss claims an increasing part of the allowed stator-dissipation.

T10

T30

T50

T80 T100

Figure 7.4.2, T10, T30, T50, T80 and T100 for a motor

We can regard eddy current losses as a viscous friction; a good approach is to say that: Peddy = d.ω2, so that the total losses are:

Q.d+S.d)()-(1+).d+(T

S=P+P

22

eddydiss max2. ≤ωωδωδ

δ 7.4.1

Qmax is the maximum allowed losses, based on the maximum allowable stator temperature. If we employ a duty cycle δ, then Tδ is related to the duty cycle δ and the rotational speed ω as follows:

.d-)1-.(1.d-.d)-Q.(S = ),(T ωδ

ωωδ

δωδ222

max 7.4.2

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J.C. Compter, Electrical drives for precision engineering designs, 2007 33

However the high iron-losses at high speeds are not a principal limitation for the ECM. The application of less powerful magnets as ferrite or a reduced NdFeB-magnet volume allows speeds up to 20000 rpm. However a reduction of the steepness S has to be accepted here. Iron losses are also related with the flux-variation caused by a varying supply current. The current from a PWM-amplifier varies per definition and this introduces additional iron losses. It is stated that the performance of an ECM is also related to the amplifier used; current ripple reduction leads to a higher performance! Reminder: beneath the reduced iron loss one also obtains a reduced copper loss, because the RMS-value of the current drops for the same value if the torque. Summary for the speed limitations of the ECM: 1) at increasing speed less torque becomes available at the shaft, because an increasing part is

absorbed by the stator iron losses, 2) the occurrence of a time delay between the current and the induced voltage by the inductance of

the stator coils. For the EC-DC motor is this effect related to the limited supply-voltage and for the AC-synchronous motor one has to add to this the phase shift over the motor-amplifier.

7.5 SINUSOIDAL OR TRAPEZOIDAL EMF AND THE AMPLIFIER It is possible to combine an amplifier, intended for an EC-DC motor, with an AC-synchronous motor. The background can be a required cost reduction or the availability of a motor with a certain performance. This combination will suffer with a torque fluctuation of at least 13 %! This also holds for the combination of an AC-synchronous amplifier and an EC-DC motor. The number of 13 % can be proven by analysing the torque production of an EC-DC motor at a three-phase sinusoidal supply system.

7.6 COMPARISON MOTOR TYPES

7.6.1 THE IRON ARMATURE MOTOR As the name suggests, it is a feature of such motors that the rotor consists of a wound lamination core. On closer examination we see that the lead-outs of the rotor teeth widen. The function of this widening is to reduce the reluctance of the air gap; in combination with the small air gap (< 0.5 mm) the result is that relatively little magnetic material is needed to achieve field strengths in the rotor plate of 1 Tesla or more.

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J.C. Compter, Electrical drives for precision engineering designs, 2007 34

Figure 7.6.1.1, The iron armature motor

The toothed structure of the rotor gives rise to cogging, which are reduced by skewing the rotor slots in the more expensive motors.

Strengths • The power density, defined as Pmech/volume, is by far the highest, because the magnets only

have a small air gap to overcome. The rest of the flux path consists of the magnetically highly conductive iron. The end result is less weight, less volume and low price.

• The iron guarantees mechanical robustness; the windings are anchored in it by means of a moulding resin. Considerable attention has generally been given to secure fixing of the winding head and to the connections to the collector. A high resistance to centrifugal forces and accelerations is achieved in this way.

• The thermal capacity is great because of the presence of the rotor iron; short-time peak powers are consequently readily absorbed. The small dimensions resulting from the high power density do generally result in the thermal resistance Rth1 being slightly higher.

• A low mechanical time constant can be achieved through the slim structural shape of the rotor in combination with the high power density; < 5 ms is feasible.

• The large number of producers means that “second sourcing” is possible at good prices.

Weaknesses • The rotating iron produces cogging in addition to eddy current and hysteresis losses. A

reduction of cogging can be obtained by applying skewing of the slots; a full removal of the cogging is rarely possible by tolerances on magnet dimensions, positioning, strength and homogeneity.

Figure 7.6.1.2, Skewing of an iron armature.

• The fact that the conductors are located in the slots of the iron rotor gives rise to a substantial

self-inductance of the rotor windings. This makes the commutation of the rotor coils difficult.

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It also gives rise to a time constant in the control loop; with fast torque changes the self-inductance means that the amplifier must have a safety margin on the voltage if it is to overcome L.di/dt.

• The drawbacks of the iron armature motor will not be found in the hollow rotor motor. In this motor the windings are located in the air gap and together form a cage that has been made into a solid entity with resins. This cage has then been fixed to the shaft of the motor.

7.6.2 THE HOLLOW ROTOR MOTOR Inside the cage there is a stationary magnet. The tolerances of the cage force suppliers to select a large air gap. Because of this structure the reluctance for the magnets is factors greater than in the iron armature motor; the magnetic field must cross four air gaps and pass the cage twice. Achieving acceptable field strength requires large magnets in these motors.

Figure 7.6.2.1, The hollow rotor motor

Strengths • No iron losses and no cogging. • A small moment of inertia J, because in this case the iron is not turning in the same direction. • The lower self-inductance moves the commutation limit to higher values. The self-inductance

is lower because there is an air gap on both sides of the windings. In addition, the magnets located on one of the two sides also behave like air. (The permeability µr of magnets is very similar to air.)

Weaknesses • The power density is a factor 3 lower than that of comparable iron armature motors, because

appreciably larger magnets must be used. • Greater vulnerability to peak torques and high speeds. The cage construction has lower limits

than the iron armature rotor. • In the absence of the rotor iron the thermal capacity is low. At the same time the thermal

resistance Rth1 is moderate because of the large air gaps and the impeded heat dissipation via the shaft. This motor is therefore better suited to short-time loads. An increase of 3* in the thermal load can be achieved by the introduction of a forced air current via a separate blower.

7.6.3 THE DISC ARMATURE MOTOR The feature of this motor is an axially oriented magnetic field and a disc-shaped rotor. The rotor disc can consist of rotor windings, which have been made into a solid whole with epoxy. Another form of construction is a printed circuit board on which a track pattern has been etched.

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Magnets are placed on both sides of the disc to obtain a sufficiently strong field.

Figure 7.6.3.1, The disc armature motor

Strengths • No cogging, no iron losses. • A small moment of inertia and a high value of S, possible through the use of a lot of magnetic

material, lead to a low mechanical time constant. • A very low self-inductance and electric time constant, because of the existence of an air gap

and magnetic material respectively on either side of the copper. The commutation forms hardly any limitation in the motor performance.

• Because of the large rotor area the thermal resistance is low. The low thermal capacity of the PCB version is partly offset by the high permissible temperature of 155ºC. The epoxy version has a temperature limit of 110ºC, but does have greater thermal capacity.

• The better controllability of the dimensioning leads to a higher permissible shaft speed, because imbalance and distortion come less into play; 8000 - 10,000 rpm occur. Extremely robust.

• The flat structural shape makes combination with an encoder, brake or tacho for example easier.

Weaknesses • The low power density and low values for S/volume and S/mass are a consequence of the

voluminous stators. Partly because of this prices are high. • Usually AlNiCo is used as magnetic material. It is a material with a high sensitivity to

demagnetisation. Peak currents must therefore be controlled, it being important to remember that these motors have hardly any damping effect by way of a high self-inductance.

• To make re-magnetisation possible by the user, the manufacturer generally has the magnetisation windings located around the magnets.

• Depending on the type of PWM amplifiers, the low self-inductance can prevent the amplifier working properly. The low self-inductance means that there are sharp variations in current. Even with a torque T = 0, this leads to unnecessary ohmic losses; it may be necessary to take the step of using (voluminous) smoothing coils.

• The good control engineering properties mean that the disc motors are frequently used for robots and in servo-controlled machines. But the latest developments in the field of brushless motors (AC synchronous and induction motors) mean that this choice is no longer as self-evident as it was.

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Aspect vs. motor type Iron rotor Hollow rotor Disk rotor Brushless

Power (Watt) 10 - 3000 1 - 200 100 - 5000 1 - 20000 S/J (relative) 100 % 30 % 70% 150 %

τm (msec) 5 - 20 1 - 30 5 - 20 1 - 30 τe (msec) 1 - 3 0.2 - 0.5 0.1 2 - 10 nmax (rpm) 3000 5000 8000 > 4000

Commutation - + ++ Torque ripple yes no no yes

Iron loss yes no no yes Lifetime (hour) 3000 3000 5000..20.000 30000

Therm. cap. + - +/- ++ Max. temp. 155 80 - 110 100 - 150 155 Price/Watt ++ - +/- + Robustness + +/- + ++

7.7 LOAD CASES

Static load If the servo loop has the task of driving a load at an almost constant torque and speed (a static load), it is easy to show in Fig. 7.7.1, where the working point of the motor is. Verification of the maximum permissible value of the torque and the speed is followed by a check on Trms, which must be less than T100. And if the point Trms/nominal speed is also between the lines of Popt and ηopt, then we have a suitable motor. If this is not the case, a transmission can provide a solution. Let’s say that we have the working point 1 in Fig. 7.7.1. For the mechanical power we have Pout = ωload.Tload. A transmission (without losses) with a transmission ratio i leads to Tmotor = i.Tload and ωmotor = ωload/i. The result is that, depending on the value of i, a different point on the curve in Fig. 7.7.1 can be used.

torque

speed

P opt

η opt

P max

UT

maxmax

ωmax

SOA

i

1

Pout =constant

Fig.7.7.1 Influence of transmission ratio

With static loads we have complete freedom to choose such a transmission ratio that the efficiency for example is at a maximum or it is possible for example to suffice with a(n) (available) supply voltage.

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Dynamic load Contrasting with this static load is the pure dynamic load, which is characterised by a load torque that is only used for the constant acceleration and deceleration of the load and motor. This means that the moments of inertia of load and motor in combination with the desired accelerations determine the motor torque required. With a transmission ratio i, the torque required is as follows:

loadloadload

motorloadloadmotormotor .i.J + i

.J = .i.J + .J = T ωω

ωω &&

&& 7.7.1

The transmission ratio i largely determines what torque the motor must deliver. Since the cost price of a motor and the maximum torque are closely linked, a low torque is preferable. The question now is what transmission ratio i must be chosen in order that a minimum torque T suffices. Depending on the load, the curve of the motor torque as a function of the transmission ratio can be a flat or a strong minimum (see Fig. 7.7.2). Differentiation of the above equation to ωmotor leads to the condition that is known as “INERTIAL MATCH". With i = ωload/ωmotor formula 7.7.1 gives us:

0 = .J+i.J

- = i d

T dloadload2

loadmotor ωω

&&

7.7.2

The solution is:

load

motor

JJ

= i 7.7.3

The fact that there is a minimum can be explained as follows: if an extremely large transmission ratio i is chosen, the term in equation 7.7.2 with the moment of inertia of the load will predominate and the torque will grow proportionately to i. This leads to an initial conclusion: reduce i. If this were to go too far, then the term in the formula with the motor moment of inertia would predominate. The reason is that with a low i the angular velocity of the motor is high, so that a lot of energy has to be expended on the acceleration/deceleration of the motor.

torque

speed

SOA T(i)

T(i)i

i

Fig.7.7.2 The curve of the motor torque and speed as

a function of the transmission ratio i for two cases The curve of the motor torque as a function of i is an hyperbola (: 1/i) at low values of i, which leads to the conclusion: increase i. Result: somewhere there is an optimum.

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Show that the motor sees as load torque T:

ωω && )..Ji+ (J = .J = T loadmotortot2 7.7.4

The procedure to follow to see whether a particular motor is suitable for this dynamic load: 1) Determine the transmission ratio that goes with the "Inertial Match". 2) Process the moment of inertia of a suitable transmission (seen from the motor side) in the total

moment of inertia and over one or more strokes see whether the "Inertial Match" can be achieved in combination with available transmissions.

3) After 2) are all the working points of the motor inside the SOA? (maximum torque, maximum speed; this point is also called "Worst-case operation point").

4) Is Trms less than T100? Since T100 falls at elevated speeds with electronically commutated motors, a safe design is obtained if the value of T100 at the maximum speed occurring in the design is compared with Trms.

It is recommended that the curve of the motor torque be drawn as a function of the transmission ratio, because depending on motor moment of inertia, mechanical time constant and required acceleration for example there can be a flat area around the optimum. Analysis of the open loop transfer function of a servo system shows that an advantage can be obtained by the selection of a certain ratio between the moments of inertia of the motor and the load plus transmission. Choosing equal ratio or simply deliberately a relatively light motor also depends on the position of the sensors. The dissertation by Mr Groenhuis (TU Eindhoven) goes into this more deeply.

Static / dynamic hybrid In a similar fashion to the one described above a method can be worked out that describes hybrids. It requires a good description of the load. Since the quantity of calculations is considerable, the use of a numerical tool is preferable, because a choice of motor and transmission ratio must be determined iteratively. The value of i that delivers a minimum torque is affected by the friction and damping present, so the ”inertial match” is not always the best solution.

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8. Voltage Brushless Motors

INTRODUCTION For each servo system containing a brushless servo motor questions arise concerning the amplifier voltage needed. This appendix describes the link between DC-based servo systems and AC synchronous motors.

8.1 DC SYSTEMS For DC voltage systems based on a classical servo motor with brushes there is a well-known formula,

(t)K.+dti(t) dL+i(t).R = u(t) ω

8.1.1

with: u terminal voltage [V] i motor current [A] R terminal resistance [Ω] L terminal inductance [H] K motor constant [N/A] ω rotational speed [rad/s] t time [s]

Taking into account the relation T(t) = K.i(t) for the torque, one can rearrange this expression as:

(t).S+dtT(t) d+T(t)

SK = u(t) ωτ

8.1.2

with S = K2/R (known as the steepness of the motor) τ = L/R, the electrical time constant.

The momentary power needed is obtained by multiplying the previous equation by the current i:

(t).T(t)+dtT(t) d

S.T(t)+

S)T(t

= K

T(t)u(t). = u(t).i(t) = P(t)2

ωτ

8.1.3

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8.2 AC SYNCHRONOUS MOTOR

Figure 8.2.1, An AC synchronous motor

The AC synchronous motor has at least three coils. There is a phase shift of 120 degrees in general between the induced voltages of those coils. First, a single coil is considered for a motor with pole pair p. The coupled flux, generated by the magnet, is given by:

)(p.. = phph θφφ sinˆ 8.2.1

For the torque produced by this single coil we have:

)(p..p. i. = d

di. = T ph

ph θφθ

φcosˆ

8.2.2

Introduction of the factor Kph according to:

phph p.=K φ 8.2.3

The principle of AC synchronous motors is based on maintaining a coil current according to the following expression:

(t))(p.(t). i = t),( i phph θθ cosˆ 8.2.4

Substitution of the current in the voltage equation of a single coil, on the assumption that the inductance does not depend on the angle θ:

dtd

+dt

.iL d+.R i = u phphph

phphph

φ

8.2.5

)(p..K

+)(p.(t).p.i ph-dt

(t)i d(pL(pR(t)i=t,u

ph

phph

phphphph

θω

θωθθθ

cos

sinˆ.ˆ

)cos)cos.ˆ)( +

8.2.6

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Here we applied ω = dθ/dt. Looking to this last expression one sees sine and cosine components.

)(p..p. i. L-)(p . .K+dtid

.L+(t).R i = t),(u phphphph

phphphph θωθωθ sinˆcosˆ

ˆ 8.2.7

To obtain the amplitude of this voltage one differentiates the previous equation to pθ ; then one finds the angle of pθ at which the maximum can be found. Subsequent substitution leads to:

).p. i. (L+). K+dti d

. L+.R i( = U 2phph

2ph

phphphphph ωω ˆ

ˆˆˆ

8.2.8

Before we continue with this expression the definition of the steepness and the motor constant for AC synchronous has to be given. The torque production of three coils together equals:

phphphph

2

0=j

iK23 =

32.j.+(p.iK = T . . . ˆ2)πcosˆ θ∑

8.2.9

The dissipation is given by:

phph .Ri23 = P 2ˆ

8.2.10

The steepness S of a motor is defined as T2/P, so in this case we have:

ph

ph

RK

23 = S

2

8.2.11

A preferred definition of the motor constant K is:

rmsiT = K

8.2.12

The background of this definition is that one can get the motor constant simply by measuring the torque and dividing the result by the rms value of a phase current. With the relation

221. i = i phrms

ˆ 8.2.13

we arrive at:

phph

2

ph

2

R 1.5K =

R 3K=S

8.2.12

with Rphph as the phase-phase resistance. The definition of the steepness S with this K-factor becomes:

ph.K223 = K

8.2.14

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Now we return to the voltage expression; we want to get the peak value of the voltage over two phases, because this voltage determines the required power supply voltage. At the same time we also substitute the motor constant K and the electrical time constant τ = Lph/Rph.

( ) .T.p. + +S.dtdT.+T

SK

32 = U 3 = U

22

phphph τωωτ ⎟⎠⎞

⎜⎝⎛ˆˆ

8.2.15

Where a linear motor has to be analysed, this equation can be derived in the same way; after the introduction of:

PF = S

2

8.2.16

and τm as pole pitch (distance N-N poles) and v as the speed, one arrives at:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ F.v.2. + +S.v

dtdF.+F

SK

32 = U

m

22

phph ττπτˆ

8.2.17

To specify the amplifier one also needs to know the peak output power. Then the power of the three phases has to be included. Let j indicate the phases with j = 0, 1 or 2. For the phase currents we now have:

))3

.j2.+(t)(p.(t). i = t),( i jjphπθθ cosˆ

, 8.2.18

The phase voltages are now:

dtd

+dt

iL d+.R i = u jphjphph

phjphjph,,

,,

φ

8.2.19

The flux of phase j is given by:

)3

.j2.+(t)(p.. = t),( jphjphπθφθφ sinˆ

,, 8.2.20

After some effort one obtains as momentary input power:

(t).T(t)+dt

tdTS

.T(t+StT = t)(x,t).i(x,u = P(t) jphjph

2

0=j

ωτ )())( 2

,,∑ 8.2.21

It is nice to recognise in this equation the similar terms as found for a DC motor given at the beginning of this chapter. When for a linear movement in the x-direction a linear motor is applied the equation becomes:

v(t).F(t)+dt

tdFS

.F(t+StF = t)(x,t).i(x,u = P(t) jfjph

2

0=j

)())( 2

,,τ∑

8.2.22

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Figure 8.2.2, Example of the link between motion profile and amplifier requirements.

CONCLUSIONS

The result of this exercise is that the determination of the supply voltage and power needed for an AC synchronous servo drive can be done by hand and the only input needed is:

• the motor constants K, S, τ and τm or p • the motion profile related force or torque in the form of F(t) , dF(t)/dt or T(t) and dT(t)/dt • and the speed (v(t) or ω(t))

An example is now given that demonstrates the use of the voltage equations for an AC synchronous machine in combination with a third order motion profile.

0 0.05 0.10

1

2Speed vs time

vi

ti

0 0.05 0.10

500

1000Force vs time

Fi

ti

0 0.05 0.10

200

400Phase-phase voltage vs time

Ui

ti

0 0.05 0.10

10

20Current vs time

Ii

ti

0 0.05 0.10

2000

4000Electrical power vs time

Pi

ti

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9. Motion profiles When a load has to be moved from position A to position B within a given time one can apply all kind of speed versus time functions. Arguments to select a certain function might be:

• maximum speed needed • maximum acceleration needed • power dissipation in the motor • a limited jerk (time derivative of the acceleration) • amplifier power, voltage or current needed.

Modern motion controllers allow the application of different motion profiles, which are called trajectories. For systems with a high demand on the accuracy one prefers the third order motion profile, which are characterized with a finite value of the jerk. The background of this jerk limitation is that a jerk energizes the mechanical vibrations in a motion system. The next figure shows three typical examples:

• a triangular speed profile • a trapezoidal speed profile • a third order speed profile

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Figure 9.1, Motion profiles

For the 3rd order speed profile holds that the acceleration pattern can be split in 9 equal time intervals with a specific acceleration function; for the trapezoidal profile holds 3 equal time intervals. When the stroke to be made equals S and one has Ts as available time then holds:

Peak acceleration Amax

Maximum speed vmax

Triangle 2s/T4.S s/T2.S

Trapezium 2s/T4,5.S s/T1,5.S

Third order 2s12.T/81.S s/T1,5.S

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Characteristic values.

acceleration (peak)

speed (peak)

acceleration (effective)

mechanical peak power (F.v)max

triangular 1 1 1 1

trapezoidal 1.13 0.75 0.92 0.85

3rd order 1.7 0.75 1.03 0.95

Normalized comparison for a load without damping and friction. The practical meaning of the preceding table is, that a lower dissipation in a motor can be obtained by applying a trapezoidal speed profile instead of a triangular profile, however then one has to take into account that:

• high jerk values are present again • the maximum acceleration in increased 13 %; this holds also for the amplifier current

The 3rd order speed profile prevents the high jerk, however then the acceleration, and so the motor current, is increased with 70 % and the dissipation with 6 % in comparison with the first speed profile. Despite the higher claims on the amplifier and motor this 3rd order motion profile is the nowadays standard for advanced production equipment. Remark:

• the relation between dissipation and the effective acceleration is quadratic • a parabolic speed profile requires leads to a minimum motor dissipation. However this profile

is hardly used based on its high initial acceleration level of 12.S/Ts2 in combination with a

very high jerk at the start of the motion. To translate the preceding table into voltages and power required from the motor amplifier one should apply the related equations given earlier. The voltage required is obtained with the worst-case operation point, given by the highest acceleration in combination with a high speed. For a triangular and second order profile this is a unique instance of time at the end of the acceleration interval. For a 3rd order profile one has to scan the time interval with a decreasing acceleration and still rising speed. The torque or force in the case of a triangular and second order profile is in general constant between time intervals as indicated in the following figure.

Speed Torque

Time0,0

Worst case OP

tct1 t2 T3 t5 t6 t7t4

T1

T2

T3

Figure 9.2, Torque vs time

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This enables a simple equation to get the effective value for the torque (or force). Instead of using:

,)(1 2∫=ctc

rms dttTt

T 9.1

one can apply:

∑−

=8

1

2 .1i

iic

rms tTt

T 9.2

It prevents integration, however it can only be applied in the case of constant torque- (or force-) levels during the time intervals.

The conclusions are that the motion profile applied influences significantly:

• peak current, voltage and power of the amplifier and • the dissipation in the motor.

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10. The motor amplifier This chapter deals with the motor amplifier with having in mind the application of this amplifier in precision engineering designs.

10.1 LOW COST It is sometimes sufficient to use a simple circuit to control a motor, as shown in Fig. 10.1.1. At the input of the operational amplifier (OpAmp) a reference for the motor current is compared with the actual current. If there is any deviation in the current, the transistor is made more or less conductive by changing the base voltage. The diode parallel to the motor provides protection for the transistor, activating if the transistor blocks so fast that the EMF and the voltage term L.di/dt together reach a value that is too high for the transistor in question. A limitation of this circuit is that only a positive current i can be controlled; this corresponds to a controlled torque in one direction. A torque in the other direction is not possible, because a transistor only allows the current to flow in one direction.

+

I ref

I

Uce

-

+-

Figure 10.1.1, Motor control

10.2 PULSE WIDTH AND FREQUENCY MODULATION The greater or lesser pinching of the transistor means a high level of dissipation for the transistor itself, because the motor current i flows through the transistor continuously and there is a voltage drop uce (determined by the base-emitter voltage) between the emitter and the collector. The transistor dissipation is P = uce.i. Extreme, but very revealing is the determination of the losses at standstill, if the motor has to deliver maximum torque. This soon leads to physically large (and expensive) transistors, which have to be equipped with a cooling block and possibly even a fan. At the same time this approach is not in the least economical of energy, which can be completely unacceptable in the case of battery supply. The solution is to make use of the self-inductance of the motor. If the transistor is driven via the base-emitter voltage alternately within a return time T, fully non-conductive during Toff and fully conductive during Ton, then little transistor dissipation will remain (see Fig.10.2.1). The current im through the motor can be determined via simple first-order differential equations, which show that the ratio Ton/T determines the mean value of the current. There are two approaches. First, to keep Toff constant and to influence Ton via the controller; the cycle time is consequently a function of the desired current; in this case we speak of frequency modulation. The second approach is to keep the cycle time T constant, in which case the ratio Ton/Toff is varied. In this case we have pulse width modulation.

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I

U ce

+

-

U b U

U b

U ce

I m

I m

I P = I U * ce

I d I m I d= + I

T T onoff

T

Figure 10.2.1, Pulse Width Modulation

Notes:

• There are clear similarities to a heating boiler. • The cycle time T is generally much less than 1 ms.

Figure 10.2.2, Example of Pulse Width Modulation Circuit

10.3 FOUR QUADRANT OPERATION Four quadrant operation means that the motor can deliver both a positive and a negative torque and in both directions of rotation. The use of a motor in the four quadrants of the torque-speed plane is possible with the standard circuit shown in Fig. 10.3.1. This circuit is called the “full H-bridge”.

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L

i

ControlI ref

+

-

Li

L

i

A B

Figure 10.3.1, H-bridge circuit Figure 10.3.2, Path of current with positive and negative torque

Fig. 10.3.2 A gives the path for the current with a torque in one direction, Fig. 10.3.2 B for the other direction. Here again one of the functions of the diodes is over voltage protection. The control of the transistors must be such that two transistors can never be conductive in the same branch simultaneously. Pulse width modulation and frequency modulation are the methods of control in this circuit. There are applications in which the motor returns energy to the power supply. In these circumstances the motor is working as a generator and will return current to the power supply via two recovery diodes. If no measures are taken, the voltage across the power supply capacitor C will rise sharply, the power supply eventually suffering damage. This can happen in a crane for example. This shortcoming can be solved by dissipating the energy released in a braking resistor Rbrake, as shown in Fig. 10.3.3. This resistor is used as the occasion arises via transistor Q. Of course it would be even better if the surplus energy went back to the mains. But the cost price of the additional electronics required often seems to be a barrier to such a step.

L

ControlI ref

+

-

C

Rbrake

Q

i

i

TT

T T

T

Figure 10.3.3, H-bridge with energy dissipation Figure 10.3.4 The four quadrants

The presence of self-inductance in the circuit is highly desirable, because it is a means of limiting the ripple in the current. This is important because the dissipation in the copper of the rotor is proportional to irms and the output torque to imean! See for example Fig. 10.3.5, where the torque T is equal to zero. It is not uncommon to place an extra coil in series with the motor to reduce the current fluctuations due to PWM.

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time

I I

time

0 0

high self-inductanceI mean = 0

low self-inductanceI mean= 0

I rms = 1/3 I top

Figure 10.3.5, The impact of self-inductance Note: the definitions of irms and imean are:

dt iT1 =idtiT

1 = iT

0mean

2T

0rms ∫∫

10.3.1

A second reason is that the fluctuations in the current also produce flux variations in the iron and the magnets of the motor. These variations induce eddy current losses, which can mean considerable dissipation if the self-inductance is too low. (Even at standstill and with Tshaft = 0!). Fig. 10.3.6 shows the measured current as a function of time for a moving coil motor in combination with a PWM amplifier without an extra coil. It also shows what we can expect if the motor details given in the product catalogue are used for calculating the current. The extreme difference is due to neglected eddy currents in the magnets and the yoke!

Current with PWM

-0.2

0

0.2

0.4

0.6

0.8

1

1 74 147 220 293 366 439 512

time in microsec.

actual current in amp.

theoretical via L/R

Figure 10.3.6, Current in a moving coil motor without external inductance

(Measurements made available by M. van der Steen, MSD.) Apart from an increase in the dissipation, we must also expect a decrease in the life of the commutation system. A supplier of moving coil motors mentioned that a PWM frequency of more than 100 kHz has to be used to prevent excessive wear, whereas 18 kHz as a usual PWM frequency. The full PWM-amplifier supply voltage is connected to the motor terminal, even at no-load conditions (then 50 % of the time in positive sense and 50 % in the negative sense). It is noticed once that this continuous high voltage resulted into silver migration from a silver plated commutator towards the shaft, leading to a short circuit to earth and finally a serious damage.

10.4 THE TRANSFER FUNCTION OF A PWM AMPLIFIER The ideal behaviour of an amplifier is got when the output voltage or output current follows the input voltage perfectly. There are two effects that we must always consider carefully, that are the bandwidth and the linearity around the zero-axis crossing.

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A standard value of the bandwidth is 500 ..1000 Hz. This may seem a lot when we think that for a speed control a bandwidth of 100 Hz is a high value. But what we have to remember is that long before the bandwidth a phase shift occurs that can be an impediment for control engineering.

Figure 10.4.1, General equivalent circuit for an amplifier/motor circuit.

The analysis of this system is simplified by the introduction of:

ampeampampamp RRRLCR /,,. === γττ

10.4.1

The transfer function are as follows:

)1)(1.(1

)1(

eamp

ampamp

amp

ss

sR

EII

ττγ

τ

+++

+−

=

10.4.2

First look to the transfer function I/Iamp.

Figure 10.4.2, I/Iamp To be concluded with the given numbers: more than 10 degrees lost below 200 Hz. Now look to I/E.

L

R

R C

E

I

I+

+

τ e 0.013= τ amp 6.283 10 4= Ramp 1 103= γ 0.01=

Ampi

fi

0 100 200 300 4000.8

1

1.2

1.4I/Iamp

Frequency

Am

plitu

de

Phasei

fi

0 100 200 300 40030

20

10

0I/Iamp

Frequency

Phas

e

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Figure 10.4.3, I/E

Conclusion is that the induced voltage causes a current via the output impedance; a rising frequency of the induced voltage leads to a rising torque or force in the case of a coupled linear motor. The translation to precision engineering design is that the torque (or force) level is influenced by vibration of the stator. In other words, frame vibrations are transferred to the load of the motor via the finite output impedance of the amplifier. (Sven Hol, ASML, thesis to be published) Another consideration is that with electronically commutated motors the phase shift occurring makes the generation of the torque less efficient at a rising speed (see Chapter 7). The last remark related to the output impedance of the current source amplifier is, that its presence leads to a rising sensitivity for the load impedance on the bandwidth of the transfer function motor current over the amplifier set point. Linearity is the second effect. Fig. 10.4.4 and Fig. 10.4.5 show the transfer of a 15 Ampere amplifier near the zero-axis crossing for a sine of 10 Hz and an output current amplitude of 0.6 A. For a control this phenomenon means that the transfer function around the zero-axis crossing is not well defined; as frequency increases, this distortion becomes more pronounced. The cause of the distortion lies in the imperfect behaviour of the power transistors of the H-bridge, which cannot be fully controlled by the internal feedback within the amplifier. So, the linearity around the zero-crossing should be a specification point.

PM3380A

ch2

ch1ch2: dT=-----s V2=- 148mV

CH1 10mV= STOP AVGCH2 0.1 V= MTB20.0ms ext+

1

2

PM3380Ach2: dY= 584mV

STOP AVGCH2 0.1 V= X= CH1 10mV=

2

Figure 10.4.2, Input voltage and the current as

function of time Figure 10.4.3, Input voltage versus

current

R 10=

Ampi

fi

0 100 200 300 4005 10 4

0.001

0.0015

0.002

0.0025I/E

Frequency

Am

plitu

de

Phasei

fi

0 100 200 300 40030

20

10

0I/E

Frequency

Phas

e

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10.5 GAIN ERRORS AND OFFSET The gain of a motor amplifier is one of the terms in the open loop gain, so variations caused by e.g. temperature dependency should be investigated in relation to the control stability. Offset in the case of a single phase amplifier for a motor with brushes or an actuator are in general no problem, because the I(ntegrator) action of the control will compensate this. A serious attention has to be given to gain errors and offsets in the case of an electronically commutating motor. In the case of a DC-brushless motor one will notice stepwise torque changes near the commutating positions when those amplifier errors exists. For an AC-synchronous motor the more complex consequences has to be described with equations.

For the torque produced by one phase ph holds:

)(p..p. .i = d

d.i = T phph

phphph θφ

θφ

cosˆ 10.5.1

Introduction of the factor Kph according to:

phph p.=K φ 10.5.2

Let us assume deviating relative gains, ∆Kamp(ph) and offsets Ioffset(ph) per phase.

(ph)I)ph(t)(p..(ph)K(t) i = ph)t,(i offsetampphph ++∆+3.2cos1ˆ, πθθ

10.5.3

The torque production of three coils together equals:

=

=

=

+∆+

⎥⎦⎤

⎢⎣⎡ ++∆+

2

0

2

0

2

0

)cos

2)cos

)cos

).cos

3.2)

3.2ˆˆ

)3.2cos1ˆ.

3.2

,3.2

phoffsetph

phphampphphph

phoffsetampphph

phph

2

0=ph

ph(phIK

ph(t) i(ph)KKiK

(phI)ph(t)(p..(ph)K(t) iphK

= ph)t,(iphK =

+(p.

+(p.23 =

+(p.

+(p.T

.

..

.

.

π

π

πθπ

θπ

θ

θ

θ

θ

10.5.4

The conclusion is that gain errors and offsets are introducing position dependency in the torque. The frequency of the gain error related disturbance is twice the frequency of the offset related disturbance. So one has to analyse carefully the consequences of these amplifier errors when precision engineering drives are designed. As an initial value one can use 2 % gain error and 2 % of the maximum current (or voltage) as offset errors. In graphical form one gets now including the output impedance Z of the current source amplifiers the following figure for a three phase system. By assuming a star point connection

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of the three phases of the motor it is allowed to take the third current of the motor as the sum of the other two currents.

Θ

Offset1(θ)

Tref

Offset2(θ)

K1

K2

Figure 10.5.1, The amplifier errors combined with a thermal model of a rotating

brushless motor. A general representation of the amplifier, taking also into account the bandwidth of the current source is as follows.

Offset( t)θ,

K( )∆,θR ( , )∆ θ m r

L m

K( , )∆ θ ω

LoadAmplifier

Bandwidth Gain Offset Impedance Figure 10.5.2, The amplifier as control loop component.

The frequency dependent behaviour of the current source and offset can be measured when the current amplifier output is short-circuited. The measurement of the output impedance should be done by adding a controllable voltage source in series with the amplifier output and load. Fix the current amplifier input on zero, make a frequency scan with the controllable voltage source and measure the current running through the source.

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10.6 CABLES When designing a system with a PWM power supply the designer must design to a short distance between the PWM power supply and the motor. The reason is that the wires to the motor are connected alternately to the + and the – voltages of the power supply respectively with a high frequency and steep edges. This causes electro-magnetic interference in the surrounding area, which can for example affect sensor lines (the incremental encoder!) or prevent the installation from meeting inspection standards (= approbation). The subsequent installation of shielding involves additional expense through time lost as a result of the fault finding and the costs of re-cabling. The minimisation of the electro-magnetic interference is just one reason; another is the electrical resistance of the cable. For a 20-metre cable (out and back), with 30 wires of 0.2 mm section as conductors, this amounts to approximately 0.7 Ohm (without taking in to account a certain temperature rise). A servomotor with currents of 10 Ampere is no exception; this means a 7 Volt voltage loss over the cable and a 70-Watt loss. For the power supply, usually a servo amplifier, this means a higher specification in terms of voltage and power to be delivered! If we also think that a power supply voltage below 50 V does not usually mean any special requirements for approbation, the selection of a sufficiently thick cable can avoid many arguments. With voltage loss in mind attention must also be paid to the resistance of thermal fuses and the number of connectors in the cable between the amplifier and the motor. This latter point can involve a conflict with the division of a drive system into separate modules for logistical or service reasons.

10.7 THE SMALLER THE BETTER? The cost of a servomotor is of course linked to its size. Also the available space in equipment is forcing designers to apply the smallest motor possible in many cases. The application of a transmission is one way as far as the backlash, additional friction, the stiffness reduction and the higher motor speed can be accepted. The result is that the motor torque required is reduced and then one can go for a smaller motor, because the motor volume is linked with the torque capability. Also in Chapter 5 it is concluded that a motor should be used between 0.5 ω0 and ω0 to combine a high efficiency and mechanical output power. So one should always look if a transmission leads to a smaller motor. But one should also include the cost of the amplifier; the smaller the motor the more power is needed to get the same mechanical output. So profit at the motor side is lost at the amplifier side. Additionally holds that a smaller motor leads to a higher operating temperature, so more heat is entering the equipment. The related drawbacks are potentially:

• shorter lifetime of the brushes and bearings • shorter lifetime of the electrical insulation • the rising heat leads to thermal expansion in the mechanics being a risk in high accuracy

systems • the positive effects of a well tuned feed-forward is partly lost by the temperature dependent

motor constant K. So a transmission should be investigated always, but additionally using a motor at its limits leads to a rising development effort.

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11 Linear motors and actuators

11.1 INTRODUCTION Linear motors and actuators are the most suited drives for advanced machines with a high accuracy and fast reaction time. An example is the lens actuator of a CD-player. This actuator type can be considered as an extreme example of a mass product with a short reaction time in combination with a high accuracy. The control loop bandwidth of this system goes up to 2 kHz.

Figure 11.1, CD-lens actuators, 1995 and 2001

Looking to data sheets of linear motors and actuators one gets the impression that the right choice for an application can be made on the base of a small set of considerations. Target of the following chapters is to teach potential users the meaning of the keywords in the linear motor and actuator technology and to link their application aspects to the characteristics of linear motors and actuators and to find a fitted amplifier.

11.2 THE APPLICATION FIELD OF LINEAR DRIVES

The linear drive has got a clear position in the servo-technology for direct driven systems in the last decade. The main characteristic of a direct driven system is the absence of a mechanical transmission between the drive and the load. A mechanical transmission is for example a set of gear wheels or a gearwheel in combination with a toothed bar. This last combination can be used to transfer a rotation into a linear movement. Wear, play, a limited stiffness, hysteresis and friction are often performance limiting factors related to mechanical transmissions and the direct drive is the answer. Examples of direct drives are:

• a pneumatic of hydraulic cylinder • a linear electric motor • an air coil actuator or so called moving coil actuator • a piezo-actuator.

Pneumatic and hydraulic solutions are preferred when force levels of several thousands of Newtons are required and when the costs and volume of the compressor and pump respectively can be accepted. The accuracy and reaction time of these systems can be brought to the same level as an electric drive by the selection of fitted sensors and control valves. However this is outside the scope of this treatise.

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Piezo technology can be considered as a mature technology for small movements; small because the contraction of piezo crystals is near 0.5 %. It is possible to realize a long stroke and accurate drive with three piezo crystals; the operation principle is like a human body, lift one foot, shift this in the required direction, put the foot on the ground and do the same with the other foot. This drive principle is highly suitable for electron microscope by the absence of interfering magnetic fields. However the limited speed and costs of those drives are arguments to search for alternatives many times. Lifetime issues by wear and fatigue are also met. Linear electrical drives and actuators are an alternative. Force levels exceeding 1000 N, speeds beyond 2 m/s and accuracies better than 1 micrometer are nearly on stock products. The combination of all these numbers is not so easy, because then the full system behaviour starts to be important. The behaviour of the bearing, sensors, mechanical dynamics, the control, power electronics starts to be important for short time intervals and thermal effects might disturb the absolute accuracy. Attention will be given here to linear motor and actuators for servo systems. The principal difference between these two is that the stroke of a linear motor can be extended with limited consequences. E.g. the extension of the magnet strip of a linear motor is all needed to get a longer stroke. The definition of a servo system is not given until now. Within the context of electrical drives we consider a drive as a servo system when the object to be moved has to follow accurately a changing (electronic or software) reference. This reference might be linked to the acceleration, speed or position. Control engineering learns that accuracy is gained when the control behaviour of the loop components is predictable. Electric motors and actuators with permanent magnet are the ideal components within this respect and the rising strength and falling costs of modern rare earth magnets (Samarium-Cobalt and especially Neodymium-Iron-Boron) have given these drive components a very competitive position.

Figure 11.2.1, A gantry with linear motors

11.3 CONSEQUENCES OF A DIRECT DRIVE

The lack of a transmission means that the force and speed have to be the same for driving motor or actuator and the load. The required force level determines highly the costs of the motor/actuator with as consequences that one should not expect reduced motor costs in comparison to a solution with a rotating motor and a rotation to linear transmission. But direct drive means also one looses the acceleration torque needed for the transmission and an example is known where the total costs of a spindle driven solution was comparable with a linear direct drive.

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Usually a linear encoder is applied as position sensor and those costs exceed highly the costs of a rotating encoder. At the other hand the advantage obtained is that less mechanics can be found between the load and the sensor, so the sensor reading reflects more accurate the load position. Standardization is brought to a high level for rotating electric motors and one can find easily alternative systems from competitors. This does not hold for linear motors and the rising integration of the motor, amplifier, bearings and position sensor makes it in general difficult to switch from one supplier to another. The design of a linear drive system should start with an analysis of mechanics to locate the reaction forces and the centre of mass, because the fast rising forces of linear motors and actuators (e.g. rising from 0 to 1000 Newton in 5 millisecond) will initiate easily all vibration modes in a mechanical construction. The transmission between a rotating motor and the load reduces the disturbance sensitivity for forces acting on the load. It is seen many times that an increased control loop bandwidth has to be used for a direct driven system to get a similar disturbance rejection. Chapter 12 will give attention to actuators; chapter 13 deals with linear motors and chapter 14 gives a critical view on the motor constants in general.

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12 Actuators The most famous electrical actuator is the electro magnet of a doorbell. However one will meet this rarely in servo systems because its main characteristic is a force that depends on the ratio of the current and the air gap to the power of two. Control engineering rules on stability are valid only for linear systems and this is the reason why actuators with a linear transfer function are preferred. Examples of linear transfer functions are:

• the force linearly related to the current, • a displacement linearly related to the voltage applied • the speed linearly related to the voltage applied.

An additional preference is that the actuator should not have preferred positions as can be seen when a supply voltage is removed. An example is the cogging torque present in fan motors and in toy motors. Those torques are external disturbances in the eyes of a control engineer, which has to be counteracted by an additional control action with at least a transient error during set point changes. Exceptions in this respect are the electronic throttle of fuel engines and hoisting equipment, where safety has to be guaranteed when the electric supply fails. First attention will be given to the electro dynamic actuator, which can be found in loudspeakers, CD-players and many servo systems.

12.1 ELECTRO DYNAMIC ACTUATORS The electro dynamic actuator is characterized by a current carrying coil in the field of a permanent magnet where holds that the coil moves relatively with respect to the magnet. The design with a moving coil is widely spread, because then one will find:

• the highest ratio possible between the force and the moving mass • a good linear relation between force and the current and • no preferred positions.

Figure 12.1.1 shows the geometry consisting of:

• a magnet and an iron yoke, which concentrates the field in a gap • a coil in this gap • a coil carrier, which transfers the force to the load and this carrier also removes the heat from

the coil to the ambient and the yoke. This description allows many forms, but only the rotational symmetric geometry will be discussed here.

Figure 12.1.1, Electro dynamic actuator

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Figure 12.1.2, Examples of electro dynamic actuators

The principle of operation is based on the Lorentz force. The power balance will be used to get the force instead of using the Lorentz approach, because the power balance is always valid, whereas the Lorentz force only holds for moving charges in free air. The voltage equation for a coil is:

dt dN. + R.i =u φ

12.1.1

The flux φ consists of two parts, the first part given by L.i and the second part induced by the permanent magnets. The values of this last part depends on the relative position x of the coil with respect to the magnet. With Figure 12.1.3 holds now:

N.B.l.x + L.i(t) = N.φ 12.1.2

X

Figure 12.1.3, A coil in the field of a permanent magnet system The time derivative of the coupled flux φ.N satisfies:

dtdiL. +

dtdx

xN.=

dti d

i +

dtx d

xN. =

dt dN.

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ φφφφ

12.1.3

One should be aware that this last results only holds in the case that the inductance L does not depend on the position x. The next step to come to the force is the multiplication of equation 12.1.1 with the current i to get the power balance, with as result:

dti di.L. +

dtx d

x i.N. + .Ri=

dt di.N. + .Ri =u.i = P

∂∂ φφ 22

12.1.4

Rearrangement gives:

dt)L.i (0.5d +

dtdx

x i.N. + .Ri =u.i

22

∂∂ φ

12.1.5

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The left hand term represents the incoming electrical power. The electrical dissipation Pdiss is given by i2.R and the change of the stored magnetic energy is linked to the last term. The second right hand term must be consequently the power going to or coming from the mechanical part of the system.

One has to follow the following line of thinking to prove that the stored magnetic energy is given by 21

2 Li . Suppose that a switch close at t=0 to connect a coil to a voltage source with U as voltage. The

current in the coil rises according ( ) Ui t tL

= . At t=T holds TUI TL

= .

The stored energy equals: 2

2 212

0

( ) . ( ) .2

T

TUE T U i t dt T L I

L= = =∫

Mechanical engineering learns:

dtx dF. = vF = Pmech .

12.1.6

The combination of the equation 12.1.5 and 12.1.6 leads to the force:

x i.N. = F

∂∂ φ

12.1.7

The ratio between the force in [Newton] and the current in [Ampere] is called the force constant Kf ; this ratio becomes:

xN. =

iF = K f ∂

∂φ

12.1.8

The voltage equation 12.1.1 can be written also as:

.dtdx

xN. +

dtdiL. + i.R = u

∂∂φ

12.1.9

This last term is well known as the EMF (the electro motoric “force”). The word motoric force is based on the fact that a movement is required to get the voltage. It should be noted that the second term is based on the change of the current level!

The EMF can be rewritten as:

dxEMF = N. = N.v.x dt xφ φ∂ ∂

∂ ∂

12.1.10

The ratio between the EMF in [Volts] and the speed in [m/sec] is usually indicated as Ke with the equation:

x N. =Ke ∂

∂ φ

12.1.11

Ke [V sec m-1] and Kf [N Amp–1] are clearly equal to each other with the given S.I. units and from now on we will use for both the motor constant K. Question: Prove with the preceding equations that the mechanical power is given by: Pmech=EMF.i .

Equation 12.1.11 gives implicitly a way to determine the motor constant K. Suppose one connects the coil with an electronic integrator and starts moving the coil while measuring the position x, starting from x0. The output of the integrator will be:

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0

int 00

( ) . . ( ( ) ( ))t x

offset offset offsetx

U t N v dt U = N dx U N x x Ux xφ φ φ φ∂ ∂

= + + = − +∂ ∂∫ ∫

12.1.12

The derivative of Uint to x has to be obtained by post processing the data and now holds:

int ( ) ( )dU t d xNdx dx

φ=

12.1.13

This means, that the motor constant can be determined without having the usual troubles with friction in the case one uses a force sensor.

12.2 FORCE AND DISSIPATION The temperature of the coil is always the performance-limiting factor in electro mechanics. The cause of the temperature rise is the dissipation in the current carrying coil in relation to a finite thermal conductivity for the heat flow to the ambient. This paragraph will spend attention to this performance limitation. The time dependent force F [N] in servo systems is clearly linked to the dissipation. For the current holds by using the motor constant K [N/A]:

( )( ) .F ti tK

= 12.2.1

With R [Ω] as the electrical resistance of the coil one gets as time averaged dissipation:

∫∫ ==cc T

c

T

cRdttF

KR

TRdtti

TP

0

22

0

2 .)(1.)(1 12.2.2

The time Tc is the cycle time of the process to be controlled. To link the requirements of the mechanics to the dissipation P we introduce the effective force Frms as:

FT

F t dtrmsc

Tc

= ∫1 2

0

( ) , 12.2.3

It is also very helpful to introduce the steepness S=K2/R. The value of Frms is determined by the load characteristics (mass, damping and a spring constant) and the movement to be realized in time (position, speed and acceleration). The steepness S is a figure of merit for each actuator. The combination of the introduced variables gives:

P FSrms=2

. 12.2.4

It should be clear by the preceding text that one always needs the peak force level and the effective force Frms to determine if an actuator is fitted for a job.

Remark: in English data sheets one will meet many times the constant Km, which is equal to S and called the specific damping. The link between damping and this constant is that a short-circuited actuator gives S [Ns/m] as damping. The induced voltage EMF leads to a current I, which gives finally with the motor constant K an opposing force according:

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.. . . . .EMF K vF K I K K S vR R

= = = =

12.2.5

This relation between force and speed can be found also for viscous dampers. So a short-circuited permanent magnet based actuator behaves like a viscous damper.

12.3 THE VOLTAGE The voltage equation of an actuator can be written as:

U t i t R L di tdt

K v t( ) ( ). ( ) . ( ).= + + 12.3.1

L [H] is the self-inductance of the actuator. For this type of actuators holds in general L/R ~ 1 millisecond. The speed of coil is given by v(t). Substitution of F=K.i leads to:

U t F t RK

LK

d F tdt

K v t( ) ( ). ( ) . ( ).= + + 12.3.2

The application determines the functions v(t) and F(t), so these are required to get the maximum voltage needed. The motion profile determines v(t); Figure 12.3.1 gives the common motion profiles. F(t) is coupled with v(t) via the mechanical behaviour of the load in terms of mass, friction and stiffness.

Triangular Trapezoid Third orderAcceleration a(t)Speed v(t)Position s(t)

t t t

Figure 12.3.1, Motion profiles

12.4 THE STROKE AND THE FACTOR K The consequence of displacements is of course a relative moment of the coil with respect to the yoke. The motor constant K is in general a function of the relative position and this position dependency can be reduced by extending the length of the coil or by extending the length of the gap in the axial direction. The first solution leads to more dissipating copper without an increase of the force and the second leads to a bigger yoke and magnet. A position dependency of K of 10 to 20 % is hardly a risk for the control loop stability. However when one applies a feed forward for e.g. the acceleration force one will over- or under compensate the acceleration force leading to a rise of the servo errors. The conclusion is that one has to specify the stroke and the allowable position dependency of the motor constant K before ordering or designing an actuator.

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12.5 HEAT TRANSFER The dissipation in the coil leads to a temperature rise, which is limited by:

• the maximum allowable temperature of the wire insulation class (90 tot 220 ºC), • the temperature dependent mechanical stability of the synthetic materials used • the decreasing strength of permanent magnets at rising temperature (0.1 .. 0.2 %/K for

modern rare earth magnets. Additionally holds that the heat flow to the ambient might be restricted and that the copper resistivity rises with 0.4%/K (so 60 degrees temperature rise means a rise of the coil resistance with 24 %). This kind of considerations leads to a certain allowable temperature rise θmax for the coil. The following figure represents the thermal model of an electro dynamic actuator, including the heat capacity of the coil and the yoke. For the sake of simplicity we will only analyse the steady state behaviour allowing us to forget the heat capacities. The assumption is also made that Re can neglected, what is allowed when the coil is not fixed to a heat conducting body or frame.

i .R( )θ Cu

2

θCu θYoke

θamb

R

RC Cth,1

th,1

th,2th,2

Re

Figure 12.5.1, Thermal model

The ambient temperature is indicated by θamb and the total thermal resistance between copper and ambient by Rth [W/K]. The temperature of the copper follows with:

,1 ,2. .( ).Cu amb th amb th thP R P R Rθ θ θ= + = + + 12.5.1

Rth,1 is the thermal resistance between the coil and the yoke, Rth,2 is the thermal resistance from the yoke to the ambient. The first one, Rth,1, is determined by:

• the width of the air gap at both side of the coil • the material and surface of the coil carrier (paper or aluminium) • the winding technology applied (“wild” or orthocyclic) • the speed of the actuator with respect to the yoke • the thermal contact between the wires and the coil carrier • the thickness of the insulation layer on the wires • the altitude

The inner and outer surface of the coil and coil carrier are transferring the heat under the assumption the Re can be neglected. When the total surface is given as A, we can introduce the specific heat conductivity λ according:

λ =1

A Rth..

12.5.2

For a vertical plate in free air holds approximately λ=10 W/m2K; other shapes like a cylinder can go up to λ=14 W/m2K. The presence of cool surface on a short distance will reduce the thermal resistance significantly. The value of Rth,2 depends on:

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• the external surface and shape of the yoke and magnet • is the yoke mounted to a heat transferring frame • the presence of a forced airflow.

The position dependency of the motor constant K is discussed in the preceding paragraph. In this paragraph is introduced the temperature dependency of the resistance. Both are influencing the voltage and power consumption, as described in Chapter 6.

12.6 MECHANICS Electrical aspects are linked now to the thermal aspects. Mechanical aspects are not requiring special attention, because the material stress remains in general far below a dangerous level. The most critical part is the coil, which can be made very robust by applying the orthocyclic winding technology in combination with melting the wires to each other by means of a thin nylon layer on top of the wire insulation. The only exception is that the shear stress on the glue layer between the coil and the coil carrier might become a danger at a maximum coil temperature. The electro dynamic actuators described are intended to linear movements. The application of a linear air bearing system is an expensive solution (investment in air equipment, maintenance and energy consumption) with as additional penalty the weight of the moving member of the air bearing. An alternative is a leaf spring construction, but this starts to be bulky when the stroke exceeds some millimetre. Many times a solution is found by applying a pivot point, with the attractive option that the position of the pivot point with respect to the actuator and load can be used to maximize the system performance. Roller bearings are nice when they are rotating over big angles, but they are not fitted for small repetitive rotations. A set of leaf spring acting as a pivot point is here the solution. Those components are commercially available and this paragraph is directed to the bearing system A weak spot of those leaf spring designs is a limited stiffness for forces, which gives in combination with the mass eigen-frequencies and position errors. So, for a design as given in the next figure, one should minimize the forces in the pivot point. For wire bonding an ultra sonic transducer has to land on a silicon dye to fix a wire on the surface of the dye. It is preferred to minimize the virtual mass as seen by the landing area to prevent any damage during landing. Now we have two design objectives. A third objective, minimizing the peak force required, will be added in the next paragraph.

J ,m

l

rc

vbbeam

Figure 12.6.1, A tool.

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Figure 12.6.2, A wire bonder Suppose that the transducer is fixed at the end of a homogenous beam with l as length and m as mass. The question is to determine the position of the pivot point giving a minimum impact force and force acting on the bearing. The inertia according Steiner equals:

2 2 2112. . . ,beamJ J m r m l m r= + = + 12.6.1

Jbeam is the inertia of the beam in its centre of mass. The impact mass at the landing zone is given by:

m Jr l

m l rr limpact =

+=

++( . ).

( . ).

12

2

2 112

2

12

2 12.6.2

Lets us assume as stiffness of the landing zone c and assume as landing speed vb. The impact force follows as the result of energy conservation:

2 2112

212

.. . . . .( . )b impact bl rF v m c v m cr l

+= =

+

12.6.3

Differentiation to r gives at r = 1/6 l a minimum impact mass mimpact = ¼ m! The force acting on the bearing is the next question. The force F de-accelerates the beam:

22 13 6. . ( .( . ) ). .beamF l J m l ϕ= + && 12.6.4

The angular de-acceleration becomes:

&& ..

.ϕ =6 Fm l

12.6.5

This means that the centre of mass get a linear de-acceleration according: 16 . .a l ϕ= && 12.6.6

The vertical force is:

. .mF m a= 12.6.7

Substitution of 12.6.5 and 12.6.6 in 12.6.7 proves F=Fm, so the landing force is fully spend to the de-acceleration of the centre of mass. No force is left on the bearing and excitation of vibrations on the bearing is prevented.

capillary

crystal

goldball

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The acceleration of the ultra sonic transducer in a wire bonder is pre-described to fulfil the requirement of e.g. 14000 components per hour. The impact mass and bearing force must be minimized and now the actuator is added and of course a minimum peak force is preferred, because this allows the smallest actuator and amplifier. There is no doubt that the costs are reduced in this way, it is also likely that the highest control loop bandwidth can be reached with this line of thinking. Give the best values of r1 and r2 is the task.

a

r r2 1

Jm

workworkmbmbm

x1x2

act 2 1 Figure 12.6.3, Actuator, beam, pivot point and the load.

The derivation of the analytical equations involved is not too complex. The following issues should be taken into account:

• the cross section of the beam is kept constant, • the mass of the moving coil is in linear relation to the peak force required and • the mass and inertia of the transducer is fixed.

The enclosed Mathcad program calculates as a function of r1 and r2 the impact mass, the peak force of the actuator and the force on the bearings. The following graphs are the result with along the horizontal axis r1 and along the vertical axis r2. Figure 12.6.4 indicates as optimum r2=3 cm and r1=8.5 cm, figure 12.6.5 r2=4 cm and r1=8 cm and finally figure 12.6.6 r2=3 cm and r1=9. It is nice to notice that the three objectives are not in conflict with each other.

Mimpact0.07 0.08 0.09 0.1

0.03

0.04

0.05

0.06

0.070.040.035

0.030.025

0.02

0.02

0.015

0.015

0.015

0.010.01

0.01

Figure 12.6.4, The impact mass in [kg] versus r1 (horizontal) and r2 (vertical) in [m].

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Fact0.07 0.08 0.09 0.1

0.03

0.04

0.05

0.06

0.071110

9

8

8

8

7

7

7

7

6

6

6

6

55

5

Figure 12.6.5, The actuator force in [N] versus r1 (horizontal) and r2 (vertical) in [m].

FLZ0.07 0.08 0.09 0.1

0.03

0.04

0.05

0.06

0.07

0

0.005

0.005

0.01

0.01

0.015

0.015

0.02

0.02

Figure 12.6.6, The bearing force in [N] versus r1 (horizontal) and r2 (vertical) in [m].

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σalu 2.7 103. kg. m 3. specific weight Alu

a 275 m. sec 2. specification tip acceleration

α r1( )ar1

resulting angular speed

x1 0.078 m. distance mass centre tool and tipx2 0.05 m. distance end of the beam and tooMwerk 0.072 kg. mass tool

Jwerk 0.25 10 4. kg. m2. inertia toolh 0.015 m. beam heightb 0.008 m. beam widthmb2 r2( ) σalu h. b. r2. beam mass left sidemb1 r1( ) σalu h. b. r1 x2( ). beam mass right side

cmot 2 10 3. kg. newton 1. mass actuator coil per newton

J r1 r2,( )Jwerk r1 x1( )2 Mwerk. 1

3mb2 r2( ) r22. mb1 r1( ) r1 x2( )2..

1 cmot α r1( ). r2.total inertia

Mimp r1 r2,( )J r1 r2,( )

r12impact-mass

F r1 r2,( )J r1 r2,( ) α r1( ).

r2actuator-force

range ........imax 40 jmax 25i 0 imax.. r1i 0.001 i. m. x1 0.8. range r1

j 0 jmax.. r2j 0.002 j. 0.025( ) m. range r2

Mimpacti j, Mimp r1i r2j, Jtoti j, J r1i r2j, Facti j, F r1i r2j,

r10 0.062 m= r1imax 0.102 m= r20 0.025 m= r2jmax 0.075 m= Fimp 0.01 Assumed impact forcemtot r1 r2,( ) cmot F r1 r2,( ). mb2 r2( ) mb1 r1( ) Mwerk Total mass

Position mass centre i.r.t. tip

x r1 r2,( )cmot F r1 r2,( ). r1 r2( ). mb2 r2( ) r1

r22

. mb1 r1( )r12

x22

. Mwerk x1.

mtot r1 r2,( )

α r1 r2,( )Fimp r1.

mtot r1 r2,( ) r1 x r1 r2,( )( )2. J r1 r2,( )Angular acceleration beam

dzz r1 r2,( ) α r1 r2,( ) r1 x r1 r2,( )( ). Z-acceleration mass centre

FL r1 r2,( ) dzz r1 r2,( ) mtot r1 r2,( ). Fimp Z-force on bearing

FLZi j, FL r1i r2j,

Figure 12.6.7, Model for bearing force calculation

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12.7 ELECTRO DYNAMIC ACTUATOR TYPES

Several types of the electro dynamical actuators will be treated in the following chapter and the most dominant differences will be indicated. 12.7.1 THE LOUDSPEAKER

The human ears are very sensitive sound; a range of more than 100 dBA is covered over a frequency range from 20 to 16.000 Hz. Distortions as clipping, crossover and inter-modulation are easily detected. The electro-dynamic actuator satisfies high specifications without problems and is considered as the best performing electro-mechanical actuator. A typical loudspeaker geometry is given in figure 15. A ring shaped, axially magnetized, ferrite magnet is clamped between two iron plates and an iron cylinder is placed in the inner bore. The iron concentrates the magnetic flux in the gap between the upper plate and the cylinder and 1 Tesla is a usual field strength here.

Figure 12.7.1.1, A loudspeaker

The turns on the coil are directed tangentially and the axially directed force is generated in the gap consequently. Only a part of the coil generates indeed the force. The axial length of the coil is related to the required stroke and this clarifies why a raising stroke leads to a decreasing efficiency. Current is brought to the coil by two Litze-wires, bundles of very thin wires to reduce their stiffness and to enhance their lifetime. The coil itself is wound with orthocyclic winding technology to increase the number of turns and to improve the heat transfer. The coil carrier is made of paper or aluminum foil. This last choice is mainly based on heat transfer improvement. One will get eddy currents in this foil, opposing the currents in the coil with a reduced force as consequence when no precautions are made. The foil thickness is chosen as less than 0.1 mm and a split in the axial direction prevents circular eddy currents. For servo applications is a general rule to minimize the moving weight as much as possible to get the highest possible bandwidth. So the dimensions of an actuator are nearly always a point of discussion. One should give attention to reluctance forces in those cases where the peak force is only required for a small part of the time, allowing a heavily overloaded actuator for a short time. Reluctance forces are described by the equation:

21 LF = .i2 z∂∂

12.7.1.1

The intended force satisfies according 12.1.7:

z d d

i.N. = F pmφ

12.7.1.2

The reluctance force is based on a position dependent self-inductance, which is clearly changing when the coils in figure 12.7.1.1 moves.

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Fig. 12.7.1.2, Example of the current and position dependency of the force of a moving coil actuator.

The theoretical background of the reluctance force is given first, starting with the voltage equation for the coil:

. pmd di L dxu = R.i + N. i.R + + N.dt dt x dt

φφ ∂=

12.7.1.3

Let us assume that the current is fixed in time. Multiplication with the current gives the power balance:

2 .. dL i dxu i i .R + i + N.dt x dt

φ∂=

12.7.1.4

The first and third term at the right hand side are treated earlier as the dissipation and the power going to the mechanics. For the second term holds under the assumed time independent current:

2. .dL i dL ii

dt dt=

12.7.1.5

Defined already earlier:

21

2 .magn

d L iPdt

= 12.7.1.6

The remaining part of the power in equation 12.7.1.5 has to go consequently to additional mechanical power.

2 2,

1 12 2mech rel rel rel

dx L dx LP F i F idt x dt x

∂ ∂= = ⇒ =

∂ ∂

12.7.1.7

Comparing the equations 12.7.1.2 and 12.7.1.7 shows a linearly and quadratically dependency on the current respectively. This explains why reluctance forces start to be important at high current levels.

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There is a very simple method to verify if reluctance forces are present by measuring the force at a positive and negative current. As a rule of thumb holds that a reluctance forces start to be significant when the current density exceeds 10 A/mm2.

Let us assume the following relation between force and current. 2

1 2( ). ( ).F c x i c x i= + 12.7.1.8

At a fixed position x holds now for +I Amp and –I Amp.: 2

1 22

1 2

1 2 2

( ). ( ).

( ). ( ).

( ) ( )

F c x I c x I

F c x I c x I

F F F Fc x c xI I

+

+ − + −

= +

= − +

− += =

12.7.1.9

So, by current reversal one can find if a reluctance force exists and if it has to be considered as relevant. An extreme improvement is reached in the last two decade as far as magnets are concerned. The most popular magnet type around 1970 was ferrite, with 0.37 T as strength. Nowadays NdFeB is the favorite material with 1.2 .. 1.4 Tesla as strength. More powerful or compacter designs can be made now. Typical examples of NdFeB based magnet systems are given in Figure 12.7.1.3. An important issue is that the magnet fields outside the yoke are highly reduced compared to a configuration as given in Figure 12.7.1.1.

Figure 12.7.1.3, Yokes for NdFeB based magnet systems

The sintering process needed during the production of the magnets does not allow the aspect ratio found in the right hand figure, so those rings has to be assembled by e.g. 16 segments. The related costs give a preference to the left solution. The advantage of the right solution at the other hand is that the force constant depends less on radial displacements.

12.7.2 A SLEDGE ACTUATOR It is hard to get a long stroke with a loudspeaker actuator. A good option is to apply the geometry given in Figure 12.7.2.1 The coil is moving here also. The magnets are extended in direction of motion and here a force is generated over the whole length of the coil. At the other hand only 50 % of the circumference of the coil is in a magnet field. A field strength of 0.5 Tesla can be expected in this geometry when NdFeB-magnets are applied.

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Figure 12.7.2.1, Sledge actuator

The linear guiding system requires attention, because an air bearing adds a significant mass and a roller bearing will introduce lifetime questions and disturbances. The self-inductance of this actuator is far higher than for a loudspeaker actuator and has to be taken into account when the amplifier voltage has to be determined. A reluctance force exists when the self-inductance depends on the position as mentioned in the preceding paragraph. This position dependency is present here significantly. The designer of this actuator can reduce the position dependency by the addition of small air gaps in the vertically drawn areas. One of the weak points of this actuator is the assembly to the load. The mounting bracket can be fixed with an epoxy compound to one coil side, but this is surely influencing the dynamical behavior in negative sense. Applications are e.g. driving the sledge of a CD-player and driving a sub-micron measurement equipment. The concepts of 12.7.2.1 can be characterized with:

• a fixed magnet • a moving coil with moving wires • no cogging forces and preferred positions • cooling by convection and some conduction to the moving member of the system • the control is rather simple, because the force is in linear relation to the current in the coil.

12.7.3 A FLAT ACTUATOR Actuators with a limited building height are required in wafer steppers. Thermal conditions in wafer steppers are very tight and volume/weight are interfering always with other features. On top of this holds that the system performance is determined by the position accuracy and the throughput, expressed as wafer per hour. For the design of these actuators one has to judge the dynamical, thermal, and control behavior, the power consumption, weight and volume. A typical configuration is given in the following figure.

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Figure 12.7.3.1, a short stroke actuator

The coils are mounted in a stainless steel housing provided with a water-cooling system. Stainless steel plates are glued on top of the coils to reduce the thermal resistance from the coil to the water channels. The stroke of this actuator is limited to some millimeters. Accuracies better than 5 nanometer within milliseconds after a force pulse of 500 N are the targets. This can be achieved only with a very well predictable behavior of the system and as far as the actuator is concerned, one has to find a good combination of magnet and coil dimensions. Mounting this actuator well in the system is much easier even when force vector has to go through the center of mass. The mass of the magnet yoke is twice the mass of the coil part, so it seems the most appropriate solution to fix the coil part to the load. However this interferes with the nanometer accuracy required, because the noisy water flow and rather stiff conductors of the coils will act as a significant disturbance for the position control. A general point of attention to investigate the positioning of the two actuator parts with respect to each other in relation to stacked mechanical tolerances in a drive system. Misalignment will introduce force in another direction than intended as indicated in Figure 12.7.3.2.

Figure 12.7.3.2, influence on the force vector by misalignment

12.7.4 MOVING MAGNET An interesting actuator for a short stroke (<2 cm) at a force level of 100 is given in Figure 20.

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+ -

Iron

Magnet

currentcurrent

Figure 12.7.4.1, Short stroke moving magnet actuator. The two coils in the inner bore are electrically connected in series in opposite direction. The two iron disks on the magnet concentrate all the flux of the magnet to guide it radially through the copper to the stator iron. The enclosed flux of each coil is changing when the magnet is moving, so a force is created when current is going through the copper. The simplicity of the production and robustness of this actuator is the major attractive side. The ideal behavior of an actuator is a position independent force in combination with a linear relation between the generated force and the current. There are four issues here to be discussed, a cogging and a reluctance force and damping. Cogging is present when the actuator has preferred positions when no current is in the coils. Here is the mid position indeed a preferred position for the magnet and this effect becomes more dominant with a shorter iron tube. This is the reason why the iron tube is extended in comparison to the coils. Reluctance is present when the inductance is position dependent and the moving iron disks give indeed a position dependency. The motor constant K [N/A] depends on the position. Damping is present when a force is opposing when the magnet is moving with a certain speed, even when the coils are disconnected. The explanation that damping exists here is based on the position dependent flux in the stator iron. The flux going through in a thin ring of the stator near the iron disk is changing when the magnet moves. The derivation of the (estimated) damping goes as follows: R 0.02 m. Radius of the magnet

Amagn π R2. Surface of the magnet

Bmagn 0.5 tesla. Assumed flux density in the magnet

Φ Amagn Bmagn. Flux

d 4 10 3. m. Thickness of the disks

Ri 26 10 3. m. Inner radius cylinder

Ro 29 10 3. m. Outer radius cylinder

dΦdxΦd

The change of the flux

EMF v( ) dΦdx v. The induced voltage

ρ 88 10 9. ohm. m. The specific resistivity of iron

The resistance of a ring with the axial length dRρ π. Ro Ri( ).

d Ro Ri( ).

The dissipated power as a function of the speeof the translatorP v( )

EMF v( )2

R

damping v( ) 2P v( )

v2. The damping found for both sides

damping 1 m. sec 1.( ) 38.945 newton sec. m 1.=

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Laminated iron is used in electric motors and transformers to reduce these eddy currents losses, but the only way to laminate the stator here is to apply strips oriented radially and this is rather expensive. Another option is to apply a cylinder of powdered iron (highly compressed iron particles with an insulating surrounding, but this technology is very young. Another option is to apply iron with 3 to 4 % silicon, known as SiFe to increase the resistivity with approximately a factor 5, but it is hard to get this as solid iron. So what remains is make a cylinder of a rolled SiFe-sheet of 0.5 mm with e.g. 6 layers; this solution is however never applied. Suppose that the shaft is made of iron. The left hand side is then a magnetic north pole and the other side a magnetic south pole. Handling this actuator starts to be difficult and not every bearing type is fitted for operation in a strong magnetic field. So stainless steel shafts are preferred. This problem can be prevented by changing the concept as given in Figure 12.7.4.2. The production of the shaft is an easy job compared to the previous design but it is very likely that the magnets has to be ordered as segments of e.g. 45 degrees.

Figure 12.7.4.2, Alternative design

Figure 12.7.4.3, linear moving magnet actuator for a wire bonder

Figure 12.7.4.3 gives a patented moving magnet actuator for a wire bonder. The operation principle is proven by the position dependent permanent flux in the coil. Both stator parts should be made of laminated SiFe to reduce eddy currents and of course both sides can be made of E-shaped SiFe provided with a coil.

+ -

Iron

Magnet

currentcurrent

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The relative permeability of ferrite and rare earth magnets is nearly equal to 1, so the self-inductance will not change when the magnet moves. Here one will not find a reluctance force consequently. A cogging force is present, but can be reduced highly by a suited dimensioning. The local flux density in the iron is dependent on the position of the magnets and changing the local flux density in iron requires a non-reversible energy flow, which is linked to the magnetic hysteris of the SiFe type used. A user of this actuator will notice this as a virtual static friction. It is further likely that the mass of the moving magnet exceeds significantly the moving mass of a moving coil actuator with the same force level; acceptable or not is determined by the application. A general remark for all linear actuators with a moving magnet and iron at the fixed world: high bearing forces when the alignment is not done well.

12.8 SUMMARY

Linear short stroke actuators are treated as components to be used as part of a drive system and the link between the system requirements and specification of the drive is made. Several actuator types are analyzed and attention is given to the strong and weak aspects.

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13. Linear motors Rotating motors are available on the market based on brushes and on electronic commutation. A lifetime of 20.000 hours for the brushes can be expected for disk motors. However the required conditions to obtain this lifetime for linear motors based on brushes cannot be fulfilled. Major points are the flatness of a long commutator and the repetitive start or stop action at a fixed position. So, linear motors with brushes are very rare, nearly all linear motors are based on electronic commutation and that means that position sensing is required to feed the electronics with position information. Two principles can found nowadays. The first consists of a magnet strip and an armature consisting of an iron core and coils and the second one has two magnet strips with and ironless coil block between the magnets strips.

Armature incl. Coils

Magnetstrip

Magnets

Figure 13.1, the iron core motor

Magnet yoke

Colis

Figure 13.2, The ironless linear motor

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13.1 ELECTRONIC COMMUTATION

Armature incl. Coils

Magnetstrip

Magnets

Figure 13.1.1, The iron core linear motor

The layout of an electronically commutating linear motor with an iron core is given in Figure 13.1.1. One part is provided with permanent magnets and the armature is provided with coils. The current values in the coils, usually a 3-phase system, are link to the armature position with respect to the magnets strip and the current amplitude is linked to the required force level. Similar as holds for rotating motors holds here as commutation principles:

DC-brushless, based on square wave currents and AC-brushless, based on sinusoidally currents.

13.2 DC-BRUSHLESS Figure 13.2.1 gives the basic circuit for a DC-brushless linear motor. Some digital circuits are controlling the transistors linked with the coils. The motor is provided with sets of three coils. Within each set are the individual coils spatially shifted that the EMF’s are shifted 120 degrees wit respect to each other. Very specific is that the EMF’s behave trapezoidally as a function of the position.

PWM-logic HallDecoder

I ref v, s

+

-

Position

S 1 S 3

Current

Coil block

θ=0 2 π

EMF

0

0

0

Coil 1 Coil 2 Coil 3

Figure 13.2.1, Basic circuit for DC-brushless The target of the logic is to put current in the motor coils only when the EMF has reached its constant value. The direction of the current is linked to the polarity of the EMF. As indicated earlier holds for the power transfer from the electrical to the mechanical part that this power is equal to EMF.i. Based on the previous philosophy holds that two coils are carrying a current, leading to:

2.EMF.i = vF = P mech . 13.2.1

Figure 13.2.2 gives the tranferred power per coil and summing the power leads to the same transferred power level at each instant of time. Assuming a constant speed leads now to a constant force level.

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θ= 0 2π

Coil 1 Coil 2 Coil 3

EMF Power Current

Figure 13.2.2, Mechanical power for a DC-brushless motor

The force level is determined by the current amplitude and its direction is determined by the current polarity. One will notice that the ideal behaviour is disturbed by the following causes:

1. deviations related to the Hall sensors (offset, drift and positioning) lead to current switching at a deviating position

2. the induced EMF is not flat over 120 degrees 3. unequal phase currents by offset and/or gain differences.

The result is that one will notice at each commutation position a disturbance in the force. Positioning in the neighbourhood of the commutation positions might give rise to a limit cycling phenomenon. The application of this DC-brushless principle is limited to ironless linear motors, because the switching of currents leads to a disturbing noise level in iron armature motors.

13.3 AC-SYNCHRONOUS MOTORS

The second group of brushless motors is based on a sinusoidally changing EMF. To get this sinusoidal behaviour one adapts the magnet dimensions and/or coils shapes and the shape of the teeth in the case of an iron armature. Figure 13.3.1 shows how the current has to behave as a function of the position in relation to the EMF values. The next figure shows the mechanical power produced. The underlying equations are the same as valid for a rotating electronically commutated motor, which are described earlier.

0 2 4 61

0

1

I1i

I2i

I3i

θ i

The three phase currents as a function of theposition in rad.

0 2 4 61

0

1

E1i

E2i

E3i

θ i

The three EMF's as a function of the position in rad.

Figure 13.3.1, Current and voltage for an AC-synchronous motor

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0 2 4 60

1

2

P1i

P2i

P3i

P1i P2i+ P3i+

θ i

The mechanical power produced per coil and the sum as a function of the position in rad.

Figure 29, The mechanical power

In line with the AC-synchronous rotating motor hold here again:

1. To operate well is the actual position information required at each instant of time. A linear encoder can provide this information. An alternative is to apply a number of Hall sensors, however the absolute accuracy is limited (most likely 0.1 mm).

2. The amplitude of the sinusoidally changing currents determines the force level produced. PWM-amplifiers are used as supply in general.

This commutation principle can be applied on ironless and iron core linear motors.

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13.4 COMPARISON IRONLESS AND IRON CORE LINEAR MOTORS A comparison in general terms is given in the following table. Ironless Iron core Short stroke + 0 Long stroke - + Volume - + Cogging (at I=0 Amp.) + - Load on the bearings + - Noise level + 0 Frms/Fpeak 0.05 0.4 Ftop/moving mass + 0 Accuracy during constant speed 3-10 nano-meter microns Explanation:

Stroke: the costs of the magnet strips start to be dominant at a long stroke. Volume: iron core motors are more compact by using the magnets more efficient. In the case of water-cooling holds, that this can be done more efficient in iron core motors. Cogging: the interaction between the teeth of the iron core and the magnets leads to preferred positions which are detectable easily at i=0. A usual level is 2 to 5 % of the continuous allowed force level. For a control loop is this noticeable as a disturbance, which can be counteracted by a (learning) feed forward. Always one should specify the amplitude of the cogging and its spatial frequency. Load on the bearings: for an iron core motor holds as load on the bearing 2 to 5 times the peak force of the motor, whereas the bearing load of an iron core can neglected usually. The consequences are that the bearing moving mass will be significantly higher for an iron core motor. In cases where a air bearing is used the presence of a high pretension is attractive. Accuracy: at standstill the controller, the amplifier and the sensor determine the accuracy and resolution for both motor types. At constant speed are cogging and the high bearing loading the causes of a limited accuracy for the iron core motors. Noise level: the teeth structure of the iron core motor moving along the magnet strip gives highly changing attraction and shear forces with noise as consequence. Frms/Fpeak: this ratio is low for ironless motors and the application (mechanics and the motion profile to be realized) might give rise to a preferred motor type. Ftop/moving mass: the iron of an iron core motor is the issue involved. The mass of the load determines to what extent this has to be considered as a selection criterium.

The market moves clearly to ironless linear motors, because the costs of the widely used magnet material, NdFeB, is dropped with a factor of ten in the last ten year. So, the efficient use of the magnet material in an iron core motor became less important.

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13.5 LINEAR MOTOR, MOVING MAGNET The previous linear motor concepts are based on moving copper and magnets fixed to the world. Figure 13.5.1 gives a concept based on moving magnets and the coils fixed to the world. This motor was the heart of a successful lithographic machine for many years with 5 nano-meter accuracy. The two moving iron yoke are fixed to each other and provided with SmCo magnets. The coils are wound around and fixed to a bar of SiFe sheets, glued well to a nearly solid body.

Magneet (4*)

IJzer

+ stroomstroom

Magneet (4*)+ stroom_ stroom Figure 13.5.1, Linear motor with moving magnets and a long stroke

The direction of the currents in the stator coils is linked to the required force direction and their strength determines the force value, as seen before in the AC-synchronous and DC-brushless linear motor. The coils are connected in two series connections, consisting the coils 1,3, 5 and 2,4,6 respectively. Two amplifiers are needed to feed the motor. Of course one has to apply also here a position sensor for the electronic commutation. The bar is made of the SiFe sheets to reduce eddy currents damping and magnetic hystersis friction to improve the servo behaviour. It is an advantage to use the double-sided construction to reduce the bearing load. The series connection means, that all coils are carrying current, so losses can be found over the total length of the motor. This is the payback for preventing moving motor cables. This motor concept is rather rare, in spite of its attractive aspects,

• no cogging • no moving motor cables • the coils can be cooled by e.g. a water channel

The weak spots are: • iron losses • copper loss over its full length.

The stroke is mainly limited by the first vibration mode of the SiFe bar and 40 cm is realistic.

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14 The ”constants” The users of motors, rotating and linear, are always selecting a motor on the base of data given in data sheets. The lack of standards in the field of servo motor specifications complicates a comparison. One has to consider carefully the test conditions (e.g. ambient temperature, frame(s) used and airflow conditions) in order to prevent the selection of a too weak motor. One should also be keen on tolerances on the motor constants; generally holds 5% to 10% as tolerance on the torque (or force) constant K and the resistance R. Already this tolerance has severe consequences for the loss and the specification for the motor amplifier. The relevant constants will be treated in the following paragraphs with having in mind linear motors.

14.1 THE K-FACTOR The following issues are involved with respect to the motor constant K:

• Current level; the load requires a certain torque or force level and a changing motor constant K means a changing current level.

• Voltage level; in the voltage equation of a motor one can find the EMF, linearly related to the speed via K and the current, for a given force (or torque) linear related to 1/K. It depends on the motor, load and motion profile whether one can find a rising or decreasing voltage in combination with a decreasing motor constant K.

• Power losses; the losses in the motor are linearly related to square of the current, so P(:)1/K2 • Control loop behavior; the motor constant is one of the components of the open loop gain.

Another open loop gain changes always the phase and gain margin. • Feed forward; a deviating motor gain leads to wrong feed forward value, so K-factor

deviations leads to a lower efficiency of the feed forward. So consequences can be found in the motor, the amplifier and the control loop behaviour. Items influencing the K-factor are:

• The magnets o The tolerance on the strength of magnets is usually –5% to +5%. With calibration one

can obtain commercially the range –2% to +2%. The sensitivity is simple, 5% higher strength means +5% in the K-factor

o The magnetisation direction can deviate from –6 to +6 degrees. The sensitivity for the magnetisation direction depends on the motor type, so only the motor designer can determine this sensitivity.

o The mechanical tolerance on the dimensions of the magnets o The accuracy of the magnet positioning in a stator bore or on a magnet strip o The temperature dependency of the modern rare earth magnets equals -0.2 %/K for

NdFeB magnets and –0.05%/K for SmCo magnets (this explains why SmCo magnets can be found in servo-motors in spite of their relatively high costs)

Based on these items we distinct as effects temperature dependency, position dependency of the K-factor and a potential variation in the mean value of the K-factor over production series. Not mentioned in the list are long-term effects; the strength of NdFeB magnets is slowly decreasing in time and this has to be investigated when stability within 1 % over 10 years has to be given.

• The coils o There exists a link between the position accuracy of the coils within the coil block of

ironless motors and shape of the coils of ironless motors respectively and the position dependency of the motor constant exists. Again holds here that only a designer can give exact numbers concerning this sensitivity.

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• The air gap o The air gap in an linear iron core motor is determined by the supporting bearings. A

smaller gap leads to a higher magnet field and consequently to a rising motor constant K and rising attracting force. It should be clear that the alignment requirements of the bearings with respect to magnets strip are high given a usual value of 0.25 mm as air gap.

o The position of the coil block between the two magnet strips of an ironless linear motor is of course depending on assembly accuracies and the extent of deformations of the strips (e.g. by gravity). One should be aware of a rising motor constant when the coil block leaves it mid-position between the strips. Simultaneously one will find then forces perpendicular on the plane of the coil block. So absorbing the stacked tolerances in equipment by uncertainty in the coil block position requires a careful analysis.

• Magnetic saturation

o The field of the coils in an iron core motor can magnetically saturate the iron of the armature. The result is that the flux density caused by the magnets decreases with finally a decreasing motor constant at a rising current level. So for iron core motors one should ask a supplier the behaviour of the motor constant as a function of the current level for a certain value of the air gap.

The items given above are based on the experience built up by designing linear drives for precision equipment. It is overdone to check all these aspects for a linear transport system! It should be noted that the constant K gives together with the current the force or torque produced in the motor; the shaft torque is got from this number by subtracting the internal motor damping, friction, cogging and that part of the force or torque needed for the acceleration. Additionally has to be mentioned, that amplifier gain errors and offsets influences the force or torque of electronically commutated motors, as described earlier.

14.2 THE RESISTANCE R The following issues are involved with respect to the resistance R:

• Voltage level; in the voltage equation of a motor one can find the term i.R, so a rising resistance means that a higher voltage should be available.

• Power losses; the losses in the motor are linearly related to square of the current times the resistance, so P=i2.R.

So consequences can be found in the motor and the amplifier.

Items influencing the resistance are: • Temperature dependency of the resistance

Copper wire is considered as the best material for motor coils, by its low specific resistivity. But this material has as temperature dependency +0.4 %/K. A temperature rise from 25 to 100 degrees, not unusual, leads to 30 % increase of the coil resistance

• Tolerances The tolerance on copper wire resistance is 2 % conform standards. In addition to this tolerance there is the elongation of the wire during the coil winding process, where up to 12 % can occur. A usual tolerance on the resistance (at a specified temperature of e.g. 30 degrees Celsius) is 5 tot 10 %.

• Eddy currents in the magnets, yoke and armature Eddy currents exist when a rapidly changing flux goes through an electrically conducting material. One will notice the presence of those eddy currents by a rising resistance at an increasing frequency. To predict to what extent one has to counteract this effect is rather

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difficult, because it is linked to the material used in the motor, its geometry and the slope of the current in time.

• Brushes Is the resistance including the resistance of the brushes? Sometimes is listed the armature resistance, sometimes the terminal resistance.

• Cabling The resistance of the cabling is not a part of the motor, but the amplifier must overcome this resistance.

14.3 THE STEEPNESS S The steepness S equals K2/R. The preceding paragraphs show the factors influencing K and R, so it should be clear, that the steepness is subject to much variation.

14.4 THE THERMAL RESISTANCE RTH Data sheets are mentioning always the thermal resistances, but the lack of standards for servomotors allows many test conditions. Questions to be answered by the supplier are at least:

• Is the motor mounted to a frame; what material is used and what are the dimensions? • What is the orientation? Is air moving freely around the motor? • What is the relation between the speed and the thermal resistance? • The thermal resistance is a function of the temperature of the heated body and the ambient

temperature by the changing ratio between convection and radiation. Consequently one should know the temperatures involved.

• What is the reference altitude? A de-rating of the allowed loss of 5%/km altitude should be applied.

Suppliers have two interests, getting the best position in comparison to competitors and to prevent customer disappointments. So open the discussion with a supplier as soon as fitted to ensure that your own conditions are in line with the test conditions.

14.5 LIFE-TIME The lifetime of electric motors are dominantly determined by the bearings and, in the case of a motor with brushes, by the latter. A lifetime indication can be listed in data-sheets; however, most likely this number is based on the motor running continuously at a constant speed. That is rarely the case in servo-systems and it remains always the responsibility of the designer to verify the lifetime. As far as the bearings are concerned we have to consider:

• radial and axial load • temperature • acceleration (noticed: slipping roller cage!) • ambient conditions (e.g. glass particles) • repetitive point-to-point movements and grease concentration at the end of the stroke • PWM supply induced capacitive currents running through the bearings.

The life of brushes are influenced by • peak currents • repetitive point to point movements • current pulses by PWM • peak voltage of the amplifier • temperature

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• vibrations • ambient conditions (e.g. dust, humidity must be >5 gr. H2O/m3 for carbon brushes) • speed • current at standstill • the materials used for the commutator • the materials used for the brushes.

The prove that lifetime is good can only be given on the base of a Weibull graph (a statistical method), based on at least 6 samples. Imagine that the target lifetime is 3 year and the sales department has to wait on the test results ….. Then one should consider the temporary application of electronically commutating motors to bring the system on the market, because the lifetime of bearings is far better predictable than the lifetime of brushes.

14.6 AMPLIFIER CHOICE Based on the contents of the preceding chapters we can summarize all relevant items related to the specification of an amplifier:

• the cable and connectors between the amplifier and the motor • the temperature dependent K-factor and resistance R • the commutation system (brush resistance and voltage drop over the contact layer between

brushes and commutator) • the voltage drop over the end-stage of the amplifier • the voltage drop over the supply at high currents. • voltage fluctuations of the mains • a margin on the peak current and voltage for control purposes • tolerances on the motor constant and resistance given by the supplier • uncertainties concerning the load data • changing ambient conditions (ambient temperature or altitude) • damping by eddy currents in the motor friction related to the bearings and brushes • a position or current level depending motor “constant” • bandwidth • output impedance • noise level • PWM induced motor losses and life time reduction • offset, drift and phase gain errors in the case of 3-phase amplifiers • non linear behavior around zero current for PWM-amplifier.

This in addition to costs, volume, standardization, interfacing, safety, serviceability, remote sensing, communication protocols, mains and radio interference, cooling, electro-magnetic compatibility, approbation, life time, stability, supply voltage, supply voltage disturbance rejection, monitoring, etc.

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15 Literature [1] P.L. Huricks,

Handbook of Electromechanical Product Design, Longman Scientific and Technical, ISBN 0-582-04083-3

[2] Kenjo, T. and S. Nagamori Permanent-Magnet and Brushless DC-Motors Oxford Science Publications

[3] H. Wayne Beaty and James L. Kirtley, Jr Electric motor handbook, Mc Graw Hill, ISBN 0-07-035971-1

[4] Hans-Dieter Stolting & Eberhard Kallenbach, Handbuch Elektrische Kleinantriebe, Hanser, ISBN 3-446-21007-5

[5] J.H.J. Boekema, A.M.C.J. Cramer, R.H. Dijken, H,J, Nanninga Aandrijfsystemen Nijgh & Van Ditmar Educatief ISBN 90-236-0363-X

[6] W.G.V. Rosser Interpretation of classical electromagnetism

Kluwer Academic Publishers ISBN 0-7923-4187-2

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