II ADVANCED APPLICATION OF INDUSTRIAL SERVO DRIVES doc

The advanced application of industrial servo drives requires the use ofdifferential equations to describe mechanical, electrical, and fluid systems.. Summarizing, the mechanical time cons

IIADVANCED APPLICATION OF INDUSTRIAL SERVO DRIVES

Background

Part I discussed the basics of industrial servo drives from a hardware point

of view Physical parameters and practical applications were discussed Part

II repeats some of the things in Part I but from a mathematical point ofview The advanced application of industrial servo drives requires the use ofdifferential equations to describe mechanical, electrical, and fluid systems

As applied to servo drives there are numerous academic techniques toanalyze these systems (e.g., root locus, Nyquist diagrams, etc.) In workingwith industrial machinery we live in a sinusoidal world with such things asstructural machine resonances Thus frequency analysis is used in Part II todescribe and analyze industrial servo systems To solve the differentialequations describing the physical systems of servo drives, transformationcalculus is used to obtain the required transfer functions for the components

of servo drives and in analyzing the servo system

There are a multitude of academic textbooks and university coursesdealing with feedback control It is the purpose of Part II to show how thefundamentals of servo drives described in the many academic sources areapplied in practice

7.2 PHYSICAL SYSTEM ANALOGS, QUANTITIES, AND VECTORS

As a beginning, analogous parameters for an electrical system, a linearmechanical system, and a rotary mechanical system are compared for futurereference In all physical systems there are scalar quantities and vectorquantities Vector quantities can be represented as complex numbers on acomplex plane, in polar form, or in exponential form as in Eq (7.2-1) to(7.2-12)

Rectangular Polar Exponential

Fcos yþ jF sin y ¼ jFjffy ¼ jFje+jy (7.2-12)

SYSTEMS

The differential equations for physical systems can be written for individualservo components such as motors and amplifiers or for complete multiloopservo drives In actual practice servo drive block diagrams can be puttogether with a combination of individual transfer functions representingthe differential equation of the separate servo drive components Theseindividual transfer characteristics can, in general, be represented by single-order or second-order blocks in the overall servo block diagram A single-order differential equation or transfer characteristic results from a circuithaving a single time-varying parameter Likewise, a second-order transfercharacteristic results from a circuit (mechanical, electrical, or fluid) havingtwo time-varying parameters Most servo drive components can berepresented by either a first-order transfer characteristic (or transferfunction) or a second-order transfer function A transfer function is, bydefinition, the ratio of the Laplace transform of the output to the Laplacetransform of the input In general a transfer function is a shorthand solutionfor solving differential equations

The derivation of a single-order electrical circuit transfer functionhaving an inductor as a single time-varying parameter is shown inFigure 1

The steady-state equations for the output voltage, based on a sinusoidal

input voltage, are 7.3-1 to 7.3-17 Replacing the jo term by the differentialoperator p or the Laplace transform operator s changes Eq (7.3-10) to thetransform function of Eq (7.3-19) This transform function can berepresented in the frequency response of Figure 2.

To illustrate a second-order circuit, the circuit of Figure 3 with twotime-varying parameters has the differential equation of Eq (7.3-20) Theoutput voltage for the unique case of a sinusoidal input voltage is Eq (7.3-25) A second-order mechanical circuit for linear translation is shown in

Figure 4.Eq (7.3-30) is the differential equation for this circuit Assuming asinusoidal input, the output displacement is Eq (7.3-35) Lastly, a rotarymechanical circuit is shown inFigure 5.The output angular motion is Eq.(7.3-44) These examples of single-order and second-order circuits are toillustrate that individual servo drive components can be representedmathematically by differential equations, transfer functions, or the absolutecase for a sinusoidal input driving source

The circuit shown in Figure 1 is further described for three cases ofabsolute, vector, or differential analysis followed by the response of thiscircuit to a step input and a ramp input

Fig 2 Single-order frequency response.

Note: aþ jb ¼ cffy:

Z Re

Zþ jXLZ

Fig 3 Inductive/capacitive/resistive circuit

Fig 4 Spring/mass diagram (linear)

Fig 5 Spring/mass diagram (rotary)

where p is the differential operator.

o2

jeij ¼ jijðR þ 2pfLÞ
eei¼
iiðR þ 2pfLÞjij ¼ jei j

ð Þ2

1þjo L R

ðT 1 sþ1Þ

Examples Absolute Case

ð0:04 s þ 1Þ

For the Case of Sinusoidal Input

o

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið25Þ2þ o2

T1¼ 0:04 sec

eiðsÞ¼ EsðT1sþ 1Þ¼

25Esðs þ 25Þ

e ¼ Eð1
e
25tÞ

For Case Where ei(t)¼ Ramp ¼ Et (Figure 6b)

FUNCTIONS AND TIME CONSTANTS

In Section 7.3 some simple first- and second-order mathematical tions were given to illustrate how the corresponding transfer functions weredeveloped In Part I, the components of servo drives were described from aphysical point of view These servo-drive components are now describedfrom a mathematical or transfer function point of view

descrip-In the analysis of electric servo-drive motors, the equations for themotor indicates the presence of two time constants One is a mechanicaltime constant and the other is an electrical time constant Commercialservo-motor specifications usually list these two time constants However, itshould be cautioned that these two time constants as given in thespecifications are for the motor alone with no load inertia connected to

Fig 6b For case where eiðtÞ¼ RAMP ¼ Et

the motor shaft Since these two time constants are part of the motor blockdiagram used in servo analysis, it is important to know the real value of thetime constants under actual load conditions.

There are two types of servo motors to consider The first is theclassical DC servo motor and the second is the AC servo motor oftenreferred to as a brushless DC motor The brushless DC motor is a three-phase synchronous motor having a position transducer inside the motor totransmit motor shaft position to the drive amplifier for the purpose ofcontrolling current commutation in the three phases of the motor windings

A derivation of the motor equations and the electrical and mechanicalmotor time constants will be discussed for the DC motor followed by adiscussion for the AC motor The DC motor equivalent diagram is shown inFigure 8, where:

eI¼ Applied voltage (volts)

ia¼ Armature current (amps)

JT¼ Total inertia of motor armature plus load (lb-in.-sec2)

Ke¼ Motor voltage constant (V/rad/sec)

KT¼ Motor torque constant (lb-in./A)

La¼ Motor winding inductance (Henries)

Ra¼ Armature resistance (ohms)

TL¼ Load torque (lb-in.)

Vm¼ Motor velocity (rad/sec)

a¼ Acceleration (rad/sec2

)The steady-state (DC) equations are:

Rearranging Eq (7.4-11) gives:

Therefore, the closed-loop motor equation can be expressed as:

d¼ 0:5 tmom¼ 0:5 tm

ffiffiffiffiffiffiffiffiffiffiffiffiffi1=tmte

LL
L¼ Motor inductance ¼ ½Henries

at the motor shaft¼ ½lb:-in:-sec2

Most manufacturers give the electrical parameters in line-to-line values.Thus some of these values must be converted to the phase values as shown inthe preceding text Summarizing, the mechanical time constant can becomputed as:

te¼Total inductive pathTotal resistive path ¼PLL
L

Another factor affecting the mechanical time constant is the temperature.Most manufacturers specify the motor parameters at 258C (cold rating).This implies that the magnet and wires are both at room temperature.However, the motors used in industry will operate hotter, which meanscould reach a magnet temperature of 808C to 908C in a 408C ambient Thewinding temperature is considerably more than that Some means must beused to compensate for the motor parameters rated at 258C For thosemanufacturers that offer the hot rating on motor specification parameters,they should be used to calculate the time constants The parameters ofmotor resistance, torque constant, and voltage constant should be adjusted,

if needed, for the hot rating The motor resistance will increase; the torqueconstant and voltage constant will decrease However, contrary to theirimplied name both time constants are not of constant value Rather, theyare both functions of the motor’s operating temperature

The electrical resistance of a winding, at a specified temperature, isdetermined by the length, gauge and composition (i.e., copper, aluminum,etc.) of the wire used to construct the winding The winding in the vastmajority of industrial servo motors are constructed using film-coated coppermagnet wire Based on the 1913 International Electrical Commissionstandard, the linear temperature coefficient of electrical resistance forannealed copper magnet wire is 0.00393/8C Hence, knowing a copperwinding’s resistance at a specified reference or ambient temperature, the

RðTÞ ¼ RðT0Þ½1 þ 0:00393ðT
T0Þ (7.4-24)where:

T¼ Winding’s Temperature ðCÞ

T0¼ Specified Ambient Temperature ðCÞ:

Using Eq (7.4-24), a 1308C rise (1558C–258C) in a copper winding’stemperature increases its electrical resistance by a factor of 1.5109.Correspondingly, the motor’s mechanical time constant increases by thissame 1.5109 factor while its electrical time constant decreases by a factor of1/1.5109¼ 0.662 In combination, the motor’s mechanical to electrical timeconstant ratio increases by a factor of 2.28 and this increase definitely affectshow the servo motor dynamically responds to a voltage command

In consulting published motor data, many motor manufacturersspecify their motor’s parameter values, including resistance, using 258C asthe specified ambient temperature NEMA, however, recommends 408C asthe ambient temperature in specifying motors for industrial applications.Therefore, pay close attention to the specified ambient temperature whenconsulting or comparing published motor data Different manufacturerscan, and sometimes do, use different ambient temperatures in specifyingwhat can be the identical motor

In the same published data servo motors are generally rated to operatewith either a 1308C (Class B) or 1558C (Class F) continuous windingtemperature Although motors with a Class H, 1808C temperature rating arealso available Assuming the motor’s resistance along with its electrical andmechanical time constants are specified at 258C, it was just demonstratedthat all three parameters significantly change value at a 1558C windingtemperature If the motor’s winding can safely operate at 1808C theresistance change is even greater because Eq (7.4-24) shows that a 1558Crise (1808C–258C) in winding temperature increases its electrical resistance

by a factor of 1.609 Hence, if the servo motor’s dynamic motion response iscalculated using the 258C parameter values then this calculation over-estimates the motor’s dynamic response for all temperatures above 258C

In all permanent magnet motors there is an additional effect thattemperature has on the motor’s mechanical time constant only As shown in

Eq (7.4-24), a motor’s mechanical time constant changes inversely with anychange in both the back EMF, Ke, and torque constant, KT Both Keand

KT have the same functional dependence on the motor’s air gap magneticflux density produced by the motor’s magnets All permanent magnet

motors are subject to both reversible and irreversible demagnetization.Irreversible demagnetization can occur at any temperature and must beavoided by limiting the motor’s current such that, even for an instant, itdoes not exceed the peak current/torque specified by the motor manufac-turer Exceeding the motor’s peak current rating can permanently reduce themotor’s Keand KTthereby increasing the motor’s mechanical time constant

at every temperature including the specified ambient temperature

Reversible thermal demagnetization depends on the specific magnetmaterial being used Currently, there are four different magnet materialsused in permanent magnet motors The four materials are Aluminum-Nickel-Cobalt (Alnico), Samarium Cobalt (SmCo), Neodymium-Iron-Boron (NdFeB), and Ferrite or Ceramic magnets as they are often called

In the temperature range,
60C < T < 200C, all four magnet materialsexhibit reversible thermal demagnetization such that the amount of air gapmagnetic flux density they produce decreases linearly with increasing magnettemperature Hence, similar to electrical resistance, the expression for thereversible decrease in both Ke(T) and KT(T) with increasing magnettemperature is given by:

Ke;TðTÞ ¼ Ke;TðT0Þ½1
BðT
T0Þ (7.4-25)

In Eq (7.4–25), the B-coefficient for each magnet material amounts to:B(Alnico)¼ 0.0001/8C

B(SmCo)¼ 0.00035/8CB(NdFeB)¼ 0.001/8CB(Ferrite)¼ 0.002/8CUsing Eq (7.4-25) it can be calculated that a 1008C rise in magnettemperature causes a reversible reduction in both Keand KTthat amounts

to 1% for Alnico, 3.5% for SmCo, 10% for NdFeB, and 30% for Ferrite orCeramic magnets Like the motor’s electrical resistance, most motormanufacturers specify the motor’s Ke and KT using the same ambienttemperature used to specify resistance However, this is not always true and

it is again advised to pay close attention as to how the manufacturer isspecifying their motor’s parameter values

Combining the effects of reversible, thermal demagnetization withtemperature dependent resistance, the equation describing how a permanentmagnetic motor’s mechanical time constant increases in value withincreasing motor temperature amounts to:

½1 þ 0:00393ðT
T0Þ

Notice in Eq (7.4-26) that the magnet’s temperature is assumed equal to themotor’s winding temperature Actual measurement shows that thisassumption is not always correct Motor magnets typically operate at alower temperature compared to the winding’s temperature However thisconservative approximation is recommended and used.

An example will be given to illustrate a change in time constants Toraise the mechanical time constant to a 1558C temperature rating inside aFerrite magnet motor, for example, the resistance increase will be thefollowing:

tmð155CÞ ¼ 1:5109=ð0:77Þ2¼ 2:54 tmð25CÞ

In Part I, the transport lag was described in the application of siliconcontrolled rectifiers (SCR) The SCR has a sinusoidal power line frequencyapplied to it in some form of circuit configuration such as a three-phase,half-wave amplifier circuit Each SCR will only conduct current in one-halfcycle of the line frequency In addition, the amount of current that willconduct in the conducting half cycle is controlled by the current at the gate

of the SCR Therefore there is a resulting dead time in the conducting halfcycle where no current flows This dead time is described mathematically asthe transport lag Transport lag has the transfer function

e
ts¼ e
jot¼ e
jy

y¼ ot ¼ phase shift of the transport lag

y ðdegÞ ¼ ot657:5 ðdegÞ

o¼ rad=secThe significance of the transport lag (dead time in the firing of anSCR) is that each type of SCR amplifier circuit will have an increasing phaselag versus increasing frequency This phase lag adds to the overall phaseshift of the servo amplifier, contributing to an unstable servo drive Thetransport lags for four different SCR servo amplifier circuits are shown inFigure 11 The relation between the transport lag of the four types of SCRcircuits and the phase lag versus frequency is shown inFigure 12.The phaselag of the SCR servo amplifier is a limiting factor in the available frequencyresponse (servo bandwidth) of this type of DC servo drive The transferfunction for transport lag does not have any amplitude attenuation withincreasing frequency