1 DESIGN AND SIMULATION OF MECHATRONIC SYSTEMS Intusoft February 1998 SUMMARY In mechatronic systems which consist of smart motion controllers, actuators and mechanisms, different technologies such as analog and digital electronics, mechanics and hydraulics, interact with each other. Through the integration of the various technologies, a more optimal product can be obtained. By the use of Computer Aided Engineering (CAE) tools such as ICAP/4 throughout the complete design process, better and more efficient products can be designed.
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DESIGN AND SIMULATION OF MECHATRONICSYSTEMS
Intusoft
February 1998
SUMMARYIn mechatronic systems which consist of smart motion controllers, actuators and mechanisms, differenttechnologies such as analog and digital electronics, mechanics and hydraulics, interact with each other.Through the integration of the various technologies, a more optimal product can be obtained. By the use ofComputer Aided Engineering (CAE) tools such as ICAP/4 throughout the complete design process, better andmore efficient products can be designed.
1.1. Mechatronic SystemsMechatronics is the synergetic combination of precision mechanical engineering, electronic control and systemsthinking in design of products and manufacturing processes [6].Motion control systems are the backbone of mechatronic machines and instruments. Motion control systemsgenerally consist of the following elements (Fig. 1) [41]:
Power source: Source that supplies the energy, e.g. electric net or battery, hydraulic tank.
Power drive: (or power amplifier) controls the amount of energy retrieved from the power source.Examples of electrical power drives are switch-mode controlled converters and thyristor controlledrectifiers. Valves are hydraulic power drives.
Actuator: converts the electrical or hydraulic energy to mechanical energy.
Transmission: converts the mechanical energy, e.g. from rotational to translational, and reduces thevelocity by means of gears, belts, screws, cams, planar or 3-D linkages, etc.
Load: the system that has to be moved.
Sensor: performs measurements of e.g. position, velocity, pressure, current, and converts it to a signalthat can be input to the controller.
Task programming unit: defines the task which is to be performed by the mechanism.
Task control: calculates the setpoint for the power drive controller; its input is the desired task and theoutputs of the sensors. The task control can be implemented in an analog (continuous) way (e.g. bythe use of analog electronic circuits or hydraulic systems) or digitally, via the use of a motioncontrol IC or an external PC or DSP board.
Power drive control: creates the signals for the switches from the power drive, based upon the setpointwhich is delivered by the task control. The power drive controller also uses sensor data, e.g. theoutput current of the power drive or the actuator velocity. Electrical power drive controllers andpower drives are often provided in the same package, which is often delivered together with theactuator.
The different components of mechatronic systems belong to different energy domains (electronics, mechanics,hydraulics, etc.). The electronic part can consist of both analog and digital electronics.Designers of mechatronic systems typically have a strong background in a few technologies. Due to the lack ofknowledge or lack of experience with the other technologies, a wide spectrum of design concepts and design
LoadTransmissionActuatorPowerDrive
PowerSource S SSS
PowerDrive
Control
TaskControl
TaskProgramming S = Sensors
Figure 1: Elements of a generic motion control system.
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problems can be ignored. The application of simulation throughout the complete design process can aid thedesigner in producing the product faster and more efficiently, and improve the information flow betweendifferent subsystem designers. Through the use of simulation, the designer can verify the correct behavior of thesubsystems in the complete mechatronic system, and the influence of design parameters on the behavior of thecomplete system.
1.2. Design LevelsDue to the different technologies and domains involved in the design of mechatronic systems, the designprocess is rather complex. To manage the design of the mechatronic product, the design process is decomposedinto several steps: conceptual design, functional design and physical design [20]:
1.2.1. Conceptual DesignThe conceptual design level deals with the design of the product architecture and the selection of the mainsystem components. At this level, rule-based algorithms are used.
1.2.2. Functional DesignThe functional design level refines the product architecture, and considers the dynamic interactions among thecomponents. The physical character of the interactions is neglected: interactions are treated as unidirectionaldimensionless signals [22]. The control design is generally performed at this level. For the functional design,block oriented diagrams are primarily used.Continuous time systems can be divided in two classes: lumped parameter models and distributed parametermodels. Lumped parameter models are described by Differential-Algebraic Equations (DAE's). Distributedparameter models are described by Partial Differential Equations (PDE's) which have to be solved with FiniteElement Methods. Most engineering systems can be treated as lumped parameter models. The generaldescription for a lumped parameter model is:
( )( )
0 f x x u
y g x u
x x
===
& ( ), ( ), ( ),
( ) ( ), ( ),
( )
t t t t
t t t t
0 0
(1)
where x is the state variable vector, u is the input variables vector, and y is the output variables vector;all are time dependent. In most cases, the first equation can be converted to an explicit form, i.e. to a setof ordinary differential equations (ODE's):
( )& ( ) ( ), ( ),x f x ut t t t= (2)
Differential equations can be solved numerically with integration rules such as Euler, Runge-Kutta andGear [27]. If the system is linear, Eq. (1) can be written using the state-space form:
& ( ) ( ) ( )
( ) ( ) ( )
( )
x Ax Bu
y Cx Du
x x
t t t
t t t
= += +=0 0
(3)
If Eq. (1) does not depend on the time derivative &x , a set of non-linear algebraic equations is obtained:
( )F x u( ), ( ),t t t = 0 (4)
Non-linear algebraic equations are often solved with Newton-Raphson techniques.In computer-controlled systems, the signals do not vary continuously; they vary only at equidistant timeintervals. Discrete-time models are characterized by sets of difference equations:
( )x f x uk k k kt+ =1 , , (5)
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1.2.3. Physical DesignThe physical design level takes the energetic interactions between the various subsystems into account. Mostengineering systems can be treated as lumped parameter systems, so that the energy flow between subsystemscan be concentrated to a few paths. There are two primary methods for modeling continuous systems at thephysical level: multiports, or bond graphs.A multiport [21] describes a system function in terms of its energetic interactions with the environment. Themultiport exchanges energy with the environment at a finite number of ports. A through variable and an acrossvariable are attached to each port. A multiport diagram of a system resembles a block diagram. In a blockdiagram, the signals are unidirectional. However, in a multiport diagram, power can flow in both directionsbetween the multiports. Analog electronic circuits are examples of multiport diagrams. Resistors and transistorsare multiports, the behavior of which is completely determined by the voltage (across variable) and the current(through variable) at the element terminals. The multiport can be constructed from basic multiport elementssuch as resistors, inductors and capacitors, or can be defined on a behavioral level, i.e. directly frommathematical equations.A Bond graph [3,12,44] is a unified graphical and topological description of the energy interaction, storageand dissipation within a dynamic system. In bond graphs, the energy flow is characterized by a flow and effortvariable. A complex system can be described by a number of bond graph basic elements which are connected bybonds. A bond is denoted by a stroke which represents the energy flow from one end of the bond to the otherend. A bond connecting two bond graph elements has a causal relationship. The construction of bond graphsrequires a high abstraction of the system, often through abstraction of the multiport model [22]. Moreover,there are not many direct bond graph simulators.Multiports and bond graphs refer to the mathematical representation, and could both be used to modelmultitechnical systems. In the mechatronic library, the multiport approach is primarily used during modelingand simulation at the physical level.
Electronic circuits are commonly simulated with circuit analysis programs such as SPICE [32] and itsderivatives, e.g. IsSpice (from Intusoft.). IsSpice is delivered with a large library of electronic components.Intusoft’s graphical pre-processor, SpiceNet (schematic editor), has been used to capture the schematics.
1.3. LibrariesFor rapid prototyping of mechatronic systems, libraries which contain models of the various components areneeded. In the ideal case, the designer only needs to select a model from the library and add the model to thesystem; and after building the complete system model, just press the button to start the simulation. Two types oflibraries can be developed: parametric component model libraries, and commercial component libraries.
1.3.1. Parametric Component Model LibrariesParametric component model libraries contain a few built-in primitives that can be called through Intusoft’sbuilt-in graphical user-interface, e.g. capacitors, resistors, stimuli, transfer function and algebraic functionblocks, etc. In addition, Intusoft now offers the multitechnical mechatronic libraries. The libraries are definedin the standard SPICE language. Since multitechnical systems can be converted to electronic systems by theenergy conservation principle, the standard can also be used to model multitechnical systems. Modeling issometimes difficult since different parameters are required in order to describe the same component at differentdesign levels: for example, motor catalog data such as maximum torque and velocity are used for theconceptual design level of electric drives, whereas the physical design level requires the rotor and statorresistances. Intusoft has added functions to SPICE in order to ease modeling tasks.
1.3.2. Commercial Component Libraries
Commercial component libraries can be constructed from primitives, or based upon parametric models. In thelatter case, the model contains only a reference to the parametric model with the corresponding set ofparameters. A variety of commercial electronic component libraries (for SPICE simulation) are available fromIntusoft.
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CHAPTER 2.
EXTENDED ANALOG CIRCUIT SIMULATION
2.1. Energy Conservation PrincipleThe total energy rate flow between a component and the environment can, for most engineering systems, beapproximated by a finite sum of some complementary physical quantities [21]. One of these quantitiesrepresents a through variable, the other an across variable. The criterion for selecting a physical quantity aseither a through or across variable is based upon the way in which it can be identified, e.g. by measurement (cf.Table 1): through variables are measured between adjacent points by first disconnecting them and including themeasuring instrument between them; across variables are measured between distant points withoutdisconnecting [21]. The reference of the across variable is related to the energy domain.
The energy conservation principle allows the conversion of multitechnical systems to a single energy domain.By translating the through and across variables in the different energy domains to the through and acrossvariables of a single energy domain, e.g. voltage and current, multitechnical systems can be modeled as electriccircuits, and can be simulated with analog circuit simulators.
2.2. Analog Circuit SimulationDue to the selection criterion for the across and through variables in the various energy domains, there is acomplete analogy between mechanical and electrical nodes and connections. The structure of the electric circuitresembles the physical structure of the model, which eases the generation of multitechnical models. Forexample, if a motor is connected to a load, the connection provides an internal feedback. This is in contrast toblock diagram modeling, in which there is no need in analog circuit modeling to build an external feedbackloop to model the effects of the load on the driver. The modeling process can be simplified by providingmacromodels for the various components, which can be added to the equivalent electric circuit in the same wayas the physical components are added to the system.
2.2.1. SPICE and IsSpiceIn analog electronics, the de facto standard for simulation is SPICE (Simulation Program with IntegratedCircuit Emphasis). All SPICE-simulators are derivatives of the SPICE2 circuit simulator which was developedat the University of California, Berkeley, in the mid-1970s. Intusoft’s version of SPICE, called IsSpice, offers awide variety of analyses, including DC analysis, transient analysis, AC analysis, etc.
Power variables Power (P)through across [W]
Electrical current [A]i
e.m.f. [V]v
i*v
Magnetic flux rate [Wb/s≡V]dφ/dt
m.m.f. [A] *dφ/dt
Mechanicaltranslational
force [N]F
velocity [m/s]v=dx/dt
F*v
Mechanicalrotational
torque [Nm]T
angular velocity [1/s]ω=dθ/dt
τ*ω
Fluidic volume flow [m3/s ]Q
pressure [Pa]p
Q*p
φ = magnetic flux x = position θ = angle
Table 1: Power variables in the different energy domains
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In SPICE, the circuit is built from primitives such as resistors, capacitors, inductors, independent andcontrolled voltage and current sources, diodes, transistors, etc. The schematic is captured, and then the SPICEinformation is automatically written to a text file, called a netlist. Circuit analysis is based upon the applicationof Kirchhoff’s current law, Kirchhoff’s voltage law and branch constitutive equations, i.e. the relation betweenthe voltage and current for the electric elements [27]. SPICE converts the circuit elements and theirinterconnections to system equations using node equations which are expressed in terms of node voltages andKirchhoff's current law [27]. The system equations are a set of nonlinear first order ordinary differentialequations. During transient analysis, a numerical integration method is used to convert the nonlineardifferential equations into a set of nonlinear difference equations, which are solved simultaneously using theNewton-Raphson method. This involves the conversion of the non-linear equations to linear equations, andtheir subsequent solution is accomplished using a sparse LU decomposition technique [32].Standard SPICE2G.6 does not allow behavioral modeling: controlled sources can only have a fixed (nonparametric) linear gain, or they can be a polynomial of a combination of node voltages. This makes thedevelopment of parametric and multitechnical models in standard SPICE very difficult.IsSpice uses either the trapezoidal or gear integration method. The results of IsSpice simulations can be viewedwith the graphical post-processor, IntuScope.In addition to the standard SPICE primitives, IsSpice allows the use of mathematical expressions. With thesecapabilities, it is possible to model complex mechatronic systems in a relatively simple way.
2.2.2. IsSpice primitivesThe basic components in IsSpice, which are used to model multitechnical systems, are:• resistor (prefix R).
• capacitor (prefix C). The initial voltage can be set.
• inductor (prefix L). The initial current can be entered as a parameter in the definition of theinductor. Core material models (including hysteresis) are provided in a library.
• voltage source (prefix V). Generates a voltage source as a function of time. Possible waveformsare: constant, exponential, pulse, piecewise linear, sinusoidal, and frequency-modulated.
• current source (prefix I). Generates a current source as a function of time. Possible waveformsare: constant, exponential, pulse, piecewise linear, sinusoidal, and frequency-modulated.
• voltage controlled voltage source (VCVS) (prefix E). Generates a voltage as a function of thevoltage between other nodes.
• voltage controlled current source (VCCS) (prefix G). Generates a current as a function of thevoltage between other nodes.
• current controlled voltage source (CCVS) (prefix F). Generates a voltage as a function of thecurrent through a voltage source.
• current controlled current source (CCCS) (prefix H). Generates a current as a function of thecurrent through a voltage source.
• voltage controlled switch (VCSW) (prefix S). Generates a switch which is controlled by a voltagebetween two controlling nodes. IsSpice treats the switch as a voltage-controlled resistance, thevalue of which changes from high to low over a voltage interval.
• transmission line (prefix T). Generates a bi-directional ideal delay line. Distributed models arealso provided in order to simulate lossy lines.
• diode (prefix D). Together with the diode, a model must be entered. An ideal diode (infinite flowfor positive voltage, no flow for negative voltage) is modeled as [5]:
D 1 2 DIDEAL.MODEL DIDEAL D(N=0.001)
• transistor (Bipolar, JFET, MOSFET, etc.).
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2.3. Mechatronic LibraryA hierarchical IsSpice library has been generated to extend the use of IsSpice to multitechnical devices. Thelibrary consists of mathematical elements and electrical, electromechanical, mechanical and hydrauliccomponents. Table 2 shows the elements contained in the library.
2.3.1. Hierarchy, Parameters and Accuracy LevelsThe library is constructed hierarchically: the component models are constructed from lower level models orphenomena. The basic phenomena for each energy domain are described by the built-in IsSpice primitives(resistance, inductance, capacitance), or by voltage controlled sources. The models are parametric.During the different phases of the design of a mechatronic product, the different models do not have to bemodeled at the same accuracy level. Modeling of the components at the highest accuracy level requires a lot ofcomputing time, and also a large number of parameters (which are not always known). The model accuracy hasto be adapted to the design level. The various (often not yet selected) components can be modeled roughly, e.g.as ideal elements, during the initial phases of the design. To check the influence of nonlinear elements on thesystem behavior, all of the components that may have an influence have to be modeled accurately. For instance,during the design of the power drive in a motion control system, the power switch behavior is of interest, andthe mechanical load can be modeled rather roughly (e.g. ignoring flexibility and backlash). During the designof e.g. the computer control loop for positioning the load, calculation of the different power switch signals isnot desired − if they do not influence the motion of the load − since they increase the computing timetremendously. The power drive is then modeled on a functional level, e.g. as a first order system with outputlimits. In order to verify the starting transients and the influence of the motor torque ripple on the motion of theload, the complete motion control system has to modeled accurately.The library therefore provides models at various accuracy levels. The accuracy level is indicated by a numeralthat is appended after the model name (e.g. GEAR0, GEAR1, GEAR2) : level 0 is the most simple (ideal) level,whereas levels 1,2, etc. are more complex models. For instance, the most simple model of a gear pair (GEAR0)is an ideal transformer. The more complex models GEAR1 and GEAR2 include the inertia of the gears,damping, friction and backlash.The different levels are compatible with each other: all of the parameters that are needed for the most complexlevel return in the definition of the more ideal levels, although they are not used there. The default values of theparameters are chosen so that they do not influence the model if they are not defined (e.g. friction force 0, gain
sources electrical voltage and current sources (IsSpice primitives, net supply)mathematical input function: (step, impulse, ramp, PWL, etc.)
control task control : PIDpower drive control: PWM, hysteretic current control, thyristor firing angle
mathematical Algebraic operations (+,-,*,/) and functions (abs, sqrt, sin, etc.), differentiator, integrator,sample and hold, first and second order transfer functions, second order filters, limit,delay, dead zone, relay, hysteresis, saturation, cubic curve segments, powercalculations, statistics, transformers, gyrators, variable impedances
electrical IsSpice primitives and libraries
mechanical translational: mass, damper, spring, friction, gravity, coupling, contact, backlashrotational: inertia, damper, spring, friction, coupling, shaft, contact, backlashmixed translational/rotational: rot.↔trans., rolling (sliding), lever, unbalanceplanar (linkages) : mass, spring, damper, point on object, rotational and translational joint
1, initial velocity 0, infinite limits, etc.).While selecting a more accurate model, the number of input ports of the model increases if the more accuratemodel describes the influence of a new time-dependent parameter.Also the complete model structure can change, since a behavioral model can consist of more than one physicalcomponent. For instance, a power amplifier is modeled at the behavioral level as a first order system. At thephysical level, the power amplifier model consists of one or more converters, the drivers for the gate signals,the control algorithm for the switch signals and the power supply.
2.3.2. ScalingIsSpice works optimally in the range 10-3...103, which is the most common rage in electrical engineering. Inmultitechnical systems, not all of the variables are in this range. For instance, hydraulic systems arecharacterized by large pressure values (in the order of 105 - 106 Pa) and small volume flow values, in SI-units.To convert the actual values to the IsSpice range, a scaling factor σ is used. The relation between the IsSpicevoltage and current values and the original values (for across variable a and through variable q) is:
va
i q
=
=σσ
(6)
The through variable is determined with respect to a reference which depends on the energy domain. Forinstance, temperature can be entered in degrees Celsius and in Kelvin; pressure can be entered asabsolute pressure or related to the atmospheric pressure.
The reference at which (e.g. temperature dependent) parameters have been determined does not have tocorrespond to the reference for the across variable. If the across variable temperature is entered in Celsius, andthe measuring reference is Tmeas =20°C, then the measuring temperature and the temperature dependence arepassed as parameters to the subcircuit. For instance, in the model of a temperature dependent resistor R=Rmeas+kT(T-T meas ), the temperature dependence kT and the measuring temperature T meas have to be entered asparameters.The scaling influences the values of the primitives (the index * indicates the values entered in IsSpice):
( )
Ra
qv
ivi
RR
Cq
da dtidv dt
idv dt
C C
La
dq dtv
di dt
vdi dt
LL
= = = ⇒ =
= = = ⇒ =
= = = ⇒ =
∆ ∆ ∆
∆ ∆ ∆
σσ
σσ
σσ σ
σ
σσ
σσ
22
22
22
1
*
*
*
(7)
In the multitechnical library, scaling has only been included for hydraulics.
2.4. User Interface: The Schematic EditorThe multitechnical library consists of a large variety of models. This results in a large variety of subcircuitdefinitions (two-node subcircuits, four-node subcircuits, mechanical, electrical and hydraulic inputs, etc.) andof subcircuit parameters. For efficient and error-free model entry, the user interface should be such that it givesthe user as much information as possible on the models and the model parameters. A graphical environment istherefore desired, in which the user can pick the component model from a menu, draw the connections betweenthe various components, and enter the model parameter values.Schematic editors have been developed for graphical input of electronic circuits. The user can create aschematic by retrieving the desired symbols from a library and connecting the symbol nodes with wires.This section will describe how Intusoft’s SpiceNet schematic editor has been used to capture the design ofmultitechnical systems.
2.4.1. Nets and BusesThe different components of the multitechnical systems are connected by nets or buses. A net between two sym-bols corresponds to a common node in the SPICE-netlist. A bus is an array of nets, and is normally only used in
Schematic
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digital electronics. They are however very useful for linking components that have large number of similarnodes.Buses are used in the mechatronic library for planar mechanics, and for the gate signals in switch-mode andthyristor power converters.
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CHAPTER 3.
MULTITECHNICAL MODELINGTable 1 gives an overview of the relationship between the electric power variables and the variables in the othertechnologies. Table 3 gives the basic electric primitives in the various energy domains.
3.1.Mathematical Model BuildingMost mathematical models which are included in the library are blocks such as integrators, differentiators,algebraic functions, transfer functions, limiters, etc. The mathematical library contains further couplingelement models and variable impedances.
3.1.1.Block diagramsThe definition of blocks allows the use of IsSpice for the functional design level. The output of the block is onlya function of the input and a set of internal state variables. The input and output impedance are infinite. Infiniteimpedances can be created by copying voltages over voltage controlled sources. For instance, the subcircuit foran amplification is:.SUBCKT AMP 101 102 K=1 ; same as gain* AMP : Amplification* input node 101* output node 102* parameters : K (gain)RIN 101 0 1GROUT1 103 102 1NROUT2 102 0 1GBE1 103 0 V= K*v(101).ENDS
For most of the blocks, equivalent electric circuits or SPICE models can be found in literature [5].
3.1.2.Coupling elementsPure coupling elements are elements which neither store nor dissipate energy, but simply transform it from oneform to another [20]. In a pure two-port coupling element, the power at both sides is the same, i.e.: i1v1=i2v2.Coupling elements are essential for modeling power transmission components and actuators, where energy istransformed from one energy domain to another.
An ideal transformer is a two-port coupling element in which the voltages at both sides areproportional to each other:
v Kv
iK
i
2 1
2 1
1=
=(8)
A transformer is modeled with two voltage controlled current sources, one in the primary loop and one in thesecondary loop [5].
Table 3: Electric primitives in the different energy domains
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Lossy transformer. At the conceptual level, losses are characterized by entering the efficiency of the powertransmission, and is expressed as a percentage of the global power. In most cases, one of the power variables iscorrectly transmitted (e.g. the velocity in a gear pair), and the losses occur in the other power variable (e.g. thetorque in a gear pair). The efficiency can be represented by ηi for the through variable, and ηv for the acrossvariable:
v Kv
iK
i
v
i
2 1
2 1
=
=
ηη (9)
If the gain, K, of the transformer is not constant, but depends on an external variable, then the couplingelement is called a modulated transformer. [12].
For an ideal gyrator, the voltage at the secondary end is proportional to the current through theprimary circuit:
v Ki
iK
v
2 1
2 1
1=
=(10)
The gyrator is modeled with two current controlled voltage sources.
3.1.3.Variable ImpedancesNonlinearities can often be modeled by the use of nonlinear impedances. The resistors, inductors and capacitorsused by IsSpice are, however, always constant. In this section the modeling of non-linear impedances isdescribed. The value of the impedance equals always a reference value multiplied with the voltage between twocontrol nodes.Variable Resistances A non-linear resistance, R, can be modeled via a controlled voltage source. The
value of the voltage source equals the actual value of the resistance R=Rref*vctrl (where Rref is areference resistor and vctrl is the voltage over the control nodes), multiplied by the current, i,through the circuit:
.subckt rvar 101 102 201 202 params: rref=1 ; 101 102: power nodes; 201 202: control nodes
rin 201 202 1T ; input resistance for control nodesbeout 101 106 v=Rref*i(vsense)*v(201,202) ; multiply current by Vctrlvsense 106 102 0 ; sense output current.ends
Variable Inductances can be defined in two ways: v=Ldi/dt (LVAR) and v=d(Li)/dt) (LVARX). Theschematic is the same for both definitions, but the values for the controlled sources differ.
For LVAR, the current, i, is sent through a reference inductor, Lref; a voltage equal to theproduct of the voltage drop over the inductor and the control voltage, vctrl, is then placed betweenthe output nodes. For LVARX, the product of the current, i, and the control voltage, vctrl, is sentthrough the reference inductor, Lref; the voltage drop over the reference inductor is copied over theoutput nodes.
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Figure 2: Schematic of the variable inductance
Variable Capacitances A variable capacitance can also be defined in two ways: i=Cdv/dt andi=d(Cv)/dt.
3.1.4. Equation Solution
Algebraic Equations The IsSpice program can be used to solve algebraic equations by application of aVCCS. The netlist for the equation x−cos(x)=0 is:
bg 0 x v=v(x)-cos(v(x))r 0 x 1G
The VCCS applies a current which is function of the voltage over the output nodes of theVCCS. The large resistance pushes the current to zero. The result can be obtained from the biaspoint calculation. If the equation is written in explicit form, (x=f(x)), an alternative model is:
bg 0 x v=cos(v(x))r 0 x 1
The current through the resistor and the VCCS should then be equal to v(x), since the value ofthe resistor is 1Ω.
Systems of Algebraic Equations can be modeled by defining a set of electric circuits, all of whichconsist of a VCCS, and large resistances, which push the current to zero. For example, a systemwith two equations (x+y=1 and xy=1) can be modeled as:
bgx 0 x v=v(x)+v(y]-1rx 0 x 1Gbgy 0 y v=v(x)*v(y)-1ry 0 y 1G
Differential Equations (in explicit form) can be solved by changing the resistor (in the explicit form ofthe algebraic equation) to a 1 Farad capacitor. For instance, the equation dx/dt=4x−2 is modeled as:
bg 0 x v=4*v(x)-2c 0 x 1 ic=1 ; capacitor with initial conditions.r 0 x 1G ; large resistor to avoid floating nodes
For Integral Equations (in explicit form) the capacitor has to be replaced by an inductance of 1 Henry.
3.2.1.Translational mechanicsIn translational mechanics, the across variable is the translational velocity, v, and the through variable is theforce, F (cf. Table 1).Mass The law of Newton for translational motion is:
Fd mv
dt=
( )(11)
In most cases, the mass, m, is constant, and the law of Newton is: F=m*dv/dt. The electric analog of amass is a capacitance C=m. A variable mass, e.g. a fuel tank, is represented by a variablecapacitance. In some models,
Gravity is added as a constant current source which is proportional to the mass.
Spring. The force in a spring is given by: F=kx=k∫vdt. The electric analog of a spring is an inductance,L=1/k. If initial conditions on the position are applied, it is more appropriate to represent the springby a position-controlled force. This method allows the inclusion of non-linear springs.
Damping. The force in a damper is: F=bv. This corresponds to a resistance, R=1/b.
Friction. Friction cannot be represented easily by electric primitives, and requires behavioral modeling.The friction force between two objects consists of three components which depend upon the relativevelocity, vr, between the objects: the viscous friction, Wb=bvr, the kinetic friction force,Wk=Wsign(vr), and the static friction, Ws, which only occurs at zero velocity.
The discontinuous functions for Wk and Ws are extremely difficult to handle by simulators, andare therefore approximated by piecewise linear functions, which have a small ramp near thediscontinuity. Friction is modeled by two tables and one mathematical expression in IsSpice (Fig.3).
The static friction force, Ws, is however often not taken into account. The library contains twofriction models: FRICT1 which models only kinetic friction, and FRICT2 which models viscous,kinetic and static friction. The FRICT1 friction model returns in many mechanical models todetermine friction losses and to check relative motion: the friction force is the maximum force thatcan be transmitted between two objects; if the force is smaller than the maximum force, then norelative motion occurs between the objects. The kinetic friction is evaluated as a function of W0v
r:
the term W0 is added to discard the evaluation of the tabular function if the W friction force is equal
to 0; this occurs often in hierarchical models in which friction is included as a parameter. Thekinetic friction force can be entered as a constant force, W,or by a friction coefficient, µ.
Contacts. Two models are provided for contacts: a rigid contact, and a flexible contact.
The rigid model is modeled as a position controlled switch and a diode. A contact can only exertpressure without tension, which can be modeled by a diode. This approach, however, causes suddenvelocity jumps, and thus infinite accelerations. A disadvantage is that the combination of diode andswitch may cause convergence errors.
The flexibility of the switch is modeled by a spring which has a limited length, damping, frictionand mass. The contact is only active if the spring is pressed.
Backlash. Many authors model backlash by opening or closing a switch as a function of the relativeposition of the two objects. This model is not correct: the switches also allow negative forces (i.e. atensional force on the contact), which is not possible. A backlash is modeled by two contacts. Twomodels have been provided: one for rigid backlash, and one for flexible backlash.
A General Load consists of a mass, damping and friction. This model is almost always called if amechanical component is included (e.g. the rotor in a motor, the piston in a hydraulic cylinder).
3.2.2.Rotational mechanicsFor rotational mechanics, the across variable is the rotational velocity, ω, and the through variable is thetorque, T (cf. Table 1). Due to the analogy between rotational and translational systems, the same models asthose for translational mechanics can be used.
3.2.3.Planar mechanicsThe analog circuit simulator can also be used to simulate the motion of mechanisms. General purposemechanism analysis programs perform advanced manipulations on the system's geometrical description inorder to obtain an optimal set of equations. SPICE uses only node equations to construct the system's equations,so that only a simple Cartesian model can be used to model planar mechanisms. This method is not computer-efficient, but complete simulation of the system in IsSpice allows the investigation of, for example, theinfluence of actuator transients on the mechanism motion.For the simulation of a planar mechanism, the following basic models have been developed: the planar motionof the mass center of an object, a model for relating forces and velocities in a point to the forces and velocitiesof the mass center, and models for rotational and translational joints.Mass For a mass, moving in a plane, three motion laws (Eq. 11) are generated. The motion of the mass
center is modeled by three electrical circuits, one for each velocity component. The voltage in eachof these nets corresponds to the absolute velocity component. Gravity is included.
Position The velocity components are integrated in order to obtain the position components. The netsfor the position components do not transport energy, however. The position components maytherefore only be used as function values in controlled sources.
Motion of a point on an object A point, P, on the object is characterized by the coordinates xp,yp withrespect to the coordinate system which is attached to the mass center, M, of the object. The absoluteposition of this point, P, is given by:
x
y
x
y
x
yx
y
x
yP
P
M
M
p
p
M
M
p
p
=
+
=
+
−
⋅
,
,
cos sin
sin cosθ
θ
θ θθ θ
(12)
where xM,yM and xP,yP are the absolute position components of resp. the mass center, M, and thepoint P, xp,yp represents the position components of point P in a coordinate system attached to themass center, M, and θ is the angle of the object's local coordinate system with respect to the globalcoordinate system. The translational velocity components of P equal:
v v yP x M x p, , ,= − ω θ (13)
16
v v xP y M y p, , ,= + ω θ (14)
where vM,x,vM,y is the absolute velocity of the mass center, M, and ω is the rotational velocity of theobject. The contribution TMP of the forces XP and YP and the torque TP in P to the torque in themass center, M is:
T T X y Y xMP P P p P p= − +, ,θ θ (15)
Equations (13), (14) and (15) are equivalent to a combination of two transformers with variableratios: one between the electric circuit for the rotation and the circuit for the x-direction with ratio -yp,θ, and one between the circuit for the rotation and the circuit for the y-direction with ratio xp,θ .
Rotational Planar Joint In a rotational planar joint (or pin), the translational velocity components atboth sides are equal (this corresponds to a direct link between the two circuits), and the torque iszero (this corresponds to a very high impedance to the ground at both sides). Macromodels havebeen generated for the rotational connection between two parts, combining both the rotational jointand the transformation between the mass centers and the connection point. A more complex modelof the pin joint includes joint damping and friction.
Translational Planar Joint In a translational joint in which a fixed point of part 2 moves in part 1, thefollowing relationships exist between the velocity components:
( )v v vP x P x r s1 2 1, , cos= + +θ θ (16)
( )v v vP y P y r s1 2 1, , sin= + +θ θ (17)
where θ1 is the angle of part 1, θs is the angle of the slider with respect to the local coordinatesystem in part 1, and vr is the relative velocity. In a frictionless slider, the force along the slider iszero, or:
( ) ( )0 1 1= = + + +F X Yr P s P scos sinθ θ θ θ (18)
Equations (16), (17) and (18) are equivalent to two transformers with variable gains: one between acircuit which represents the relative motion and the circuit for the x-motion with ratio cos( )θ θ1 + s ,and one between the circuit for the relative motion and the circuit for the y-motion with ratiosin( )θ θ1 + s .
Initial Velocity If, for a mechanism with one degree of freedom, the initial velocity of the driver is notzero, all of the initial velocities of all mass centers (translational and rotational) should be entered.To avoid this, an initial velocity component has been generated; it applies the desired initial velocityduring a small starting period (e.g. 1 µs or 1 ms). The initial velocity is applied by a constantvoltage source over a switch which is opened after the starting period, and does not influence thecalculations afterwards.
The use of busses in the Schematic Editor The models for planar mechanics have many nodes, e.g. fora translational joint, the x- and y-translational velocity components and the rotational velocity andposition of the mass center of both parts are required. The resulting schematic for a simplemechanism consists of a large number of nets; connection errors are easy to make.
Through the use of busses, the schematic for planar mechanic systems becomes much moresimple. The dimension of the bus is 6: x-position, y-position, rotational velocity, x-position, y-position and rotational position. For each subcircuit for planar mechanics, a subcircuit has beenbuilt; this subcircuit converts the bus-format of the subcircuit to the original library format. Buscomponents, which are not used in the original subcircuit, are connected over a very largeimpedance to ground, or connected to equivalent bus components at other input ports. Positioncomponent nets do not transmit power. The position components are always considered to beinputs, except in the case of the mass model (which gives the position of the mass center) and forthe point on object model.
17
3.2.4. Mixed Rotational-TranslationalMixed Rotational-Translational systems have one translational and one rotational component, like rolls, andcan be treated as special cases of planar mechanisms.Rolls An ideal roll is a transformer from rotational to translational: vroll-vbasis=-Rωroll, with vroll
and ωroll are the translational and rotational velocity of the roll center, and vbasis is the velocity ofthe reference. A higher level model includes slippage: slippage occurs if the force between the rolland the reference at the contact point is larger than the maximum transmittable friction. Atslippage, the velocity of the contact point will differ from the velocity of the reference, and only themaximum friction force is transmitted.
Unbalance Due to unbalances of rotors, vibrations are passed to the frame on which the system ismounted. The unbalance is defined as a mass which is added eccentrically to the rotor. The forcethat is passed to the system frame depends on the position of the rotor. An unbalance is a specialcase of the planar mechanical model “point on an object” (the centric rotating disk). The “point onthe object” is the unbalance where an additional mass is added. Only 1 velocity component is used(the system is supposed to be very stiff in the other direction, so that no power is transmitted). Theunbalance transforms the rotating energy of the mass to vibration energy of the motor support, andis modeled as a transformer (Eq.(13) and Eq. (15)).
3.2.5.TransmissionThe ideal power transmission element is a pure coupling element, since no power is dissipated. Models havebeen provided at three levels: level 0 corresponds to an ideal coupling element, level 1 includes inertia and jointfriction, and level 2 includes other nonlinear effects.
Gears. An ideal gear is a transformer with a ratio k=(ω2-ωref)/(ω1-ωref)=-n1/n2, where n1 is thenumber of teeth on the first gear. This is valid for all kinds of gears: spur gears, epicyclical gears,helical gears, rack and pinion, etc. Models have been provided for gears with fixed axes and forplanetary gears. More advanced models include gear inertia, friction and backlash.
The Friction Drive is equivalent to a gear. Slippage occurs if the driving torque is larger than themaximum friction force.
Lead Screws transmit a rotation to a translation, with a ratio of k=v2/ω1=p/2π, where p is the pitch perround. A distinction is made between conventional screws and ball screws.
The higher level model for the normal screws includes the friction in the screw, which dependsupon the motion direction and the geometry of the screw. The tangential force, F2, is a function ofthe driving moment, T1 [16]:
( )( )F
r p v
p r vT2
2
2
1
2
2=
+
−
π µ
π µ
' sign
'sign(19)
where µ' is the friction coefficient, r is the screw radius, and v2 is the translational velocity.
For ball screws, a constant friction force is taken. A preload is included, which aims to avoidbacklash [52]. The total friction force becomes:
( )W W F Ft N pre= + +µ (20)
where Wt is the translational friction force in the joint, FN is the normal force, and Fpre is the
preload.
Belt drives. An ideal belt drive is a transformer with a ratio k=ω2/ω1=r1/r2, where r1 is the radius ofthe first pulley. At belt drives, slippage can occur if the driving torque is larger than the maximumtorque transmitted by friction. The friction along the belt is derived from the equation of Eythelwein[16]:
18
F F
F Fe F Av qvf
ff
1
2
2 2−−
= = =µβ ρ with (21)
where F1 and F2 are the forces acting along the belt sides, β is the arc of contact, µ is the frictioncoefficient, and Ff is the centrifugal force, with q being the mass per unit length for the belt. Twopossible pretension methods have been included: a constant force at the slack side, andpretensioning by adjustment of the center distance. Models have been developed for flat belts, forV-belts and for synchronous belts.
The model of synchronous belts is based upon the chain model, since no slippage can occur.Synchronous belts can also be used for transmission of rotational to translational energy. Atranslational node and load have been included in the model.
Chains are analogous to belts, but no slippage can occur. The model includes the flexibility of thechain. A model which uses the polygon effect [16] has also been included.
Clutches are modeled by two inertia, in between which a friction force is applied.
Brakes are a special case of clutches, where one inertia corresponds to the fixed frame.
Freewheels are the equivalent of diodes. In one direction, they transmit only a small friction torque, anda large torque can be transmitted in the other direction.
Cams Ideal cams are modulated transformers. In most cases, the pressure angle is not zero, and thetotal driving torque is not transmitted to the follower. The cam model calls two external tables (oralgebraic functions): the first table is for the velocity of the follower as a function of the position ofthe cam (at unit cam speed) (i.e. the derivative dx/dθ where x is the position of the follower and θ isthe position of the master; the second table defines the relationship between the torque delivered bythe master and the force on the follower (which depends upon the pressure angle and the momentarm). IsSpice cannot not jump from one end of a table to the other end without passing through allof the other elements, as needed at the crossing 2π-0.
Cardans allow the rotation to pass to non-aligned shafts. The rotation speed of the output shaft dependsupon the angular position, θ1, of the input shaft [16]:
ωω
αθ α
2
12
121
=−
cos
sin sin(22)
where α is the angle between the two shafts.
Levers convert a rotation to a translation. If the rotations are small, the lever corresponds to atransformer with a fixed ratio. Models with fixed and moving pivots have been provided. Themodels are only valid for small rotations; for larger rotations, the planar mechanics models must beused.
Pulley. The ideal pulley reverses the sign of the velocity (relative to the motion of the axis of the pulley).Models have been provided with fixed, moving and driven axis. If the force between the belt orcable and the pulley is larger than the allowable friction, slippage occurs. The maximum frictionforce is derived from the equation of Eythelwein (Eq. 21).
3.3. Electro-mechanical and magnetic systemsElectro-mechanical systems convert energy from the electrical to the mechanical energy domain. In most cases,energy is transferred in two steps: first from electric to magnetic, and then from magnetic to mechanic. In themagnetic system, the across variable is the magnetomotive force, , and the through variable is the flux rate, dφ/dt.
19
3.3.1. Magnetic elements
Reluctance The only element that occurs in magnetic systems is the magnetic material which ischaracterized by its reluctance, = /φ, which is constant if the material is not saturated.
F R= = =lA
lArµ
φµ µ
φ φ0
(23)
where is the reluctance, l is the path length, A is the section, and µ is the permeability (µ0 isthe permeability of air, and µr is the relative permeability). In the electric-magnetic equivalence,the reluctance corresponds to a capacitor, C=1/ , since it is the ratio between the across variableand the integral of the through variable.
Permanent Magnet Materials are, in a first approximation, treated as elements that generate a constantflux φ (the flux rate is thus zero, which is the same as a ground element for an electric circuit and afixed point in translational mechanics).
Hysteresis To model the saturation of the core, and the losses due to hysteresis, the model of Chua andStromsmoe [4] is used. In this model, the hysteresis is described by a differential equation:
[ ]d tdt
g t f tφ
φ( )
( ) ( ( ))= −F (24)
which can be rewritten as:
F ( ) ( )t gddt
f=
+−1 φ
φ (25)
This equation can be transformed into an electric circuit: the function g −1 corresponds to anonlinear resistance, and the function f −1 corresponds to a nonlinear capacitance in series with theresistance. The capacitance models the energy which is stored in the magnetic field, and theresistance models the hysteresis losses. Both the resistance and the inductance are modeled by acubic curve in the IsSpice model.
Electro-magnetic Conversion The element that converts energy from the electric circuit to themagnetic system is a gyrator. The electric circuit generates a magnetomotive force, =Ni, in themagnetic system. In the electric circuit, the magnetic system generates a back e.m.f. voltage whichequals:
Vddt
Nddtg = =
ψ φ(26)
where ψ=Nφ is the flux linkage, and N is the number of windings per stator pole.
Magneto-mechanical Conversion The energy in the gap is converted partially to mechanical energy,and partially to an increasing magnetic field energy in the gap. The total energy in the gap(magnetic coenergy) equals [9,13]:
W dgap gap gap
gap
= ∫φ θ( , )F F
F
0
(27)
where gap is the magnetomotive force over the air gap, which depends upon the position, θ, of therotor. The torque which is produced by the rotating magnetic field is equal to the partial derivativeof the magnetic coenergy with respect to the angle [9]:
TW
TW
ggap
const
ggap
constgap
= − == =
∂∂θ
∂∂θ
φ
or F
(28)
20
Electro-mechanical Conversion The inductance, L , is defined as the ratio between the flux linkage, ψ,and the input current, i, and depends generally upon the rotor angle, θ, and the current, i (in thecase of saturation). For a linear system (i.e. L independent of i), Eq. (26) becomes:
( ) ( ) Vd i
dt ididt
ddt
Ldidt
idL
dL
didt
idLdg r s r s= = + = + − = + −
ψ θ ∂ψ∂
∂ψ∂θ
θθ
θθ
ω ω θθ
ω ω( , )
( )( )
( ) (29)
The first term in Eq. (29) is the self-inductance, the second term is the back e.m.f. Eq. (28)becomes [29]:
( ) ( ) ( )( )
TW
d L id Ni L idii dL
dgap
gap gap N
i i
= = = =
=∫ ∫ ∫∂
∂θ
∂ φ θ
∂θ
∂ θ
∂θ
∂ θ
∂θθ
θ
φ
F F, ( )0
1
0 02
2(30)
The dependence of the inductance upon the position increases the simulation time for motorswhich operate at high velocities. In many cases, a position-independent motor model can then bechosen (e.g. a permanent magnet DC-motor model).
Behavioral model In many cases, not all of the motor model parameters are available. The models thathave been developed model each electric phase separately, and contain position-dependentinductances. This makes simulation speed very slow. Behavioral models are used to speed up thesimulation; these models use parameters that can be retrieved from catalogues.
The motor is combined with the power amplifier, allowing the control of the torque between aminimum and a maximum torque. A model has been developed for the combination of motor andamplifier, with an input node for the desired velocity, and two mechanical nodes. The model uses aPI controller to calculate the desired torque, and limits the output torque according to a maximumtorque-velocity relationship.
3.3.2. DC-motorsIn DC motors, the orientation of the magnetic field is constant with respect to the stator. In this case, the torquecan be derived from the force on a conductor, which carries a current, i, in a magnetic field [11]: F=Bileff,where B=φ/A is the flux density and leff is the effective length of the conductor. The torque, Tg, which isdeveloped in the motor, therefore equals:
T C ig t a= ψ (31)
where Kt is the torque constant, a factor that depends upon the design of the motor, ψ is the rotor fluxlinkage, and ia is the armature current.
The back-e.m.f. (electromotive force) voltage in a conductor, which moves with a velocity, v, through amagnetic field, equals [11]: Vemf=Bvleff. For the armature windings in the DC-motor, this becomes:
( )V Cemf e r s= −ψ ω ω (32)
where Ke is the voltage constant, ωr is the rotational velocity of the rotor and ωs is the rotationalvelocity of the stator frame.
Separately Excited DC-motor. The equations for a separately excited DC-motor, including fieldsaturation, are [19]:
V R id i
dtR i
d i
di
di
dtf f ff
f ff
f
f= + = +ψ ψ( ) ( )
(33)
( )V R i Ldidt
Ca a a aa
e r s= + + −ω ω ψ (3433)
( )T C i Jddt
b Tl t a mr
m r s fric= − + − +
ψω
ω ω (35)
21
where Vf is the field voltage, Va is the armature voltage, Tl is the torque at the motor shaft, ψ is themagnetic flux linkage, ia is the armature current, ωr and ωs are the rotor and stator velocity, Raand Rf are the armature and the field resistance, La is the armature inductance, Jm is the rotorinertia, bm is the damping coefficient, and Tfric=Wmsign(ωr-ωs) is the friction torque between therotor and stator.
If the magnetic circuit is not saturated, the flux linkage, ψ, is proportional to the field current, if: ψ=Lfif, where Lf is the field inductance. Eq. (33), (33) and (33) are then simplified to:
V R i Ldi
dtf f f f
f= + (36)
( )V R i Ldidt
K ia a a aa
e r s f= + + −ω ω (37)
( )T K i i Jddt
b Tl t a f mr
m r s fric= + − +
−ω
ω ω (38)
where Ke=CeLf is the back-emf constant and Kt=CtLf is the torque constant. If Ke and Kt are
equal, all of the electrical power is converted to mechanical power. By making Kt smaller than Ke,
losses in the power conversion can be modeled.
22
Figure 4: Schematics for the separately excited DC-motor model and its subcircuits.SUBCKT DCSE1 1 2 3 4 5 6 Ra=5.5 La=6.2M Rf=5.5 Lf=6.2M Jm=56U Bm=.1N+ Ke=62M Kt=62M w0=0 Wm=0 Ifsat=1 Lfsat=.6M di=.2* DCSE : Separately exited shunt direct current motor model* Level 1 : linear model* nodes : 1 2 : armature nodes* 3 4 : field nodes* 5 6 : mechanical nodes* params: Ra, La : armature resistance and inductance* Rf, Lf : field resistance and inductance (non-saturated)* Jm, Bm, Wm : rotor inertia, damping and friction* Ke : back emf constant (V= Ke*w*if)* Kt : torque constant* w0 : initial velocity* Ifsat : saturation field current* Lfsat : inductance at saturation (tangent to flux curve)* di : blend (symmetrical) of saturation curve between Ifsat-* di/+di* A+ A- F+ F- SHAFT+ SHAFT-xa 1 8 winding R=Ra L=La ; armature windingVIa 9 2 ; armature current sensingxf 3 11 winding R=Rf L=Lf ; field windingVIf 11 4 ; field current sensingBemf 8 9 v=Ke * i(Vif) * v(5,6) ; back emfBgtrq 6 5 i=Kt * i(VIa) * i(VIf) ; generated torquexrot 5 6 load m=Jm B=Bm w=Wm v0=w0 ; mech. load.ENDS*------.subckt WINDING 101 102 R=1 L=1m* WINDING : model of a motor winding* nodes : 101 : input node* 102 : output node* parameters : R : resistance* L : inductanceR1 101 1 RL1 1 102 L.ends*------.subckt load 1 2 m=1 b=1n w=0 v0=0 g=0* LOAD : Load (i.e. mass with connection to ground)* input node : 101* output node : 102 (reference, e.g. ground)* parameters : m : load inertia* b : load drag* w : load friction* v0 : initial mass velocity* g : gravityxm 1 massg m=m v0=v0 g=gxd 1 2 damper b=bxw 1 2 frict1 w=w.ends*------.SUBCKT MASS 101 m=1 v0=0* MASS : Mass (1 node)* input node 1* parameter : m : mass* v0 : initial velocityR1 101 0 1G ; dummy resistance to avoid floating nodeC1 101 0 m ic=v0.ENDS*------.SUBCKT DAMPER 101 102 B=1* DAMPER : Viscous damping* input node 101* output node 102* parameter : B : dampingRx 101 102 1/Br1 101 0 1G ; input and output resistancer2 102 0 1G.ENDS
23
*------.SUBCKT frict1 1 2 W=0* FRICT : Friction* LEVEL 1 : constant friction force* nodes : 101 102 : nodes between which force has to be added* parameters : W : kinetic friction forcerin 1 2 1GBEFRIC 111 0 V= V(1,2)*W^0 < -.1m ? -1 :+ V(1,2)*W^0 > .1m ? 1 :+ 1e4*V(1,2)*W^0RDUMMY 111 0 1ohmBGW 1 2 I=W* V(111)
Fig.4 shows the schematic and Fig. 5 shows the netlist for the IsSpice model of the separatelyexcited DC-motor and its subcircuits. The models of the actuators always include a model of theelectrical subsystem (the field and armature windings), a power conversion part, and a mechanicalsubsystem (the load). The electromotive force is represented by a voltage controlled voltage source,and the generated torque is represented by a current controlled current source. For both thewindings and the mechanical load, lower-level models are provided.
If the machine is operating near its rated load, the linearmodel is not valid anymore, due to saturation of themagnetic field. The flux linkage, ψ, is then modeled as aparabolic blended piecewise linear function of the fieldcurrent, if (Fig. Error! Bookmark not defined.). The
expressions for the controlled sources in the powerconversion part are adapted. The derivative of the fluxlinkage in Eq.(33) is modeled as a variable inductancewhich is a function of the field current, if .
In a Shunt DC-motor, the same voltage is applied over thearmature and the field winding. The model is therefore thesame as the model of the separately excited motor, exceptthat it has only two electric terminals.
In a Series DC-motor, the field winding is in series with the armature winding, so the field current isequal to the armature current. Eq. (31) and (32) then become (for the non-saturated case):
T C i i K ig t a a t a= =ψ( ) 2 (34)
( ) ( )V C K iemf e r s e a r s= − = −ψ ω ω ω ω (35)
The IsSpice model is analogous to the model of the shunt and the separately excited DC-motor.
Compound DC motors. The flux in a compound DC-motor is generated both in a shunt winding, andin winding in series with the armature winding. Eq. (31) and (32) become:
( )T C ig t a sh se= +ψ ψ (36)
( )( )V Cemf e sh se r s= + −ψ ψ ω ω (37)
Permanent magnet DC-motors. For a permanent magnet DC-motor, the rotor flux, φ, is constant, andthe torque is therefore only a function of the current through the windings. The equations for amotor, including winding losses and mechanical losses, are [9]:
.ENDS
Figure 5: Netlist for the separately excited DC-motor model
Figure 6: Saturation of magnetic fieldfor a permanent magnet DC-motor
24
( )
( )
( )
V R i Ldidt
V
T T Jddt
b T
V K
T K i
T W
a a aa
emf
l g mr
m r s fric
emf e r s
g t a
fric m r s
= + +
= − + − +
= −== −
ωω ω
ω ω
ω ω with
sign
(38)
Brushless DC-motors have a permanent magnet rotor and a salient stator [29]. The back-e.m.f. voltageand the generated torque are functions of the position, and may be approximated by a sinusoidalfunction [40]. The back-e.m.f. in the j-th phase and the torque generated by the j-th phase are:
( )
V K n jp
T K i n jp
emf j e r s s
g j t j s
,
,
sin ( )
sin ( )
= − − −
= − −
ω ω θπ
θπ
12
12
(39)
where ns is the number of north poles on the stator, p is the number of phases, and j=1...p is thephase number. An IsSpice model for a 3-phase brushless DC-motor has been developed. It includesarmature losses and mechanical losses, just like the other motor models. The mutual inductance ofthe phases has been included. This motor also needs a commutator model.
3.3.3. Stepping Motors and Switched Reluctance MotorsModels have been developed for Permanent Magnet (PM) and Variable Reluctance (VR) stepping motors, andfor Switched Reluctance (SR) motors. SR-motors need feedback, while stepping motors do not.Permanent Magnet Stepping motors have a round permanent magnet rotor and a salient stator (with
concentrated windings). The back-e.m.f. in the j-th phase and the torque generated by the j-th phaseare:
( ) ( )( )
V K p j
T K i p jemf j e r s
g j t j
,
,
sin ( )
sin ( )
= − − −= − − −
ω ω θ λθ λ
1
1(40)
where p is the number of pole pairs, and λ=π/2 for a two-phase and λ=2π/3 for a three-phasemotor. An IsSpice model has been generated for both a 2-phase and a 3-phase motor.
Variable Reluctance Stepping motors have both a salient stator and rotor. The back-e.m.f. in the j-thphase and the torque generated by the j-th phase are:
( ) ( )( )
V K i p j
T K i p jemf j e j r s
g j t j
,
,
sin ( )
sin ( )
= − − −= − − −
ω ω θ λθ λ
1
12 (41)
The mutual inductance between the phases has been neglected in these equations [18]. An IsSpicemodel has been generated for both a 2-phase and a 3-phase motor.
Switched Reluctance (SR) Motors have a salient rotor and a salient stator. The equations are thereforesimilar to the equations for the VR stepping motor. The efficiency of the SR-motor can beimproved using saturation. Saturation can be modeled in two ways: by applying a physical model,based on the reluctances of the gap and the rotor and stator core, or by using a more behavioralmodel, which is based upon current-dependent inductances.
Behavioral model The model is based upon the model of Miller and McGilp [30]. The fluxlinkage, ψ, varies between the unaligned flux, ψ u uL i= , and the aligned flux, ψ a i( ) . Forintermediate positions, the function is a linear function of the angle, θ (except for a transition zone):ψ θ ψ θ ξ( , ) ( )i ku a= + − 0 , where ka and ξ0 are both dependent upon the current, i. Thetransition zones are modeled with a rational function (Fröhlich curve). For each zone (and each
25
phase), the relations for the induced voltage and the generated torque are derived Eq.(29) andEq.(30).
Physical model The core is modeled as a magnetic circuit. The inductance is replaced by thegap reluctance, gap, which remains constant, since the gap is not saturated. The gap reluctance is
derived from the inductance: if the rotor is not saturated, the core reluctance is very small comparedto the gap reluctance, so the inductance, L, is inversely proportional to the gap reluctance, gap:
Li
NN
NN N
core gap gap
= = = =+
≈ψ φ φ
F F R R R/2
2 2
(42)
The expression for the generated torque becomes:
TW
d dd
dgap
gap gap
gap
gap= = = =
=∫ ∫∂
∂θ
∂ φ θ φ
∂θ
∂ θ φ φ
∂θ
∂θ φ
∂θφ θ
θ
φ φ
F ( , ) ( )( )
( )0 0
2
22
2
RR
R
(43)
The SR-motor requires position feedback and commutation. An IsSpice model has beendeveloped for an SR motor with a 3-phase stator.
3.3.4. AC-motors
Three-phase induction motor The 3-phase induction motor is characterized by the equations:
( )r rr
rr r
u R iddt
R i Ldidt
Md i e
dts s ss
s s ss r
j
= + = + +ψ ε
(44)
( )r rr
rr r
u R iddt
R i Md i e
dtL
didtr r r
rr r
sj
rr= + = + +
−ψ θ
(45)
The model is based upon the generalized machine theory [33,37]. Two stationary coils are attachedto the stator (direct axis, d, and quadrature, q), and two coils are attached to the rotor (which canbe seen as the real and the imaginary component of one complex coil). This leads to the followingequations [37]:
( )( )
u R i Ldidt
Mdidt
u R i Ldi
dtM
di
dt
u R i Mdidt
Ldidt
ddt
Mi L i
u R i Mdi
dtL
di
dtddt
Mi L i
ds s ds sds dr
qs s qs s
qs qr
dr r drds
rdr
qs r qr
qr r qr
qs
r
qr
ds r dr
= + +
= + +
= + + + +
= + + − +
ε
ε
(46)
where uds is the direct component of the stator voltage, and iqr is the quadrature component of therotor current. Rs is the stator phase resistance, Lr is the rotor phase self inductance and M is themutual inductance between rotor and stator phases. The angle ε is the electrical angle which isrelated to the mechanical angle, θ=Npε/2, where Np is the number of poles. The electric circuit isshown in Fig.5, which is analogous to the steady state circuit of an induction motor [19], but in therotor branch there’s a controlled source which is dependent upon the velocity and currentcomponents. The rotor circuit is short circuited, so that udr=0 and uqr=0.
26
The generated torque is calculated from [19] :
( )[ ] ( ) ( )TN M
i i eN M
i iN M
i i i ig
p
s rj p
s r
p
sd rq sq rd= = ⊗ = −3 3 3
Im*r r r r
ε (47)
The direct and quadrature components of the stator voltage are derived from the phase voltages:
( )
( )
u u u u
u u u
sd s s s
sq s s
= − +
= −
1 2 3
2 3
1
23
2
(48)
The stator phase currents and the direct and quadrature current components are related by:
i i i
i i e i i
i i e i i
s s sd
s s
j
sd sq
s s
j
sd sq
1
2
2
3
3
2
3
2
3
2
32
3
2
3
1
2
3
2
2
3
2
3
1
2
3
2
= =
= = − −
= = − +
−
+
Re( )
Re( )
Re( )
r
r
r
π
π
(49)
Single phase induction motor The equivalent scheme for the three phase motor is used. The net isconnected to two stator windings; the third stator winding is not connected.
Synchronous motor with field winding The model is also based upon the generalized machine theory.The machine is modeled with three stator windings (the field windings and distributed winding), andtwo rotor windings [19,37]. If the field winding is placed on the rotor, as is typically the case, rotorand stator are switched in the generalized machine model.
Synchronous motor with permanent magnet rotor This motor is more commonly called a brushlessAC (or synchronous) motor. This motor differs from the brushless DC motors since the windingsare not concentrated, but are equally spread over the stator. The flux in the rotor, and hence therotor current, is constant. The equations are [19]:
u R i Ldidt
ddt
u R i Ldi
dtddt
sd s sd ssd
M
sq s sq s
sq
M
= + −
= + +
3
23
2
φε
ε
φε
ε
sin
cos(50)
where φM is the rotor flux. The torque equals [19]:
Figure 7: Equivalent circuit for the induction motor.
27
( )TN
u ug
p M
sq sd= −φ
ε ε2
cos sin (51)
3.3.5. SolenoidsThe solenoid is the only translating electromechanical motor which is included in the library [9]. The
magnetic field is generated by a current in a winding. A cylindrical plunger moves in a cylindricalsolenoid magnet. The flux is guided to two air gaps. The air gap width depends upon the position of theplunger, and is equal to l0±x, where x is the displacement (if x=0, both gaps are equal to l0). The widthof the air gaps is limited between 0 and a, the width of the poles. The plunger is assumed to be largerthan the magnet, so that the width of one of the air gaps is equal to a. The inductance of the system is:
( )L
N N
ka l x
k a l x
a l xR
L= =+
+
=+
+ +
2 2
0
0
01 1R ( )(52)
28
3.4. Modeling of Electrical Power Drives and their ControlsThe power drive regulates the power flow from the power supply to the actuator. Power drives can be hydraulicor electrical. This section will discuss the electrical power drives; hydraulic power drives (valves) will bediscussed in the next section.The basic module of the power amplifier is the power electronic converter, which is built from powersemiconductor devices (or “power switches”) such as transistors, MOSFETs, thyristors, etc. [31]. The energyflow from the power supply to the actuator is controlled by opening or closing the power switches, i.e. bysending an on-off signal to the gate of the power switch. The behavior of power drives is highly nonlinear dueto the switching, and this can cause simulation convergence problems.The complete power drive consists of one or two converters and the circuits for the control signals for the powersemiconductors. The power drive consists of the following elements:• One or two converters which are built with power electronic devices. The converter architecture is
limited to a few types, but there is a wide variety in the types of power electronic devices. Powerelectronic systems are subdivided according to how the devices are switched:
Line Frequency Converters: The devices are switched on at the line frequency of 60 Hz. Linefrequency converters can be uncontrolled (diode rectifiers), or controlled (thyristor bridges).
Switch-mode converters: The controllable switches (e.g. MOSFETs, BJTs, IGBTs) in theconverter are turned on and off at frequencies that are higher than the line frequency.
Power converters often have two converters: a controlled or uncontrolled rectifier to convert thenet supply to a DC-voltage, and a switch-mode converter to generate the pulsed signals for theactuator.
• A switch control algorithm to determine the state of the switch, e.g. a PWM or hysteretic currentcontrol algorithm. The algorithm is relatively independent of the type of the power switches.
• A gate driver for the power semiconductor devices. This circuit converts the output signal to avoltage or a current. The driver depends upon the type of the switches.
3.4.1. Power Drive ModelsThe power drive has a discontinuous behavior, so an accurate model of a switch-mode converter requires a lotof computing time. An accurate model of the power drive is, however, not always required: the mechanical timeconstant of the system is generally much larger than the switching interval of the power converter, so abehavioral model of the power drive - which does not take switching into account - can be used for thesimulation of the mechanical system. Behavioral power switch models have been developed at different levelsof abstraction:
Behavioral model of the complete power amplifier The most simple and most abstract model of thepower drive is a first order transfer function, i.e. a voltage or current amplifier with a time constant.The output voltage is limited. The model is almost linear, and therefore does not take switching intoaccount. The model represents the power converters (i.e. rectifier and switch-mode converter), theswitch control algorithm, the gate drivers, and the net supply. Models have been developed forcurrent controlled power drives (i.e. the input variable is current, the output is voltage), voltagecontrolled power drives and frequency controlled power drives (i.e. the output signal is a sinusoidalsignal with a frequency which is proportional to the input signal). Models have been developed for1-phase and 3-phase output.
Behavioral model of the Power Converter This is a model of the power converter, the net supply, andthe driver for the switch signals. The output signals of the switch control algorithm are the input tothe model. The system is modeled as a first order system, a limiter and a voltage source. Asopposed to the first model, this model takes the switching into account, but still models theconverter as an input/output block with infinite impedances.
29
Behavioral model of the Power Switches In this model, the power switches are assumed to be ideal:the power converters are modeled as variable gain (mathematical) transformers. The transformergain (±1) is determined by the gate control algorithm. Models have been provided for VSI (VoltageSource Inverter), CSI (Current Source Inverter), VCR (Voltage Controlled Rectifier) and CSR(Current Source Inverter).
Electrical Converter Models The least abstract model corresponds to the actual electrical model. Thetopology of the converters is limited to a few types which all use a number of identical powerswitches, of which there is a large variety. In order to avoid the writing of a new subcircuit for eachpossible type of switch, the IsSpice library contains a user-defined subcircuit for the powerswitches and the power diodes.
Models have been developed for diode and thyristor rectifiers (1 phase and 3 phase) and for 1-quadrant, 2-quadrant DC-drives and full bridge switch-mode converters (DC, 1-phase and 3-phase).
3.4.2. Switch Control AlgorithmTwo switch control algorithms have been modeled: PWM and hysteretic current control. Both algorithms havebeen modeled on a behavioral level.PWM: The PWM model consists of two parts: the PWM triggering signal (triangular or ramp), and the
PWM-signal generation. Depending upon the sign of the difference between the control signal andthe triggering signal, either a positive or negative constant voltage is output. The control signal,which is the input to the PWM model, is generated by the task controller.
Hysteretic Current Control The model for hysteretic current control consists of a Schmitt-trigger: ifthe current, i > iref+∆i, (where iref is a reference current and ∆i is its tolerance) then a negativeconstant voltage is output; if i < iref-∆i, then a positive constant voltage is output. The output isanalogous to the output of the PWM.
The switch control algorithms are blocks which have infinite input and output impedance. The output of thealgorithm is a single signal for a DC or 1-phase motor; there are three signals for a 3-phase motor.
3.4.3. Power Switch DriversThe power switch driver models generate the gate signal from the control algorithm (e.g. PWM). Behavioraldriver models have been developed for MOSFETs, BJTs and thyristors.MOSFET Drivers The gate signal is a voltage source. The value of the voltage is about 10 V higher
than the voltage between the drain and the source of the MOSFET.
BJT Drivers The gate signal is a current source.
Thyristor Drivers The gate signal is a current pulse. The commutation of the thyristor is characterizedby the firing angle, which determines the moment when the current pulse starts. The value of thefiring angle, α, is determined by the desired output voltage V:
απ
=
= −
−
arccos arctanV
kV
VkV
VkV
m
m
m
21
2. (53)
where Vm is the amplitude of the input signal, and k is the relationship between the maximumoutput voltage and Vm (2/π for a 1-phase rectifier and 3/π for a 3-phase full bridge rectifier [7]).
The pulse for the first thyristor starts when the phase of the input signal is equal to the firing angle.The model assumes a constant input frequency, so the phase of the input signal can be determinedby a triangular pulse wave. The pulse is stopped after the next thyristor is ignited.
30
The input for the model is the output of the switch control algorithm. The output is the voltage or current signalwhich is sent to all of the power switches. Busses are used in the schematic editor between the power switchdriver and the converter model, in order to reduce the number of nets that have to be entered by the designer.
3.4.4. DiscussionWith the models that have been provided, normal motor control methods can be modeled.The abstraction level of the power drive model has a significant effect upon the simulation time: the differencein simulation time between the least and the most abstract model of the power converter is on the order of 100xto 10,000x.The switching of the power switches causes a discontinuity in the voltage over the motor, which will make itmore difficult for the simulator to find a convergent solution. Power controllers can be modeled withoutproblems if the switching occurs according to a predetermined pattern. If the switching is controlled by afeedback signal, convergence problems may occur. In order to find a convergent solution, the convergenceparameters (voltage and current tolerance, relative tolerance, number of iterations) can be changed, but thisdoes not guarantee that a solution will be found, and that this solution will be correct. Another possibility is toadd a delay in order to cut the feedback loop in the system of equations. This may, however, cause a ripple inthe output results, which can be reduced by decreasing the delay. But this, in turn, will increase the simulationtime, since the simulation step is always smaller than half of the delay.
3.5. Modeling of SensorsSensors convert a signal from electrical, mechanical or hydraulic energy to an analog or digital electric signal.Therefore, the model of a sensor consists of two parts:• a model in the sensing energy domain.
• an electric model. The sensor typically consists of the actual sensor and a signal conditioning devicewhich converts the measurement to a usable form. An encoder typically generates a pulse trainwhich has to be further processed in order to derive the position. In the multitechnical library, theelectrical part is often modeled on a behavioral level, for example, as a voltage source that isproportional to the measured position.
The following models have been developed with behavioral output:Encoder: the model consists of a mechanical load. The output signal is proportional to the position.
LVDT: the model consists of a mechanical load. The output signal is proportional to the position.
Tacho: the model consists of a mechanical load. The output is proportional to the velocity.
Current Sensing Resistor: the model is a resistor. The output is proportional to the current through theresistor.
Vibrometer: the mechanical model is a mass which is connected with a damper and spring to thesystem. The output is proportional to the displacement of the spring.
Accelerometer: the mechanical model is the same as the model of the vibrometer. Typically, the spring(representing a piezoelectric element) is much stiffer. The output is proportional to the forcebetween the two masses.
The following models have an electric output (at least two electric output nodes).Potentiometer: the mechanical model is a load, the electric model is a potentiometer circuit.
Resolver: is modeled as two transformers, the gains of which depend upon the position [41].
Synchro: is modeled as three transformers, the gains of which depend upon the position [41].
Position controlled switch: a switch is closed if the relative position between the measured object andthe reference object is larger than a predetermined value.
31
3.6. Modeling of hydraulic systemsIn hydraulic systems, the across variable corresponds to the pressure, p, and the through variable corresponds to
the volume flow rate, Q. In the IsSpice model, the reference pressure, (v=0), corresponds to 0 Pa or 105 Pa, thepressure for which the reference parameters are determined.
3.6.1. Basic Hydraulic Laws
Compressibility: Bulk modulus Hydraulic fluids are compressible: the density of the fluid, ρ, dependsupon the pressure, p and the temperature, T. The compressibility is characterized by the bulkmodulus, E [28]:
( ) ( )ρ ρ α= + − − −
0 0 01
1
Ep p T T (54)
where p0 is the measuring reference pressure (i.e. 105 Pa if v=0 corresponds to 0 Pa, 0 if v=0corresponds to 105 Pa), and α is the cubic expansion coefficient. The cubic expansion coefficient issmall for liquids, and its effect on fluid density is small. Temperature dependency is not included inthe library models.
Continuity Law For hydraulic systems, the continuity equation for a flow through a chamber becomes[28]:
Q QdVdt
VE
dpdtin out∑ ∑− = +0 0 (55)
where V0 is the volume of the chamber.
Flow Through Conduits Flow can be laminar or turbulent, depending upon the Reynolds number. Inthe IsSpice modeling, the type of flow is predetermined.
Orifice Flow is characterized by the relationship :
Q C Ap
d= 0
2∆ρ
(56)
where Cd is the discharge coefficient, A0 is the orifice area, and ∆p is the pressure drop over theorifice. The orifice flow is modeled as a controlled current source.
Scaling Hydraulic systems are characterized by large pressure values (in the order of 105 − 106 Pa) andsmall volume flow values, in SI-units. To convert the hydraulic SI-values to this range, a scalingfactor of σ=106 Pa/V is used. The relationship between the IsSpice values and the hydraulic SI-values is:
vp p
i Q
ref=−
=σ
σ(57)
3.6.2. Hydraulic Primitives
Tank For a constant volume tank with one input port, Eq. (55) becomes:
QVE
dpdtin = 0 (58)
which corresponds to a capacitance, C=V0/E, which is connected to the ground.
Hydraulic inertia Due to the law of Newton, a force, or a pressure difference, will generate anacceleration of a fluid in a pipe:
32
∆pl
AdQdt
=ρ
(59)
where l is the length, and A is the section of the pipe. This corresponds to an inductance, L=ρl/A.
Viscous flow During laminar flow in a pipe, a pressure drop is generated, and it is proportional to thevelocity, e.g. for a circular pipe [28]:
∆pl
dQ=
1284
µπ
where l is the length and d is the diameter of the pipe. This corresponds to a resistance, R=128µl/d4.
Leaks are also modeled by a resistance (flow proportional to pressure difference)
3.6.3. Hydromechanical ActuatorsIn an ideal hydromechanical actuator, the flow is proportional to the rotational (or translational) velocity of theshaft (with respect to the plunger). An ideal hydromechanical actuator corresponds to a gyrator, due to therelationship between force and pressure (F=pA) and between velocity and volume flow (Q=vA).Models have been provided at two levels: Level 0 corresponds to the ideal gyrator, level 1 includes losses.Pump A pump converts rotational energy to hydraulic energy. Models have been provided for fixed and
variable displacement pumps. The models include a mechanical load, leaks (both internal betweenlow and high pressure compartment and external leaks), and a cavitation loss [28]. The mechanicalload consists of the rotating mass, and friction (consisting of a constant term, a term which isproportional to the velocity, and a term which is proportional to the pressure).
Hydraulic motor The models for the hydraulic motor are the same as those for the pump. Models areprovided for both fixed and variable displacement motors.
Cylinder A cylinder consists of two hydraulic chambers and a mechanical load. The model of the loadis the same as that for the pump. The influence of the changing volume of the chambers on thepressure (Eq. 55) is modeled by a variable capacitance. The force which is created through theacceleration of the fluid on the piston is modeled by a variable capacitance (to the mechanical load).The model also includes internal and external leakage.
3.6.4. Hydraulic Power Drives : ValvesThe physical model of a 4-way valve is a network of 4 orifices, with a mechanical load. The flow through theorifice is determined by a nonlinear equation (Eq. 56). The forces from the fluid on the plunger include aninertia force and the steady state force which is derived from the approximated law: F=0.43wx∆p [28], where wis the area gradient of the orifice and x is the plunger displacement. Problems may occur during simulation,especially after the closing of the valve. The model parameters are physical parameters such as the gap width,the length between the ports, and the contraction coefficients, which are difficult to determine.Linear models have been developed for the valve. The flow to the load is:
Q K x K pq c L= − ∆ (60)
where Kq is the flow gain, Kc is the flow-pressure coefficient and ∆p is the pressure difference between
the input and output port of the load. The number of inputs and outputs to this model differs from thephysical model since this model does not include the supply and return pressure, and it has nomechanical load.
3.6.5. Other Hydraulic Elements
Pipe A pipe is modeled by a resistance (for the pressure drop due to viscous friction), a capacitor atboth pipe ends (for the compressibility), and an inductance (for the inertia). The flexibility of thepipe is taken into account in the capacitance [2].
33
Accumulator Two accumulator models have been developed for gas loaded accumulators and forspring loaded accumulators. The accumulators are modeled as hydraulic chambers (variablecapacitors). The pressure of the fluid is the same as the pressure of a gas volume for the gas loadedaccumulator, and in equilibrium with the spring force for the spring loaded accumulator.
Hydraulic Losses A model has been added to simulate hydraulic losses at bends, fittings, and entries.At bends, the pressure loss is described by [28]:
∆p KQA
=
ρ2
2
(61)
where K is the loss coefficient and A is the section of the pipe.
Pressure Relief Valve An ideal pressure relief valve is a limiter: the valve is always closed until thepressure exceeds a preset maximum pressure. Above the maximum pressure, it acts like a resistor.Both a behavioral model and a physical model have been generated.
Positive Pressure Element Negative IsSpice voltages indicate pressures which are smaller than thereference. The pressure cannot be smaller than a minimum value. A model has been provided tokeep the pressure above the minimum value. Eq.(54) is, however, a linearization around thereference pressure, and is not valid for low pressures. For low pressures, the results will beinaccurate.
3.7. Commercial Motor LibrariesAll of the model libraries which have been discussed thus far contain parametric component models. In order tosimulate a commercially available component, the parameters must be obtained from manufacturer data sheetsand entered as the model’s parameter values.
3.7.1. Additional Intusoft device librariesIn addition to the libraries which are shipped with all Intusoft software packages, Intusoft also offers optionallibraries: Power library devices (PWM ICs, etc.); RF library devices (RF BJTs, MMICs, etc.); and IC Vendorlibraries (Operational Amplifiers, Comparators, etc.).
3.8. Digital Behavioral ModelingModern motion control systems consist not only of analog components, but also of digital electroniccomponents. The digital electronics can be a digital network on a printed circuit board, but can also be anadvanced IC (e.g. motion control IC) or be coded as a computer program.Digital systems can be modeled on several levels: the electrical (analog) level, the switch level, the gate (orlogic) level, the register transfer level, and the behavioral level [36].• In the switch level, transistors are modeled as gate-controlled switches which are “on” or “off”.
• At the gate or logic level, transistors are grouped into logic gates. Rather than dealing withvoltages and currents at signal nodes, discrete logic states are defined, and Boolean operations areused to determined the new logic value at each node.
• The Register Transfer Level (RTL) is a higher level of abstraction, and is used for data pathdesign. At the RTL, related bits of information are grouped into ordered sets of words or busses.The set of statements for the RTL model involves a sequence of register transfers and arithmeticoperations that are similar to data-flow descriptions.
• The behavioral level allows the definition of functional blocks. The statements are often written ina hardware description language (HDL).
The possibility of modeling multitechnical systems at the behavioral level is desired for the modeling of digitalcontrollers. These are often software programs or motion control ICs, from which the internal structure is not
34
known - only the behavior is known. Software programs must be modeled as sampled systems in which theinput and output signals change at equidistant time intervals.
35
CHAPTER 4.
DESIGN INTEGRATIONDesign of mechatronic systems is a complex process. At the conceptual design level, different designalternatives are compared and the optimal components are selected, mostly by the use of some design rules. Atthe functional design level, the aim is to design the controller for optimum dynamic behavior. The tools that areused at the conceptual design level typically have no advanced possibilities for simulation, whereas the physicaland functional design tools have no provision for selecting commercial components efficiently. By integratingconceptual and physical (or functional) design, different design alternatives can be verified rapidly.
4.1. From Conceptual to Physical
4.1.1. Physical Model of the Electric DriveDue to the selection criterion for across and through variables for the different energy domains, the topology ofthe model in IsSpice resembles the physical system. The nodes in the physical model correspond to mechanicalconnection points.
If we want to generate a model of the complete motion control system, we also need to include a model of thecontrol system.
4.1.2. Physical Model Parameters from Catalog DataConceptual models are based upon catalog data, whereas physical models are based upon physical properties.For example, the conceptual model of the AC induction motor uses the rated and the maximum torque, whereasthe physical torque uses rotor and stator resistances and inductances.If the physical model of a product is available at an early stage of the design, different design alternatives canbe verified rapidly and decisions on the selection of the other components can be made early. If the physicalparameters cannot be measured, they must be derived from the catalog data.
4.1.2.1. DC-motors
The databases for DC-motors include the torque and the gain constant, the armature resistance and inductance,and the rotor inertia. All DC-motors are modeled as permanent-magnet DC-motors. Major motormanufacturers use these models in their guidelines for motor selection [50].
4.1.2.2. AC-motors
In general, AC motors are treated as 3-phase induction motors. The rotor and stator resistances and inductancescan be derived from the catalog data via a rather simple conversion routine. The following equations may beused:
Figure 8: An electric drive (left) and the equivalent SpiceNet drawing (right).
36
( )( )
TN V
RRs
X
Rs
TN V
R R X
f
V VXX
s nnn
nN
fN
X X X
X L M
X L M
X
ratp
e
e
sr
rat
r
rat
peakp
e
e
s
e
e sm
rat synrat
syn
synp
e
p
e s
e r
m e
s
= ⋅
+
+
⋅
= ⋅+ +
=
= −
= −
=
= += −= −=
3
4
3
8
2
1
1
30 120
2
2
2
2
2 2
1
1 2
1
2
ω
ω
ω π
ωπ
ωωω
σ
σ
σ
with
=2
M
(67)
The first 2 equations yield the rated torque, Trat, at rated speed, nrat rpm, and the peak torque, Tpeak.
The following assumptions are made:
L L M X X
R Xs r m
s
= = ⇒ ==
105 0 05
0 051. .
.
σ(68)
With these assumptions, the rated torque can be written as:
( )( )
( )( )T n
V
XX n
V
Tpeak
syn
s
syn
s
peak
= ⋅+
⇒ = ⋅+
3
860
0 95
0 05 10025
3
860
0 95
0 05 10025
2 2
π πσ
σ
.
. .
.
. . (69)
From the reactance, Xσ, the stator resistance, Rs , and the inductances, M, Ls and Lr are calculated:
( ) ( )[ ] [ ]X L M L M M MX
e s r ee
σσω ω
ω= − + − = ⋅ ⇒ =2 0 05 10. (70)
The rotor resistance is approximated as: R s R Xr peak s= +2 2σ , where speak is the slip corresponding
to the maximum torque. At the rated speed, the torque is approximately proportional to the slip, or:
s sT
Tpeak rat
peak
rat
=
This leads to: R sT
TR X s
T
TXr rat
peak
rats rat
peak
rat
= + =2 2 10025σ σ.
These assumptions are approximate. The calculated theoretical starting torque does not correspond to thecatalog data. Another iterative approach has been used, one which also uses the value of the starting torque andthe power factor, (cosϕ). However, it is not always possible to find values for rotor and stator resistances andinductances which match all of these parameters.
4.1.2.3. Behavioral motor models
These motor models represent the combination of motor, motor control and power amplifier. The parametersfor the control (PI-velocity control) are entered as parameters to the model.
4.1.2.4. Reducers
The reducer is modeled as a single gear pair. The two gears are supposed to have the same width, b, andmaterial density, ρ. The radius of the output gear is equal to the product of the radius of the input gear and thetransmission ratio, i. The inertia of each of the gears is calculated to compute the combined inertia at the entryshaft. The ratio of the inertia is:
37
( )( )
JJ
m rm r
br r
br r
rr
i2
1
2 22
1 12
22
22
12
12
24
14
4= = = =πρ
πρ(71)
The total inertia at the input shaft, Jtot, equals: ( )J J J i Jtot = +
= +1
2
1
2
22
11ωω
. For higher transmission
ratios, however, this assumption is not valid. If there is no backlash, it is not necessary to divide the inertia overthe two gears.The efficiency is a parameter that also occurs in the physical model. In the catalog data, the efficiency isentered at four different velocities. A variable efficiency gear pair model has been added to the library.
38
CONCLUSIONSExtended circuit simulation is based upon the energy conservation principle, which allows the modeling ofcomplete multitechnical systems in one single technology dedicated physical simulator, e.g. an analog circuitsimulator. IsSpice, a SPICE-based analog/mixed signal circuit simulator, is used to model multitechnicalsystems. A large multitechnical library containing mathematical, electric, electromechanical, mechanic(translational, rotational and planar mechanics) and hydraulic component models, has been developed for rapidprototyping of mechatronic systems. Component models are available at different accuracy levels; the simplemodels are less accurate but simulate quickly, and the more complex models are more accurate but requiremore simulation time.
39
REFERENCES
[1] P.J. Ashenden, The VHDL Cookbook, University of Adelaide, 1990
[2] J.F. Blackburn, G. Reethof, J.L. Shearer, Fluid Power Control, Technology Press of M.I.T. and JohnWiley & Sons, New York, 1960
[3] F.E. Cellier, Continuous System Modeling, Springer Verlag, New York, 1991
[4] L.O. Chua, K.A. Stromsmoe, “Lumped-Circuit Models for Nonlinear Inductors Exhibiting HysteresisLoops”, IEEE Trans on Circuit Theory, Vol. CT-17, No. 4, November 1970, pp. 564-574
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