EE 410/510: Electromechanical Systems Chapter 4


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1 EE 410/510: Electromechanical Systems Chapter 4 Chapter 4. Direct Current Electric Machines and Motion Devices Permanent Magnet DC Electric Machines Radial Topology Simulation and Experimental Studies Generator Driven by a Motor Electromechanical Systems with Power Electronics Axial Topology Permanent Magnet DC Electric Machines Device Fundamentals Axial Topology Hard Drive Actuator Electromechanical Motion Devices: Synthesis and Classification 5/21/ All figures taken from primary textbook unless otherwise cited.
2 Radial Topology Permanent magnetic DC Electric Machines DC electric machines guarantee: High power High torque densities Efficiency Affordability Reliability Ruggedness Overloading capabilities Power range of modern DC electric machines W 100 kw Dimensions i for modern devices 1mm in diameter and approx 5 mm long to 1 m in diameter Used widely in aerospace, automotive, marine, power, robotics, etc. Only permanent magnet synchronous machines which don t have brushes surpass the use of DC machines in the field 5/21/2010 2
3 Radial Topology Permanent magnetic DC Electric Machines Permanent magnet DC (PMDC) machines are rotating energy transforming electromechanical motiondevices thatconvert energy Motors convert electrical to mechanical Generators convert mechanical to electrical The same PMDC can serve as either a motor or a generator Electric machines will always have stationary and rotating components separated by an air gap The armature winding is placed on the rotor slots connected to a rotating commutator which rectifies the voltage One supplies the armature voltage, u a, to the rotor windings The rotor windings and permanent magnets on the stator are magnetically coupled The brushes ride on the commutator which is connected to the armature windings 5/21/2010 3
4 Radial Topology Permanent magnetic DC Electric Machines The armature winding consist of identical uniformly distributed coils Excitation of the magnetic field is produce by permanent magnets The commutator, armature windings, and permanent magnets produce stationary mmfs which are displaced by 90 electrical degrees The armature re magnetic force is along the rotor magnetic axis while the direct axis stands for a permanent magnet magnetic axis Torque is produced as a result of the interaction of these mmfs 5/21/
5 Operation of a Radial PM DC Motor
6 The equation for the electronic circuit in the motor described is u a r i a a d rai dt a L a dia dt k Equations of Motion d rai dt E L Where the mechanical coupling term comes from the constant, k a, which depends on factors such as the number of turns in the armature winding, and the permeability of the magnet a a a a dia dt Yielding the following ODE for the electronic circuit Likewise the mechanical ODE is derived from the energy differential Neglecting a the spring force one can write the following: 5/21/2010 6
7 Equations of Motion q Governing equations of motion These same equations can be written in Laplace form as i r s L u Equations set into a set of linear algebraic functions a r m L e a a a a T k B Js T T i r s L u 1 L m a m a a a r T B Js u B Js r s L 5/21/2010 7
8 Steady State Torque Speed Characteristics di DC operation stipulates that 0 dt Thus one operates a DC motor under the following steady state operation Torque speed characteristics are mapped below for different applied voltages less than the maximum rated voltage for the motor Where For the mechanical side, the electromagnetic torque must equal the applied torque load for steady state operation o T k i T e Thus the torque speed of the system is described entirely by a a Thus, angular velocity is Increased the applied armature voltage Decreased with applied torque with a slope of r a /k 2 a L Under constant load, velocity is decreased by reducing the applied voltage. The angular velocity at which h the motor rotates tt is found at the intersection of the two curves Furthermore, if one neglects friction, then Newton s second law states that And in stead state, Te=TL providing a constant angular velocity with no load. ie
9 Torque Speed Example Calculate and plot the torque speed characteristics for a 12V PM DC motor with the following parameters. N m V s r a 2 k a A rad The load is a nonlinear function of angular velocity: TL Solution: Torque speed characteristics are governed by: N m r One can use different values of the armature voltage to plot the steady state characterisitics
10 Practice Applications Angular velocity of a PM DC motor is regulated by the applied armature voltage Note that one can use power converter electronics from the previous chapter to regulate the voltage of a PM DC motor To rotate the motor clockwise or counterclockwise, the bipolar voltage should be applied to the armature winding Large motors require high end electronics capable of driving multiple amps through a circuit Small motors (1 10W) can be driven using dual op amps as shown
11 PM DC Electric Generators Assume a resistive electrical load, R L The following equation is used for the electric circuit The induced emf is In steady state operation, the induced terminal voltage is proportional to the angular velocity. Voltage is therefore generated by applying torque, T pm, by aerodynamic, thermal, or hydrodynamic forces. The resulting differential equation is: Note the sign change in the first and last terms of the angular acceleration equation. Thisrepresents the flowof of currentbackintothe the circuitinstead instead fromthe applied torque instead of driving the system forward as achieved in motor opreration
12 Simulation and Experimental Studies PM DC electric machines are among a very limited class of EM motion devices which can theoretically be described by linear differential equations The majority of EM motion devices are solved using nonlinear differential equations which h can be solved using Matlab ODE solvers. (ex. ODE45 solver) Even though the equations of motion are described by linear ODE s, linear theory may not always be applied to PM DC machines b/c of various applied voltage constraints For our immediate purposes, we will apply the state space model previously developed to analyze a simple system
13 Step Input Simulation and Experimental Studies
14 Simulation and Experimental Studies Square wave input
15 Applying a square wave using Matlab Homework: assign an applied armature voltage u a =10rect(0.5t) with a load torque T L =0 Use the following Matlab code to aid in generating the plots presented in the previous slide
16 Loses and Efficiencies of EM Devices Loses associated with these devices are the sum of the resistive and drag effects Efficiency can be determined by the ratio of input to output power
17 Simulation and Experimental Studies
18 Example of a Physical System Let us stop for a moment and examine the performance of a real world motor For this example we will examine the JDH2250 PM DC motor The following figure documents the acceleration of the unloaded motor at an applied armature voltage of 7.5 and 15 V. As the torque is applied, the angular velocity decreases as described by our steady state torque speed characteristics
19 Example of a Physical System For this example we will examine the JDH2250 PM DC motor The deceleration dynamics of a loaded vs. unloaded motor are presented Note that the experimental results presented did not match the numerical simulation derived for this system b/c of complex friction phenomenon that we will did not sufficiently describe.
20 PM DC Generator Driven by a PM DC Motor Let us now analyze two PM DC electric machines integrated as a motor generator system. In this case, the prime mover (The PM DC motor) will drive the generator pm = prime mover g = generator Assume that a resistive load, RL, is inserted in series with the generator armature winding. Kirchhoff s voltage law yields The torsional dynamics of the generator prime mover system is
21 PM DC Generator Driven by a PM DC Motor The applied torque on the PM DC motor is Where i apm is the armature current in the prime mover K apm is the torque constant of the prime mover While the load torque on the prime mover is that created by the generator Thus one obtains the torsional mechanical dynamics by the following differential equation The dynamics of the electric circuit in the prime mover is given by pm = prime mover g = generator
22 PM DC Generator Driven by a PM DC Motor The resulting three differential equations must therefore govern the system pm = prime mover g = generator
23 Example: PM DC Motor The following parameters are used to model an electric machine: Prime Mover Generator r apm L apm k apm B apm J pm = 0.4 Ohm = 0.05 H = 0.3 Vsec/rad = Nms/rad = 0.04 kgm2 r ag L ag k ag B ag J g = 0.3 Ohm = 0.06 H = 0.25 Vsec/rad = Nms/rad = 0.05 kgm2
24 Example: PM DC Motor The following parameters are used to model an electric machine: Prime Mover Generator r apm L apm k apm B apm J pm = 0.4 Ohm L =005H 0.05 = 0.3 Vsec/rad = Nms/rad = 0.04 kgm2 r ag L ag k ag B ag J g = 0.3 Ohm L =006H 0.06 = 0.25 Vsec/rad = Nms/rad = 0.05 kgm2 Simulate and examine the state and dynamic operation of a PM DC generator driven by 100V PM DC motor. Study the transient dynamics and the voltage generation, u apm, for different resistive loads, R L, and angular velocities, rpm Using the state equation: Assuming steady state operation and an infinite resistive load, R L = One then finds:
25 Example: PM DC Motor A Simulink model can be created for the system using system using the Matlab inputs provided as: Simulink help file for creating custom blocks
26 Example: PM DC Motor for constant R L A Simulink model can be created for the system using system using the Matlab inputs provided as:
27 Example: PM DC Motorfor constant R L Where:
28 Example: PM DC Motor for constant R L Where:
29 Example: PM DC Motor with R L = 5 Ohms p L
30 Example: PM DC Motor with R L = 25 Ohms INCORRECT FIGURE IN TEXTBOOK? MAKE THIS SIMULATION AND PROVIDE THE CORRECT FIGURE FOR HOMEWORK
31 Example: PM DC Motor with R L = 100 Ohms p L
32 Example: PM DC Motor driven at constant u apm
33 Example: PM DC Motor driven at u apm = 50 V p apm
34 Example: PM DC Motor driven at u apm = 75 V p apm
35 Example: PM DC Motor driven at u apm = 100 V p apm
36 PM DC Motor Driven by a Buck Converter Let us now examine the application of a high frequency step down switch (buck) converter to control a PM DC motor The duty ratio of the converter is The equations for the buck converter developed in Chapter 3 are Yielding the following 4 differential equations for the system
37 PM DC Motor Driven by a Buck Converter Recall that duty ratio is regulated by the signal level control voltage, u c, which is bound between u tmax and u tmin. Assume in these systems that u tmin =0. u tmax = max voltage rating for the motor u c = input contol voltage for the motor = drive voltage With the nonlinear term
38 PM DC Motor Driven by a Buck Converter Recall that duty ratio is regulated by the signal level control voltage, u c, which is bound between u tmax and u tmin. Assume in these systems that u tmin =0. u tmax = max voltage rating for the motor u c = input contol voltage for the motor = drive voltage With the nonlinear term
39 PM DC Motor Driven by a Buck Converter
40 PM DC Motor Driven by a Buck Converter
41 PM DC Motor Driven by a Buck Converter
42 PM DC Motor Driven by a Boost Converter The resulting four differential equations govern the system
43 PM DC Motor Driven by a Cuk Converter The resulting six differential equations govern the system
44 Axial Topology PM DC Electric Machines Motors using planar segmented permanent magnet arrays that are driven by windings above or below the magnet We know that a planer current loop of any size and shape generates Torque, T, in a uniform magnetic field Where i is the current, s is the area of the loop, B is the magnetic field, and m is the magnetic dipole moment generated Using the relation one can show that the torque generated causes motion II to the plane of the coil One can also write the force applied to the rotor as Rotor North and South PM poles Rotor Stator It is important to note that this type of motor design works for both linear translation and rotary motors alike Axial motors are used in hard disk heads, cooling fans, linear axis drive systems, etc. etc. etc. Stator
45 Axial Topology Example: Assume a current of 10 Amps is applied around a square loop with dimensions 10 x 20 cm Parallel to the loop and located slightly below, is a permanent magnet generating the following magnetic flux density: Tesla Using the equation for torque, where the vector s is normal to the surface aˆ aˆ x y aˆ z T 10*0.1*0.2* N m a ˆxN m
46 1D Axial Topology Linear Motor Using the previous equations. Consider a series line filament, l, each carrying current in or out out of the page. Now consider a series of magnetic poles on the rotor facing up or down from top to bottom of the page The force generated by current in the line element and the magnetic field generated by the magnet generate motion horizontally along the length of the page. pg Rotor Stator td d /d / / td t i t d/ch ll th i df
47 1D Axial Topology Rotational Motor By wrapping the linear motor into a circular shape, the linear motion becomes that of a rotation about a central axis. Motion is then described by angular velocity with an effective flux density that depends on angular displacement, r. The magnetic flux density, B( r ) applied depends on the magnet magnetization, geometry, and shape of the rotor/stator system. For permanent magnets, the flux density is viewed from the windings as a periodic function of angular displacement. If the rotor design is produced such that there are no gaps between magnet segments, then one may use the following relation to accurately describe the magnetic flux density relation Rotor North and South PM poles Stator Where B max is the maximum effective flux density produced by the magnets from the winding N m is the number of magnets (segments) n is the integer function of the magnet magnetization, geometry, shape, thickness, separation, and so on.
48 1D Axial Topology Rotational Motor Example Consider the three different magnetic flux density values given Where B max = 0.9 T and N m = 4. max m We can plot B( r ) using the following statements
49 1D Axial Topology Rotational Motor Example Consider the three different magnetic flux density values given Where B max = 0.9 T and N m = 4. max m We can plot B( r ) using the following statements
50 1D Axial Topology Rotational Motor Example Consider the three different magnetic flux density values given Where B max = 0.9 T and N m = 4. max m We can plot B( r ) using the following statements
51
52 Rotary PM DC Motor The electric circuit equation for torque can be derived as where l eq is the effective length, which includes the winding filament length and the lever arm, and N is the number of turns in the coil One can also derived dthe expression for magnetic energy, where A eq is the effective area that takes into account the number of turns, magnetic field no uniformity, etc. Applying One obtains:
53 Hard Drive Actuator Consider an axial topology PM hard drive actuator assembled with two permanent magnet segments in an array Rotation of the motor is achieved by applying voltage across the current loop. The polarity voltage applied sets the current and therefore the direction of the motor The relative change in magnetization of the two motor segments also contributes to the direction of the rotation In hard drive actuators, a mechanical limiter restricts the angular displacement to Typical hard drives operate with displacement limiters of Applying Kirchhoff s Voltage law to the problem
54 Rotational Hard Disk Motor Example The equation of the circuit must be further limited by use of two (left and right) filaments Likewise, Newton s 2 nd law of motion results in Consider two practical cases when two magnetic strips are magnetized to ensure For these cases, we will let k = 1 and a =10 and 100 for a maximum magnetic flux density of 0.7 Tesla
55 Rotational Hard Disk Motor Example For the case where The electromagnetic torque can then be described as where L and R are the left and right angular displacements respectively. We will solve for the system using the following limiting factors Torque is then expressed as One can further match the system by applying a nonideal Hook s law (for spring forces) to the system
56 Rotational Hard Disk Motor Example For this example, let us use the following parameters
57 Rotational Hard Disk Motor Example Case Number 1:
58 Rotational Hard Disk Motor Example Case Number 2:
59 Rotational Hard Disk Motor Example Case Number 2:
60 Rotational Hard Disk Motor Example Case Number 2:
61 Rotational Hard Disk Motor Example Case Number 2:
62 Rotational Hard Disk Motor Example Case Number 3: Simplified Linear Model
63 Geometrical Variations of Electromechanical lmotion Devices
64
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