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Lecture 9: 1. Finish purely electrical systems • Modeling in the Laplace domain • Loading of cascaded elements 2. Modeling electromechanical systems • Introduction • Sensors and the measurement system • Actuators (continue with DC motors next class) ME 431, Lecture 9 Modeling Electromechanical Systems 1 Modeling in the Laplace Domain E ( s) complex impedance Z ( s) I ( s) • Complex impedance is treated like resistance • Ohm’s law • Equivalent components in series and parallel ME 431, Lecture 9 • Idea: model circuits directly in terms of Laplace transformed equations 2 Modeling in the Laplace Domain • Resistors: • Capacitors: E ( s) 1 1 1 I ( s) Z (s) L [e i dt ] E ( s) I ( s) Cs C C s • Inductors: E ( s) di sL L [e L ] E ( s ) LsI ( s ) Z ( s) I (s) dt ME 431, Lecture 9 E ( s) R L [e iR ] E ( s ) I ( s ) R Z ( s) I ( s) 3 Example • Find the transfer function Eo(s)/Ei(s) Example (con’t) • Note: this approach is only valid if initial conditions are zero Loading of Cascaded Elements ME 431, Lecture 9 • Consider the two loops separately 6 Loading of Cascaded Elements • Consider the two loops separately • Note that G1(s) Ec(s) G2(s) Eo(s) Ec Eo Eo G1G2 Ei Ec Ei ME 431, Lecture 9 Ei(s) • The second loop loads the first loop • Can add an isolating amplifier to decouple • This is addressed by some simulation software 7 Electromechanical Systems • Most control systems (including automobiles) include electrical and mechanical components • Need components that convert between the domains Example: mechanical domain electrical domain angle Transducer R+ E - Controller voltage Actuator voltage voltage Sensor U torque Y Plant speed Electromechanical Systems • Piezoelectric materials produce charge when deformed (ex: accelerometer, microphone, load cell, etc.) • Electrical properties of many materials change with temperature, deformation, etc. (ex: thermistor, pressure transducer, strain gage) • Motion of a conductor through a magnetic field can induce an emf (ex: LVDT, resolver) • Special materials produce charge in response to light (digital camera, optical encoder) ME 431, Lecture 9 • Sensors/Transducers: often convert mechanical quantities into electrical ones 9 The Measuring System measurand Stage 1 Sensor transduced signal Stage 2 Signal Conditioning filter amplify integrate differentiate DAC ADC analogous driving signal Stage 3 Readout/ Computation ME 431, Lecture 9 • Measurement process generally involves multiple elements that each may have dynamics that need to be modeled gauge LED display LCD display speaker computer 10 The Measuring System Example Stage 2 Stage 1 accel signal w/o noise voltage sensor filter Stage 3 analogous to velocity integrator increased voltage amplifier ADC digital Computer Numerical Integration • Error accumulates if there is a bias in the measurement ME 431, Lecture 9 • Simple approach calculates the area of a series of rectangles 12 Numerical Differentiation • Large error when noisy measurements are differentiated ME 431, Lecture 9 • Simple approach calculates the slope between two points 13 Analog to Digital Conversion • Can introduce time lag and quantization error ME 431, Lecture 9 • An analog signal is sampled at discrete intervals of time and is held 460 455 450 445 14 440 1.47 1.48 1.49 1.5 1.51 1.52 Electromagnetic Induction • Converts electricity into force/torque • Examples: solenoids, speakers, electric motors • Lorentz’s law: A current carrying conductor in a magnetic field generates a force ME 431, Lecture 9 • Like many sensors, some actuators employ electromagnetic induction 15 • Current through coil generates a magnetic field (Ampere’s law) • Magnetic field imparts a force on the iron core V • Spring return • Examples include valves and switches, like a car starter ME 431, Lecture 9 Solenoid Actuator 16 • Current moving through a magnetic field induces a force F i B ME 431, Lecture 9 DC Motor • Parts: • Stator: stationary part (includes the magnet) • Rotor: rotating part (wire coil) • Commutator: half rings that connect to the current source through brushes 17 • Multiple coils and multiple magnetic pairs ensure current carrying wire near magnet for higher proportion of time • Armature wrapped around iron core so that magnetic field doesn’t have to cross a large air gap ME 431, Lecture 9 DC Motor 18 DC Motor 1. Armature control: change torque by changing current in the armature (rotor) 2. Field control: change torque by changing the strength of the magnetic field (by changing current through an electromagnet) ME 431, Lecture 9 Two approaches to DC motor control ( F i B ) 19 • Armature-controlled DC motor model • Model resistance and inductance of the coil as lumped parameters • Same with mechanical inertia and friction Fixed field Ra La + ea + _ Armature circuit ia ME 431, Lecture 9 DC Motor T eb θ 20 J _ Rotor b