Amplifier-Motor Modeling
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Elements to be modeled: 1. 2. 3. 4. Amplifier Motor & Load Position Sensor Controller Amplifier-Motor Modeling Amplifier- Current or Torque Mode Amplifier In this type of operation mode, amplifier output current, I that is directly proportional to the input voltage V the proportionality factor Ka πΌ = πΎπ π------- 1 Torque ππ = πΎπ‘ πΌ--------- 2 where πΎπ‘ → π‘ππππ’π ππππ π‘πππ‘ Moment of Inertia of a solid disc of mass ‘m’ and radius ‘r’: π½= ππ 2 2 πππππ’π = πΉππππ π₯ πΏππ£ππ π΄ππ π = πΉπ If J is total moment of inertia of load & motor, Tf is opposing friction, π½πΌ = ππ + ππ ---------- 3 Where α is acceleration, πΌ= ππ ππ‘ π ππ π‘βπ π£ππππππ‘π¦, π = ∫ πΌ ππ‘ → ππππππ πΏππππππ πππππ ππππ → π = π= 1 πΌ−−−− 4 π ππ 1 → π = ∫ π ππ‘ → ππππππ πΏππππππ πππππ ππππ → π = π − − − − 5 ππ‘ π Combining equations 1 through 5 1 π= π π 1 1 = ( ) ( πΌ) π π = 1 ππ π 2 π½ ππ ππ ππ πππππππ‘ππ →= π= 1 πΎπ‘ πΌ π 2 π½ 1 πΎπ πΎπ‘ π π 2 π½ π πΎπ πΎπ‘ = −−−− 6 π π½π 2 Position Sensor Modeling Motor position is indicated by position sensor as signal ‘c’. Kf proportionality factor, Kf equals the number of units of feedback per one radian of rotation. Encoder provides the position, suppose an incremental encoder generates N pulses per revolution, that the encoder generates output. Channels A & B produces 1000 pulses for each encoder rotation. As two signals are shifted by one quarter of a cycle, the controller can divide each encoder cycle into four quadrant counts resulting in an effective resolution of 4N counts per revolution or turn. Since each revolution is 2π radians, the resulting encoder gain is πΎπ = 4π -2π 7 Model for Incremental Encoder: πΎπ = = 4π −− 7 2π 4 π₯ 1000 πππ’ππ‘π = 636.537 2π πππ£ Thus, 1000 pulse per rev encoder has an equivalent gain of 636 counts/ revolution. ο Another common type of position sensor is the one of binary representation. Total number of positions per revolution is 2π the model for this sensor is: πΎπ = 2π − − − −8 2π For example- For Absolute encoder or resolver with 16-Bit binary position signal has a gain of: πΎπ = 216 = 10,435 πππ’ππ‘π /πππππππ 2π When a sensor is linear, if the resolution of a sensor is 1 micron and if the length is measured in meters, the encoder gain equals 106 πππ’ππ‘π /πππ‘ππ. Modeling a Controller The desired position signal is R(t) or simply ‘R’ actual position is ‘C’. Thus position error ‘E’: πΈ =π −πΆ This position error is used to generate the output signal that drives the motor. The proportional term xp , π₯π = (π)(πΈ) Gain of the proportional part of the controller Error, input The derivative term xd , π₯π = (π π·)(πΈ) Gain of derivative controller Error, input The integral term xi , 1 π₯π = ( πΌ) πΈ π Gain of the Integral controller Error, input Sum of all three outputs, π₯ = π₯π + π₯π + π₯π The Transfer function F(s), relating the output ‘x’ to position error E is, πΉ(π ) = π₯ 1 = π + π π· + πΌ − − − − − ππΌπ· πΆπππ‘ππππππ πΈ π Digital to Analog converter resolution, 8-16 bit. DAC having output voltage range -10 V to +10 V: Output- -10 V to +10 V Gain of DAC ‘K’, equal to the number of volts it produces per unit of ‘x’ input signal. DAC Resolution in ‘n’ bits, the DAC Gain equals: A 12-Bit DAC has K=0.0048 V/unit πΎ= 20 20 = = 0.0048 π/π’πππ‘ 2π 212 ππ π½πΌ = ππ ππ πΌ = π½ 1 π= πΌ π 1 π= π π π= 11 πΌ π π π= 1 ππ π 2 π½ = = 1 πΎπ‘ πΌ π 2 π½ 1 π πΎπ‘ πΎπ 2 π π½ π 1 πΎπ‘ πΎπ = 2 π π π½ π πΎπ‘ = 2 ∪ π π½πΏ + π π½π + πΎπ πΎπ‘ Encoder Gain: πΎπ = 2π Kf 4π 2π 4N 4 π₯ 1000 ( ) = 636 πππ’ππ‘π /πππ (2)(3.1428)
