Lab 2 Manual

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University of Calgary *

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561

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Electrical Engineering

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Dec 6, 2023

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Uploaded by ProfessorHawk2150

QUBE-SERVO2Workbook-Student Hardware Interfacing, Filtering, Block Diagram Modeling, And State-Space Modeling Lab adapted from Quanser labs Hardware Interfacing, Filtering, Block Diagram Modeling, and Stat-space Modeling. 1 Background 1.1 DC Motor Direct-current (DC) motors are used in a variety of applications. As discussed in the QUBE-Servo 2 User Manual, the QUBE-Servo 2 has a brushed DC motor that is connected to a PWM amplifier. See the QUBE-Servo 2 User Manual for details. 1.2 Encoders An encoder is a measurement device that is used to find the position of a spinning shaft. There are many types of encoders but one of the most common is the rotary incremental optical encoder, shown in Figure 1.1. Unlike potentiometers, encoders come in two different types, absolute or relative. Incremental encoders are relative. The angle they measure depends on the last position and when it was last powered. Figure 1.1: US Digital incremental rotary optical shaft encoder Attached to the shaft is a set of bands that change if light can be seen by a photo sensor which combined with a decoding algorithm result in a count-based output giving the rotational position of the shaft. This count can then be converted into either radians or degrees based on how many counts there are in a revolution of the shaft. 1.3 Filtering A low-pass filter can be used to block out the high-frequency components of a signal. A first-order low-pass filter transfer function has the form , (1.1) QUBE-SERVO 2 Workbook - Student 2

QUBE-SERVO2Workbook-Student v1.2 where ω f is the cut-off frequency of the filter in radians per second (rad/s). All higher frequency components of the signal will be attenuated by at least −3 dB ≈ 50 %. 1.4 Motor Modeling The Quanser QUBE-Servo 2 is a DC rotary servo system. Its motor armature circuit schematic is shown in Figure 1.2 and the electrical and mechanical parameters are given in Table 1.1. The DC motor shaft is connected to the load hub. The hub is a metal disk used to mount the disk or rotary pendulum and has a moment of inertia of J h . A disk load is attached to the output shaft with a moment of inertia of J d . Figure 1.2: QUBE-Servo 2 DC motor and load The back-emf (electromotive) voltage e b ( t ) depends on the speed of the motor shaft, ω m , and the back-emf constant of the motor, k m . It opposes the current flow. The back emf voltage is given by: e b ( t ) = k m ω m ( t ). (1.1) Symbol Description Value DC Motor R m Terminal resistance 8 . 4Ω k t Torque constant 0 . 042 N . m / A km Motor back-emf constant 0 . 042 V /( rad / s ) Jm Rotor inertia 4 . 0 × 10 −6 kg . m 2 Lm Rotor inductance 1 . 16 mH m h Load hub mass 0 . 0106 kg rh Load hub radius 0 . 0111 m J h Load hub inertia 0 . 6 × 10 −6 kg . m 2 Load Disk m d Mass of disk load 0 . 053 kg r d Radius of disk load 0 . 0248 m Table 1.1: QUBE-Servo 2 system parameters

QUBE-SERVO2Workbook-Student Using Kirchoff’s Voltage Law, we can write the following equation: . (1.2) Since the motor inductance L m is much less than its resistance, it can be ignored. Then, the equation becomes v m ( t ) − R m i m ( t ) − k m ω m ( t ) = 0 . (1.3) Solving for i m ( t ) , the motor current can be found as: . (1.4) The motor shaft equation is expressed as J eq ω ˙ m ( t ) = τ m ( t ) , (1.5) where J eq is the total moment of inertia acting on the motor shaft and τ m is the applied torque from the DC motor. Based on the current applied, the torque is τ m = k t i m ( t ). (1.6) The moment of inertia of a disk about its pivot, with mass m and radius r , is . 1.5 Linear State-Space Representation The standard state-space representation of a multi-input multi-output (MIMO) continuous linear-time invariant (LTI) system with n state variables, r input variables, and m output variables is x ˙( t ) = Ax(t) + Bu(t), (1.7) y ( t ) = Cx ( t ) + Du ( t ), (1.8) where x is the vector of state variables ( n ×1) , u is the control input vector ( r ×1) , y is the output vector ( m ×1) , A is the system matrix ( n × n ) , B is the input matrix ( n × r ) , C is the output matrix ( m × n ) , and D is the feed-forward matrix ( m × r ) . The block diagram representation of the state-space Equation 1.7 and Equation 1.8 is shown in Figure 1.3. Figure 1.3: State-Space Block Diagram QUBE-SERVO 2 Workbook - Student 2

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