# Brushed DC Motor System

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1 Brushed DC Motor System Pittman DC Servo Motor Schematic Brushed DC Motor Brushed DC Motor System K. Craig 1

2 Topics Brushed DC Motor Physical & Mathematical Modeling Hardware Parameters Model Hardware Correlation H-Bridge Operation Brushed DC Motor System K. Craig 2

3 Pittman DC Servo Motor 8322S001 Brushed DC Motor System K. Craig 3

4 Pittman DC Servo Motor 8322S001 Encoder 500 counts/rev Wire Function Color Pins 1 GND Black GND 2 Index Green - 3 CH A Yellow 4 Vcc Red 5V 5 CH B Blue Brushed DC Motor System K. Craig 4

5 For a permanent-magnet DC motor i f = constant. Physical Modeling Brushed DC Motor System K. Craig 5

6 Physical Modeling Assumptions The copper armature windings in the motor are treated as a resistance and inductance in series. The distributed inductance and resistance is lumped into two characteristic quantities, L and R. The commutation of the motor is neglected. The system is treated as a single electrical network which is continuously energized. The compliance of the shaft connecting the load to the motor is negligible. The shaft is treated as a rigid member. The total inertia J is a single lumped inertia, equal to the sum of the inertias of the rotor and the driven load. Brushed DC Motor System K. Craig 6

7 There exists motion only about the axis of rotation of the motor, i.e., a one-degree-of-freedom system. The parameters of the system are constant, i.e., they do not change over time. The damping in the mechanical system is modeled as viscous damping B, i.e., all stiction and dry friction are initially neglected. The optical encoder output is decoded in software. Position and velocity are calculated and made available as analog signals for control calculations. The motor is driven with a PWM control signal to a H- Bridge. The time delay associated with this, as well as computation for control, is lumped into a single system time delay. Brushed DC Motor System K. Craig 7

8 Mathematical Modeling Steps Define System, System Boundary, System Inputs and Outputs Define Through and Across Variables Write Physical Relations for Each Element Write System Relations of Equilibrium and/or Compatibility Combine System Relations and Physical Relations to Generate the Mathematical Model for the System Brushed DC Motor System K. Craig 8

9 Physical Relations P out Tm Kti m Vb Kb P T K i P V i K i P out m t m in b m b m in K K K t b m dil VL L VR Ri R TB B dt T J J J J J J motor load t P P out in t t Brushed DC Motor System K. Craig 9 K K t b K (oz in / A) K (V / krpm) K (Nm / A) b K b(v / krpm) K (Nm / A) K (V s / rad) b

10 System Relations + Equations of Motion KVL Vin VR VL Vb 0 Tm TB TJ 0 ir il im i di d Vin Ri L Kb 0 J B Kti 0 dt dt d B Kt dt J J 0 V di K 1 b R i L dt L L Newton s Law Brushed DC Motor System K. Craig 10 in

11 Steady-State Conditions di Vin Ri L Kb 0 dt T Vin R Kb 0 K t K t K tkb T Vin R R K t Ts V Stall Torque in R Vin 0 No-Load Speed K b Brushed DC Motor System K. Craig 11

12 Transfer Functions di d Vin Ri L Kb 0 J B Kti 0 dt dt V s (Ls R)I(s) K (s) 0 Js B (s) K I(s) 0 in b t (s) Kt Kt V (s) Js B Ls R K K JLs BL JR s BR K K 2 in t b t b s K t JL B R BR KK s J L JL JL 2 t b Brushed DC Motor System K. Craig 12

13 Block Diagram V in Ls R i K t T m 1 Js B K b Brushed DC Motor System K. Craig 13

14 Simplification J L m >> e B R d Vin Ri Kb 0 J B Kti 0 dt d 1 K t J B K ti K t Vin Kb Vin Kb dt R R d 1 dt d K tkb B K t V dt RJ J RJ d 1 1 dt motor m K t V in since m motor motor RJ Brushed DC Motor System K. Craig 14 K t RJ V in in

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23 MatLab M-File Brushed DC Motor System K. Craig 23

24 H-Bridge Operation For DC electric motors, a power device configuration called an H-Bridge is used to control the direction and magnitude of the voltage applied to the load. The H- Bridge consists of four electronic power components arranged in an H-shape in which two or none of the power devices are turned on simultaneously. A typical technique to control the power components is via a PWM (Pulse Width Modulation) signal. A PWM signal has a constant frequency called the carrier frequency. Although the frequency of a PWM signal is constant, the width of the pulses (the duty cycle) varies to obtain the desired voltage to be applied to the load. Brushed DC Motor System K. Craig 24

25 The H-Bridge can be in one of the four states: coasting, moving forward, moving backward, or braking, as shown on the next slide. In the coasting mode, all four devices are turned off and since no energy is applied to the motor, it will coast. In the forward direction, two power components are turned on, one connected to the power supply and one connected to ground. In reverse direction, only the opposite power components are turned on supplying voltage in the opposite direction and allowing the motor to reverse direction. In braking, only the two devices connected to ground are tuned on. This allows the energy of the motor to quickly dissipate, which will take the motor to a stop. Brushed DC Motor System K. Craig 25

26 Brushed DC Motor System K. Craig 26

27 The four diodes shown in anti-parallel to the transistors are for back-emf current decay when all transistors are turned off. These diodes protect the transistors from the voltage spike on the motor leads due to the back-emf when all four transistors are turned off. This could yield excessive voltage on the transistor terminals and potentially damage them. They must be sized to a current higher than the motor current and for the lowest forward voltage to reduce junction temperature and the time to dissipate the motor energy. Brushed DC Motor System K. Craig 27

28 Diodes for back-emf protection are shown. The solid line is the current flow when the transistors on the upper left corner and on the lower right corner are turned on. The dashed line shows the motor current when all transistors are turned off. Brushed DC Motor System K. Craig 28

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