# Motor Control. Suppose we wish to use a microprocessor to control a motor - (or to control the load attached to the motor!) Power supply.

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1 Motor Control Suppose we wish to use a microprocessor to control a motor - (or to control the load attached to the motor!) Operator Input CPU digital? D/A, PWM analog voltage Power supply Amplifier linear, PWM voltage, current Motor torque, speed, position Load Convert discrete signal to analog voltage - D/A converter - pulse width modulation (PWM) Amplify the analog signal - power supply - amplifier Sensor strain gauge, potentiometer, tachometer, encoder Types of power amplifiers - linear vs. PWM - voltage-voltage vs. transconductance (voltage-current) DC Motor - How does it work? What to control? - electrical signals: voltage, current - mechanical signals: torque, speed, position Sensors: Can we measure the signal we wish to control (feedback control)? EECS46, Lecture 6, updated September 7, 28

2 Outline Review of Motor Principles - torque vs. speed - voltage vs current control - with and without load D/A conversion vs. PWM generation - harmonics - advantages and disadvantages - creating PWM signals power amplifiers - linear vs PWM - voltage vs transconductance Control - choice of signal to control - open loop - feedback References are [5], [3], [], [4], [8], [7], [6], [9] EECS46, Lecture 6, updated September 7, 28 2

3 Motor Review Recall circuit model of motor: R I L V V B =K V Ω - T M, Ω T L J Suppose motor is driven by a constant voltage source. Then steady state speed and torque satisfy Torque-speed curve Ω = K MV RT L K M K V + RB T M = K M(V B + K V T L ) K M K V + RB T M increasing V Ω EECS46, Lecture 6, updated September 7, 28 3

4 Voltage Control Suppose we attempt to control speed by driving motor with a constant voltage. With no load and no friction (T L =, B = ) Ω = V K V T M = Recall that torque is proportional to current: T M = K M I. Hence, with no load and no friction, I =, and motor draws no current in steady state. Current satisfies I = V V B R In steady state, back EMF balances applied voltage, and thus current and motor torque are zero. With a load or friction, (T L and/or B ) Ω < V K V T M > Speed and torque depend on load and friction - friction always present (given in part by motor spec, but there will be additional unknown friction) - load torque may also be unknown, or imprecisely known EECS46, Lecture 6, updated September 7, 28 4

5 Issue: Open Loop vs Feedback Control Using constant voltage control we cannot specify desired torque or speed precisely due to friction and load - an open loop control strategy - can be resolved by adding a sensor and applying closed loop, or feedback control add a tachometer for speed control Ω* (volts) Ω error V DC controller motor - Ω volts tachometer rad/sec add a current sensor for torque (T M = K M I) control I* I error V controller - DC motor I Will study feedback control in Lecture 7. EECS46, Lecture 6, updated September 7, 28 5

6 Issue: Steady State vs. Transient Response Steady state response: the response of the motor to a constant voltage input eventually settles to a constant value - the torque-speed curves give steady-state information Transient response: state is achieved. the preliminary response before steady The transient response is important because - transient values of current, voltage, speed,... may become too large - transient response also important when studying response to nonconstant inputs (sine waves, PWM signals) The appropriate tool for studying transient response of the DC motor (or any system) is the transfer function of the system EECS46, Lecture 6, updated September 7, 28 6

7 System A system is any object that has one or more inputs and outputs input System output Input: applied voltage, current, foot on gas pedal,... Output: other variable that responds to the input, e.g., voltage, current, speed, torque,... Examples: - RC circuit R v i (t) + - C v o (t) Input: applied voltage, Output: voltage across capacitor - DC motor R I L V V B =K V Ω - T M, Ω T L J Input: applied voltage, Output: current, torque, speed EECS46, Lecture 6, updated September 7, 28 7

8 Stability We say that a system is stable if a bounded input yields a bounded output If not, the system is unstable Consider DC Motor with no retarding torque or friction - With constant voltage input, the steady state shaft speed Ω is constant the system from V to Ω is stable - Suppose that we could hold current constant, so that the steady state torque is constant. Since dω dt = T M J, the shaft velocity Ω and velocity increases without bound the system from I to Ω is unstable Tests for stability - mathematics beyond scope of class - we will point out in examples how stability depends on system parameters EECS46, Lecture 6, updated September 7, 28 8

9 Frequency Response A linear system has a frequency response function that governs its response to inputs: u(t) H(jω) y(t) If the system is stable, then the steady state response to a sinusoidal input, u(t) = sin(ωt), is given by H(jω): y(t) H(jω) sin(ωt + H(jω)) We have seen this idea in Lecture 2 when we discussed antialiasing filters and RC circuits The response to a constant, or step, input, u(t) = u, t, is given by the DC value of the frequency response: y(t) H()u EECS46, Lecture 6, updated September 7, 28 9

10 Bode Plot Example Lowpass filter, H(jω) = /(jω + ) H(jω) = /(jω+) gain, db phase, degrees frequency, rad/sec Steady state response to input sin(t) satisfies y ss (t) =. sin(t 85 )..8 response of H(jω) to sin(t) input output MATLAB file bode plot.m time, seconds EECS46, Lecture 6, updated September 7, 28

11 Frequency Response and the Transfer Function To compute the frequency response of a system in MATLAB, we must use the transfer function of the system. (under appropriate conditions) a time signal v(t) has a Laplace transform V (s) = Z v(t)e st dt Suppose we have a system with input u(t) and output y(t) u(t) H(s) y(t) The transfer function relates the Laplace transform of the system output to that of its input: Y (s) = H(s)U(s) for simple systems H(s) may be computed from the differential equation describing the system for more complicated systems, H(s) may be computed from rules for combining transfer functions To find the frequency response of the system, set s = jω, and obtain H(jω) EECS46, Lecture 6, updated September 7, 28

12 Transfer Function of an RC Circuit RC circuit - Input: applied voltage, v i (t). - Output: voltage across capacitor, v o (t) R v i (t) + - C v o (t) differential equation for circuit - Kirchoff s Laws: v i (t) I(t)R = v o (t) - current/voltage relation for capacitor: I(t) = C dv o(t) dt - combining yields RC dv o(t) dt + v o (t) = v i (t) To obtain transfer function, replace - each time signal by its Laplace transform: v(t) V (s) - each derivative by s times its transform: dv(t) dt sv (s) - solve for V o (s) in terms of V i (s): V o (s) = H(s)V i (s), H(s) = RCs + To obtain frequency response, replace jω s H(jω) = RCjω + EECS46, Lecture 6, updated September 7, 28 2

13 Transfer Functions and Differential Equations Suppose that the input and output of a system are related by a differential equation: d n y dt n + a d n y dt n + a d n 2 y 2 dt n a dy n dt + a ny = Replace d m y/dt m with s m Y (s): d n u b dt n + b d n 2 u 2 dt n b du n dt + b nu s n + a s n + a 2 s n a n s + an Y (s) = b s n + b 2 s n b n s + bn U(s) Solve for Y (s) in terms of U(s) yields the transfer function as a ratio of polynomials: Y (s) = H(s)U(s), H(s) = N(s) D(s) N(s) = s n + a s n + a 2 s n a n + a n D(s) = b s n + b 2 s n b n s + b n The transfer function governs the response of the output to the input with all initial conditions set to zero. EECS46, Lecture 6, updated September 7, 28 3

14 Combining Transfer Functions There are (easily derivable) rules for combining transfer functions - Series: a series combination of transfer functions u(t) G(s) H(s) y(t) reduces to u(t) G(s)H(s) y(t) - Parallel: a parallel combination of transfer functions u(t) H(s) Σ y(t) G(s) reduces to u(t) G(s)+H(s) y(t) EECS46, Lecture 6, updated September 7, 28 4

15 Feedback Connection Consider the feedback system u(t) Σ e(t) G(s) y(t) -/+ H(s) Feedback equations: the output depends on the error, which in turn depends upon the output! (a) y = Ge (b) e = u Hy If we use negative feedback, and H =, then e = y u - the input signal u is a command to the output signal y - e is the error between the command and the output Substituting (b) into (a) and solving for y yields u(t) G(s) +/-G(s)H(s) y(t) The error signal satisfies u(t) +/-G(s)H(s) e(t) EECS46, Lecture 6, updated September 7, 28 5

16 Motor Transfer Function, I Four different equations that govern motor response, and their transfer functions - Current: Kirchoff s Laws imply I(s) = L di dt + RI = V V B «(V (s) V B (s)) () sl + R - Speed: Newton s Laws imply Ω(s) = J dω dt = T M BΩ T L «(T M (s) T L (s)) (2) sj + B - Torque: T M (s) = K M I(s) (3) - Back EMF: V B (s) = K V Ω(s) (4) We can solve for the outputs T M (s) and Ω(s) in terms of the inputs V (s) and T L (s) EECS46, Lecture 6, updated September 7, 28 6

17 Combine ()-(4): Motor Transfer Function, II V V-V B - sl+r I K M T M T L - sj+b Ω V B K V Transfer function from Voltage to Speed (set T L = ): - First combine ()-(3) Ω(s) = K M (sj + B) (sl + R) (V (s) V B(s)) - Then substitute (4) and solve for Ω(s) = H(s)V (s): Ω(s) = = K M (sj+b) (sl+r) + K M K V (sj+b) (sl+r) K M V (s) (sl + R)(sJ + B) + K M K V «V (s) K Similarly, T M (s) = M (sj+b) (sl+r)(sj+b)+k M K V (s) V The steady state response of speed and torque to a constant voltage input V is obtained by setting s = (cf. Lecture 5): Ω ss = K M V RB + K M K V, T Mss = K M BV RB + K M K V EECS46, Lecture 6, updated September 7, 28 7

18 Motor Frequency Response DC Motor is a lowpass filter 2 DC motor frequency response 5 2 gain, db phase, degrees frequency, Hz Parameter Values - K M = N-m/A - K V = V/(rad/sec) - R = ohm - L =. H - J =. N-m/(rad/sec) 2 - B =.28 N-m/(rad/sec) Why is frequency response important? - Linear vs. PWM amplifiers... 2 Matlab m-file DC motor freq response.m EECS46, Lecture 6, updated September 7, 28 8

19 Linear Power Amplifier Voltage amplifiers: D/A V Voltage Amplifier V Motor - output voltage is a scaled version of the input voltage, gain measured in V/V. - Draws whatever current is necessary to maintain desired voltage - Motor speed will depend on load: Ω = K M V RT L K M K V +RB Current (transconductance) amplifiers: D/A V Current Amplifier I Motor - output current is a scaled version of the input voltage, gain measured in A/V. - Will produce whatever output voltage is necessary to maintain desired current - Motor torque will not depend on load: T M = K M I Advantage of linearity: Ideally, the output signal is a constant gain times the input signal, with no distortion - In reality, bandwidth is limited - Voltage and/or current saturation Disadvantage: - inefficient unless operating full on, hence tend to consume power and generate heat. EECS46, Lecture 6, updated September 7, 28 9

20 Pulse Width Modulation Recall: - with no load, steady state motor speed is proportional to applied voltage - steady state motor torque is proportional to current (even with a load) With a D/A converter and linear amplifier, we regulate the level of applied voltage (or current) and thus regulate the speed (or torque) of the motor. PWM idea: Apply full scale voltage, but turn it on and off periodically - Speed (or torque) is (approximately) proportional to the average time that the voltage or current is on. PWM parameters: - switching period, seconds - switching frequency, Hz - duty cycle, % see the references plus the web page [2] EECS46, Lecture 6, updated September 7, 28 2

21 PWM Examples 4% duty cycle 3 :.2 duty cycle = 4%, switching period = sec, switching frequency = Hz.8 PWM signal % duty cycle: time,seconds.2 duty cycle = %, switching period = sec, switching frequency = Hz.8 PWM signal % duty cycle: time,seconds.2 duty cycle = 9%, switching period = sec, switching frequency = Hz.8 PWM signal Matlab files PWM plots.m and PWM.mdl time,seconds EECS46, Lecture 6, updated September 7, 28 2

22 PWM Frequency Response, I Frequency spectrum of a PWM signal will contain components at frequencies k/t Hz, where T is the switching period PWM input: switching frequency Hz, duty cycle 4% 4 : duty cycle = 4%, switching period =. sec, switching frequency = Hz.8 PWM signal time,seconds Frequency spectrum will contain - a nonzero DC component (because the average is nonzero) - components at multiples of Hz 4 duty cycle = 4%, switching period =. sec, switching frequency = Hz frequency, Hz 4 Matlab files PWM spectrum.m and PWM.mdl EECS46, Lecture 6, updated September 7, 28 22

23 PWM Frequency Response, II PWM signal with switching frequency Hz, and duty cycle for the k th period equal to.5( + cos(.2πkt )) (a. Hz cosine shifted to lie between and, and evaluated at the switching times T =. sec) 5.2. Hz sinusoid,.5(+cos(.2π t)) mostly on mostly off time, seconds Remove the DC term by subtracting.5 from the PWM signal PWM signal shifted to remove DC component time, seconds 5 Matlab files PWM sinusoid.m and PWM.mdl EECS46, Lecture 6, updated September 7, 28 23

24 PWM Frequency Response, III Frequency spectrum of PWM signal has - zero DC component - components at ±. Hz - components at multiples of the switching frequency, Hz 3 frequency response of PWM signal frequency, Hz Potential problem with PWM control: - High frequencies in PWM signal may produce undesirable oscillations in the motor (or whatever device is driven by the amplified PWM signal) - switching frequency usually set 25 khz so that switching is not audible EECS46, Lecture 6, updated September 7, 28 24

25 PWM Frequency Response, IV Suppose we apply the PWM output to a lowpass filter that has unity gain at. Hz, and small gain at Hz low pass filter, /(.jω + ) gain, db phase, degrees frequency, Hz Then, after an initial transient, the filter output has a. Hz oscillation. filtered PWM output time, seconds EECS46, Lecture 6, updated September 7, 28 25

26 PWM Generation Generate PWM using D/A and pass it through a PWM amplifier CPU D/A V/V PWM amplifier Motor techniques for generating analog PWM output ([6]): - software - timers - special modules Feed the digital information directly to PWM amplifier, and thus bypass the D/A stage duty cycle CPU V/V PWM amplifier Motor PWM voltage or current amplifiers must determine direction - normalize so that * 5% duty cycle represents * % duty cycle represents full scale * % duty cycle represents negative full scale * what we do in lab, plus we limit duty cycle to 35% 65% - use full scale, but keep track of sign separately EECS46, Lecture 6, updated September 7, 28 26

27 References [] D. Auslander and C. J. Kempf. Mechatronics: Mechanical Systems Interfacing. Prentice-Hall, 996. [2] M. Barr. Introduction to pulse width modulation. [3] W. Bolton. Mechatronics: Electronic Control Systems in Mechanical and Elecrical Engineering, 2nd ed. Longman, 999. [4] C. W. desilva. Control Sensors and Actuators. Prentice Hall, 989. [5] G.F. Franklin, J.D. Powell, and A. Emami-Naeini. Feedback Control of Dynamic Systems. Addison-Wesley, Reading, MA, 3rd edition, 994. [6] S. Heath. Embedded Systems Design. Newness, 997. [7] C. T. Kilian. Modern Control Technology: Components and Systems. West Publishing Co., Minneapolis/St. Paul, 996. [8] B. C. Kuo. Automatic Control Systems. Prentice-Hall, 7th edition, 995. [9] J. B. Peatman. Design with PIC Microcontrollers. Prentice- Hall, 998. EECS46, Lecture 6, updated September 7, 28 27

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