Pulse Width Modulation

Transcription

1 ECEn 621" Computer Arithmetic" Project Notes Week 1 Pulse Width Modulation 1

2 Pulse Width Modulation A method of regulating the amount of voltage delivered to a load. The average value of the voltage fed to the load is controlled by switching between a low and high voltage at a high switching frequency f sw. The duty cycle D is the ratio of 'on' time to the switching period T = 1/f sw. Average value = kd. T ON D = T ON / T T Pulse Width Modulation 2

3 1 Zero Centered PW Modulation PWM Average Value f (t) = y max for y min for 3

4 Frequency Power Spectrum signal (low freq) f s (high freq) Filtering to Get the Output 4

5 Some Applications of PWM Modulate, transmit and/or store analog signals like audio/voice telecommunications and music. The output power and/or voltage of switching power supplies can be controlled digitally. The torque and speed of a DC motor can be controlled by altering the voltage or the duty cycle of the PWM. PWM is widely used in dimmer circuits that could control a variety of lamps including LEDs. H-Bridge Circuit that enables a voltage to be applied across a load in either direction. Used to implement an Inverter circuit that converts DC to AC. 5

6 Operational States of Inverter ON OFF ON PWM OFF ON OFF OFF OFF High side transisters switch at low-frequency. Low side transistors switch at high-frequency Change Direction OFF - + ON PWM OFF - + ON ON OFF OFF OFF Free-wheeling Diodes PIC Microprocessor Control Pic18 8-Bit microcontroller with multiply instruction and 2 built-in PWM controllers. 6

7 Basic PIC PWM Mode CCP2 PR2 CCPR2 TMR = Timer Register PR = Period Register CCPR = Comparator Period Register If (TMR2==PR2) { TMR2 = 0 ;` CCP2 = 1 ; CCPR2H = CCPR2L; } If (TMR2==CCPR2) { CCP2 = 0 ; } Enhanced PIC PWM Mode 7

8 Task: PWM Control for Sinudoid Tasks Control PWM to output a sinusoid for voltage inversion Period Input period input n = number of PWM periods in one quarter of a sinusoid period. Each PWM period advances the angle of the sinusoid by Challenges No sine or cosine functions to call No division instructions, only multiplication Frequency is variable, so a table approach won t work Control summation error. Those little summation errors can add up fast. Recurrence for Sine and Cosine This has to be precomputed given input n. 8

9 Controlling Summation Error In each iteration of the sine/cosine recursion, a new summation error is accumulated. If not controlled, these errors build up and can corrupt the significant part of the result. To sum k items, log2(k) extra guard bits have to be computed in addition to the bits required for accuracy. But if k is not bounded... neither are the number of guard bits. After n iterations of the sine/cosine iteration, the angle is at a multiple of 90, and sine and cosine are whole numbers 0, 1, or -1. A simple round will clear any accumulated error. Computing sin( ) and cos( ) Taylor Series Polynomial 9

10 Taylor Series Approximations Accuracy of Taylor Approximations n Sine Sine Sine 1 Term 2 Term 3 Term in number of bits n Cosine Cosine Cosine 1 Term 2 Term 3 Term

11 Division For Taylor series computation, division by constants can be done by precomputing the reciprocols of the constants and multiplying. To compute Δ, an actual division has to be computed. But the PIC microcontroller doesn t have a divide instruction... Project #2 PWM on a PIC 11

12 H-Bridge Chip H-Bridge Chip 12

13 PIC Controlling the H-Bridge 24V + Sine Wave For the positive part of the sine wave, Pin 22 = FIN = PWM Pin 21 = RIN = L For the positive part of the sine wave, Pin 22 = FIN = L Pin 21 = RIN = PWM PIC18F24J11 To H-Bridge Other Pins: MCLR Reset OSC1,OSC2 Crystal Oscillator SCL1, SDA1 I 2 C Port RX1, TX1 Serial Port PGD, PGC In-Circuit Programming 13

14 Clocks 12 MHz 48 MHz Basic PIC PWM Mode CCPR1L CCP2CON<5:4> CCP2 CCPR1H R overrides S PR2 TMR2 R Q S CCP1 pin CCPR2H PR2 TMR = Timer Register PR = Period Register CCPR = Capture/Comparator Period Register (Duty Cycle Register) If (TMR2==PR2+1) { TMR2 = 0 ;` CCP1 = 1 ; CCPR1H = CCPR1L; <Trigger Interrupt> } else TMR2 = TMR2 + 1; If (TMR2==CCPR1H) { CCP1 = 0 ; } 14

15 PWM Registers Setup Continued 1/3 Set the PR2 register = the value of a comparison register to which the timer will be compared, in this case we set it to 254 which means the timer will count from 0 to 254. If the PWM duty cycle (CCPR1) is 255 the timer will never reach that value and the PWM output will stay permanently high - just as we need for full power. If the PWM duty cycle (CCPR1) is zero, the comparator will equal that value as it starts, so the PWM output will remain permanently low - again, just as we need for zero power. This means the sine computation must peak at ±0.FFC instead of 1. 15

16 Setup Continued 2/3 T2CON sets the carrier frequency of the PWM derived from the 48 MHz system clock divided by 4. The prescaler divides the frequency before the timer and the postscaler afterwards. For this example we set the prescaler to divide by 1, this gives us a timer frequency of 12 MHz. The timer counts 0 to 254 (PR2). The PWM frequency is thus 12 MHz / 255 = 47 KHz. To get higher frequency, you must reduce PR2 and trade-off precision in duty cycle. Set the CCP1CON register to operate as PWM, CCP1 can operate in various modes, so we need to specifically set it as PWM. T2CON Bit Descriptions 16

17 CCP1CON Bit Descriptions Setup continued 3/3 Set the PWM output low, so it is off when it starts up. Start the PWM system by turning TMR2 on. Once this line executes the PWM operates independent;y of the rest of the code. Unless we alter the register settings, the PWM will continue to operate independently. 17

18 Initialise: BANKSEL ADCON1 ;turn off A2D MOVLW 0x06 MOVWF ADCON1 BANKSEL PORTA BANKSEL TRISC MOVLW 0 ;set PORTC as all outputs MOVWF TRISC BANKSEL PORTC MOVF CCP1CON,W ;set CCP1 as PWM ANDLW 0xF0 IORLW 0x0C MOVWF CCP1CON MOVLW 0xFE BANKSEL PR2 MOVWF PR2 BANKSEL TMR2 CLRF T2CON ;set highest PWM value ;over this is permanently on PWM Code CLRF CCPR1L ;set PWM to zero BSF T2CON, TMR2ON ;and start the timer running /* C CODE */ // Set CCP1 pin as PWM CCP1CON = (CCP1CON & 0xF0) 0x0C; // Set PWM period to 254 PR2 = 254; // Set prescaler and post scaler to 1 T2CON = 0; // Set postscaler to 1 T2CON = ( T2CON & 0x07 ) 0x00; // set PWM duty cycle to zero CCPR1L = 0; // start timer T2CON = T2CON (1 << TMR2ON); Set Polarity Setup for positive half cycle Set pin 21 (RIN) low Program PIC to route PWM to pin 22 (FIN) Setup for negative half cycle Set pin 22 (FIN) low Program PIC to route PWM to pin 21 (RIN) 18

19 Setting the Duty Cycle CCPR1L may be loaded with the duty-cycle for the next cycle anytime during the PWM period. When the TMR2 counter is reset to 0, CCPR1H will be loaded from CCPR1L and the duty cycle will be changes for that period. (CCPR1H is read-only) At the end of the PWM period, an interrupt will be generated which starts the next duty-cycle computation. The resulting duty-cycle is loaded into CCPR1L. PIC18 Arithmetic The PIC18 is an 8-bit Microcontroller extended with an 8 x 8-bit multiply instruction. To control error, more bits may need to be computed. Higher bit-width arithmetic (such as 16-bit, 24- bit, or 32-bit arithmetic) can still be done by combining 8-bit operations together. Trick is to do computations in the least time possible. If you can t do it fast enough, it is time to move up to a PIC24 (a 16-bit that processor that costs more). 19

20 PIC18 Arithmetic Unit PIC Add 20

21 PIC Multibyte Add PIC Subtract 21

22 Negate PIC Multiply 22

23 PIC Multibyte Multiply [0x10] WREG [0x11] * WREG PRODH, PRODL PRODH [0x21] PRODL [0x20] 16 x 16 = 32 from four 8 x 8 = 16 23

24 PIC Multibyte Multiply How to Divide without Division N D = N d 1 D d 1 = N d 1 d 2 D d 1 d 2 = = Q 1 2 D : k =1 D k = D k 1 ( 2 D k 1 ) : k "1/x" "2-x" N : k =1 N k = N k 1 ( 2 D k 1 ) : k 2 D k 1 N k Q = N D D must first be normalized to between 0.5 and 1. Use this method to compute π 2N. 24

25 Diagramming out Division N = Π/2 = x. xxxx xxxx xxxx xxxx (fixed point) D = n = xxxx xxxx. (integer, max 255) Normalize D by shifting MS-1 to MSB D norm = 0. 1xxx xxxx * 2 e Remember e. Iteration (1) (0)x. xxxx xxxx xxxx xxxx D =(0)x. xxxx xxxx xxxx xxxx d x. xxxx xxxx xxxx xxxx (N or D) * x. xxxx xxxx xxxx xxxx d =0x. xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx = x. xxxx xxxx xxxx xxxx (Round -> new N or D) Final N = x.xxxx xxxx xxxx xxxx * 2 e = 0.000x xxxx xxxx xxxx (shift right e and Round) sin( Δθ ) r = sin( Δθ ) ± 1 2 ulp Error Analysis cos( Δθ ) r = cos( Δθ ) ± 1 2 ulp sin( θ 1 ) r = sin( 0) cos( Δθ ) ± 1 2 ulp cos( θ 1 ) r = cos 0 sin θ 2 ( ) cos( Δθ ) ± 1 2 ulp ( ) r = sin( θ 1 ) ± 1 2 ulp cos θ 2 = sin θ 2 + cos 0 sin 0 cos Δθ ( ) ± 1 2 ulp + cos θ 1 ( ) sin( Δθ ) ± 1 2 ulp ( ) sin( Δθ ) ± 1 2 ulp ( ) ± 1 2 ulp ( ) ± ( cos( θ 1 ) + sin( θ 1 ) + cos( Δθ ) + sin( Δθ )) 1 2 ulp ( ) r = cos( θ 1 ) ± 1 2 ulp = cos θ 2 cos Δθ ( ) ± 1 2 ulp sin θ 1 ( ) ± 1 2 ulp ( ) ± ( cos( θ 1 ) + sin( θ 1 ) + cos( Δθ ) + sin( Δθ )) 1 2 ulp = sin θ 1 = cos θ 1 sin Δθ ( ) ± 1 2 ulp = sin θ 2 sin Δθ ( ) ± 1 2 ulp = cos θ ulp 1 2 ( ) ± 1 2 ulp ( ) ± sin( θ 1 ) + cos( θ 1 ) ( ) ± 1 2 ulp e ulp 1 2 ( ) ± sin( θ 1 ) + cos( θ 1 ) e 2 25

26 1 ( ) r = sin( θ k 1 ) ± e k 1 sin θ k cos θ k = sin θ k = sin θ k Error Analysis 2 ulp cos Δθ ( ) ± 1 2 ulp + cos θ k 1 1 ( ) ± e k 1 ( ) 1 2 ulp ( ) + e k 1 ( sin( Δθ ) + cos( Δθ )) +sin( θ k 1 ) + cos( θ k 1 ) 1 2 ulp ulp cos( Δθ ) ± 1 2 ulp sin θ k 1 ( ) + e k 1 +sin( θ k 1 ) + cos( θ k 1 ) 1 ( ) r = cos( θ k 1 ) ± e k 1 = cos θ k = cos θ k 1 ( ) ± e k 1 ( ) 1 2 ulp ( ) + e k 1 ( sin( Δθ ) + cos( Δθ )) +sin( θ k 1 ) + cos( θ k 1 ) 1 2 ulp 1 2 ( ) + e k 1 +sin( θ k 1 ) + cos( θ k 1 ) 2 ulp sin Δθ 2 ulp sin Δθ ( ) ± 1 2 ulp ( ) ± 1 2 ulp Error Analysis n = 20 delta theta = theta sine cosine sin+cos ek

27 Fixed-Point Arithmetic sin(δθ) = 0.xxxx xxxx xxxx xxxx cos(δθ) = 0.xxxx xxxx xxxx xxxx sin(θ k ) = s.xxx xxxx xxxx xxxx cos(θ k ) = s.xxx xxxx xxxx xxxx sin(θ k ) = s.xxx xxxx xxxx xxxx Absolute Value + Round 10-bit Duty Cycle Project #2 Rewrite your program for project #1 Only use integer arithmetic (char, int, unsigned) Integer logical ops, addition, subtraction and multiplication Compute division using iterative approximation Use fixed-point interpretations of your numbers Control error as best as possible using guard bits resetting summation error after n steps Write C-code that could be compiled for a PIC microprocessor. Measure actual error by comparing results to project #1 27