1. Delta Sigma Modulator For my Mom

2. What is Analog? Analog is a quantity that may have any value in a given range Pick a voltage between 0 and 7 volts: 23.45.86.21464321 These are all valid analog voltages

3. What is Digital? Digital is a quantity that may only have a defined set of values in a given range The value is defined by a number of bits that may be either 1 or 0 Let’s define the range from 0 to 7 volts by 3 bits 000 = 0 001 = 1 010 = 2 011 = 3 100 = 4 101 = 5 110 = 6 111 = 7

4. What is Digital? Pick a voltage between 0 and 7 volts: 23.45.86.21464321 This is the only valid digital voltage (with 3 bits)

5. Why use Digital? Noise You are a computer. You need record a voltage. What are the voltages below? Analog Digital (1 or 0) This is a 1This is 5.26383

6. So Digital is better? For computers, yes. For noiseless record of data, yes. The world is actually analog We see, hear, smell, taste, feel, analog

7. Analog to Digital Converter So we need a way to convert back and forth between analog and digital ADC A to D 2010 DAC D to A 010 2

8. Delta Sigma Modulator One type of ADC

9. Analog to Digital Converter So we need a way to convert back and forth between analog and digital ADC A to D 2.12 0000111101 011100… DAC D to A 0000111101 011100… 2.12

10. Story problem A farmer has a 10 gallon tank of water. It leaks 1 gallon per minute 1 gal/min Water is put in the tank at an unknown rate ? gal/min The tank indicates if it is more than half full The rate of input water is between 0 and 1 gal/min

11. Story problem Every minute, the farmer checks the half-full indicator. More than half-full = 1 Less than half-full = 0 If the tank is less than half-full, the farmer adds a gallon to the tank.

12. Amount in tank Indicator Amount leaking Amount input Amount added Net change 5.011.00.70.0-0.3 4.701.00.71.00.7 5.411.00.70.0-0.3 5.111.00.70.0-0.3 4.801.00.71.00.7 5.511.00.70.0-0.3 5.211.00.70.0-0.3 4.901.00.71.00.7 5.611.00.70.0-0.3 5.311.00.70.0-0.3 5.011.00.70.0-0.3 4.701.00.71.00.7 Amount in tank Indicator Amount leaking Amount input Amount added Net change

13. Story problem Now we have a bit-stream 101101101110 To find the unknown rate of water put into the tank, take the average 1+0+1+1+0+1+1+0+1+1+1+0 12 = 0.67 ≈ 0.7 More iterations will give you more accuracy

14. Circuit Equivalent Φ1Φ1 Φ2Φ2 Φ2Φ2 Φ1Φ1 DQ Q

15. $ 1 5.0E-6 13.741202019156521 50 5.0 50 159 160 224 192 224 0 159 256 224 288 224 0 159 192 272 192 304 0 159 256 272 256 304 0 w 192 224 192 272 0 w 256 224 256 272 0 w 176 240 240 288 0 w 176 288 272 240 0 c 192 224 256 224 0 1.0E-5 2.4747768778269292 R 256 304 224 336 0 0 40.0 2.5 0.0 a 352 240 464 240 0 5.0 0.0 c 352 160 464 160 0 9.999999999999999E-6 0.1979719629491683 w 464 240 464 160 0 w 352 160 352 224 0 R 352 256 352 304 0 0 40.0 2.5 0.0 w 288 224 352 224 0 R 144 400 96 400 1 2 200.0 2.5 2.5 I 144 400 144 480 0 0.5 w 176 240 144 272 0 w 144 272 144 400 0 w 176 288 176 480 0 w 176 480 144 480 0 a 512 272 656 272 1 5.0 0.0 w 512 256 464 240 0 R 512 288 512 336 0 0 40.0 2.5 0.0 155 720 272 768 272 0 5.0 w 720 272 656 272 0 w 144 400 720 400 0 w 720 304 720 400 0 I 880 544 800 544 0 0.5 w 720 544 608 544 0 w 320 224 352 224 0 w 880 272 880 544 0 w 880 272 944 272 0 R 160 224 80 224 0 4 10.0 2.5 2.5 w 816 272 880 272 0 w 192 304 192 544 0 w 192 544 608 544 0 I 800 544 720 544 0 0.5 o 33 16 0 3 5.0 1.220703125E-5 0 o 34 16 0 2 5.0 0.1 0