Delta Sigma Modulator

For my Mom

What is Analog?

• Analog is a quantity that may have any value in a given range

Pick a voltage between 0 and 7 volts:

2 3.4 5.8 6.21464321

These are all valid analog voltages

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 = 0001 = 1010 = 2011 = 3100 = 4101 = 5110 = 6111 = 7

What is Digital?

Pick a voltage between 0 and 7 volts:

2 3.4 5.8 6.21464321

This is the only valid digital voltage(with 3 bits)

Why use Digital?

• Noise

You are a computer. You need record a voltage.What are the voltages below?

0

1

2

3

4

5

6

7

1 21 41 61 81 101 121 141 161 181 2010

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 21 41 61 81 101 121 141 161 181 201

Analog Digital (1 or 0)

This is a 1This is 5.26383

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

Analog to Digital Converter

• So we need a way to convert back and forth between analog and digital

ADC

A to D2 010

DAC

D to A010 2

Delta Sigma Modulator

One type of ADC

Analog to Digital Converter

• So we need a way to convert back and forth between analog and digital

ADC

A to D2.12 0000111101

011100…

DAC

D to A

0000111101011100… 2.12

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

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.

Amount in tank Indicator Amount

leakingAmount

inputAmount added

Net change

5.0 1 1.0 0.7 0.0 -0.34.7 0 1.0 0.7 1.0 0.75.4 1 1.0 0.7 0.0 -0.35.1 1 1.0 0.7 0.0 -0.34.8 0 1.0 0.7 1.0 0.75.5 1 1.0 0.7 0.0 -0.35.2 1 1.0 0.7 0.0 -0.34.9 0 1.0 0.7 1.0 0.75.6 1 1.0 0.7 0.0 -0.35.3 1 1.0 0.7 0.0 -0.35.0 1 1.0 0.7 0.0 -0.34.7 0 1.0 0.7 1.0 0.7

Amount in tank Indicator Amount

leakingAmount

inputAmount added

Net change

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+012 = 0.67 ≈ 0.7

• More iterations will give you more accuracy

Circuit Equivalent

Φ1Φ2

Φ2Φ1 D Q

Q

Circuit Equivalent$ 1 5.0E-6 13.741202019156521 50 5.0 50159 160 224 192 224 0159 256 224 288 224 0159 192 272 192 304 0159 256 272 256 304 0w 192 224 192 272 0w 256 224 256 272 0w 176 240 240 288 0w 176 288 272 240 0c 192 224 256 224 0 1.0E-5 2.4747768778269292R 256 304 224 336 0 0 40.0 2.5 0.0a 352 240 464 240 0 5.0 0.0c 352 160 464 160 0 9.999999999999999E-6 0.1979719629491683w 464 240 464 160 0w 352 160 352 224 0R 352 256 352 304 0 0 40.0 2.5 0.0w 288 224 352 224 0R 144 400 96 400 1 2 200.0 2.5 2.5I 144 400 144 480 0 0.5w 176 240 144 272 0w 144 272 144 400 0w 176 288 176 480 0w 176 480 144 480 0a 512 272 656 272 1 5.0 0.0w 512 256 464 240 0R 512 288 512 336 0 0 40.0 2.5 0.0155 720 272 768 272 0 5.0w 720 272 656 272 0w 144 400 720 400 0w 720 304 720 400 0I 880 544 800 544 0 0.5w 720 544 608 544 0w 320 224 352 224 0w 880 272 880 544 0w 880 272 944 272 0R 160 224 80 224 0 4 10.0 2.5 2.5w 816 272 880 272 0w 192 304 192 544 0w 192 544 608 544 0I 800 544 720 544 0 0.5o 33 16 0 3 5.0 1.220703125E-5 0o 34 16 0 2 5.0 0.1 0