5 (Transducer) (Sensors)

5.1 () 5.1 4 5.1

5.1 ()

(Primary Sensors)

A / DMicrocomputer

mV

Input

Output

Primary Secondary

5 2

(Secondary Sensor) (Sensing) (Pressure Transducer) (Output) 5.1.1

1. - (Active Sensors) - (Passive Sensors) 2. - (Variable Capacitance Transducer) - (Variable Inductance Transducer) - (Variable Resistance Transducer) 3. - - - - 4. - - (ON OFF) -

5 5 3

5. - (Input Transducer) - (Output Transducers) 6. - 5.1.2

(Capacitive and Inductive)

(Capacitive) 5.2 )

C = dA

ro (5.1) o = Permittivity = 8.85 pF / m r = Relative Permittivity ( 5.1) A = (m2) d = (m) (5.1) C d, A r

1. 5.2 ) x d + x

C = xdA

ro + (5.2) 2. 5.2 ) x A = wx w =

C = d)wxA(

ro (5.3)

5 4

5.2

5.1

(Material) (Dielectric Constant : r) Vacuum 1.0 Air 1.0006 Teflon 2.0 Wax 2.25 Paper 2.5 Amber 2.65 Rubber 3.0 Oil 4.0 Mica 5.0 Ceramic (low) 6.0 Bakelite 7.0 Glass 7.5 Water 78.0 Ceramic (high) 8000.0

d

.) .)

x

dx

xl

21.)

d A

.)

ary

dt

.) (Pressure) .)

F1

F2M

B2dd - x

d + xA

hl

ba

.) .)

(Glass substrate) (Tantalum)

(Folymer)

(Chromium layer)

5 5 5

3. 5.2 ) x 2 ( 2 > 1 ) A1 1 A2 2 C = d

A1ro + d

A2ro

A1 = wx

A2 = w (l x) w = ; l = C = ]x)([d

w122

o l (5.4) 4. 5.2 ) ( ) ( 1) P y r

y = P)ra(Et

)1(163 222

3

2 (5.5)

a = t = E = Youngs Modulus = Poissons Ratio C C

C = Pa

Edt16)1( 4

3

2 (5.6) d () C = oa2/d P = 0 (Zero) 5. (Differential or Push pull) () 3 5.2 ) F1 F2 x

AB C1 C2 MF1 MF2

C1 = xdA

ro + (5.7) C2 = xd

Aro (5.8)

5 6

C1 C2 x C1 C2 (Deflection) x 6. (Capacitive Level Sensor) 5.2 ) h ( 0.1 mho/cm3) b a ( b > a) r 2or/loge(b/a)

Ch = )a/b(log)h(2

)a/b(logh2

eo

ero + l

Ch = ]h)1([)a/b(log2

reo + l (5.9)

(Deflection Bridge) 5.3

) )

5.3

5.3 5.3 ) 5.1 ) Ch (5.9)

Z1 = 1/(jCo) ; Z2 = R2 Z4 = 1/(jCh) ; Z3 = R3

Z2

Z3

Vs

I1I2CO

Ch

Vth

R

R

Vs

I1I2

Vth

L1

L2

5 5 7

Vth = Vs

+

+23

h0

RR1

1

CC1

1

Vth = 0 hmin Co = Chmin(R3/R2)

Vth = Vs

+

+23

23

hminh

RR1

1

RR

CC1

1 (5.10)

R3/R2 1 (5.10)

Vth Vs

1CC

RR

minhh

32

5.2 ) C1 = xdA

ro + (5.7) C2 = xd

Aro (5.8)

Z1 = 1/(jC1) ; Z2 = R Z4 = 1/(jC2) ; Z3 = R Vth = Vs

+ 2

1CC

C21

2 (5.11)

Vth = xd2VS (5.12)

Vth x (Pushpull) 7. (Capacitive Lumidity Sensor) 5.2 ) ()

5 8

0 100 % RH 0 % 375 pF 1.7 pF / %RH C = 375 + 1.7(RH) pF (5.12) (5.12) 2 % 1 % (Oscillator) 100 k 100 kHz Loss Tangent tan tan = CR

1 (5.13) C = 500 pF tan 0.03 5.4) (Pure) (Q)

Q = CRn = R LC

5.4 RC R C C Q R fn

105 Hz Q 30

Vi = 0

VF L

RCH(s)

G(s)+-

VO

5 5 9

5.5

(e.m.f.) e.m.f. = Current Resistance (5.14) 5.5 ) i n (Ferromagnetic Material) (m.m.f.)

in

in

(air gap)

R

L

d air gap

2 rt

)

)

2 a a + x

a - xx

L1

L2

)

Central flux pathc r

5 10

m.m.f. = Flux Reluctance = (5.15) m.m.f. = ni

= ni (5.16) 1 N n

N = n = in2

(5.17)

L

L = iN =

2n (5.18) (5.18)

= Arol (5.19)

l = o = Permeability = 410-7 H/m r = Relative Permeability A = 5.5 ) (Air Gap) 1 5.5 ) (Variable Reluctance ) 3 (Armature) total = core + gap + armature (5.20) R r2 core = 2

co rR

= 2co r

R (5.21)

5 5 11

2d 1

gap = 2o r

d2 (5.22)

2R 5.5 ) 2rt

armature = rt2R2ao = rt

Rao (5.23)

total = 2co r

R + 2o r

d2 + rt

Rao (5.24)

o =

+ t

1r

1r

Raco

= total = o + kd (5.25) k = 2

o r2

n = 500 R = 2 . r = 0.5 . t = 0.5 . c = a = 100 , o = 1.3 107 / H, k = 2 1010 /Hm L = 19 (mH) () L = 7.6 1 (5.18) (5.25)

L = kdn

o

2

+ = d1L0+ (5.26)

(5.18) (5.26) Lo = n2/ o = = k/ o

(5.26) Lo L d 5.5 ) 2a (5.26) 5.5 ) - L1 L2

L1 = )xa(1L0 + (5.27)

5 12

L2 = )xa(1L0 ++ (5.28)

L1, L2 x 5.4 ) Z1 = jL1) ; Z2 = R Z4 = jL2) ; Z3 = R Vth = Vs

+ 2

1LL

L21

1 (5.29)

Vth = )a1(2xVS +

(5.30)

x Vth x (Pushpull) 0.5 Lo 25 mH 70 0.5 R 5.6

5.6 LC R L

Vi = 0

VF

L

R

C

G(s)+-

VO

H(s)

5 5 13

LVDT (Linear Variable Differential Transformer)

5.7 L.V.D.T.

LED

5.8 LED

5.9 LED

LED

Vo sin(2pfSt +j )V1

V2

Phase-sensitivedemodulator

+ Low pass filterVDCtf2sinV SS

primarysecondaries

x = 0

x = l

5 14

5.9 LED R2 LED LED TIL139 Taxas Instruments 5.10 LED

5.10

(Opto isolator)

(LED) 5.11 5 12

R1 270 W

R2 10 kW

LED 1Q1

+ 5V

LED

) )

5 5 15

5.11

1. (Inductive Sensors)

5.12 (www.balluff.com/solutions/inductive.htm)

5.12 5.13

680 W 4.7 kW

+ 5V

+ 12V

5 16

1) () (Active Zone : Coil) 2) (Oscillator) 3) (Evaluator) 4) (Trigger) 5) (Status display)

5.13

6) (Internal Constant Voltage Supply) 7) (External Voltage) 8) (Output and Protective) 9) ( ON-OFF)

5.13 1 2 3 4 5 8 ON OFF (Mild Steel) (Factor) = 1 = 0.35 = 0.25 10 3.5 2.5

G1

2 3 4 5

6

7

89

5 5 17

(Specification)

5.14

5.14 5.14 - (Sensing Range) (ON) (OFF) - (Norminal Sensing Range : Sn) - (Real Sensing Range; Sr) 90 % 110 % (Sn) - (Useful Sensing Range : Su) EN 50010 81 %121 % - (Working Sensing Range : Sw) - (Sn) (Mild Steel) (Factor)

0.81 Sn

Sr

S WS U

1.21 SnSn

0.9 Sn1.1 Sn

d

5 18

(Eddy Current) - (Repeatability) EURO-NORM - (Switching Hysteresis) (ON) (OFF) 2. (Capacitive Sensor)

5.15

(www.balluff.com/solutions/inductive.htm)

1. Active Electrode 2. 3. Earth Electrode 4. 5.

5.16

123

4

5

5 5 19

Active Earth Electrode

5.16 (Dielectric Constant) () (ON) (OFF) 2

5.2

()

1.0 1.0

0.3 ... 0.5 0.3 ... 0.6 0.3 ... 0.5 0.2 ... 0.7 0.1 ... 0.3

(Potentiometer)

5 20

5.17 (www.ifmefector.com/ifmus/web/capacitive.htm)

17 5.2

5.2.1. ( 4 ) 3 1) VS 2) R4

3) 23

RR

1) (Range) (Output)

5 5 21

2) (Linearity)

3) (Sensor)

IMIN = IMAX = RIMIN = RIMAX =

VMIN VMAX

VMIN =

+

+23

IMIN4S

RR1

1

RR1

1V (5.31)

VMAX =

+

+23

IMAX4S

RR1

1

RR1

1V (5.32)

VMIN = 0 I = IMIN (5.31) (5.32)

IMIN

4R

R =

23

RR

(5.33)

3 1) (Sensor) (Self Heating) PD = i2RI

PD 24I

I2S )RR(

RV + (5.34) PD = i = ( i2) 2) (Linearity) Eth () VMIN = 0 V I

VIDEAL MINMINMAXMAX

MINMAXMAX III

VIIIV

(5.34)

5 22

N(I) = Eth VIDEAL (5.35) ( N )

N 100VVE

MAXIDEALth (5.36)

R3/R2 (5.31)

(5.32) R4 = IMIN23 RR

R (5.37)

Sth

VE

=

23

IIMIN

23

RR1

1

RR

RR1

1

+

+ (5.37)

Sth

VE

= v , r = 23

RR

x = IMIN

IR

R (5.37)

v = r11

xr1

1+

+

v = r11

rxx

++ (5.38) v x 5.18 x 0.1 2.0 r = 0.1, 1.0 , 10.0 100

5.18 (v = r11

rxx

++ )

5 5 23

v = 0 x = 1 I = IMIN v(x) r 5.1 0-10 (Bar) 120 120 3 (Output) (Input) (Infinite) 30 (mA) 1) 2) 338 / ( 1) 10 1) R1 = R2 = R3 = 120

I1 ADC

Vi = I1(Ru + R3)

Ru = 120 I1 0.03

Vi = 0.03 (120 + 120 ) = 7.2 V 7.2 2) 10 = (338 m/bar ) (10 bar) = 3.38 Ru = 120 + 3.38 = 123.38

VO =

++ 21

13u

ui RR

RRR

RV

5.2 100 50 60 0 2 1 (Sensor Transducer) 100 High

5 24

5.19 RTD 30 R 65 150 65 0.004 / 1 30 / 50, 80 100 R(t) = Rt0(1 + 0 T) ... 50 C = 150[1 + 0.004 (50 65 )] = 141 ... 80 C = 150[1 + 0.004 (80 65 )] = 159 ... 100 C = 150[1 + 0.004 (100 65 )] = 171 1 RTD

T = DPP

T = ( Self Heating) P = RTD PD = RTD (W/1C) P = PD T = (30 mW/C) (1C)

100OC

Temperaturetransducer

heater

5 5 25

= 30 mW ( 80 C )

I = RP =

159mW30

= 13.7 mA RTD 50 C 5.0 RTD R4 R2 13.7 mA RTD 80 C V = IR = (13.7 mA)( 159 ) = 2.17 V R2

R2 = mA7.13)17.25(

= 206.5 R2 = 220 50 C R1 = 220 R3 141

5.20 5.1

5 26

50 C V = 1412201415141220

1415 ++ = 0 () 80 C V = 141220

14151592201595 ++ = 0.1447

100 C V = 1412201415171220

1715 ++ = 0.2338 5 Vref = 13.8 (0.2338) = 3.23

5.21

() 5.3

(ADC) (DAC) ADC

5 5 27

ADC () ADC ADC (Accuracy and Resolution)

1 100 99 101 100 100 mV (0.1 volt) 10.6 10.65 ( 50 mV) 10.1 10.2 10.5 1 1 10 9.2 9.8 0.1 9.2 9.8 (Conversion Time)

(AD) ( (Counter Stage)) 5.22

5 28

5.22 5.3 1.024 MHz 10 ? = T 2n T = n = = (1/1.024 106)(1024) = 1000 s = 1 ms 1 1 1 1,000 ( 5.22) (Gray Code Encoder) R2R 2 R2R R2R R2R 10 1000 () 10 / (1000 ) = 0.01 /

DC

5 5 29

5.23 R-2R

10,000 0.001 1 5.4 8 R 2R = 28 = 256 1 256 = (1/ 250) 100 % ( 1 / 256) = 0.4 % 5.5 8 R2R 10 ? 5.4 = 0.4 % 100 % 10 = 0.4 % (10 ) = (0.4 / 100) (10) = 0.04 V = 40 mV

( 0)

()

R-2R

5 30

5.6 10 kHz 8 ? = 28 = 256

/ = 1 /(10 103) = 100 s = 100 s 256 = 25,600 s = 25.6 ms (Per Conversion) 5.7 8 10 kHz ? 10 kHz 25.6 ms ( 5.5) 1 = 1 / (25.6 10-3) = 1000 / 25.6 40 5.24 (ADC) (DAC)

5.24

R2R (Summing Opamp) R2R 5.3.1 (DAC)

n BCD R2R 5.25

ADC

DAC

5 5 31

5.25 R2R R2R

R2R 5.25 R 2R 1 k 2 k R2R +16 0 ( 1 0) R2R

5.26 R2R 1,000 5.26 0 () 0 0 0 0 0 0 +16 1 23 = 8 +16 0 5.26 ) 8 ( 5.26 ), ) )) 5.26 5.26 ) 1 .. 2R R 5.26 ) R R 2R 2R 2 5.26 ) 2

R R R R

R

R

R

R

R

R

R

R

R

R

23 22 21 20

+16 V

1R

23 22 21 20

R R R

2R2R2R2R

()

+16 V

R

R

23 22 21

R R R

2R2R2R2R

234 4 3 2

+16 V

R

23 22 21 20

R R 2R

2R2R2R

4 3 2

1

+16 V

23

R

2R2R

4

2221 20

()

() ()

5 32

... 2R 2R .. R 3 2 2R 2R 3 R ... R 4 3 2R 5.26 ) 5.26 ) ) 1 2 3 4 2R 4 +16 2R 2R 8 8 16 ( 24) 8 /16 16 8 0100 5.27 ) 2R 5.27 ) 3 +8 ... R 5.27 ) R 4 3 5.27 ) 4 +4 4 0100 4 16 (4/16 = 1/4) 0010 2 0001 1

5.27 R2R 0100 1 (Superposition) 5.3

+16 V

1R

23 22 21 20

R R R

2R2R2R2R

()

R

2322

R R

R2R2R2R

234 4 3 2

+8 V

R

23 22

R

R2R

4 3

23

R

2R2R

4

22+8 V

+8 V

()

() ()

5 5 33

5.3 ( 16 )

(Volts) (Volts) 0000 0 1000 8 0001 1 1001 9 0010 2 1010 10 0011 3 1011 11 0100 4 1100 12 0101 5 1101 13 0110 6 1110 14 0111 7 1111 15

20/() 2 1/2 10 20 / 210 1 / 1024 29 / 210 512 / 1024 = 1 / 2 10.24 1 00 0000 0001 (1 / 1024) 10.24 = 10 10 0000 0000 (512 / 1024) 10.24 = 5.12 10 0000 0001 5.12 + 0.010 5.13 5.8 5 +6.4 1 0 0 ) 1 0 0 0 0 ) 0 0 0 0 1 ) 0 1 0 0 0 ) 0 1 1 0 1 ) 1 0 0 1 0 ) 25 = 32 ; 1 0 0 0 0 = 16 1 0 0 0 0 = (16/32) (6.4) = 3.2 ) 0 0 0 0 1 = 1 0 0 0 0 1 = (1 / 32) (6.4) = 0.2 ) 0 1 0 0 0 = 8 0 1 0 0 0 = (1 / 4) (6.4) = 1.6

) 0 1 1 0 1 4.6321

320

324

328

320

++++ = (13 / 32) (6.4) = 2.6

) 0 1 1 0 1 4.6320

322

320

320

3216

++++ = (18 / 32) (6.4) = 3.6

5 34

R -+ VO

LSB

R

2R2R2R

R

2R

2R

MSB

Vref

R-2R

5.9 8 51.2 ) 1 0 1 1 0 1 0 0 ) 1 0 0 1 1 1 0 1 ) 0 0 0 1 1 1 0 0

) (Vout) = 82457

22222 +++ (51.2)

= 25641632128 +++ (51.2) = 36

) (Vout) = 802347

222222 ++++ (51.2)

= 25614816128 ++++ = 31.4

) (Vout) = 8234

2222 ++ (51.2)

= 2564816 ++ (51.2) = 5.6

R 2R

R-2R 5.28

5.28 R 2R

0 0 1 Vref

5 5 35

(Unibipolar) 5.28 (Sign Bit) 5.28 10

= 000977.0024,11

21

21

10n === (Part Per Million (ppm)) 0.000977 100 % = 0.0977 % 0.000977 1,000,000 ppm = 977 ppm 5.4

5.4

(Resolution) (n)

% ppm 4 6.2500 62,500 8 0.3906 3,906

10 0.0977 977 12 0.0244 244 14 0.0061 61 16 0.0015 15 18 0.0004 4

8 0.5 16 15 ppm BCD (BCD Resolution)

BCD

5 36

5.10 BCD 8 () ? = ppm000,10%1100

1 == 5.11 BCD 12 BCD 12 4 3

= %1.0%10010001

10001 ==

= ppm000,000,110001 =

5.4 12 244 ppm BCD 12 1,000 ppm (Weighted Resistance DAC) 5.29 5.5

5.29

100 k200 k400 k800 k

100 k

-+

VO

D3D2

D1D0

( 0 + 5 )

5 5 37

5.5 4 (+5 V = 1 , 0 V = 0 , )

D3 D2 D1 D0 V0(Volts) 0 0 0 0 0.000 0 0 0 5 0.625 0 0 5 0 1.250 0 0 5 5 1.875 0 5 0 0 2.500 0 5 0 5 3.125 0 5 5 0 3.750 0 5 5 5 4.375 5 0 0 0 5.000 5 0 0 5 5.625 5 0 5 0 6.250 5 0 5 5 6.875 5 5 0 0 7.500 5 5 0 5 8.125 5 5 5 0 8.750 5 5 5 5 9.375

5.29 (0 5 ) +5 D0

)V5(k800k100 +

= 0.625 V

D3 +5

)V5(k100k100 +

= 5 V

5.5 5 = 1 0 = 0 5.30 ... R 2n-1R

5 38

5.30

5.3.2 (ADC)

R2R R2R R2R ADC (Digital Ramp (Staircase) ADC)

R2R 5.31 1. ( BCD) ( 0 ) 2. () 3. ( 5.31 ))

R

-+

VO

(Vref)

R

2R

4R

2n-1R

()

5 5 39

5.31 ADC ) )

5.12 ADC 8 2 MHz 28 256 2 MHz

T = 1/f = 1/(2 106) = 0.5 s T N = 0.5 s 256 = 128 s ADC

= 1/2n 100 % n = () 5.13 ADC 12 = 1/212 100 % = 0.0244 % = 244 ppm ADC ADC 1

-

+

)

)

5 40

= .fsn V121

Vfs = ADC n = 5.14 ADC 12 10.24 ? = .fsn V12

1

= mV5.2)V24.10(12

112 =

ADC 2n 1/f n = () f =

= f/122f/12 1nn =

5.15 ADC 10 2.5 MHz ?

= f/122f/12 1nn =

= S8.204105.2

12 69 =

(Successive Approximation Converter)

5 5 41

(Vi)

LSB

(SAR)

MSB

DAC

-+

5.32

5.33

Vi MSB (0) 1

1 Vi DAC 5.34 )

5 42

5.13 ) ) SAR 3 011 3.4 V ( 8.0 V)

) 3.4 V ( 8.0 V) ADC0804

ADC0804 ADC0804 ( Stand-alone operation) 5.35

5 5 43

5.35 ADC0804 Stand-alone Vcc = +5 Vdc R C

f )RC1.1(1 (5.39)

5.38 9 Vref

Vref = 2Vcc 6 7

7 0.0 5 ADC 0804 IC CPU 8080A 8080A 8228 Clock 8224 8048 MPU (Microprocessor Unit) WR , INTR , CS RD WR INTR (Pushbutton) 100 (s) 8 11 18 11 MSB 0.0 0000 0000 (00H) 1111 1111 (FFH) +5.0

1112131415161718

3521

194

6

7

9

20

10 8

R10 k

C 150 pF

+Vcc

MSB

LSB

Vi(+)

Vi(-)

Vcc/2

ADC0804

WRINTR

CS

RD

CLK R

CLK INDigitalOutput

Analoginput

Start

5 44

1 LSB +5.0 28 = 256 1 ( LSB) 256

V0.5 = 19.53

5.16 5.35 ADC0804 1 LSB 1) 2.5 2) 0010 0010 (22H) 1) 2.5 1000 0000 1 bit 27 = 128 128 19.53 mV = 2.5 V 2) ( 25 + 21 ) 19.53 mV = (32 + 2) 19.53 mV = 0.664 V (Span Adjust)

ADC0804 0.0 5.0 0.0 2.0 2.0 5.0 Vref 9

(Span) Vref = 2.0 V / 2 = 1.0 Vdc

Vref = 2 = 2

(5.40)

5.36 Vref (5.41)

Vref = RRR2RR

V21

2CC ++

+ (5.41)

(5.41) 5.36

Vref = +5 Vdc +++

k5.0k7.4k1k25.0k1 = 1.01 Vdc

Vref =1.0 Vdc 0.0 2.0 2.0 1111

1111 (FFH) LSB 256V0.2 7.8

5 5 45

5.36 Vref ADC0804

(Offset Zero Shift)

ADC0804 +1.5 +4.0 6 7 2.5 (4.0 1.5 ) (5.40)

Vref = 2 = 2

V5.2 = 1.25 Vdc

Vref 9 7 (Lower Limit) (Offset)

OFFSET Vi- = (5.42)

Vref = 1.25 Vdc Vi- = 1.5 Vdc 1.5 0000 0000 4.0 1111 1111

LSB 256V5.2 9.77

(Positive and Negative Input Voltages)

5.37 5 +5 (R R) 6 7

1112131415161718

3521

194

6

7

9

20

10 8

R10 k

C 150 pF

MSB

LSB

Vi(+)

Vi(-)

Vcc/2

ADC0804

WRINTR

CS

RD

CLK R

CLK INDigitalOutput

Analoginput

Start

+Vcc = + 5 Vdc

R1 4.7 k

+Vcc = + 5 Vdc

R 500 k

R2 1 k

5 46

5.37 1. Vi = -5 Vi+ = 0.0 0000 0000 (00H)

2. Vi = 0.0 Vi+ = 2.5 1000 0000 (80H) 3. Vi = +5.0 Vi+ = +5.0 1111 1111 (FFH) Vi+ 5.0 Vi+ 0.0 + 5.0 Vi MSB 0 (

0.0) Vi 0 MSB 1 ( 0.0) LSB 10 V/256 = 39.01

(Testing)

ADC0804 A/D

1112131415161718

3521

194

6

7

9

20

10 8

R10 k

C 150 pF

MSB

LSB

Vi(+)

Vi(-)

Vcc/2

ADC0804

WRINTR

CSRD

CLK R

CLK INDigitalOutput

Start

+Vcc = + 5 Vdc

+Vcc = + 5 Vdc

Vref

RR

+Vi-

5 5 47

5.38 LED VCC = 5.120 Vref VCC / 2 = 2.560 LSB 5.120 V /256 = 20

5.38 LED 1. 1010 1010 (AAH) (128 + 32 + 8 + 2) (20 mV) = 3.400 2. 0101 0101 (55 H) (64 + 16 + 4 +1)(20 mV) = 1.700 LED A/D 1 (high) LED

0 (Low) LED

LED low 0 =

1112131415161718

3521

194

6

7

9

20

10 8

R10 k

C 150 pF

+ 5 Vdc

LSB

Vi(+)

Vi(-)

VCC/2 = +2.560 Vdc

ADC0804

WRINTR

CSRD

CLK R

CLK INDigitalOutput

Start

+VCC = + 5.120 Vdc

Vref

R =1.5 kMSB

5 48

LED high 1 = 1011 0010 5.39 MSB LSB

5.39

(Shaft Position Conversion)

360 () 5.38 3 8 360 8 360 / 8 = 45 10 15 360 /215 360 / 32768 ~ 1 /60 ( ) 5.17 10 ? = 360 / 210 = 360/1024 = 0.36 1/3 0 0 0 0 0 1 20

0 0 0 21 22 0 0 1 3 16 ( 1 ) 1 1 1 1 0 0 0 20 21 1 1 1 22 000 011

5 5 49

1 ( 180 ) 180 1 1 (Code Disk) Gay tobinary 1

Decimal Gray Code Binary Code Decimal Gray Code Binary Code 0 0000 0000 8 1100 1000 1 0001 0001 9 1101 1001 2 0011 0010 10 1111 1010 3 0010 0011 11 1110 1011 4 0110 0100 12 1010 1100 5 0111 0101 13 1011 1101 6 0101 0110 14 1001 1110 7 0100 0111 15 1000 1111