1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification...
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Transcript of 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification...
![Page 1: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/1.jpg)
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Lecture 11Combinational Design Procedure
![Page 2: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/2.jpg)
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Overview
° Design digital circuit from specification
° Digital inputs and outputs known• Need to determine logic that can transform data
° Start in truth table form
° Create K-map for each output based on function of inputs
° Determine minimized sum-of-product representation
° Draw circuit diagram
![Page 3: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/3.jpg)
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Design Procedure (Mano)
Design a circuit from a specification.
1. Determine number of required inputs and outputs.
2. Derive truth table
3. Obtain simplified Boolean functions
4. Draw logic diagram and verify correctnessA00001111
B00110011
C01010101
R00000001
S01111111
S = A + B + CR = ABC
![Page 4: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/4.jpg)
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Previously, we have learned…
° Boolean algebra can be used to simplify expressions, but not obvious:• how to proceed at each step, or
• if solution reached is minimal.
° Have seen five ways to represent a function:• Boolean expression
• truth table
• logic circuit
• minterms/maxterms
• Karnaugh map
![Page 5: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/5.jpg)
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Combinational logic design
° Use multiple representations of logic functions
° Use graphical representation to assist in simplification of function.
° Use concept of “don’t care” conditions.
° Example - encoding BCD to seven segment display.
° Similar to approach used by designers in the field.
![Page 6: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/6.jpg)
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BCD to Seven Segment Display
° Used to display binary coded decimal (BCD) numbers using seven illuminated segments.
° BCD uses 0’s and 1’s to represent decimal digits 0 - 9. Need four bits to represent required 10 digits.
° Binary coded decimal (BCD) represents each decimal digit with four bits
a
b
c
g
e
d
f
10019
00018
11107
01002
10001
00000
![Page 7: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/7.jpg)
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BCD to seven segment display
0 a,b,c,d,e,f1 b,c2 a,b,d,e,g3 a,b,c,d,g4 b,c,f,g5 a,c,d,f,g6 a,c,d,e,f,g7 a,b,c8 a,b,c,d,e,f,g9 a,b,c,d,f,g
a
b
c
g
e
d
f
° List the segments that should be illuminated for each digit.
![Page 8: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/8.jpg)
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BCD to seven segment display
.0111110019
.1111100018
.0011111107
.
.1101101002
.0011010001
.1111100000
.edcbazyxwDec
° Derive the truth table for the circuit.
° Each output column in one circuit.
Inputs Outputs
![Page 9: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/9.jpg)
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BCD to seven segment display
1 0
10
1 1
1 1
11
yz
wx
10
11
01
00
10110100
For segment “a” :
Note: Have only filled in ten squares, corresponding to the ten numerical digits we wish to represent.
° Find minimal sum-of-products representation for each output
![Page 10: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/10.jpg)
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Don’t care conditions (BCD display) ...
1 0
10
1 1 X X
X X X X
1 1
11
yz
wx
10
11
01
00
10110100
For segment “a” :
Put in “X” (don’t care), and interpret as either 1 or 0 as desired ….
° Fill in don’t cares for undefined outputs.• Note that these combinations of inputs should never happen.
° Leads to a reduced implementation
![Page 11: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/11.jpg)
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Don’t care conditions (BCD display) ...
For segment “a” :
yFa1 1 0
10
1 1 X X
X X X X
1 1
11
yz
wx
10
11
01
00
10110100
° Circle biggest group of 1’s and Don’t Cares.
° Leads to a reduced implementation
![Page 12: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/12.jpg)
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Don’t care conditions (BCD display)
For segment “a” :
wFa2 1 0
10
1 1 X X
X X X X
1 1
11
yz
wx
10
11
01
00
10110100
° Circle biggest group of 1’s and Don’t Cares.
° Leads to a reduced implementation
![Page 13: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/13.jpg)
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Don’t care conditions (BCD display) ...
For segment “a” :
zxFa3
1 0
10
1 1 X X
X X X X
1 1
11
yz
wx
10
11
01
00
10110100
xzFa4
1 0
10
1 1 X X
X X X X
1 1
11
yz
wx
10
11
01
00
10110100
° Circle biggest group of 1’s and Don’t Cares.
° All 1’s should be covered by at least one implicant
![Page 14: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/14.jpg)
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Don’t care conditions (BCD display) ...
For segment “a” :
xzzxwyF 1 0
10
1 1 X X
X X X X
1 1
11
yz
wx
10
11
01
00
10110100
° Put all the terms together
° Generate the circuit
![Page 15: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/15.jpg)
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![Page 16: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/16.jpg)
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BCD to seven segment display
.0111110019
.1111100018
.0011111107
.
.1101101002
.0011010001
.1111100000
.edcbazyxwDec
° Derive the truth table for the circuit.
° Each output column in one circuit.
Inputs Outputs
![Page 17: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/17.jpg)
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BCD to seven segment display
1 1
01
1 1
1 0
11
yz
wx
10
11
01
00
10110100
For segment “b” :
° Find minimal sum-of-products representation for each output
![Page 18: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/18.jpg)
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1 1
01
1 1
1 0
11
yz
wx
10
11
01
00
10110100
For segment “b” :
X X X X
X X
![Page 19: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/19.jpg)
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1 1
01
1 1
1 0
11
yz
wx
10
11
01
00
10110100
For segment “b” :
X X X X
X X
![Page 20: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/20.jpg)
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1 1
01
1 1
1 0
11
yz
wx
10
11
01
00
10110100
For segment “b” :
X X X X
X X
![Page 21: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/21.jpg)
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1 1
01
1 1
1 0
11
yz
wx
10
11
01
00
10110100
For segment “b” :
X X X X
X X
![Page 22: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/22.jpg)
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1 1
01
1 1
1 0
11
yz
wx
10
11
01
00
10110100
For segment “b” :
X X X X
X X
![Page 23: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/23.jpg)
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BCD-to-Excess-3 Code converter
° BCD is a code for the decimal digits 0-9
° Excess-3 is also a code for the decimal digits
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Specification of BCD-to-Excess3
° Inputs: a BCD input, A,B,C,D with A as the most significant bit and D as the least significant bit.
° Outputs: an Excess-3 output W,X,Y,Z that corresponds to the BCD input.
° Internal operation – circuit to do the conversion in combinational logic.
![Page 25: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/25.jpg)
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Formulation of BCD-to-Excess-3
° Excess-3 code is easily formed by adding a binary 3 to the binary or BCD for the digit.
° There are 16 possible inputs for both BCD and Excess-3.
° It can be assumed that only valid BCD inputs will appear so the six combinations not used can be treated as don’t cares.
![Page 26: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/26.jpg)
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Optimization – BCD-to-Excess-3
° Lay out K-maps for each output, W X Y Z
° A step in the digital circuit design process.
![Page 27: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/27.jpg)
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Placing 1 on K-maps
° Where are the minterms located on a K-Map?
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Expressions for W X Y Z
° W(A,B,C,D) = Σm(5,6,7,8,9) +d(10,11,12,13,14,15)
° X(A,B,C,D) = Σm(1,2,3,4,9) +d(10,11,12,13,14,15)
° Y(A,B,C,D) = Σm(0,3,4,7,8) +d(10,11,12,13,14,15)
° Z(A,B,C,D) = Σm(0,2,4,6,8) +d(10,11,12,13,14,15)
![Page 29: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/29.jpg)
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Minimize K-Maps
° W minimization
° Find W = A + BC + BD
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Minimize K-Maps
° X minimization
° Find X = BC’D’+B’C+B’D
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Minimize K-Maps
° Y minimization
° Find Y = CD + C’D’
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Minimize K-Maps
° Z minimization
° Find Z = D’
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Two level circuit implementation
° Have equations• W = A + BC + BD = A + B(C+D)
• X = B’C + B’D + BC’D’ = B’(C+D) + BC’D’
• Y = CD + C’D’
• Z = D’
° Factoring out (C+D) and call it T
° Then T’ = (C+D)’ = C’D’• W = A + BT
• X = B’T + BT’
• Y = CD + T’
• Z = D’
![Page 34: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/34.jpg)
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Create the digital circuit
° Implementing the second set of equations where T=C+D results in a lower gate count.
° This gate has a fanout of 3
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![Page 35: 1 Lecture 11 Combinational Design Procedure. 2 Overview °Design digital circuit from specification °Digital inputs and outputs known Need to determine.](https://reader031.fdocuments.us/reader031/viewer/2022013004/5697bfbc1a28abf838ca14c6/html5/thumbnails/35.jpg)
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Summary
° Need to formulate circuits from problem descriptions1. Determine number of inputs and outputs
2. Determine truth table format
3. Determine K-map
4. Determine minimal SOP
o There may be multiple outputs per designo Solve each output separately
o Current approach doesn’t have memory.o This will be covered next week.