Combinational Logic Chapter 4 1 Combinational Circuits Combinational
Combinational Logic Chapter 4 1
Combinational Circuits
Combinational Circuits • Adders • Subtractors • Comparators • Decoders • Encoders • Multiplexers Available as MSI Circuits and as Standard Cells in VLSI (Bonus Assignment: Get one example of each type of combinational circuits in the CMOS family, 5 points for the second exam)
Analysis Procedure • Given a logical diagram, determine or more of the following: – Boolean functions; – Truth table; – Explanation of circuit operation • Make sure the circuit is combinational, not sequential (No feedback loops)
Analysis Procedure 1. Label all gate outputs that are a function of input variables. Determine the Boolean function for each gate output 2. Label the gates that are a function of input variables and previously labeled gates. Find the Boolean functions for these gates 3. Repeat step 2 until output of circuits are obtained 4. By repeated substitution of previously defined functions, obtain the output Boolean functions in terms of input variables
Analysis Procedure
Analysis Procedure Step 1 Step 2 Steps 3 and 4
Analysis Procedure •
Analysis Procedure
Design Procedure 1. From the specifications of the circuit, determine the required number of inputs and outputs and assign a symbol to each 2. Derive the truth table that defines the required relationship between inputs and outputs 3. Obtain the simplified Boolean functions for each output as a function of the input variables 4. Draw the logical diagram and verify correctness of the design
Code conversion example
Code conversion example
Code conversion example
Code conversion example
Binary Adder-Subtractor Half adder
Half adder
Full adder
Full adder Implementation of full adder in sum-of-products
Full Adder
Full adder Implementation of full adder using two half adders and one or gate
Binary adder
Carry propagation Carry Generate
Carry propagation Carry lookahead generator
Four-bit adder with carry lookahead Carry lookahead generator
Four-bit adder with carry lookahead
Binary subtractor
Overflow • Occurs only when adding two positive numbers or two negative numbers; • Overflow produces change in result sign Example: eight-bit adder Carry bits 0 1 0 +70 0 1000110 -70 1 0111010 +80 0 1010000 -80 1 0110000 +150 1 0010110 -150 0 1101010
Decimal Adder • Consider adding two decimal digits in BCD • Output sum cannot exceed 9+9+1=19 (the last 1 is the carry from previous digit)
Decimal Adder Need correction Carry Condition for correcting result
Decimal Adder
Binary Multiplier Explain how you carried the multiplication out. How many bits at the output?
Binary Multiplier Two-bit multiplier
Binary Multiplier Exercise: With your neighbor classmate discuss its operation.
Magnitude Comparator
Magnitude Comparator Does this circuit correspond to what you wrote down in the exercise? Discuss again with your neighbor classmate the comparison of your result with this circuit.
Decoders Exercise: Minimize the functions for two of the eight output lines.
Decoders Exercise: Compare your function with the circuit. Three-to-eight-line decoder
Decoders 0
Decoders Exercise: Explain how this decoder works.
Combinational logic implementation
Combinational Logic Implementation Compare in terms of propagation time and number of gates this Full Adder with the previously studied implementation.
Encoders
Priority encoder Highest priority Valid output
Priority encoder
Priority encoder
Multiplexers •
Multiplexers Two-to-one-line multiplexer
Multiplexers Four-to-one-line multiplexer
Multiplexers Quadruple two-to-one-line multiplexer
Boolean function implementation •
Boolean function implementation •
Boolean function implementation • Selected input line 0 1 2 3 Implementation of a three-input Boolean function
Boolean function implementation 0 1 2 3 4 5 6 7
Three-state gates
Three-state gates Multiplexers with three-state gates
Homework Assignment • • • 4. 3 4. 9 4. 17 4. 27 4. 33 Using Logic. Works, simulate a 4 -bit full adder with and without carry-lookahead.
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