Combinational Logic By : Ali Mustafa
Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer
How to Analyze any combinational circuit like this?
Analysis Procedure To obtain the output Boolean functions from a logic diagram: 1.Label all gate outputs that are a function of input variables. 2.Label the gates that are a function of input variables and previous labeled gates with different arbitrary symbols. 3.Repeat step 2 until the outputs of the circuit are obtained in terms of the input variables.
Analysis Procedure
Analysis Procedure
Analysis Procedure
Analysis Procedure Derivation of Truth Table
Analysis Procedure
Analysis Procedure
Summary : How to design a combinational circuit? 1. Determine the required number of inputs and outputs. 2.Derive the Truth Table. 3.Obtain the simplified Boolean functions. 4.Draw the logic diagram.
Example Design a combinational circuit with three inputs and one output. The output must equal 1 when the inputs are less than three and 0 otherwise. [use only NAND gates]
Example (cont.) 1-The system have three inputs and one output
Example (cont.) 2- Derive the truth table
Example (cont.)
Circuits to be Implemented Arithmetic circuits Half Adder Full Adder Binary Adder/Subtractor Binary Multiplier Magnitude Comparator
What's the example of Arithmetic Circuit? One of the famous Digital Logic Circuits is the calculator.
Arithmetic circuits An arithmetic circuit is a combinational circuit that performs arithmetic operations such as addition, subtraction, multiplication and division with binary numbers or with decimal numbers in a binary code. A combinational circuit that performs the addition of two bits is called a Half Adder.
Half adder It is required to design a combinational circuit that adds two binary numbers and produce the output in the form of two bits sum and carry Solution 1- We need to determine the inputs and output of the system and give letters for all of them our system has two inputs (X, Y) and two outputs (S, C)
Half adder 2-Derive the truth table according to the given relation between outputs and inputs In the half adder block the output equals the sum of two binary inputs
Half adder (cont.) 3- Obtain the simplified Boolean functions for each output as a function of the input variables using K-map 4- Draw the logic diagram
Full Adder It is required to add three binary numbers Solution 1. From the specifications of the circuit, determine the required number of inputs and outputs and assign a letter (symbol) to each.
Full Adder (cont.) 2. Derive the truth table according to the given relation between outputs and inputs
Full Adder (cont.) 3- Obtain the simplified Boolean functions for each output as a function of the input variables using K-map
Full Adder (cont.) 4- Draw the logic diagram
4-Bit Binary Adder (Ripple Carry Adder)
Ripple Carry Adder
Binary Subtractor The subtraction of binary number can be done most conveniently by means of complements The subtraction A-B is done by taking the 2 s complement of B and adding it to A. The 2 s complement can be obtained by taking the 1 s complement and adding 1 to the least significant bit. The 1 s complement can be implemented easily with inverter circuit and we can add 1 to the sum by making the initial input carry of the parallel adder equal to 1.
Adder/Subtractor
Subtractor A B D B 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 A B C D B 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 HALF SUBTRACTOR FULL SUBTRACTOR
Subtractor HALF SUBTRACTOR D = A B + AB B = A B FULL SUBTRACTOR D = A B C + A BC + AB C + ABC B = A C + A B + BC
4- Bit Subtractor
Assignment # 3 Design 4-Bit Adder - Subtractor
4-Bit Adder - Subtractor
Binary multiplier
Self Study 3 x 4 Bit Multiplier
Magnitude Comparator It is required to design a circuit to compare between two inputs A={A1,A0} and B={B1,B0} both inputs consists of two binary bits the circuit has three outputs Greater, Less, Equal
Magnitude Comparator (cont.) 1. Determine the required number of inputs and output.
Magnitude Comparator (cont.) 2. Derive the Truth Table that defines the required relationship between inputs and outputs.
Magnitude Comparator (cont.) 3- Get the simplified logic function of the outputs using k-map
Magnitude Comparator (cont.) 4- Draw the circuit
4 Bit Comparator
4 Bit Comparator