# Chapter 4: Combinational Logic

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Chapter 4: Combinational Logic
Dr Mohamed Menacer Taibah University

Multiplexer

The Multiplexer (MUX)

4-1 Multiplexer Principle: Use the A and B inputs to select a pair of minterms. The value applied to the MUX input is selected from {0, 1, C, C} to pick the desired behavior of the minterm pair.

Multiplexer Function Multiplexer Y = A • S + B • S´ OR AND NOT

8-1 Multiplexer

The Demultiplexer (DEMUX)

The Demultiplexer is a Decoder with an Enable Input

Half-Adder Truth Table x y C S 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0
Logic Equations C = x • y S = x  y Schematic

Full-Adder Truth Table x y z C S 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1
Logic Equations C = x’yz + xy’z + xyz’ + xyz = z • (x’y+xy’) + xy • (z+z’) = z • (x  y) + x • y = MAJ (x,y,z) S = x’y’z + x’yz’ + xy’z’ + xyz = x’yz’ + xy’z’ + x’y’z + xyz = z’(x’y + xy’) + z(x’y’ + xy) = z’(x  y) + z(x  y)’ = (x  y)  z = x  y  z

Negative Numbers and Subtraction
Several different codes for negative numbers 2's complement 1's complement signed magnitude others For 2's complement, subtraction is implemented with the same hardware by negating the subtrahend. This is done by inverting each bit and adding one. The one can be added by setting the carry-in to the first stage equal to 1, saving an operation.

Decoders

Decoder Fundamentals Route data to one specific output line.
Selection of devices, resources Code conversions. Arbitrary switching functions implements the AND plane

Decoder 1 1 1 0 0 1 00 What happens when the inputs goes from 01 to 10?

Why Encode? A Logarithmic Relationship
Lets say we have N inputs to a decoder, the number of outputs will be equal to 2^N

Decoder with Enable Demultiplexer Decoder with Enable

3 to 8 Decoder

Encoder An encoder translates a set of inputs into a binary encoding,
Can be thought of as the converse of a decoder. A priority encoder imposes an order on the inputs.

Sequential Logic

Sequential Logic The combinational logic circuits we have been studying so far have no memory. The outputs always follow the inputs. There is a need for circuits with a memory, which behave differently depending upon their previous state. Sequential circuits use current input variables and previous input variables by storing the information and putting back into the circuit on the next clock (activation) cycle.

Sequential Circuit

Synchronous & Asynchronous Sequential Circuits

The Clock Paces the System
In a positive logic system, the “action”happens when the clock is high, or positive. The low part of the clock cycle allows propagation between subcircuits, so their inputs are stable at the correct value when the clock next goes high.

Clock Pulse Definition
Positive Pulse Positive Edge Negative Negative Pulse Positive Edge Negative Edges can also be referred to as leading and trailing.

A NOR Gate with a Lumped Delay
This delay between input and output is at the basis of the functioning of an important memory element, the flip-flop.

Flip-Flops Flip-flops are the first stage in sequential logic design which incorporates memory (storage of previous states).

Types of Flip-Flops SR type Flip-flop or Set / Reset
D type Flip-flop or Data / Delay JK type Flip-flop T type Flip-flop or Triggered /Toggle

The S-R (Set-Reset) Flip-Flop
The operation of an SR flip-flop is as follows: The Set input will make Q goto 1 The Reset input will make the output Q goto 0 i.e. reset the output. Both Set and Reset at logic 1 is not allowed as this is not a logical pair of inputs. The S-R flip-flop is an active high (positive logic) device.

A Clocked S-R Flip-Flop
The clock signal, CLK, turns on the inputs to the flip-flop.

The Clocked D (Data) Flip-Flop
The operation of the D type flip-flop is as follows: Any input appearing (present state) at the input D, will be produced at the output Q in time T+1 (next state). The operation of the D type delays any input by exactly one clock cycle. Cascading several D type flip-flops together can produce delaying circuits The clocked D flip-flop, sometimes called a latch, has a potential problem: If D changes while the clock is high, the output will also change. The Master-Slave flip-flop solves this problem:

8 Bits D (Data) Flip-Flop
One D type flip-flop can store one bit of information for one clock cycle. To have eight bits of information, simply arrange eight flip-flops in parallel with a common clock.

The Master-Slave Flip-Flop
The rising edge of the clock clocks new data into the Master, while the slave holds previous data. The falling edge clocks the new Master data into the Slave.

Parallel Registers

4-Bit Parallel Register (4 Flip Flops)

4-Bit Register With Enable

Memory Devices

Magnetic Core Memory Register Decoder (AND plane)

Semiconductor Memory Decoder (AND plane) OR plane

UT4090 RAM Module Dual-port 1152 bits per cell Four configurations
64 X 18 128 X 9 256 X 4 512 X 2 WA(8:0) RE WD(17:0) RCLK WE RA(8:0) WCLK RD(17:0) MODE(1:0) ASYNCRD

Packaging