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# Logic Gates Flip-Flops Registers Adders

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Logic Gates Flip-Flops Registers Adders
Digital Electronics Logic Gates Flip-Flops Registers Adders Jordan Nash - Microprocessor Course

Numbers in a computer: In the Microprocessor and memory Bits are stored electronically Each bit is represented by an electrical signal which is either a high or low voltage level. high  1 and low  0 high  0 and low  1 Normal Logic Inverse Logic Jordan Nash - Microprocessor Course

Transistor Transistor
High and Low Transistor Transistor Logic (TTL) NO MAN’S LAND 74LSxx 0.8 – 0.4 Volts 2.0 – 2.4 Volts CMOS NO MAN’S LAND 74HCxx 0.1 – 1.0 Volts 3.5 – 4.9 Volts 1:‘High’ 0:‘Low’ It is actually a bit more complicated than this since there are different thresholds for inputs and outputs and their noise margins (indicated here in RED).. Jordan Nash - Microprocessor Course

Making a Zero or and One How do you actually make a 0 or 1 ?
It is clear that depending upon the position of the J1 switch the line will be either ‘0’ or ‘1’ Jordan Nash - Microprocessor Course

Binary Logic Once the bits are stored, we want to be able to perform various operations on the bits The operations which the microprocessor carries out can all be constructed with a few simple circuits The basic building blocks are logic “gates” Jordan Nash - Microprocessor Course

Binary Logic truth tables
You may remember from the first year lab the gates AND, OR, NOT Any digital device can be made out of either ORs and NOTs or ANDs and NOTs. A B AND 1 A B OR 1 A NOT 1 Jordan Nash - Microprocessor Course

DeMorgan’s Theorem You can swap ANDs with ORs if at the same time you invert all inputs and outputs : Exercise: Write truth table for both and prove that this is correct Jordan Nash - Microprocessor Course

An AND out of NOTs and ORs
Exercise: Test the claim that you can make any logic device exclusively out of NOTs and ORs by making an AND out of NOTs and ORs: Jordan Nash - Microprocessor Course

Answer: Show that this device has an identical truth table as the AND gate. Jordan Nash - Microprocessor Course

Exercise: Exclusive OR
Construct an exclusive OR (XOR) gate using OR, ANDs, and NOT: A B XOR 1 Jordan Nash - Microprocessor Course

The Exclusive OR Solution: Jordan Nash - Microprocessor Course

Storing Zeros and Ones: Registers
Registers are electronic devices capable of storing 0’s or 1’s D-FLIP-FLOPs are the most elementary registers can store one bit 8 DFFs clocked together make a one byte register Capable of storing 8 bits Jordan Nash - Microprocessor Course

SR Latch: Set and Reset Gate level design of an SR latch
When S* is low and R* High, Q is high (Set) When S* is high and R* is low Q is cleared (Reset) Q and Q* are complements S* R* Q Q* 1 Jordan Nash - Microprocessor Course

SR Latch: Hold Gate level design of an SR latch
When S* and R* are high both states for Q are possible (and stable) The latch remembers the last state it was in, and holds Q in that state Gate level design of an SR latch S* R* Q Q* 1 Jordan Nash - Microprocessor Course

SR Latch: Don’t do this…
When S* and R* are low Q and Q* are no longer complementary The latch can also enter a race condition where it oscillates Gate level design of an SR latch S* R* Q Q* 1 Jordan Nash - Microprocessor Course

SR Latch: 1 bit memory Gate level design of an SR latch
Q Q* H L Ambiguous but stable Gate level design of an SR latch S/R high is ambiguous, but stable This circuit “remembers” that S went low Jordan Nash - Microprocessor Course

Gated D Latch Ensures that S and R are always complementary.
When the Clock is high – D is set on the outputs of the latch and it is held there when the clock is low Exercise: Fill out the truth table for this circuit. Jordan Nash - Microprocessor Course

D latch timing The input Data to the gated D latch appears on the output (Q)while the clock is high. Sometimes we want to transfer the data into the register only at a specified moment (for example when the clock changes) The D Flip-Flop uses two d-latches to latch the data on the edge (depending on the design either positive or negative) of the input clock Jordan Nash - Microprocessor Course

Master Slave – D Flip Flop
Data is stored in the “Master” latch (Qm) when the clock is high When the clock goes from high to low The data is held in the Master The Data is stored in the slave latch output The Qs output can only change when the clock goes from high to low Jordan Nash - Microprocessor Course

A real D-Flip-Flop (DFF)
CLK D S R Q Q-bar X L H One can Set or Reset (Clear) the DFF using S or R When S and R are High, on the rising edge of the clock the data are transferred and stored in Q. Jordan Nash - Microprocessor Course

A register that stores 4 bits
4-bit Register A register that stores 4 bits Data Stored on rising edge of clock in Q0,Q1,Q2,Q3 Data coming in D0,D1,D2,D3 Jordan Nash - Microprocessor Course

Byte Register 8 bit register in a single package 74F574
Byte coming in 8 bit register in a single package 74F574 Also contains tri-state outputs CLK Byte Stored Jordan Nash - Microprocessor Course

Addition with Binary Logic Gates
Computers carry out addition with binary addition of bits stored in registers We can build additions of large numbers of bits out of units which add two bits \$3+\$4 \$3+\$C \$D+\$4 \$D+\$C 0011 1101 + 0100 + 1100 = 0111 = 1111 =10001 =11001 \$7 \$F \$11 \$19 Example: Adding two 4 bit numbers results in a 4 bit number plus one carry bit or effectively a 5 bit number Lets construct this adder from gates J. Nash - Microprocessor Course

Additions of two bits: The half adder
Adding two bits generates two bits of output 1 “Sum Bit” S 1 “Carry Bit” C The truth table for this operation is shown along with an implementation using two gates A B C S 1 Jordan Nash - Microprocessor Course

Full Adder In order to add larger numbers, we need to be able to bring the carry from the lower order bits, and add this to the sum The inputs are: the two bits to be added (A,B) The Carry Bit (C) The outputs are: The Sum Bit (S) The Carry Out Bit (C+) We can build this from two half adders and an XOR gate A B C C+ S 1 Jordan Nash - Microprocessor Course

Two bit adder We can construct logic that adds more than 1 bit together by using multiple full adder circuits Exercise: Construct a circuit that adds two two bit words (Ao,A1) and (B0,B1) and produces three Sum Bits (S0,S1,S2) Jordan Nash - Microprocessor Course

Solution With the Full adder building block we can generalize to produce a circuit which adds larger numbers Jordan Nash - Microprocessor Course

Four plus Four bit addition
This can clearly be generalized to any number of bits There is a problem in building circuits this way as the carry bits need to propogate through the circuit before the answer is correct There are other ways to construct adder circuits which avoid this problem Jordan Nash - Microprocessor Course

Arithmetic Logic Unit (ALU)
Centre of every computer Composed of building blocks we have been learning about Adders Flip Flops Registers We also need to have a data bus to move data in and out - we will learn about this soon Jordan Nash - Microprocessor Course

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