n Combinational Binary Addition nbit addition n Full

Περίληψη n Combinational Binary Addition n-bit addition n Full Adder n n-bit Adder n n Sequential Circuits Storage elements n Sequential n-bit Adder n n Computer Design n From transistor to computer 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 2

n-bit Addition n n Design an n-bit binary adder which performs the addition of two n-bit binary numbers and generates a n-bit sum and a carry out. Example: Let n=4 Cout n C 3 C 2 C 1 Cin A 3 A 2 A 1 A 0 +B 3 B 2 B 1 B 0 -------S 3 S 2 S 1 S 0 1 1 0 1 +1 1 0 1 -----1 0 This requires 3 -bit addition! 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 4

Full Adder Full adder (for higher-order bit addition) n Combinational circuit that performs the additions of 3 bits (two bits and a carryin bit) n Ai Ci+1 Bi 1 bit full adder Ci Block diagram of a Full Adder Si 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 5

Full Adder (cont. ) n. Boolean n Ci+1 = Ai. Bi + Ai. Ci + Bi. Ci n n equations: Ci+1 is 1 if at least two out of the three inputs are 1 Si = Ai. Bi’ Ci’ + Ai’Bi’Ci + Ai’Bi. Ci’ + Ai. Bi. Ci = A i B i Ci n Si is 1 if an odd number of the three inputs are 1 n. You can design full adder circuit directly from the above equations (requires 3 ANDs and 1 OR for Ci+1 and 2 XORs for Si) n. Can we do better? 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 6

Full Adder using 2 Half Adders n n n A full adder can also be realized with two half adders and an OR gate, since Ci+1 can also be expressed as: Ci+1 = Ai. Bi + (Ai Bi)Ci Si = A i B i C i A logic diagram of a Full Adder Ai Si Bi Ci+1 Ci 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 7

n-bit Combinational Adders Perform parallel multi-bit addition n Ripple Carry Adder n Constructed using n 1 -bit full adder blocks in parallel. n Cascade the full adders so that the carry out from one becomes the carry in to the next higher bit position. n Simple design n Time consuming. Why? (you’ll see in a bit!) n 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 8

Example: 4 -bit Ripple Carry Adder C 4 13/6/2021 C 3 C 2 C 1 C 0 A 3 A 2 A 1 A 0 +B 3 B 2 B 1 B 0 -------S 3 S 2 S 1 S 0 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 9

Ripple Carry Adder Delay n n Circuit delay in an n-bit ripple carry adder is determined by the delay on the carry path from the LSB (C 0) to the MSB (Cn). Let the delay in a 1 -bit FA be Δ. Then, the delay of an n -bit ripple carry adder is nΔ. 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 10

Sequential Circuits n Combinational Logic: Output depends only on current input n Able to perform useful operations (add/subtract/multiply/…) n Require cascading of many structures n Costly and inflexible n 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 11

Sequential Circuits (cont. ) n Sequential Logic: Output depends not only on current input but also on past input values n Store information between operations (no need for cascading) n Need some type of memory to remember the past input values n 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 12

Sequential Circuits (cont. ) Circuits that we have learned so far Information Storing Circuits Timed “States” 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 13

Sequential Logic: Concept Sequential Logic circuits remember past inputs and past circuit state. n Outputs from the system are “fed back” as new inputs (usually with delay). n The storage elements are circuits that are capable of storing binary information: memory. n 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 14

Synchronous Sequential Circuits: Flip flops as state memory n The flip-flops receive their inputs from the combinational circuit and also from a clock signal with pulses that occur at fixed intervals of time, as shown in the timing diagram. 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 15

Storing Elements Can’t change the stored value! Inverters Buffers 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 16

SR latch (NOR version) -- SR: “set-reset”, bistable element with two extra inputs; note the “undefined” output for S=R=1. -- Reading the logic: § Q = (R+Q’)’ and Q’ = (S+Q)’ 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 17

R=S=1 ? ? n Undefined output, because When S=R=1, both outputs go to zero. n If both inputs now go to 0, the state of the SR flip flop is depends on which input remains a 1 longer before making transition to 0. n Hence, “undefined” state. MUST be avoided. n 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 18

D Latch n n One way to eliminate the undesirable indeterminate state in the RS flip flop is to ensure that inputs S and R are never 1 simultaneously. This is done in the D latch: Observe that this is a NAND-implementation of the SR-latch 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 19

Sequential Binary Addition n How can we add two n-bit binary numbers using ONLY one full adder and a 1 -bit storage element? Ai Ci+1 Bi 1 bit full adder Ci Q 1 bit Memory D Clock Si 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 20

Serial vs. Parallel Addition n n The parallel adder is a combinational circuit, whereas the serial adder is a sequential circuit. The parallel adder has n full adders for n-bit operands, whereas the serial adder requires only one full adder. The serial circuit takes n clock cycles to complete an addition. In summary, the parallel adder in space is n times larger than the serial adder, but it is n times faster. The serial adder, although it is n times slower, is n times smaller in space. 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 21

Computer Design n Where/How do we start? Which materials, how to proceed in the design process, how to integrate very large components, etc ? q Most basic electrical components q Transistor q Design process: Bottom-up or Up-down ? q Integration: parts, processes, human knowledge 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 22

Bell Labs Museum The First Point-Contact Transistor 1947 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 25

Bell Labs The First Junction Transistor 1951 M 1752 Outside the Lab model 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 26

Central Processing Unit (CPU) 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 29

Basic Computer Structure Stores programs, I/O data, and intermediate data Supervises the flow of info. among all units Performs arithmetic and other dataprocessing operations 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 30

Texas Instrument’s First IC -- 1958 Jack Kilby Robert Noyce Fairchild Intel 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 31

Originally with Transistors 13/6/2021 Εισαγωγή στην Τεχνολογία - Διάλεξη 18 32