Combinational and Sequential Circuits in Computer Science

1. CS 3501 - Chapter 3 (3A and 10.2.2) Part 6 of 8 Dr. Clincy Professor of CS • Exam 1 Review Today • Exam 2 Part 1 on Thursday (closed book) Dr. Clincy Lecture Slide 1

2. Half Adder - Combinational Circuits • Combinational logic circuits give us many useful devices. • One of the simplest is the half adder, which finds the sum of two bits. • We can gain some insight as to the construction of a half adder by looking at its truth table, shown at the right. Lecture

3. Full Adder - Combinational Circuits • We can change our half adder into to a full adder by including gates for processing the carry bit. • The truth table for a full adder is shown at the right. Lecture

4. Adders - Combinational Circuits • Just as we combined half adders to make a full adder, full adders can be connected in series. • The carry bit “ripples” from one adder to the next; hence, this configuration is called a ripple-carryadder. Today’s systems employ more efficient adders. Lecture

5. Decoder - Combinational Circuits • Among other things, they are useful in selecting a memory location according to a binary value placed on the address lines of a memory bus. • This is what a 2-to-4 decoder looks like on the inside. If x = 0 and y = 1, which output line is enabled? Output - Decoded message Input - Encoded message Lecture

6. Decoder – another example Lecture

7. Multiplexer - Combinational Circuits • A multiplexer does just the opposite of a decoder. • It selects a single output from several inputs. • This is what a 4-to-1 multiplexer looks like on the inside. Depending the “select input” combination, 1 of 4 data inputs is chosen for output If S0 = 1 and S1 = 0, which input is transferred to the output? Lecture

8. Multiplexer - Combinational Circuits Can also use multiplexers to implement logic functions Given this truth table, group X1,X2 being 00, 01, 10 and 11 – notice what happens with X3 • 3-input truth table can be done with a 4-input mux • 4-input truth table can be done with a 8-input mux • 5-input truth table can be done with a 16-input mux • Etc.. Also explain how the Mux is used to implement data comm’s FDM and TDM Lecture

9. 10.2.2 - Programmable Logic Devices (PLD) All possible combinations of inputs ANDed       ••• All possible combinations of ANDed inputs ORed       Re-explain Sums of Products and relationship to PLDs Lecture

10. 10.2.2 - Programmable Logic Array (PLA) Ability to program a PLD, is called a PLA Lecture

11. 10.2.2 - Programmable Array Logic (PAL) For a PLA, both the AND array and OR array are programmable For a PAL, the AND array is programmable and the OR array is fixed Lecture

12. 10.2.2 - Complex Programmable Logic Devices (CPLDs) CPLDs are comprised of 2 or more PALs Lecture

13. 10.2.2 - Field Programmable Gate Arrays (FPGAs) PAL chips are somewhat limited in size due to the fact they have output pins for each sum-of-product circuit FPGA overcome this size limitation by using a general interconnection. General interconnection PAL Lecture

14. CS 3501 - Chapter 3 (3A and 10.2.2) Part 7 of 8 Dr. Clincy Professor of CS Dr. Clincy Lecture Slide 14

15. Previous State or Output Previous State or Output Circuit Flip Flops Current State or Output New Input Circuit New Input Current State or Output Sequential Circuits Vs Combinational Circuits Sequential Logic Current State or output of the device is affected by the previous states Combinatorial or Combinational Logic Current State or output of the device is only affected by the current inputs Lecture

16. NOTEYour book doesn’t do a good job in showing you how to derive or design sequential circuits (using state and state assignment tables) – the lecture will do so – please pay close attention to the lecture in understanding how to derive sequential circuits. Dr. Clincy Lecture Slide 16

17. Clock - Sequential Circuits • State changes are controlled by clocks (clock ticks). • Circuits can change state on the rising edge, falling edge, or when the clock pulse reaches its highest voltage – edge triggered. • Level-triggered circuits change state when the clock voltage reaches its highest or lowest level. Lecture

18. Previous State or Output Previous State or Output Circuit Flip Flops Current State or Output New Input Flip Flops - Sequential Circuits Notice how the output feeds the input Think of: Given R=0 and Qa=0, what can this be ? • S and R stand for set and reset respectively • constructed from a pair of cross-coupled NOR gates • the stored bit is present on the output marked Qa • If S and R inputs are both low, maintains the Qa and Qb in constant state, • If S (Set) is pulsed high while R is held low, then the Qa output is forced high,and stays high even after S returns low; • if R (Reset) is pulsed high while S is held low, then the Qa output is forced low, and stays low even after R returns low. Lecture

19. Gated SR Latch or Flip Flop • The time at which the latch is SET or RESET is controlled by a CLOCK input • Called Gated SR Latch Lecture

20. Gated D Latch • Inputs S and R are derived from a single input D • Clock pulse controls when the output is triggered • Samples the D input at the time the clock is HIGH and stores that info until the next clock pulse During the time the clock is high, the input changed,causing the output to change – this is the problem Lecture

21. Potential Problem • Thus far, the assumption has been the inputs S and R (or D) not changing while CLK is HIGH • What would happen if S, R and/or D changed ? The output would change immediately • This could be a problem • To fix this (next ppt) During the time the clock is high, the input changed,causing the output to change – this is the problem Lecture

22. Two Flip Flop Use To Fix Clock Issue FF1 FF2 Q Q m s D D Q D Q Q Clock Clk Q Clk Q Q Use 2 D flip flops – the FF2 clock is set to zero – therefore, if there was a change in FF1 input, D, it wouldn’t effect the FF2 Q value – FF2 holds the value (a) Circuit Clock D Q m Q = Q s (b) Timing diagram Clock’s negative edge causes change • If D changes while FF1 CLK is HIGH, Qm changes immediately - Qs stays the same because FF2 CLK=0 • Once the CLK goes LOW, FF2 reacts because its CLK=1 – so it thens reflects D Q D The arrow only symbolizes “positive edge” clock - the arrow with the NOT symbolizes “negative edge” clock Q (c) Graphical symbol Lecture

23. T Flip Flop T Flip Flops are good for counters – changes its state every clock cycle, if the input, T, is 1 • Positive-edge triggered flip flop • Since the previous state of Q was 0, it complements it to 1 Lecture

24. JK Flip Flop Combines the behavior of the SR and T flip flops • First three entries are the same behavior as the SR Latch (when CLK=1) • Usually the state S=R=1 undefined – for the JK Flip Flop, for J=K=1, next state is the complement of the present state Can store data like a D Flip Flop or can tie J & K inputs together and use to build counters (like a T flip flop) Lecture

25. Registers and Shift Registers A Flip Flop can store ONE bit – in being able to handle a WORD, you will need a number of flip flops (32, 64, etc) arranged in a common structure called a REGISTER. • All flip flops are synchronized by a common clock • Data written into (loaded) flip flops at the same time • Data is read from all flip flops at the same time F F F F 1 2 3 4 In Out D Q D Q D Q D Q Clock Q Q Q Q A simple shift register. • Want the ability to rotate and shift the data • Clock pulse will cause the contents of F1, F2, F3 and F4 to shift right (serially) • To do a rotation, simply connect OUT to IN Lecture

26. Registers and Shift Registers • Can load either serially or in parallel • When clock pulse occurs, • Serial shift takes place if Shift’/Load=0 or • if Shift’/Load=1, parallel load is performed Lecture

27. Counters • 3-stage or 3-bit counter constructed using T Flip Flops • With T Flip Flips, when input T=1, the flip flop toggles – changes state for each successive clock pulse • Initially all set to 0 • When clock pulse, Q0=1, therefore Q’=0 disabling Q1 and Q1 disables Q2 (have 1,0,0) • For the 2nd clock pulse, Q0=0, therefore Q’=1, causing Q1=1 and therefore Q’=0 disabling Q2 (have 0,1,0) • For the 3rd clock pulse, Q0=1, therefore Q’=0 disabling Q2 and therefore disabling Q3 (have 1,1,0) • Etc…. LSB 000 001 010 011 100 101 110 111 Hmmm Lecture Called a Ripple Counter

28. CS 3501 - Chapter 3 (3A and 10.2.2) Part 8 of 8 Dr. Clincy Professor of CS Dr. Clincy Lecture Slide 28

29. NOTEYour book doesn’t do a good job in showing you how to derive or design sequential circuits (using state and state assignment tables) – the lecture will do so – please pay close attention to the lecture in understanding how to derive sequential circuits. You can print out the slides in this lecture only for the next exam. You should NOT add any notes to the printed slides. You will receive a penalty if personal notes are written on the slides Dr. Clincy Lecture Slide 29

30. Sequential Logic Current State or output of the device is affected by the previous states Previous State or Output Previous State or Output Circuit Flip Flops Current State or Output New Input Combinatorial or Combinational Logic Current State or output of the device is only affected by the current inputs Circuit New Input Current State or Output Recall Examples: Decoders Multiplexers Examples: Shift Registers Counters Lecture

31. x = 0 ¤ z = 0 S0 S1 x = 1 ¤ z = 0 x = 1 ¤ z = 0 x = 0 ¤ z = 0 x = 0 ¤ z = 0 x = 1 ¤ z = 1 x = 1 ¤ z = 0 S3 S2 x = 0 ¤ z = 1 State diagram of a mod-4 up/down counter that detects the count of 2. Sequential Circuit – State Diagram If x=0, count up, If x=1, count down Interested when 2 is realized – z=1 when reach 2, else z=0 If at 0 and x=0, count up to 1 (and z=0) If at 0 and x=1, count down to 3 (and z=0) State diagram describes the functional behavior without any reference to implementation Lecture

32. x = 0 ¤ z = 0 S0 S1 x = 1 ¤ z = 0 x = 1 ¤ z = 0 x = 0 ¤ z = 0 x = 0 ¤ z = 0 x = 1 ¤ z = 1 x = 1 ¤ z = 0 S3 S2 x = 0 ¤ z = 1 State diagram of a mod-4 up/down counter that detects the count of 2. Sequential Circuit – State Table Can represent the info in the state diagram in a state table Lecture

33. Sequential Circuit – Equation Inputs – y2,y1,x Outputs –Y2, Y1 Lecture

34. Sequential Circuit – Circuit Design D Flip Flops used to store values of the two state variables between clock pulses Output from Flip Flops is the present-state of the variables Input, D, of the Flip Flops is the next-state of the variables Lecture

35. Finite State Machine Model The example we just implemented is an example of a “Finite State Machine” - is a model or abstraction of behavior composed of a finite number of states, transitions between those states, and actions Lecture

36. Chapter 3 Review If any time remains, we can review any topic or concept for Chapter 3 that was covered in the lectures. Please keep your questions geared towards the concepts covered in Ch 3 and the lectures (in realizing some value-added for the up-and-coming exam) Exam 2 will consists of two parts: Part 1 on Monday and Part 3 on Wednesday. Part 1 will be closed book (can use calculator, can not use phone or laptop). Part 2 will be open book (can use calculator and book, can’t use phone or laptop). If you miss either part of the exam, it can not be made up. Dr. Clincy Lecture Slide 36

37. CS3501 Exam 1 Results Average Score = 41 (Average Grade = 75) Score SD = 24 (very large) Grading Scaled Used: 100-79 A-grade (3 students) 78-54 B-grade (6 students) 53-29 C-grade (7 students) 28-4 D-grade (10 students) 3-0 F-grade (0 students) In getting your grade logged, be sure and pass back the exam after we go over them Dr. Clincy 37