Appendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring

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1 - Principles of Computer rchitecture Miles Murdocca and Vincent Heuring 999 M. Murdocca and V. Heuring

2 -2 Chapter Contents. Introduction.2 Combinational Logic.3 Truth Tables.4 Logic Gates.5 Properties of oolean lgebra.6 The Sum-of-Products Form, and Logic Diagrams.7 The Product-of-Sums Form.8 Positive vs. Negative Logic.9 The Data Sheet. Digital Components. Sequential Logic.2 Design of Finite State Machines.3 Mealy vs. Moore Machines.4 Registers.5 Counters 999 M. Murdocca and V. Heuring

3 -3 Some Definitions Combinational logic: a digital logic circuit in which logical decisions are made based only on combinations of the inputs. e.g. an adder. Sequential logic: a circuit in which decisions are made based on combinations of the current inputs as well as the past history of inputs. e.g. a memory unit. Finite state machine: a circuit which has an internal state, and whose outputs are functions of both current inputs and its internal state. e.g. a vending machine controller. 999 M. Murdocca and V. Heuring

4 -4 The Combinational Logic Unit Translates a set of inputs into a set of outputs according to one or more mapping functions. Inputs and outputs for a CLU normally have two distinct (binary) values: high and low, and, and, or 5 V and V for example. The outputs of a CLU are strictly functions of the inputs, and the outputs are updated immediately after the inputs change. set of inputs i i n are presented to the CLU, which produces a set of outputs according to mapping functions f f m. i i i n... Combinational logic unit... f (i, i ) f (i, i 3, i 4 ) f m (i 9, i n ) 999 M. Murdocca and V. Heuring

5 -5 Truth Table Developed in 854 by George oole. Further developed by Claude Shannon (ell Labs). Outputs are computed for all possible input combinations (how many input combinations are there?) Consider a room with two light switches. How must they work? GND Inputs Output Hot Light Z Z Switch Switch 999 M. Murdocca and V. Heuring

6 -6 lternate ssignment of Outputs to Switch Settings We can make the assignment of output values to input combinations any way that we want to achieve the desired input-output behavior. Inputs Output Z 999 M. Murdocca and V. Heuring

7 -7 Truth Tables Showing ll Possible Functions of Two inary Variables Inputs Outputs The more frequently used functions have names: ND, XOR, OR, NOR, XOR, and NND. (lways use upper case spelling.) False ND XOR OR Inputs Outputs NOR XNOR + + NND True 999 M. Murdocca and V. Heuring

8 -8 Logic Gates and Their Symbols Logic symbols shown for ND, OR, buffer, and NOT oolean functions. F F Note the use of the inversion bubble. F = F = + (e careful about the nose of the gate when drawing ND vs. OR.) ND F OR F F = F = uffer NOT (Inverter) 999 M. Murdocca and V. Heuring

9 M. Murdocca and V. Heuring Logic Gates and their Symbols (cont ) F NND F NOR F = F = + F Exclusive-OR (XOR) F = F Exclusive-NOR (XNOR) F =.

10 - Variations of Logic Gate Symbols C F = C F = + (a) (b) + + (c) (a) 3 inputs (b) Negated input (c) Complementary outputs 999 M. Murdocca and V. Heuring

11 VO Output Voltage V - Transistor Operation of Inverter OUTPUT VOLTGE vs. INPUT VOLTGE 4. V CC V CC = 5 V R L = 4 Ω V CC = +5 V GND = V ase V CC Collector Emitter V in R L V out (a) (b) (c) V I Input Voltage V (d) (a) Inverter showing power terminals; (b) transistor symbol; (c) transistor configured as an inverter; (d) inverter transfer function. 999 M. Murdocca and V. Heuring

12 -2 ssignments of and to Voltages +5 V +5 V Logical Logical 2.4 V.4 V V Forbidden Range Logical 2. V.8 V V Forbidden Range Logical (a) (b) 999 M. Murdocca and V. Heuring

13 -3 Transistor Operation of Logic Gates V CC (a) NND; (b) NOR V out V CC V V out + V 2 V V 2 (a) (b) 999 M. Murdocca and V. Heuring

14 -4 Tri-State uffers Outputs can be,, or electrically disconnected. C F ø ø C F ø ø C F = C or F = ø C F = C or F = ø Tri-state buffer Tri-state buffer, inverted control 999 M. Murdocca and V. Heuring

15 -5 Properties of oolean lgebra Principle of duality: The dual of a oolean function is obtained by replacing ND with OR and OR with ND, s with s, and s with s. Theorems Postulates Relationship Dual Property = ( + C) = + C = = = = + = + + C = ( + ) ( + C) + = + = + = + = Commutative Distributive Identity Complement Zero and one theorems Idempotence ( C) = ( ) C + ( + C) = ( + ) + C ssociative = Involution = + + = DeMorgan s Theorem + C + C = + C ( + ) = ( + )( +C)( +C) = (+)( +C) + = Consensus Theorem bsorption Theorem 999 M. Murdocca and V. Heuring

16 -6 DeMorgan s Theorem = = + + DeMorgan s theorem: + = + = F = + F = 999 M. Murdocca and V. Heuring

17 -7 ll-nnd Implementation of OR NND alone implements all other oolean logic gates M. Murdocca and V. Heuring

18 -8 Sum-of-Products Form: The Majority Function The SOP form for the 3-input majority function is: M = C + C + C + C = m3 + m5 + m6 + m7 = Σ (3, 5, 6, 7). Each of the 2 n terms are called minterms, ranging from to 2 n -. Note relationship between minterm number and boolean value. Minterm Index C F -side -side balance tips to the left or right depending on whether there are more s or s. 999 M. Murdocca and V. Heuring

19 -9 ND-OR Implementation of Majority C Gate count is 8, gate input count is 9. C C F C C 999 M. Murdocca and V. Heuring

20 -2 Notation Used at Circuit Intersections Connection No connection Connection No connection 999 M. Murdocca and V. Heuring

21 -2 OR-ND Implementation of Majority C + + C + + C F + + C + + C 999 M. Murdocca and V. Heuring

22 -22 Positive/Negative Logic ssignments Positive logic: logic is represented by high voltage; logic is represented by low voltage. Negative logic: logic is represented by high voltage; logic is represented by low voltage. Gate Logic: Positive vs. Negative Logic Normal Convention: Postive Logic/ctive High Low Voltage = ; High Voltage = lternative Convention sometimes used: Negative Logic/ctive Low F Voltage Truth T able Positive Logic Negative Logic low low high high low high low high F low low low high F F ehavior in terms of Electrical Levels Two lternative Interpretations Positive Logic ND Negative Logic OR Dual Operations 999 M. Murdocca and V. Heuring

23 -23 Positive/Negative Logic ssignments (Cont ) Voltage Levels Positive Logic Levels Negative Logic Levels low low high high low high low high F low low low high F F Physical ND gate F F = F = + Voltage Levels Positive Logic Levels Negative Logic Levels low low high high low high low high F high high high low F F Physical NND gate F F = F = M. Murdocca and V. Heuring

24 -24 ubble Matching Positive logic Positive logic x Positive Negative logic x x Logic Negative logic x Negative Logic (a) (b) Negative logic Negative logic x x ubble mismatch (c) Negative Logic Negative logic Negative logic x x ubble match ubble match (d) Negative Logic 999 M. Murdocca and V. Heuring

25 -25 SN74 UDRUPLE 2-INPUT POSITIVE-NND GTES description Example Data Sheet These devices contain four independent 2-input NND gates. function table (each gate) INPUTS H L X H X L OUTPUT Y L H H package (top view) Y 2 2 2Y GND V CC 4 4 4Y 3 3 3Y 4 kω schematic (each gate).6 kω 3 Ω Y V CC Simplified data sheet for 74 NND gate, adapted from Texas Instruments TTL Databook [Texas Instruments, 988] absolute maximum ratings Supply voltage, VCC 7 V Input voltage: 5.5 V Operating free-air temperature range: C to 7 C Storage temperature range 65 C to 5 C logic diagram (positive logic) Y = Y 2Y 3Y 4Y recommended operating conditions V CC V IH V IL I OH I OL T Supply voltage High-level input voltage Low-level input voltage High-level output current Low-level output current Operating free-air temperature kω MIN NOM MX GND UNIT V 2 V.8 V.4 m 6 m 7 C electrical characteristics over recommended operating free-air temperature range V OH V OL I IH I IL I CCH I CCL V CC = MIN, V IL =.8 V, I OH =.4 m V CC = MIN, V IH = 2 V, I OL = 6 m V CC = MX, V I = 2.4 V V CC = MX, V I =.4 V V CC = MX, V I = V V CC = MX, V I = 4.5 V MIN TYP MX UNIT V.4 V 4 µ.6 m 8 m 22 m switching characteristics, V CC = 5 V, T = 25 C PRMETER FROM (input) TO (output) TEST CONDITIONS MIN TYP MX UNIT t PLH R 22 ns or Y L = 4 Ω t C L = 5 pf PHL 7 5 ns 999 M. Murdocca and V. Heuring

26 -26 Digital Components High level digital circuit designs are normally created using collections of logic gates referred to as components, rather than using individual logic gates. Levels of integration (numbers of gates) in an integrated circuit (IC) can roughly be considered as: Small scale integration (SSI): - gates. Medium scale integration (MSI): to gates. Large scale integration (LSI): -, logic gates. Very large scale integration (VLSI):,-upward logic gates. These levels are approximate, but the distinctions are useful in comparing the relative complexity of circuits. 999 M. Murdocca and V. Heuring

27 Data Inputs -27 Multiplexer D D D 2 D 3 F F D D D 2 D 3 Control Inputs F = D + D + D 2 + D M. Murdocca and V. Heuring

28 -28 ND-OR Implementation of MUX D D D 2 F D M. Murdocca and V. Heuring

29 -29 MUX Implementation of Majority Principle: Use the 3 MUX control inputs to select (one at a time) the 8 data inputs. C M F C 999 M. Murdocca and V. Heuring

30 -3 4-to- MUX Implements 3-Var Function Principle: Use the and inputs to select a pair of minterms. The value applied to the MUX data input is selected from {,, C, C} to achieve the desired behavior of the minterm pair. C F C C C C F 999 M. Murdocca and V. Heuring

31 -3 Demultiplexer D F = D F = D F F F 2 F 3 F 2 = D F 3 = D D F F F 2 F M. Murdocca and V. Heuring

32 -32 Gate-Level Implementation of DEMUX F D F F 2 F M. Murdocca and V. Heuring

33 -33 Decoder Enable = Enable = D D Enable D 2 D 3 D D D 2 D 3 D D D 2 D 3 D = D = D 2 = D3 = 999 M. Murdocca and V. Heuring

34 -34 Gate-Level Implementation of Decoder D D D 2 D 3 Enable 999 M. Murdocca and V. Heuring

35 -35 Decoder Implementation of Majority Function Note that the enable input is not always present. We use it when discussing decoders for memory. C M 999 M. Murdocca and V. Heuring

36 -36 Priority Encoder n encoder translates a set of inputs into a binary encoding. Can be thought of as the converse of a decoder. priority encoder imposes an order on the inputs. i has a higher priority than i+ 2 3 F F 2 3 F F F = F = M. Murdocca and V. Heuring

37 -37 ND-OR Implementation of Priority Encoder F 2 3 F 999 M. Murdocca and V. Heuring

38 -38 C Programmable Logic rray OR matrix PL is a customizable ND matrix followed by a customizable OR matrix. lack box view of PL: C PL F F Fuses ND matrix F F 999 M. Murdocca and V. Heuring

39 -39 C Simplified Representation of PL Implementation of Majority Function C C C C F (Majority) F (Unused) 999 M. Murdocca and V. Heuring

40 -4 Example: Ripple-Carry ddition Carry In Operand Operand Carry Out Sum Carry Operand Example: Operand Sum M. Murdocca and V. Heuring

41 -4 Full dder i i C i S i C i+ i i C i+ Full adder S i C i 999 M. Murdocca and V. Heuring

42 -42 Four-it Ripple-Carry dder Four full adders connected in a ripple-carry chain form a four-bit ripple-carry adder. b 3 a 3 b 2 a 2 b a b a c 3 c 2 c c Full adder Full adder Full adder Full adder c 4 s 3 s 2 s s 999 M. Murdocca and V. Heuring

43 -43 C in PL Realization of Full dder Sum C out 999 M. Murdocca and V. Heuring

44 -44 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 memory, which behave differently depending upon their previous state. n example is a vending machine, which must remember how many and what kinds of coins have been inserted. The machine should behave according to not only the current coin inserted, but also upon how many and what kinds of coins have been inserted previously. These are referred to as finite state machines, because they can have at most a finite number of states. 999 M. Murdocca and V. Heuring

45 -45 Classical Model of a Finite State n FSM is composed of a combinational logic unit and delay elements (called flip-flops) in a feedback path, which maintains state information. Inputs Machine i o i k Combinational logic unit D... n D n... s Synchronization n signal Delay elements (one per state bit) s... f o f m Outputs State bits 999 M. Murdocca and V. Heuring

46 -46 NOR Gate with Lumped Delay τ + + Timing ehavior The delay between input and output (which is lumped at the output for the purpose of analysis) is at the basis of the functioning of an important memory element, the flip-flop. τ 999 M. Murdocca and V. Heuring

47 -47 S-R Flip-Flop The S-R flip-flop is an active high (positive logic) device. t S t R t i+ S S R (disallowed) (disallowed) R τ 2 τ τ 2 τ Timing ehavior 999 M. Murdocca and V. Heuring

48 -48 NND Implementation of S-R Flip-Flop S S S R R R R S 999 M. Murdocca and V. Heuring

49 -49 Hazard C C S S R τ Glitch caused by a hazard R τ Timing ehavior It is desirable to be able to turn off the flip-flop so it does not respond to such hazards. 2 τ 999 M. Murdocca and V. Heuring

50 mplitude -5 Clock Waveform: The Clock Paces the System Time Cycle time = 25ns 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 settle at the correct value when the clock next goes high. 999 M. Murdocca and V. Heuring

51 -5 Scientific Prefixes For computer memory, K = 2 = 24. For everything else, like clock speeds, K =, and likewise for M, G, etc. Prefix bbrev. uantity milli m micro µ nano n pico p Prefix bbrev. uantity Kilo K Mega M Giga G Tera T femto f 5 Peta P 5 atto a 8 Exa E M. Murdocca and V. Heuring

52 -52 Clocked S-R Flip-Flop S S R CLK CLK R τ 2 τ Timing ehavior The clock signal, CLK, enables the S and R inputs to the flip-flop. 999 M. Murdocca and V. Heuring

53 -53 Clocked D Flip-Flop 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 (next slide) addresses this problem. D CLK Circuit D CLK Symbol D C τ 2 τ 2 τ Timing ehavior τ 999 M. Murdocca and V. Heuring

54 -54 Master-Slave Flip-Flop The rising edge of the clock loads new data into the master, while the slave continues to hold previous data. The falling edge of the clock loads the new master data into the slave. Master Slave D D D M D S CLK CLK C Circuit C S M S Symbol D S τ 3 τ 2 τ 2 τ 2 τ Timing ehavior τ 999 M. Murdocca and V. Heuring

55 -55 Clocked J-K Flip-Flop The J-K flip-flop eliminates the disallowed S=R= problem of the S-R flip-flop, because enables J while disables K, and vice-versa. However, there is still a problem. If J goes momentarily to and then back to while the flip-flop is active and in the reset state, the flip-flop will catch the. This is referred to as s catching. The J-K Master-Slave flip-flop (next slide) addresses this problem. J J CLK K K Circuit Symbol 999 M. Murdocca and V. Heuring

56 -56 Master-Slave J-K Flip-Flop J J CLK K K Circuit Symbol 999 M. Murdocca and V. Heuring

57 -57 Clocked T Flip-Flop The presence of a constant at J and K means that the flip-flop will change its state from to or to each time it is clocked by the T (Toggle) input. J T T K Circuit Symbol 999 M. Murdocca and V. Heuring

58 -58 Negative Edge-Triggered D Flip-Flop When the clock is high, the two input latches output, so the Main latch remains in its previous state, regardless of changes in D. Stores D R When the clock goes high-to-low, values in the two input latches will affect the state of the Main latch. CLK S Main latch While the clock is low, D cannot affect the Main latch. D Stores D 999 M. Murdocca and V. Heuring

59 -59 Example: Modulo-4 Counter Counter has a clock input (CLK) and a RESET input. Counter has two output lines, which take on values of,,, and on subsequent clock cycles. Time (t) RESET q Time (t) 3-bit q Synchronous s Counter D CLK s D s s 999 M. Murdocca and V. Heuring

60 -6 State Transition Diagram for RESET Output state / q q / / Output state Mod-4 Counter / / / / C / D Output state Output state 999 M. Murdocca and V. Heuring

61 -6 State Table for Mod-4 Counter Present state Input RESET / / C/ / C D/ / D / / Next state Output 999 M. Murdocca and V. Heuring

62 -62 State ssignment for Mod-4 Counter Present state (S t ) Input RESET : / / : / / C: / / D: / / 999 M. Murdocca and V. Heuring

63 -63 Truth Table for Mod-4 Counter RESET r(t) s (t) s (t) s s (t+) q q (t+) s (t+) = r(t)s (t)s (t) + r(t)s (t)s (t) s (t+) = r(t)s (t)s (t) + r(t)s (t)s (t) q (t+) = r(t)s (t)s (t) + r(t)s (t)s (t) q (t+) = r(t)s (t)s (t) + r(t)s (t)s (t) 999 M. Murdocca and V. Heuring

64 -64 Logic Design for Mod-4 Counter RESET CLK D s q D s q 999 M. Murdocca and V. Heuring

65 -65 Example: Sequence Detector Example: Design a machine that outputs a when exactly two of the last three inputs are. e.g. input sequence of produces an output sequence of. ssume input is a -bit serial line. Use D flip-flops and 8-to- Multiplexers. Start by constructing a state transition diagram (next slide). 999 M. Murdocca and V. Heuring

66 -66 Sequence Detector State Transition Diagram / Design a machine that outputs a when exactly two of the last three inputs are. / / / D E / / / / / C / F / / / G / 999 M. Murdocca and V. Heuring

67 -67 Sequence Detector State Table Present state Input X / C/ D/ E/ C F/ G/ D D/ E/ E F/ G/ F D/ E/ G F/ G/ 999 M. Murdocca and V. Heuring

68 -68 Sequence Detector State ssignment Input and state at time t Next state and output at time t+ Present state Input X S 2 S S S 2 S S Z S 2 S S Z : / / : / / C: / / D: / / E: / / F: / / G: / / (a) s 2 s s x s 2 d d s d d s d d z d d (b) 999 M. Murdocca and V. Heuring

69 -69 Sequence Detector Logic Diagram x x x D S 2 x x x x x x x D S x x x x x x x D S x x x Z CLK 999 M. Murdocca and V. Heuring

70 -7 Example: Vending Machine Controller Example: Design a finite state machine for a vending machine controller that accepts nickels (5 cents each), dimes ( cents each), and quarters (25 cents each). When the value of the money inserted equals or exceeds twenty cents, the machine vends the item and returns change if any, and waits for next transaction. Implement with PL and D flip-flops. 999 M. Murdocca and V. Heuring

71 -7 Vending Machine State Transition / dime is inserted Diagram / = Dispense/Do not dispense merchandise N/ D/ / = Return/Do not return a nickel in change / = Return/Do not return a dime in change N/ D / D/ 5 5 / / D/ D/ N/ C N/ N = Nickel D = Dime = uarter 999 M. Murdocca and V. Heuring

72 -72 Vending Machine State Table and State ssignment P.S. Input N D / C/ C/ D/ C D/ / D / / (a) / / / / Input N D x x x x x x P.S. s s s s / z 2 z z : / / / : / / / C: / / / D: / / / (b) 999 M. Murdocca and V. Heuring

73 -73 PL Vending Machine Controller x x 5 5 PL z 2 z z s s x x (a) s D s D CLK 2 4 ase equivalent Present state Coin s s x x (b) Dispense Next Return nickel state Return dime s s z 2 z z d d d d d d d d d d d d d d d d d d d d (c) s s z 2 z z 999 M. Murdocca and V. Heuring

74 -74 Moore Counter Mealy Model: Outputs are functions of Inputs and Present State. Previous FSM designs were Mealy Machines, in which next state was computed from present state and inputs. Moore Model: Outputs are functions of Present State only. z z x 4-to- MUX D S z 4-to- MUX D S z CLK 999 M. Murdocca and V. Heuring

75 -75 Four-it Register Makes use of tri-state buffers so that multiple registers can gang their outputs to common output lines. D 3 D 2 D D Write (WR) CLK D D D D Enable (EN) WR D 3 D 2 D D 3 2 EN M. Murdocca and V. Heuring

76 -76 Left-Right Shift c Register with c Shift left output Shift right input D 3 D 2 D D Shift right input Parallel Read and c Write c CLK Enable (EN) D D D D Shift right output 3 2 Control Function c c No change Shift left Shift right Parallel load Shift right input Shift left output c c D 3 D 2 D D 3 2 Shift right output Shift right input 999 M. Murdocca and V. Heuring

77 -77 Modulo-8 Counter Note the use of the T flip-flops, implemented as J-K s. They are used to toggle the input of the next flip-flop when its output is. CLK Enable (EN) J K J K J K RESET 2 CLK ENLE RESET MOD(8) COUNTER 2 2 Timing ehavior 999 M. Murdocca and V. Heuring

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