# Karnaugh Maps. Example A B C X each 1 here gives a minterm e.g.

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1 Karnaugh Maps Yet another way of deriving the simplest Boolean expressions from behaviour. Easier than using algebra (which can be hard if you don't know where you're going). Example A B C X each 1 here gives a minterm e.g. A'BC CSC9R6 Computer Design. Spring 2006 Slide 52

2 Karnaugh Maps A Karnaugh map sets out the minterms pictorially. Example for 3 variables Not like a truth table set out differently (each column/row differs in 1 variable only from its neighbours). A BC BC A Like a truth table each 1 represents the presence of that minterm in the CSOP form numbers correspond to rows in a truth table CSC9R6 Computer Design. Spring 2006 Slide 53

3 Mechanics and Semantics Different algebraic expressions are generated by recognising patterns and grouping adjacent cells. Loop adjacent cells in 2 x sized groups (i.e. 2,4,8,.) Try to form as few groups as possible (i.e. groups are as large as possible) Adjacency can wrap round the edges, so e.g. the four corners are all adjacent. Algebraically equivalent to eliminating terms of the form A + A'. (recall a common simplification is to use DISTR to take out a common factor, and OR to make B + B' = 1, e.g. AB + AB' = A ) CSC9R6 Computer Design. Spring 2006 Slide 54

4 Example 1 A BC 0 B positive (confirm this by Boolean algebra) A positive C positive before grouping after grouping CSC9R6 Computer Design. Spring 2006 Slide 55

5 Example 2 (Karnaugh map in 2 vars) A'B' + AB' + AB A B 0 1 B positive A positive CSC9R6 Computer Design. Spring 2006 Slide 56

6 Example 3 (Karnaugh map in 4 vars) A'B'C'D' + A'B'C'D + A'B'CD' + A'BC'D + ABC'D + AB'C'D' + AB'C'D + AB'CD' D positive CD AB B positive C positive A positive CSC9R6 Computer Design. Spring 2006 Slide 57

7 Sequential Logic Systems The defining characteristic of a sequential logic system (SLS) is that the outputs of the system depend not only on the present inputs to the system, but also on the past history of the inputs and outputs of the system i.e. SLSs have a form of memory. CSC9R6 Computer Design. Spring 2006 Slide 58

8 Example Consider a counting system which receives input pulses and provides an output representing the total number of pulses received so far. input counting system binary output representing number of pulses When a new pulse arrives, the output must have '1' added to it. This requires knowledge of the preceding output, i.e. the system is sequential - it depends on input and previous outputs. CSC9R6 Computer Design. Spring 2006 Slide 59

9 SLS The basic building block of the sequential logic system is the flip-flop. A typical scheme of an SLS is inputs combinational circuit clock flip-flops outputs The clock is included to control the flip flops - it ensures that the outputs change at certain instants of time, which may be required to synchronise the SLS with other parts of the whole system. CSC9R6 Computer Design. Spring 2006 Slide 60

10 Flip-Flops and Latches Latches and flip-flops are both bistable devices (two stable states). Latch - level triggered Flip flop - edge triggered I.e. to do with how state changes are triggered. But the terminology vague and mixed! There are a number of different kinds of flip flops and latches CSC9R6 Computer Design. Spring 2006 Slide 61

11 Set/Reset (S - R) latch S R Q Q' Broadly, S = 1 causes Q = 1, R = 1 causes Q = 0. CSC9R6 Computer Design. Spring 2006 Slide 62

12 Latch Operation The latch operation is described in terms of its characteristic table (like a truth table but with state) S R Q 0 0 Q no change clear/reset set to 1 1 1? undefined For active high latch CSC9R6 Computer Design. Spring 2006 Slide 63

13 Latch application SR latch can also be used to combat switch debouncing. When a mechanical switch is opened or closed the contacts may bounce and a dirty transition results (instead of a nice clean one). results in clean transition - small bounces on S make no difference to Q after initial 1. CSC9R6 Computer Design. Spring 2006 Slide 64

14 S-R Gated Latch Controlled by an enable. En Steering gates latch If LOW enable signal then Q does not change, regardless of values of S and R. Only when enable changes to HIGH can the values of S and R be used to affect the output Q. CSC9R6 Computer Design. Spring 2006 Slide 65

15 S-R Gated Latch The latch operation is described in terms of its truth table When En = 1 S R Q 0 0 Q no change clear/reset set to 1 1 1? undefined CSC9R6 Computer Design. Spring 2006 Slide 66

16 S-R Gated Latch The operation can also be seen by looking at the timing (waveform) diagram. S R En Q Q' CSC9R6 Computer Design. Spring 2006 Slide 67

17 D type latch Removes problem of undefined output. One input D and an enable, changes in D are reflected at the output when enable = 1. (i.e. latch is transparent when enable is high) D En Q Q' En D (t) Q (t+1) Q' (t+1) X no change In the characteristic table (t) is time now, and (t+1) is the next time step. D stands for Delay (output is delayed while enable = 0) CSC9R6 Computer Design. Spring 2006 Slide 68

18 D type latch D = S and D' = R therefore never have S = R CSC9R6 Computer Design. Spring 2006 Slide 69

19 Example: a register LSB D 0 D 1 D 7 MSB D Q D Q... D Q En En En write Q 0 Q 1 Q 7 Data D 0 D 0. D 7 is loaded in parallel into the D latches on a write signal. The register may also include a special reset to clear all simultaneously. CSC9R6 Computer Design. Spring 2006 Slide 70

20 Power Up A system may contain a number of flip flops, counters, shift registers, etc. What is their initial state immediately after power is applied? We need a known (and reproducible) state at power up, e.g. flip-flops reset, counters zeroed, shift registers clears and so on. Use a simple switch (interactive input) to reset all elements simultaneously CSC9R6 Computer Design. Spring 2006 Slide 71

21 Edge Detection Problem: when enable = 1 the output varies depending on the input. We might prefer that on the transition 0-1 the input is sampled once and the output fixed, so changes of input are not reflected until the next 0-1 transition. Difference between a gated latch and a flip-flop is the use of edge triggering. Flip-flop changes on 0-1 transition (or 1-0 transition) Latch changes on HIGH level (or LOW level) CSC9R6 Computer Design. Spring 2006 Slide 72

22 Edge triggered flip flops D flip flop JK flip flop Rising edge triggered ie 0-1 transition. Also clear and preset inputs. The clock is dynamic input Falling edge triggered (1-0 transition). Also clear and preset inputs. CSC9R6 Computer Design. Spring 2006 Slide 73

23 How does edge triggering work? On the SR flip flop, a pulse transition detector (PTD) is added to the clock line. PTD The PTD produces a short spike (rather than a square pulse) from the clock. This means that only the data on S and R during the spike will be sampled. CSC9R6 Computer Design. Spring 2006 Slide 74

24 PTD The Pulse Transition Detector is implemented by the following gates. clock delay Gates suffer from propagation delay; it takes time to change the output to reflect new inputs. Therefore, both C and C' will be high for a few nanoseconds. Sometimes propagation delay can be useful (as above) but sometimes it just leads to (momentary) wrong answers. In clocked synchronous circuits all components are guaranteed to change at the same instant. CSC9R6 Computer Design. Spring 2006 Slide 75 spike

25 Presentation of Data Time constraints on presentation of data (for it to be reliably clocked into the flip flop). The set-up time is the minimum time between the leading edge of an input data pulse and the triggering clock pulse. The hold time is the time between the clock transition changing the output and the end of the input pulse. After the hold time the inputs may change with no effect on the output. The smaller these times the better. (e.g. setup times vary from 2 to 20 ns, hold times from 0.5 to 3 ns) clock input setup time hold time CSC9R6 Computer Design. Spring 2006 Slide 76

26 S-R Flip Flop Timing Diagram Note that the output Q only changes at the 0-1 clock transition, not when S or R first changes. S R C Q Q' CSC9R6 Computer Design. Spring 2006 Slide 77

27 Propagation Delays Consider the following circuit (R = A.B' + A'.C) Let A = 1, B = 0, C = 1. What happens when A changes to 0? Logically the value of R should not change, but physically it does. CSC9R6 Computer Design. Spring 2006 Slide 78

28 Static Hazards Let propagation delay of all gate be δ, and A changed at time t 0 time A B C B' A' A.B' A'.C R < t t * 1* 0* 1* t 0 + δ * 1* t 0 + 2δ * t 0 + 3δ * means old values used - change has not yet propagated These are called static hazards (when we have A and A') A hazard is likely if 2 adjacent cells in a Karnaugh map are not looped together (may give larger terms, but eliminates hazard) CSC9R6 Computer Design. Spring 2006 Slide 79

29 J - K Flip Flops Another attempt to solve the problem of undefined output on the SR latch (when S = R = 1) J K Q (t+1) 0 0 Q (t) no change clear set 1 1 Q (t) ' toggle like SR, but Q fed back to K gate, Q' back to J gate If J = K = 0 enabling does nothing If J K enabling causes Q (t+1) = J If J = K = 1 enabling causes toggle CSC9R6 Computer Design. Spring 2006 Slide 80

30 J-K Flip Flops (inside) CSC9R6 Computer Design. Spring 2006 Slide 81

31 Example: Divide by 2 J = K = 1 therefore the circuit toggles on the falling clock input. The frequency of the output waveform is 1/2 that of the input (clock) waveform. Q C T Q Q repeat C time T c T Q = 2 x T c CSC9R6 Computer Design. Spring 2006 Slide 82

32 Binary Counters (Ripple counters) Output Clock Q0 Q1 CSC9R6 Computer Design. Spring 2006 Slide 83

33 Binary Counters (Ripple counters) Represent the output of the circuit as a table: Clock pulse number Q1 Q counts 0, 1, 2, 3 repeats The system counts 4 clock pulses and then repeats. Counters for any number (2 n ) can be created by adding more J-K flip flops. The Q output is connected to the clock input for the next stage. CSC9R6 Computer Design. Spring 2006 Slide 84

34 N-bit Binary counter Q0 Q1 Q2 Qn LSB MSB The output sequence repeats every 2 n clock pulses The effect of the clock ripples through the flip-flops Propagation delay means each ff changes slightly later. -ve edge triggered flip flops are used +ve edge triggered flip flops cause the count to proceed in reverse (ie 3, 2, 1, 0..) CSC9R6 Computer Design. Spring 2006 Slide 85

35 Divide by N Counter The binary counter above only counts up to 2 m where M is the number of flip flops. How can we make a counter for N, where N is not a power of 2? The Divide by N counter detects when the upper limit has been reached and resets the counter to zero. A simple combinational logic system is used. clock M bit counter reset Q o Q 1 Q M CLS CSC9R6 Computer Design. Spring 2006 Slide 86

36 Example: A decade counter Decade counter counts from 0 to 9 and repeats. Count sequence is pulse number Q 3 Q 2 Q 1 Q reset How many flip flops? 2 3 = 8, 2 4 = 16, therefore 4 flip flops required. Reset occurs when Q 0 = 0, Q 1 = 1, Q 2 = 0, Q 3 = 1 Assume clr is active high (ie 1 on clr means reset, 0 means do nothing) and -ve edge triggered (1-0 transition) Clr = (Q 0 '. Q 1. Q 2 '. Q 3 ) CSC9R6 Computer Design. Spring 2006 Slide 87

37 Decade Counter Implementation Problems: Redundancy - can count to 15! Not practical - state 10 exists - there s a momentary glitch Ripple counters slow(er) CSC9R6 Computer Design. Spring 2006 Slide 88

38 Decade Counter Implementation LSB MSB Problems: Redundancy - can count to 15! Not practical - state 10 exists - there s a momentary glitch Ripple counters slow(er) CSC9R6 Computer Design. Spring 2006 Slide 89

39 Synchronous Counter Consider clocking all flip flops at the same time. 1 A B C D How do we make the flip-flops toggle at the right time? clock Flip flop A toggles on every clock pulse Flip flop B toggles only when A is HIGH and there s a clock pulse Flip flop C toggles only when A and B are HIGH and there s a clock pulse Flip flop D toggles only when A and B and C are HIGH and there s a clock pulse CSC9R6 Computer Design. Spring 2006 Slide 90

40 Synchronous Counter: Implementation The JK inputs come from previous Q inputs anded together. LSB MSB How can this be altered to make a decade counter? CSC9R6 Computer Design. Spring 2006 Slide 91

41 Synchronous Decade Counter More complicated arrangement to trigger each bit LSB MSB CSC9R6 Computer Design. Spring 2006 Slide 92

42 Synchronous Decade Counter Recall the table pulse number Q 3 Q 2 Q 1 Q reset Q 1 triggers only on 1, 3, 5, 7 (but not 9) i.e. when Q1 is 1 but Q3 is 0 Q 2 triggers only on 3 and 7 i.e. when both Q1 and Q0 are 1 Q 3 triggers only on 7 and 9 CSC9R6 Computer Design. Spring 2006 Slide 93

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