1. 1 Lecture 22 Sequential Circuits Analysis

2. 2 Combinational vs. Sequential  Combinational Logic Circuit  Output is a function only of the present inputs.  Does not have state information.  Does not require memory.  Sequential Logic Circuit (Finite State Machine)  Output is a function of the present state and at times present state and input.  Has state information  Requires memory.  Uses Flip-Flops to implement memory.

3. 3 Synchronous vs. Asynchronous Synchronous Sequential Logic Circuit  Clocked  All Flip-Flops use the same clock and change state on the same triggering edge. Asynchronous Sequential Logic Circuit  No clock  Can change state at any instance in time.  Faster but more complex than synchronous sequential circuits.

4. 4 General Models for Sequential Circuits A sequential circuit can be divided conveniently into two parts -- the flip-flops which serve as memory for the circuit and the combinational logic which realizes the input functions for the flip-flops and the output functions. The combinational logic may be implemented with gates, with a ROM, or with a PLA.

5. 5 Sequential Logic (Why) ? °Sequential circuit has additional dimension which is time °Combinational logic only depends on current input °Sequential circuit output depends on previous input other than current input °More powerful than combinational logic °Able to model condition that can’t be accommodated by combinational logic

6. 6 Analysis of Clocked Sequential Circuits °Analysis of a sequential circuit consists of obtaining a table or a diagram for the time sequence of inputs, outputs, and internal states. °Sequential circuit behavior is determined from the inputs, the outputs, and the state of its flip-flops °Boolean expressions that describe the behavior of the sequential circuit °Outputs and the next state are both a function of the inputs and the present state °A logic diagram is recognized as a clocked sequential circuit if it includes flip-flops. °Logic diagram may or may not include combinational circuit gates.

7. 7 The Current “State” °It is inconvenient, and often impossible, to describe the behaviour of a sequential circuit by means of a table that lists outputs as a function of the input sequence that has been received up until the current time. °To know where you are going next, you need to know where you are now. °With the TV channel selector, it is impossible to determine what channel is currently selected by looking only at the preceding sequence of presses, whether we look at the preceding 10 presses or the preceding 1000. °More information, the current “state” of the channel selector, is needed.

8. 8 State °The state of a sequential circuit is a collection of state variables whose values at any particular time contain all the information about the past necessary to account for the circuit’s future behaviour. °In the channel-selector example, the current channel number is the current state. °Inside the TV, this state might be stored as seven binary state variables representing a decimal number between 1 and 9. °Given the current state (channel number), we can always predict the next state as a function of the inputs (up/down pushes).

9. 9 “Finite-State Machines” °In a digital circuit, state variables have binary values. °A circuit with n binary state variables has 2 n possible states. °2 n is always finite, so sequential circuits are sometimes called finite-state machines.

10. 10 D Flip-Flop with Clock input Q(t+1) = Q.D + Q.D = D.(Q +Q) = D.1 = D

11. 11 Boolean equation for D Flip-Flop

12. 12 Sequential Circuit Analysis °Given sequential circuit diagram, behavioral analysis from state table and also state diagram °Need state equations to get flip-flop input and output functions for circuit output other than flip- flop (if any) °A(t) and A(t+1) are used to represent current state and the next state for flip-flop. °A and A + can also be used in order to represent current state and the following state

13. 13 Sequential Circuit Analysis °Example (using D flip-flop) State equation Output Function

14. 14 Sequential Circuit Analysis °From the state equations and output function, state table can be derived that contains all combined binary combination for the current condition (present state) and input °State table The same as Truth Table Input and condition pad on the left Output and next condition on the right Combined binary combination available for current state and input °M flip-flop and n input => 2 m+n line

15. 15 Sequential Circuit Analysis State table for circuit in Example 1 °From the state equations and output function, state table can be derived that contains all combined binary combination for the current condition (present state) and input °State table The same as Truth Table Input and condition pad on the left Output and next condition on the right combined binary combination available for current state and input °M flip-flop and n input => 2 m+n line State equationOutput function

16. 16 Sequential Circuit Analysis Other method

17. 17 Sequential Circuit Analysis °From the truth table, we can draw state diagram °State diagram Each state is represented by circle Each arrow (between two circle) represent transfer for sequential logic (i.e. line transition in truth table) a/b label for each arrow where a represent inputs and b represent output for circuit in transition °Each flip-flop value combination represent state. Therefore, m flip-flop=> until 2 m state.

18. 18 Sequential Circuit Analysis State diagram for circuit in previous example °Each state is represented by circle °Each arrow (between two circle) represent transfer for sequential logic (i.e. line transition in truth table) °a/b label for each arrow where a represent inputs and b represent output for circuit in transition

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20. 20 Flip-flop Input Function °Output of sequential circuit is a function of the current state of the flip-flop and the input. This is explained using algebra by circuit output function In previous example : y= (A+B)x’ °Circuit part that generate input to flip-flop is represented by using Boolean equation and is known as flip-flop input’s function °Flip-flop input function determine next state °From flip-flop input function and criteria table for flip- flop, next state of the flip-flop is obtained

21. 21 Flip-flop Input Function °Example: circuit with JK flip flop °2 characters are used in order to represent flip-flop input: first character represents the flip-flop input (J or K for JK flip-flop, S or R for SR flip-flop, D for D flip-flop, T for T flip-flop respectively) and the second character represents the name of the flip- flop

22. 22 Analysis: Example °Given a sequential circuit with two JK flip-flop, namely A, B and one input x °Flip-flop input function obtained from the circuit is

23. 23 Analysis: Example °Input flip-flop function °Fill the state table with the above function using criteria table for the used flip-flop

24. 24 Analysis: Example °Draw state diagram from the state table

25. 25 Flip-flop Excitation Tables Analysis Vs Design °Analysis: Start from circuit diagram, build state table or state diagram °Design: Start from specification set (i.e. in state equation form, state table or state diagram) build logic circuit. °Criteria table is used in analysis °Excitation tables is used in design

26. 26 Flip-flop Excitation Tables °Excitation tables : it give transition characteristic between current state and next state to determine flip- flop input

27. 27 Designing Sequential Circuit Design steps °Start with circuit specification – characteristic of circuit °Build state table °Perform state reduction if required °State assignment °Determine number of flip-flop ( that has to be used) °Build circuit excitation and output table from state table °Build circuit output function and flip-flop input function °Draw logic diagram

28. 28 Design: Example °Given state diagram as follows, get the sequential circuit using JK flip-flop

29. 29 Design: Example °State/excitation table using JK flip-flop For example, in the first row of Table (bottom right), we have a transition for flip- flop A from 0 in the present state to 0 in the next state. In Table (excitation table), we find that a transition of states from 0 to 0 requires that input J= 0 and input K = X

30. 30 Design: Example °Block diagram

31. 31 Design: Example °From state table, get input flip-flop function

32. 32 Design: Example °Input flip-flop function °Logic Diagram

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34. 34 Design: Example °Design, using D flip-flop, circuit is based on state table below.

35. 35 Design: Example °Determine input expression for flip-flop and y output

36. 36 Design Example °From Boolean expressions built, draw logic diagram

37. 37 Design: Example °How if using JK flip-flop (Homework)

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39. 39 Design a Synchronous Counter °Counter: sequential circuit cycle through state sequence °Binary counter: follow binary sequence. n-bit binary counter (with n flip-flop) able to count from 0 to 2 n -1. °Example: 3-bit binary counter (using T flip-flop)

40. 40 Design a Synchronous Counter °3-bit binary counter (cont)

41. 41 Design a Synchronous Counter °3-bit binary counter

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43. 43 Sequential Circuit Design °Sequential circuit consists of A combinational circuit that produces output A feedback circuit -We use JK flip-flops for the feedback circuit °Simple counter examples using JK flip-flops Provides alternative counter designs We know the output -Need to know the input combination that produces this output -Use an excitation table –Built from the truth table

44. 44 Sequential Circuit Design (cont’d)

45. 45 Sequential Circuit Design (cont’d) °Build a design table that consists of Current state output Next state output JK inputs for each flip-flop °Binary counter example 3-bit binary counter 3 JK flip-flops are needed Current state and next state outputs are 3 bits each 3 pairs of JK inputs

46. 46 Sequential Circuit Design (cont’d) Design table for the binary counter example

47. 47 Sequential Circuit Design (cont’d) Use K-maps to simplify expressions for JK inputs

48. 48 Sequential Circuit Design (cont’d) °Final circuit for the binary counter example Compare this design with the synchronous counter design

49. 49 Thanks