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EE345: Introduction to Microcontrollers Synchronous Sequentional Logic

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EE345: Introduction to MicrocontrollersSynchronous Sequentional Logic

Prof. Ahmad Abu-El-Haija

- This presentation is a modified version of lecture notes prepared by Dr. Nader Mohamed, Stevens Institute of Technology, and original slides from the publisher.

Digital System Design

- Sequential Circuits
- Storage Elements: Latches
- Storage Elements: Flip-Flops
- Analysis of Clocked Sequential Circuits
- State Reduction and Assignment
- Design Procedure

EE345 - Introduction to Microcontrollers

- The inputs and present state of the storage elements determine the value of the outputs.
- The next state of the storage elements is also a function of external inputs and the present state.

EE345 - Introduction to Microcontrollers

Their classification depends on the timing of their signals.

- Synchronous sequential circuit: its behavior can be defined from the knowledge of its signals at discrete instants of time.
- Asynchronous sequential circuit: its behavior depends upon the input signals at any instant of time and the order in which the inputs change. Storage elements are used as time-delay devices.

EE345 - Introduction to Microcontrollers

- A synchronous sequential circuit employs signals that affect the storage elements only at discrete instants of time.
- Synchronization is achieved by a clock generator that provides periodic clock pulses.
- Clock pulses are distributed throughout the digital system in such a way the storage elements are affected only with the arrival of each pulse.
- Storage elements are called flip-flops.

EE345 - Introduction to Microcontrollers

Synchronous Clocked Sequential Circuit

EE345 - Introduction to Microcontrollers

- A flip-flop circuit can maintain a binary state.
- There are different types of flip-flops.
- Basic type of flip-flops are Latches.
- Latches are basic circuits from which all types of flip-flops are constructed.
- Latches are used more in asynchronous sequential circuits.

EE345 - Introduction to Microcontrollers

- The SR Latch is a circuit with two cross-coupled NOR gates or two cross-coupled NAND gates, as will be explained next.

EE345 - Introduction to Microcontrollers

- SR Latch has two useful states:
- Set state: when output Q=1 and Q’=0, and
- Reset state: when output Q=0 and Q’=1.
- These states can be used to store 1-bit information. Output Q and Q’ are normally complement of each other.
- Undefined state: when Q=0 and Q’=0, occurs when both inputs R and S are equal to 1 at the same time.

EE345 - Introduction to Microcontrollers

- Under normal conditions, both inputs of the latch (R and S) remain at 0 unless the state has to be changed.
- To let latch in the set state, S must be 1
- To let latch in the reset state, R must be 1
- The inputs S and R must go back to 0 before any other changes, to avoid the occurrence of the undefined state.
- The latch goes to the set state or reset state and stays there, even after both inputs return to 0.

EE345 - Introduction to Microcontrollers

- When both inputs Sand Rare equal to 0, the latch can be in either the set or the reset state, depending on which input was most recently a 1.

EE345 - Introduction to Microcontrollers

As before, the SR Latch has two useful states:

- Set state: when output Q=1 and Q’=0.
- Reset state: when output Q=0 and Q’=1.
Output Q and Q’ are normally complement of each other.

- Undefined state: when Q=1 and Q’=1, occurs when both inputs R and S are equal to 0 at the same time.

EE345 - Introduction to Microcontrollers

- Under normal conditions, both inputs of the latch (R and S) remain at 1 unless the state has to be changed.
- To let latch in the set state, S must be 0
- To let latch in the reset state, R must be 0
- The inputs S and R must go back to 1 before any other changes to avoid the occurrence of the undefined state (when Q=1 and Q’=1)
- The latch goes to the set state or reset state and stays there, even after both inputs return to 1.

EE345 - Introduction to Microcontrollers

- When both inputs S and R are equal to 1, the latch can be in either the set or the reset state, depending on which input was most recently a 0.
- Because the NAND latch requires a 0 signal to change its state, it is sometimes referred to as an S’R’ latch.

EE345 - Introduction to Microcontrollers

- An indeterminate condition occurs when all three inputs are equal to 1. It is difficult to ensure that both S and R are not equal to 1 at the same time.
- Operation of the basic SR latch can be modified by providing an additional control input that determines when the state of the latch can be changed.

EE345 - Introduction to Microcontrollers

- D latch has only two inputs D (Data) and En (Enable). This circuit ensures that inputs S and R are never equal to 1 at the same time.

EE345 - Introduction to Microcontrollers

- Designated by a rectangular block.
- Normal output and complemented output (bubble)
- For NAND gates latch, set and reset by logic zero, hence the bubbles and bars at inputs.

EE345 - Introduction to Microcontrollers

- State transitions of the latches start as soon as the clock pulse changes to logic 1 level.
- The new state of a latch appears at the output while the pulse is still active.
- Combinational circuit will generate new outputs and the state of the latch will change again within the same clock cycle.

EE345 - Introduction to Microcontrollers

- State of a latch or flip-flop is switched by a change in the control input. The momentary change is called a trigger, and transition it causes is said to trigger the flip-flop.
- The problem with the latch is that it responds to a change in the level of a clock pulse.
- A clock pulse goes through two transitions: from 0 to 1, and returns from 1 to 0.
- The Solution: by changing the operation of a flip-flop to trigger it only during a single transition.
- Two types of transitions: the positive transition and the negative transition.

EE345 - Introduction to Microcontrollers

EE345 - Introduction to Microcontrollers

- Master-Slave D Flip-Flop (Negative-edge-triggered flip-flop)
- The circuit stores D in the master latch when CLK=1 and changes its output Q only at the negative-edge of the controlling clock.

EE345 - Introduction to Microcontrollers

If CLK=0 → S=1 and R=1 (present state).

If D=0 and CLK=1 → R changes to 0 (the reset state)

If there is a change in D while CLK=1 → R remains at 0 (the flip-flop is unresponsive to further changes in the input).

If CLK=0 → S=1 and R=1 (present state)

If D=1 and CLK=1 → S changes to 0 (set state)

If there is a change in D while CLK=1 → S remains at 0 (the flip-flop is unresponsive to further changes in the input).

EE345 - Introduction to Microcontrollers

- When the input clock is in the positive-edge-triggered flip-flop, the value of D is transferred to Q.
- The output Q is not affected:
- During transition of CLK from 1 to 0.
- When CLK is in the steady logic 1 level.
- When CLK is in the logic 0 level.
- Characteristic Equation: Q(t+1) = D

EE345 - Introduction to Microcontrollers

Two types:

- Master-Slave D Flip-Flop (Negative-edge-Triggered)
- D-Type Positive-Edge-Triggered Flip-Flop
Advantages:

- Requires small number of gates to construct.
- Can be used to construct other types of flip-flops.

EE345 - Introduction to Microcontrollers

- Two Inputs: J and K
- Three Operations:
- Set
- Reset
- Complement
- Q(t+1) = JQ’+ K’Q

EE345 - Introduction to Microcontrollers

One Input: T

D = T ⊕ Q=TQ’+T’QQ(t+1) = TQ’+T’Q

EE345 - Introduction to Microcontrollers

EE345 - Introduction to Microcontrollers

D FF: Q(t+1) = D

JK FF: Q(t+1) = JQ’+ K’Q

T FF: Q(t+1) =T⊕Q = TQ’+T’Q

EE345 - Introduction to Microcontrollers

- Some flip-flop have asynchronous inputs that are used to force the flip-flop to a particular state independent of the clock. This operation is used for initialization.
- The input that sets the flip-flop to 1 is called preset or direct set.
- The input that clears the flip-flop to 0 is called clear or direct reset.
- For a positive-edge-triggered D ff with asynchronous reset, when R=0, the output is reset to 0. This state is independent of the values of D or C. Normal clock operation can proceed only after the reset input goes to logic 1.

EE345 - Introduction to Microcontrollers

EE345 - Introduction to Microcontrollers

- Analysis of a sequential circuit:
a circuit diagram a state table or state diagram

- Design of a sequential circuit:
a set of specifications a logic diagram

EE345 - Introduction to Microcontrollers

Algebraic Expressions:

A(t+1)=A(t)x(t)+B(t)x(t)

B(t+1)=A’(t)x(t)

A(t+1)=Ax+Bx

B(t+1)=A’x

y(t)=[A(t)+B(t)]x’(t)

y=(A+B)x’

EE345 - Introduction to Microcontrollers

- State table for the circuit in the previous slide.

EE345 - Introduction to Microcontrollers

States are represented by circles and the transitions between states are indicated by directed lines connecting the circles.

EE345 - Introduction to Microcontrollers

DA= Ax + Bx

DB= A’x

y = (A+B)x’

We have D flip-flops:

Q(t+1) = DQ

A(t+1) = Ax + Bx

B(t+1) = A’x

EE345 - Introduction to Microcontrollers

DA= A⊕x⊕y

A(t+1) = DA

A(t+1) = A⊕x⊕y

EE345 - Introduction to Microcontrollers

JA = B

KA = Bx’

JB = x’

KB = A’x+Ax’ = A⊕x

We have Q(t+1) = JQ’+K’Q

A(t+1) = JAA’+KA’A

= BA’+(Bx’)’A

= A’B+AB’+Ax

B(t+1) = JBB’ + KB’B =x’B’+(A⊕x)’B =B’x’+ABx+A’Bx’

EE345 - Introduction to Microcontrollers

EE345 - Introduction to Microcontrollers

Characteristic equation:

Q(t+1) = T⊕Q = T’Q+TQ’

TA = Bx , TB = x , y = AB

A(t+1) = (Bx)’A + (Bx)A’

= AB’ + Ax’ + A’Bx

B(t+1) = x ⊕ B

EE345 - Introduction to Microcontrollers

State-reduction problem: the process of reduction of the number of flip-flops in a sequential circuit, while keeping the external input-output requirement unchanged.

A sequential circuit with m flip-flops has 2m states.

A reduction in the number of states may (or may not) result in a reduction in the number of flip-flips.

EE345 - Introduction to Microcontrollers

Assume circuit is in state a, and let the input sequence be 01010110100.

The complete input/output sequence:

State a a b c d e f f g f g a

Input 0 1 0 1 0 1 1 0 1 0 0

Output 0 0 0 0 0 1 1 0 1 0 0

EE345 - Introduction to Microcontrollers

Definition: Two states are said to be equivalent if, for each member of the set of inputs, they give exactly the same output and send the circuit either to the same state or to an equivalent state.

- Algorithm for the state reduction:
- Find the completely specified state table for the circuit.
- When two states are equivalent, one of them can be removed
- When the removed states are used in the next-state columns, change them with the equivalent state symbols.

EE345 - Introduction to Microcontrollers

States g and e are equivalent:

States d and f are equivalent:

EE345 - Introduction to Microcontrollers

Same output sequence results, although state sequence is different.

Reduced State Diagram

From reduced state diagram

From original state diagram

EE345 - Introduction to Microcontrollers

State assignment: process of assigning coded binary vales to the states.

EE345 - Introduction to Microcontrollers

Set of specifications a logic diagram

The procedure:

1. From the word description and specifications of the desired operation, derive a state diagram for the circuit.

2. Reduce the number of states if necessary.

3. Assign binary values to the states.

4. Obtain the binary-coded state table.

5. Choose the type of flip-flops to be used.

6. Derive the simplified flip-flop input equations and output equations (combinational circuit design).

7. Draw the logic diagram.

EE345 - Introduction to Microcontrollers

Design a circuit that detects three or more consecutive 1’s in a string of bits coming through an input line. Use D ff’s.

1. State Diagram for Sequence Detector: (We will implement a Moore model sequential circuit)

EE345 - Introduction to Microcontrollers

2. Reduce the number of states if necessary. N/A here.

3. Assign binary values to the states.

S0=00, S1=01, S2=10, and S3=11.

4. Obtain the binary-coded state table.

EE345 - Introduction to Microcontrollers

5. Choose the type of flip-flops to be used. We will use D flip-flops. We need two flip-flops.

6. Derive the simplified flip-flop input equations and output equations.

A(t+1) = DA

B(t+1) = DB

EE345 - Introduction to Microcontrollers

A(t+1) = Ax + Bx

B(t+1) = Ax + B’x

y = AB

7. Draw the logic diagram

EE345 - Introduction to Microcontrollers

When-D type flip-flops are employed, the input equations are obtained directly from the next state since Q(t+1) = DQ. This is not the case for JK and T flip-flops. In order to determine the input equations for these flip-flops, it is necessary To derive a function relationship between the state table and the input equations.

JK ff: Q(t+1) = JQ’ + K’Q T ff: Q(t+1) = TQ’ + T’Q

EE345 - Introduction to Microcontrollers

EE345 - Introduction to Microcontrollers

JA= Bx’

KA= Bx

JB= x

KB= (A⊕x)’

EE345 - Introduction to Microcontrollers

Example: Design a binary counter

State Diagram of 3-Bit Binary Counter

EE345 - Introduction to Microcontrollers

EE345 - Introduction to Microcontrollers

EE345 - Introduction to Microcontrollers