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EE 261 – Introduction to Logic Circuits

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EE 261 – Introduction to Logic Circuits Textbook Reading Assignments Practice Problems Graded Components of this Module

Module #7 – State Machines

- Topics
- Sequential Logic
- Finite State Machines
- State Variable Encoding
- Other Flip-Flop Devices

- 7.1-7.8, 7.12

- 7.4, 7.12, 7.18, 7.44
- VHDL 2-bit Binary Counter Design & Simulation (see M7_HW3 handout)

- 3 homeworks, 3 discussion, 1 quiz(2 homeworks will be uploaded to the course Dropbox, 1 homework plus the discussions & quiz are online)

EE 261 – Introduction to Logic Circuits

Module #7 – State Machines

- What you should be able to do after this module
- Understand the operation of sequential logic storage devices (Latches & Flip-Flops)
- Draw a state diagram describing the logical operation of a finite state machine
- Create the State/Output tables for a finite state machine
- Synthesize the logic diagram of a state machine
- Understand the design trade-offs of various state encoding techniques
- Use VHDL to describe and simulate finite state machines

Sequential Logic

- Combinational Logic- until now, we've covered logic whose outputs depend on the current values of the input this is the definition of "Combinational Logic"
- Sequential Logic- Now we move to logic circuits whose outputs depend on : - the current values of the inputs - the past values of the inputs- this is the definition of "Sequential Logic"- in order to make logic circuits based on the previous values of inputs, we need a storage device

Sequential Logic

- Feedback- consider this circuit- the outputs are "fed back" to the inputs- this gives the following relationships: Q = Vin1' Qn = Vin2' Vin1 = Qn Vin2 = Q Vin1 = Vin2' = Q' = Vin1 Vin2 = Vin1' = Qn' = Vin2 - this circuit will HOLD or STORE a logic value- feedback gives us the ability to build "Sequential Storage Devices"

Sequential Logic

- Metastability- what if the input is VDD/2- in an ideal world, the outputs would be driven to VDD/2- we know that noise exists in the world (thermal, shot, etc…)- noise will add a small Δv to the nodes in the system

Sequential Logic

- Metastability- let's consider a + Δv being superimposed on input Vin1 who starts out at VDD/2 - this causes a - Δv to be superimposed on output Q due to the inverting nature of the inverter- this - Δv is fed back to input Vin2, which in turn causes a + Δv on output Qn

VDD/2 - Δv

VDD/2 + Δv

VDD/2 - Δv

VDD/2 + Δv

Sequential Logic

- Metastability- this new + Δv is fed back to the original Vin1 voltage in an additive nature- this drives Vin1 even further positive, which in turn starts the loop all over again- this feedback loop continues until the inputs and outputs are driven to either a 0 or a 1

VDD/2 - Δv - Δv

VDD/2 + Δv + Δv

VDD/2 - Δv - Δv

VDD/2 + Δv + Δv

Sequential Logic

- Metastability- Metastability is the situation where the inputs cause an indeterminate output in a feedback circuit - i.e., VILmax < Vin < VIHmin- using feedback, we know that the circuit will always be driven to a final state- this is also called a "Bi-Stable" element meaning that the inputs and outputs will be driven to one of two final states (i.e., a 1 or a 0)- the final state that the circuit is driven to is unknown, but we know it will go there eventually

Sequential Logic

- Recovery time- manufactures can specify the maximum amount of time that a bi-stable element will take to reach its final value- the time that the output is unknown is called the "Metastability Region"- the time it takes to exit the Metastable region is called the "Recovery Time" (trecovery)
- Summary- feedback gives us the ability to build digital storage devices using gates- anytime we use feedback, we create a bi-stable element and can have Metastability

Sequential Logic

- SR Latch- consider the following circuit which is called an "SR Latch"- To understand the SR Latch, we must remember the truth table for a NOR Gate AB F 00 1 01 0 10 0 11 0

Sequential Logic

- SR Latch- when S=0 & R=0, it puts this circuit into a Bi-stable feedback mode where the output is either:Q=0, Qn=1 Q=1, Qn=0AB FAB F 00 1 (U2) 00 1 (U1) 01 0 01 0 (U2) 10 0 (U1) 10 0 11 0 11 0

0

0

0

1

1

0

0

1

1

0

0

0

Sequential Logic

- SR Latch- we can force a known state using S & R:Set (S=1, R=0)Reset (S=0, R=1) AB FAB F 00 1 (U1) 00 1 (U2) 01 0 01 0 (U1) 10 0 (U2) 10 0 11 0 (U2) 11 0 (U1)

0

1

1

0

0

1

1

0

0

1

1

0

Sequential Logic

- SR Latch- we can write a Truth Table for an SR Latch as followsS R Q Qn . 0 0 Last Q Last Qn - Hold 0 1 0 1 - Reset 1 0 1 0 - Set 1 1 0 0 - Don’t Use- S=1 & R=1 forces a 0 on both outputs. However, when the latch comes out of this state it ismetastable. This means the final state is unknown.

Sequential Logic

- SR Latch- Remember the Truth Table for an SR Latch :
S R Q Qn . 0 0 Last Q Last Qn - Hold 0 1 0 1 - Reset 1 0 1 0 - Set 1 1 0 0 - Don’t Use- there is delay associated with changes on the input causing a change on the outputs:tPLH(SQ) = Δt for a LOW-to-HIGH transition on S to cause an output change on QtPHL(RQ) = Δt for a HIGH-to_LOW transition on R to cause an output change on Q- there is also a specification on how small of a pulse width we can have and be recognizedtPW(min) = minimum input pulse width that can be recognized

Sequential Logic

- S’R’ Latch- we can also use NAND gates to form an inverted SR LatchS’ R’ Q Qn . 0 0 1 1 - Don’t Use 0 1 1 0 - Set 1 0 0 1 - Reset 1 1 Last Q Last Qn - Hold

Sequential Logic

- SR Latch w/ Enable- we then can add an enable line using NAND gates- remember the Truth Table for a NAND gateAB F 00 1 - a 0 on any input forces a 1 on the output 01 1 - when C=0, the two EN NAND Gate outputs are 1, which forces “Last Q/Qn” 10 1 - when C=1, S & R are passed through INVERTED 11 0

Sequential Logic

- SR Latch w/ Enable- the truth table then becomesC S R Q Qn . 1 0 0 Last Q Last Qn - Hold 1 0 1 0 1 - Reset 1 1 0 1 0 - Set 1 1 1 1 1 - Don’t Use 0 x x Last Q Last Qn - Hold

NANDAB F00 101 110 111 0

Sequential Logic

- D Latch- a modification to the SR Latch where R = S’ creates a D-latch- when C=1, Q <= D- when C=0, Q <= Last ValueC D Q Qn . 1 0 0 1 - track 1 1 1 0 - track 0 x Last Q Last Qn - Hold

Sequential Logic

- Flip-Flops- a "Latch" is a device that "tracks or holds" the input depending on a control signal (i.e., C or CLK)- a "Flip-Flop" is a device that will "acquire and hold" the input when a transition is present on C or CLK- Flip-Flops are commonly used in sequential logic due to their speed

Sequential Logic

- D-Flip-Flops- we can combine D-latches to get an edge triggered storage device (or flop)
- the first D-latch is called the “Master”, the second D-latch the “Slave”MasterSlave CLK=0, Q<=D “Open” CLK=0, Q<=Q “Close”

CLK=1, Q<=Q “Closed” CLK=1, Q<=D “Open”

- on a rising edge of clock, D is “latched” and held on Q until the next rising edge

Sequential Logic

- D-Flip-Flops in VHDL- In VHDL, we use a “process” to model the behavior of a D-Flip-Flop

Qn

R

entity DFF is

port (Clock : in BIT;

Reset : in BIT;

D : in BIT;

Q, Qn : out BIT);

end entity DFF;

architecture DFF_arch of DFF is

begin

D_FLIP_FLOP : process (Clock, Reset) -- sensitivity list

begin

if (Reset = '0') then

Q <= '0'; Qn <= '1';

elsif (Clock'event and Clock='1') then

Q <= D; Qn <= not D;

end if;

end process D_FLIP_FLOP;

end architecture DFF_arch;

Sequential Logic

- D-Flip-Flop Operation- Below is a timing diagram of a D-Flop

Qn

R

D Changes but it does not effect Q.

Q & Qn are updated on the rising edge of clock.

Finite State Machines

- Synchronous- we now have a way to store information on an edge (i.e., a flip-flop)- we can use these storage elements to build "Synchronous Circuitry"- Synchronous means that events occur on the edge of a clock
- State Machine- a generic name given to sequential circuit design- it is sequential because the outputs depend on : 1) the current inputs 2) past inputs - state machines are the basis for all modern digital electronics

Qn

Finite State Machines

- State Memory- a set of flip-flops that store the CurrentState of the machine- the Current State is due to all of the input conditions that have existed since the machine started- if there are "n" flip-flops, there can be 2n states- a state contains everything we need to know about all past events- we define two unique states in a machine 1) Current State 2) Next State
- Current State - the state that the machine is currently in- this is found on the outputs of the flip-flops (i.e., Q)

Qn

Finite State Machines

- Next State- the state that the machine "will go to" upon a clock edge- this is found on the inputs of the flip-flops (i.e., D)- the Next State depends on - the Current State - any Inputs- we call the Next State Logic "F"

Finite State Machines

- State Transition- upon a clock edge, the machine changes from the "Current State" to the "Next State"- After the clock edge, we reassign back the names (i.e., Q=Current State, D= Next State)
- State Table- a table where we list which state the machine will transition to on a clock edge- this table depends on the "Current State" and the "Inputs"- we can use the following notation when describing Current and Next StatesCurrent Next S S*ScurSnxt SC SNAcurBnxt QC QN

Finite State Machines

- State Table cont…- we typically give the states of our machine descriptive names (i.e., Reset, Start, Stop, S0)ex) State Table for a 4-state counter, no inputs Current State Next State S0 S1 S1 S2 S2 S3 S3 S0

Finite State Machines

- State Table cont…- when there are inputs, we list them in the tableex) State Table for a 4-state counter with directional input Current State Dir Next State S0 Up S1 Down S3 S1 Up S2 Down S0 S2 Up S3 Down S1 S3 Up S0 Down S2 - we don't need to exhaustively write all of the Current States for each Input combination (i.e., Direction) which makes the table more readable

Finite State Machines

- State Variables- remember that the State Memory is just a set of flip-flops- also remember that the state is just the current/next binary number on the in/out of the flip-flops- we use the term State Variable to describe the input/output for a particular flip flop- let's call our State Variables Q1, Q0, …. Current State Dir Next State Q1 Q0 Q1* Q0* 0 0 Up 0 1 Down 1 1 0 1 Up 1 0 Down 0 0 1 0 Up 1 1 Down 0 1 1 1 Up 0 0 Down 1 0

Finite State Machines

- State Variables- we call the assignment of a State name to a binary number "State Encoding"ex) S0 = 00 S1 = 01 S2 = 10 S3 = 11- this is arbitrary and up to use as designers- we can choose the encoding scheme based on our application and optimize for 1) Speed 2) Power 3) Area

Finite State Machines

- Next State Logic "F"- for each "Next State" variable, we need to create logic circuitry- this logic circuitry is combinational and has inputs: 1) Current State 2) any Inputs ex) Q1* = F(Current State, Inputs) = F(Q1, Q0, Dir) Q0* = F(Current State, Inputs) = F(Q1, Q0, Dir) - the logic expression for the Next State Variable is called the "Characteristic Equation" - this is a generic term that describes the logic for all flip-flops- when we write the logic for a specific flip-flop, we call this the "Excitation Equation" - this is a specific term that describes logic for your flip-flop (i.e., DFF)- there will be an Excitation Equation for each Next State Variable (i.e., each Flip-Flop)

Finite State Machines

- State Variables and Next State Logic "F"- we already know how to describe combinational logic (i.e., K-maps, SOP, SOP)- we simply apply this to our State Tableex) State Table for a 4-state counter, no inputs Current State Next State Q1 Q0 Q1* Q0* 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 - we put these inputs and outputs in a K-map and come up with Q1* = Q1 Q0

NextState Variable

F inputs

F outputs

Finite State Machines

- State Variables and Next State Logic "F"- we continue writing Excitation Equations for each Next State Variable in our Machine- let's now write an Excitation Equation for Q0*ex) State Table for a 4-state counter, no inputs Current State Next State Q1 Q0 Q1* Q0* 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 - we simply put these values in a K-map and come up with Q0* = Q0'

NextState Variable

F inputs

F outputs

Finite State Machines

- Logic Diagram- the state machine just described would be implemented like this:

Q1*

Q0*

Finite State Machines

- Excitation Equations- we designed this State Machine using D-Flip-Flops- this is the most common type of flip-flop and widely used in modern digital systems- we can also use other flip-flops- the difference is how we turn a "Characteristic Equation" into an "Excitation Equation"- we will look at State Memory using other types of Flip-Flops later

Finite State Machines (Mealy vs. Moore)

- Output Logic "G"- last time we learned about State Memory- we also learned about Next State Logic "F"- the last part of our State Machine is how we create the output circuitry- the outputs are determined by Combinational Logic that we call "G"- there are two basic structures of output types 1) Mealy = the outputs depend on the Current State and the Inputs 2) Moore = the outputs depend on the Current State only

Finite State Machines (Mealy vs. Moore)

- State Machines“Mealy Outputs” – outputs depend on the Current State and the Inputs- G(Current State, Inputs)

Finite State Machines (Mealy vs. Moore)

- State Machines“Moore Outputs” – outputs depend on the Current State only - G(Current State)

Finite State Machines (Mealy vs. Moore)

- Output Logic "G"- we can create an "Output Table" for our desired results- most often, we include the outputs in our State Table and form a hybrid "State/Output Table"ex) Current State In Next State Out Q1 Q0 Q1* Q0* 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1

Finite State Machines (Mealy vs. Moore)

- State Machine Tables- officially, we use the following terms:State Table - list of the descriptive state names and how they transitionTransition Table - using the explicitly state encoded variables and how they transition Output Table - listing of the outputs for all possible combinations of Current States and Inputs State/Output Table - combined table listing Current/Next states and corresponding outputs

Finite State Machines (Mealy vs. Moore)

- Output Logic "G"- we simply use the Current State and Inputs (if desired) as the inputs to G and form the logic expression (K-maps, SOP, POS) reflecting the output variable- this is the same process as creating the Excitation Equations for the Next State Variablesex) Current State In Next State Out Q1 Q0 Q1* Q0* 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1- plugging these inputs/outputs into a K-map, we get G=Q1·Q0·In

OutputVariable

G outputs

G inputs

Finite State Machines

- State Diagrams- a graphical way to describe how a state machine transitions states depending on : - Current State - Inputs- we use a "bubble" to represent a state, in which we write its descriptive name or code- we use a "directed arc" to represent a transition- we can write the Inputs next to a directed arc that causes a state transition- we can write the Output either near the directed arc or in the state bubble (typically in parenthesis)

Finite State Machines

- State DiagramsEx)

Finite State Machines

- Timing Diagrams- notice we don't show any information about the clock in the State Diagram- the clock is "implied" (i.e., we know transitions occur on the clock edge)- note that the outputs can change asynchronously in a Mealy machine

Clock

In

Found

S0

S1

S2

STATE

t

Finite State Machines

- State Machines- there is a basic structure for a Clocked, Synchronous State Machine 1) State Memory (i.e., flip-flops) 2) Next State Logic “F” (combinational logic) 3) Output Logic “G” (combinational logic)

Finite State Machines

- State Machines- the steps in a state machine design are: 1) Word Description of the Problem 2) State Diagram 3) State/Output Table 4) State Variable Assignment 5) Choose Flip-Flop type 6) Construct F 7) Construct G 8) Logic Diagram

Finite State Machines

- State Machine Example “Simple Gray Code Counter”1) Design a 2-bit Gray Code Counter. There are no inputs (other than the clock). Use Binary for the State Variable Encoding2) State Diagram

Finite State Machines

- State Machine Example “Simple Gray Code Counter”3) State/Output Table Current State Next State Out CNT0 CNT1 0 0 CNT1 CNT2 0 1 CNT2 CNT3 1 1 CNT3 CNT0 1 0

Finite State Machines

- State Machine Example “Simple Gray Code Counter”4) State Variable Assignment – binary Current State Next State Out Q1 Q0 Q1* Q0* 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 1 05) Choose Flip-Flops - let's use DFF's

Finite State Machines

- State Machine Example “Simple Gray Code Counter”6) Construct Next State Logic “F” Q1* = Q1Q0’ + Q1’Q0 = Q1 Q0 Q0* = Q0’

Q1

Q0

0

1

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2

1

0

1

1

0

3

1

Q1

Q0

0

1

0

1

2

1

0

1

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0

1

Finite State Machines

- State Machine Example “Simple Gray Code Counter”7) Construct Output Logic “G” Out(1) = Q1 Out(0) = Q1 Q0

Q1

Q0

0

1

0

0

2

1

0

0

1

1

3

1

Q1

Q0

0

1

0

0

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1

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1

1

3

0

1

Finite State Machines (Example)

- State Machines- there is a basic structure for a Clocked, Synchronous State Machine 1) State Memory (i.e., flip-flops) 2) Next State Logic “F (combinational logic) 3) Output Logic “G” (combinational logic)

Finite State Machines

- State Machines- the steps in a state machine design are: 1) Word Description of the Problem 2) State Diagram 3) State/Output Table 4) State Variable Assignment 5) Choose Flip-Flop type 6) Construct F 7) Construct G 8) Logic Diagram

Finite State Machines

- State Machine Example “Sequence Detector”1) Design a machine by hand that takes in a serial bit stream and looks for the pattern “1011”. When the pattern is found, a signal called “Found” is asserted2) State Diagram

Finite State Machines

- State Machine Example “Sequence Detector”3) State/Output Table Current State In Next State Out (Found) S0 0 S0 0 1 S1 0 S1 0 S2 0 1 S0 0 S2 0 S0 0 1 S3 0 S3 0 S0 0 1 S0 1

Finite State Machines

- State Machine Example “Sequence Detector”4) State Variable Assignment – let’s use binary Current State In Next State Out Q1 Q0 Q1* Q0* Found 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 15) Choose Flip-Flop Type - 99% of the time we use D-Flip-Flops

Finite State Machines

- State Machine Example “Sequence Detector”6) Construct Next State Logic “F” Q1* = Q1’∙Q0∙In’ + Q1∙Q0’∙In Q0* = Q0’∙In

Q1

Q1 Q0

In

00

01

11

10

0

0

2

1

6

0

4

0

0

0

1

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3

7

0

1

5

1

In

Q0

Q1

Q1 Q0

In

00

01

11

10

0

0

0

2

6

0

4

0

0

1

1

0

3

7

0

1

5

1

In

Q0

Finite State Machines

- State Machine Example “Sequence Detector”7) Construct Output Logic “G” Found = Q1∙Q0∙In8) Logic Diagram

Q1

Q1 Q0

In

00

01

11

10

0

0

0

2

0

6

0

4

0

0

1

3

0

7

1

0

5

1

In

Q0

Found

Q1*

Q0*

State Variable Encoding

- State Variable Encoding- we design State Machines using descriptive state names- when we’re ready to synthesize the circuit, we need to assign values to the states- we can arbitrarily assign codes to the states in order to optimize for our application

State Variable Encoding

- Binary- this is the simplest and most straight forward method- we simply assign binary counts to the states S0 = 00 S1 = 01 S2 = 10 S3 = 11- for N states, there will be log(N)/log(2) flip flops required to encode in binary- the advantages to Binary State Encoding are: 1) Efficient use of area (i.e., least amount of Flip-flops)- the disadvantages are: 1) Multiple bits switch at one time leading to increased Noise and Power 2) The next state logic can be multi-level which decreases speed

State Variable Encoding

- Gray Code- to avoid multiple bits switching, we can use a Gray Code S0 = 00 S1 = 01 S2 = 11 S3 = 10- for N states, there will be log(N)/log(2) flip flops required- the advantages of Gray Code Encoding are: 1) One-bit switching at a time which reduces Noise/Power - the disadvantages are: 1) Unless doing a counter, it is hard to guarantee the order of state execution 2) The next state logic can again be multi-level which decreases speed

State Variable Encoding

- One-Hot- this encoding technique uses one flip-flop for each state.- each flip-flop asserts when in its assigned state S0 = 0001 S1 = 0010 S2 = 0100 S3 = 1000- for N states, there will be N flip flops required- the advantages of One-Hot Encoding are: 1) Speed, the next state logic is one level (i.e., a Decoder structure) 2) Suited well for FPGA’s which have LUT’s & FF’s in each Cell - the disadvantages are: 1) Takes more area

Other Flip-Flops Devices

- Flip-Flops- we have learned about the following sequential storage devices - SR Latch - D Latch - D Flip-Flop- we can make state machines out of any sequential storage device- each flip-flop has a “Characteristic Equation” that governs how the Next State Logic is synthesized ex) TypeCharacteristic Eq SR Latch Q* = S + R’∙Q D Latch Q* = D D Flip Flop Q* = D - we use the Characteristic Equation & the Next State Logic to form the “Excitation Signals” that are fed into the State Memory

Other Flip-Flops Devices

- D Flip-Flop w/ Enable- a regular DFF transfers D to Q on a clock edge and then holds it- adding an enable gives the DFF the ability to continue holding the data while ignoring clock edgesENCLKQ / Qn 1 D / D’ 1 0 Last Q / Qn 1 1 Last Q / Qn 0 x Last Q / Qn- the Characteristic Eq is : Q* = EN∙D + EN’∙Q

Module Overview

- TopicsFeedback, MetastabilityLatches & Flip Flops - SR Latch - S’R’ Latch - D Latch - D Flip FlopSynchronous State Machines - State Memory - Next State Logic “F” - Output Logic “G” - Mealy, Moore - State Diagrams - State Tables - Characteristic Equation - Excitation Equation - State Variable Encoding

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