Ch. 6: Digital Circuits

What is a D flip-flop?

(also called a data flip-flop) a flip-flop with a single input D, whose value is stored and presented at the output Q at the edge of a clock pulse

What is an analog signal? Draw an image of one.

A signal that changes in a continuous manner. Analog = continuous

What is important to note about the operation of sequential logic devices?

All sequential logic devices depend on feedback and propagation delays for their operation.

What is the basis of all semiconductor information storage and processing in digital computers?

Flip-flops

What's the alternative form of the equations for De Morgan's Laws? How to remember?

How to Remember: Take the original form of De Morgan's Laws and shift all NOT's to one side.

6.2: Digital Representations

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6.9: Flip-Flops

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What does a dot at the input or output of a digital device denote?

It denotes signal inversion. Example: - A 0 becomes a 1, and - A 1 becomes a 0

What does the 7404 integrated circuit consist of ?

It is a hex inverter (INV) gate IC. Thus, it contains 6 INV gates.

What does the 7408 integrated circuit consist of?

It is an a quad AND gate IC. Thus, it contains 4 AND gates.

What is the clock (CK) in a sequential logic device?

The trigger signal in these type of devices. What typically causes the logic device to respond to the inputs i.e. when the clock changes value. - Can be a periodic wave or an aperiodic collection of pulses

How do logic gates come packaged on IC's?

They come in groups of 4 (quad), 6 (hex), or 8 (oct), all of the same type of gate.

What is a good way to verify a Boolean identity?

Truth tables where each term in the identity is evaluated for all components.

What is the mathematical basis for logic representation and manipulation?

boolean algebra

What are the Distributive Laws of boolean algebra?

distributive

What is the T (toggle) flip-flop? Draw the schematic for a positive-edge-triggered T flip-flop and its equivalent JK flip-flop.

flip flop that toggles the output every time it is triggered. - Preset and clear are necessary to control the output.

What code are Karnaugh maps written in? Why?

gray code b/c it allows for patterns to be easily seen in the logic.

What is finding a boolean expression given a truth table?

it is an alternative method to solving a logic problem. Instead of writing out the logic and writing quasi-logic statements, you list all of the possible input and output combinations in a truth table.

What are logic high and logic low?

logic high (1) logic low (0) - These outputs are usually specified at a given voltage range. It depends on the device.

What are synchronous operations? Example?

operations where the changes in the output state are all coordinated by clock pulses Ex. a microprocessor - all system changes are coordinated by a common clock signal

What is the base of a numbering system? Examples?

simply, the number of different symbols we can use to represent a digit Example: - Base 10: can represent a digit with 10 symbols: the numbers 0-9. - Base 2: can represent a digit with 2 symbols: 0 or 1

What is the foundation for the design of all digital devices?

boolean logic and binary number representation

1) What is binary coded decimal (BCD)? 2) In what situations is it beneficial to use BCD? 3) Where is BCD a poor choice? Why? 4) What is important to note about BCD?

1) Another type of number representation that is typically used for the input and output of numerical data. - In BCD, 4 bits are used to represent a single, base 10 digit 2) Good For: - Representing decimal numbers in a binary number format. 3) Bad For: - Inefficient for storing or transmitting multiple-digit numbers because only 10 of the 16(2^4) possible states of the 4-bit number are used. term-26 4) BCD representation ≠ Binary representation

1) What is the hexadecimal number system? 2) What symbols are used for the digits in the hex system? 3) Why it is used as an alternative to binary?

1) Base 16 number system (0-15). 2) 0-9 and A-F The hexadecimal number system is often used as an alternative to the binary representation because binary numbers can be long and cumbersome to write and display.

1) What are sequential logic devices? 2) What devices are encompassed in this category? 3) When do sequential logic devices usually respond to inputs?

1) Timing of the sequencing of the inputs signals are important. 2) Examples of Sequential Logic Devices: - Flip-flops - Counters - Monostables - Latches - Microprocessors 3) When a separate trigger signal transitions. Trigger signal is usually called the clock signal(CK).

How does the RS flip-flop operate? What applications is the RS flip-flop used for?

1. As long as the inputs S and R are both 0, the outputs of the flip-flop will remain unchanged. 2. When S is 1 and R is 0, the flip-flop is set to Q = 1, Qbar =0. 3. When S is 0 and R is 1, the flip-flop is reset to Q = 0, Qbar = 1. 4. It is "not allowed" to place a 1 on S and R simultaneously because the output will be unpredictable. Applications: The RS flip-flop is rarely used in designs because of the limitation that it cannot have a 1 on S and 1 on R at the same time. Not as versatile as other designs.

How do you design logic networks (6 Steps)? What are the steps necessary to facilitate the design of a digital circuit?

1. Define the problem. 2. Write quasi-logic statement in English that can be translated into Boolean logic. 3. Write the statements in step 2 in Boolean logic. 4. Simplify and optimize Boolean expression in step 3 if possible. 5. Write an all-AND, all-NAND, all-OR, or all-NOR version of the circuit to minimize the number of required logic IC's. 6. Draw the logic schematic for the circuit.

How do you convert a binary number to hexadecimal? Convert 123 (base 10) to binary and then to hexidecimal.

1. Divide the binary number into groups of 4 digits beginning with the LSB. 2. Replace each group with its hexadecimal equivalent. 123 = 0111 1011 = 7B

How does the edge-triggered RS flip-flop operate?

1. If S and R are both 0 when the clock edge is encountered, the output state remains unchanged. 2. If S is 1 and R is 0 when the clock edge is encountered, the flip-flop output is set to 1. If the output is at 1 already, there is no change. 3. If S is 0 and R is 1 when the clock edge is encounter, the flip-flop output is reset to 0. If the output is at 0 already, there is no change. 4. S and R should never both be 1 when the clock edge is encountered.

What are the 7 combinational logic operators? What basic operation does each perform? Write out the gate name, operation, symbol, mathematical logic expression, and the truth table for each of the 7 logic gates/combinational logic devices.

1. Inverter (INV, NOT) also called a NOT gate - Operation: Inverts signal 2. AND gate - Operation: AND logic 3. NAND gate - Operation: Inverted AND logic - How to Remember: NAND = NOT AND 4. OR gate - Operation: OR logic 5. NOR gate - Operation: Inverted OR logic - How to Remember: NOR = NOT OR 6. XOR gate - Operation: Exclusive OR logic - How to Remember: X-clusive 7. Buffer - Operation: Increase output signal current - Connection: same as a buffer amplifier

How does a latch function?

1. Output Q tracks the input D as long as the clock ,CK, is high. Because the output follows the input when the clock is high, we say the latch is transparent during this time. 2. When a negative edge occurs (when CK goes low), the flip-flop will store (latch) the value that D had at the negative edge, and that value will be retained at the output.

How do you generate gray code?

1. Write 0,1 in a column. 2. Draw a mirror line under the column. 3. Mirror across the line (i.e. 1,0 in the same column). 4. Add leading zeros to numbers above the mirror line. 5. Add leading ones to the numbers below the mirror line. 6. Result = 2-bit gray code Note: for gray code larger than 2-bits, just repeat the process above, minus step 1.

Write out the truth table for the RS flip-flop. Show all possible combinations.

1st Row = memory state where flip-flop retain last value (either set or reset) Q_0 = value of the output Q before the indicated input conditions were established. 1 = logic high 0 = logic low

In what ways can you find a boolean expression given a truth table? (Alternative method to designing logic networks)

2 Methods: 1. Sum-of-products method 2. Product-of-sums method 3. Karnaugh mapping

What is ASCII and what does it stand for? Examples? How to remember?

7-bit codes used to denote all of the alphanumeric characters. Stands For: American Standard Code for Information Interchange How to Remember: 1. One is bun: Hamburger in a bun with the American flag on it. 2. Two is canoe: a Ron Swanson canoe: the "highest standard" 3. Three is debris: a broken enigma code machine floating in the water in sunken u-boat debris 4. Four is door: a door that leads to the information desk 5. Five is hive: a bee hive and flow where pollen is being interchanged.

What is a digital device? How are they characterized? Types?

A device that takes a digital input and converts it to one or more digital outputs. They are characterized according to their function. Characterization: 1) Sequential logic devices 2) Combinational logic devices

What are timing diagrams? Why are they useful?

A diagram that show the simultaneous levels of the inputs and outputs in a digital logic circuit vs. time. Why?: 1. Make it easy to analyze complex logic circuits. 2. Provides an illustration of every possible combination of input values and corresponding outputs. 3. Provides a graphical/visual representation of input/output relationships.

What is a digital signal? Draw an image of one.

A signal that only exists at specific levels and changes its level in discrete steps - Most only have 2 states: high and low Digital = discrete

Given multiple inputs (i.e. more than 2), what are the logic tables AND, NAND, OR, NOR, and XOR gates?

AND - 1 if and only if all inputs are 1 - Otherwise, output = 0 NAND - Output = 0 if and only if all inputs are 1 - Otherwise, output = 1 OR - Output = 0 if and only if all inputs are 0 - Otherwise, output = 1 NOR - Output = 1 if and only if all inputs are 0 - Otherwise, output = 0 XOR - Output = 0 if all inputs are 1 or all inputs are 0 - Otherwise = 1

What are the 9 Fundamental Laws of boolean algebra?

AND'ed, and OR'ed w: 1. 0 2. 1 3. Itself 4. Its inverted self

What is the product-of-sums method? Give and example. How to remember?

Base on the fact that we can represent an output as a product of sums containing combinations of the inputs. Method: 1. Make truth table. 2. Form a sum of the inputs of every row that has an output of 0. 3. Take the product of the sums from step 2. 4. Invert any input that appears as one in the truth table and is in the equation in step 3. How to Remember: product-of-sums: multiples to zero

What is the sum-of-products method? How do you perform the sum-of-products method on a boolean truth table? Give an example.

Based on the principal that we represent the outcomes as a sum of the products containing combinations of the inputs. Method: 1. Make truth table. 2. Form a product of the inputs of every row that has an output of 1. 3. Sum the products from step 2. 4. Invert any input that appears as zero in the truth table and is in the equation in step 3. How to Remember: sum-of-products: adds to 1

Why are logic gates called gates?

Because they control the flow of signals from the inputs to the single output. Connection: irrigation gate - controls the amount of water that flow from the inlet to the outlet

What the difference between combination logic devices and sequential logic devices? Give examples of each.

Big difference: signal timing Combinational Logic: outputs exclusively depend on the instantaneous value of the inputs. - Generate output based on inputs, independent of timing. - Example: a quad AND IC Sequential Logic: the timing/"sequencing history" of the input signals plays a role in determining the output - Example: flip-flops

6.3: Combination Logic and Logic Classes

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6.4: Timing Diagrams

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6.5: Boolean Algebra

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6.6: Design of Logic Networks

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6.7: Finding a Boolean Expression Given a Truth Table

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6.8: Sequential Logic

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What are De Morgan's Laws? What are the equations for De Morgan's Laws? How to remember?

De Morgan's Laws - They are boolean algebra laws that are useful for rearranging or solving longer boolean expressions. - They are also useful for converting btw OR and AND gates. Equations: 1) Whole NOT of all inputs OR'ed together = all inputs individually NOT'ed and all AND'ed together. 2) Whole NOT of all inputs AND'ed together = all inputs individually NOT'ed and all OR'ed together How to Remember: 1) Total NOT of all inputs OR'ed = individual NOTs of all inputs AND'ed 2) Total NOT of all inputs AND'ed = individual NOTs of all inputs OR'ed

What are combinational logic devices/logic gates? Give some examples.

Digital devices that convert binary inputs into binary output by using mathematical logic. combinational logic device = logic gate

What is the base 10 numbering system? Equation?

Each digital is a place holder for a different power of 10. Note: fractions can also be included if you include negative digits n = number of digits d_i = symbol for the i'th digit

How does binary arithmetic work? Examples: solve the following - 1) 1001 + 0011 2) 1001 * 0011 3) 1001 - 0011 4) 1001 / 0011

Exactly the same way as base 10 arithmetic. Examples: 1) 9+3 = 12, 1001 + 0011 = 1100 2) 9*3 = 27, 1001*0011 = 11011 3) 9- 3 = 6, 1001 - 0011 = 0110 Note w/ Subtraction: when you borrow, you carry 2 over rather than 10 like in base 10. 4) 9 / 3 = 3, 1001 / 0011 = 0011 Note: Remember how to do long division

What's the difference between a hardware solution and a software solution?

Hardware: An example is the alarm example given in section 6.6 in the textbook. The logic of the circuit is provided by integrated circuits in this type of solution. Software Solution: logic in this solution is implemented using a program running on a microcontroller

What is a buffer used for? What is a situation where you would want to use a buffer?

It is used to increase the current supplied at the output while retaining the digital state. - Important if you want to drive multiple digital inputs from a single output. - Overcomes the limitations associated with fan-out by increasing the output current

What is Karnaugh mapping? What is a Karnaugh map? When would you use it?

Karnaugh Mapping: A technique for simplifying boolean logic, just like boolean algebra. - Results in a simplified Boolean expression through truth table manipulation. Karnaugh Map: special form of truth table When to Use: - when you have 3-6 variables - Boolean algebra is faster for 1-2 variables

Compare and contrast a latch and a D flip-flop. How are they similar? How are they different?

Latch: 1) Exhibits transparency - Output tracks input when clock is high 2) NOT edge triggered D: 1) Does NOT exhibit transparency - Output only changes when triggered by the appropriate clock edge 2) Edge triggered

In a binary number, what is the most significant bit? What is the least significant bit? Why are they label in this way? How to Remember?

MSB: the first or leftmost bit - Why?: Because it is the largest power of two (i.e. most significant) LSB: the last or rightmost bits - Why?: Because it is the smallest power of two (i.e. least significant) How to Remember?: Most significant bit = largest bit (most important) Least significant bit = smallest bit (least important)

How do you convert from a base 10 number to a binary number (base 2)? Method of Loci? Example: Convert 123 in base 10 to its binary equivalent

Method of Loci Location: Apt. G148 1. Successively divided the number by 2 and record the remainders after each division. - Location: Kitchen: dividing peanut m and m's into half groups and recording on a sticky note. Hands swelling. 2. Write the remainders in reverse order to form the digits of the new binary number. Meaning, the last remainder is the MSB, and the first remainder is the LSB. - Location living room: "Cha Cha Slide" playing on the sound bar- "Reverse, Reverse!" Ex. 123 in binary is 1111011

Once you have determined the Boolean expression for you digital circuit, what is the next important step? Why is this step important?

Next, it's important to convert all the gates to a single type of gate. W/ more complex Boolean systems, having one type of gate is likely to reduce the number of IC's needed to complete the circuit.

Draw the timing diagram for a RS flip-flop, specifically the reset operation. Assume Q_o = 1 and thus Q_o(bar) = 0.

Order of Change: R then Q_bar then Q

What type of signal transitions do positive-edge-triggered devices respond to? What type of signal transitions do negative-edge-triggered devices respond to? Draw both these clock pulse edges on a timing diagram.

Positive: low to high (0-1) transition Negative: high to low (1-0) transition

What is a fundamental flip-flop, an RS flip-flop? Draw its schematic and define each input and output.

S = set input R = reset input Q = complementary output Qbar = complementary output It is common for most flip-flops to include both outputs, where one output is the inverse of the other.

What is a latch? Draw a schematic symbol of a latch.

Special sequential logic device that is not edge triggered. - When the clock is high, the output (Q) matches the input (D) (also called being transparent) - When the clock goes low, the output(Q) locks the value of the input. Also called a positive-level-triggered device. Note: this is different than a positive-edge-triggered device.

What does TTL stand for? Imagery? What does CMOS stand for? Imagery?

TTL (transistor-transistor logic): - Note: in electronics the T in an acronym is often transistor CMOS (complementary metal-oxide semiconductors): Imagery: sounds like "sea moss" - Image: algae that looks like pn junction that is glowing/conducting electricity on a rusty sunken metal ship

Write out the positive-edge-triggered RS flip-flop truth table.

Up-arrow = positive-edge transition from 0 to 1 X symbols show that there the inputs (S and R) have no effect on the output as long as there is not positive-edge transition of the clock. 0,1,down-arrow = shows 3 different states of the clock, none of which cause the input to effect the output

Draw a timing diagram of the positive-edge-triggered RS flip-flop.

What is shows: - Changes in output only occur when a positive-edge of the clock is hit. - When positive-edge is encountered, whether or not Q changes is dependent on the values of R and S.

What applications does the JK flip-flop serve?

Wide range of applications. All flip-flops can be constructed from it w/ proper wiring.

What is a bistable device? Give an example of one.

a device that has only 2 possible, stable output states Example: - Flip-flop: 2 output states: 1 (high) and 0 (low)

What is a multiplexer? Draw a schematic of a multiplexer. How to remember?

a device that passes the value of one of the two inputs based on the value of a third input. For Example in the figure: - If P = 1, X = B - If P = 0, X =A Connection: Multiplex = complex that houses multiple movie theaters (for example, say 2) - Inputs = 2 different movies being input/played in each theater - Outputs = what the you (the viewer) see - Third input the theater number on your ticket (if you have theater 1, output = input/movie of theater 1) (If you have theater 2, output = input/movie of theater 2)

What is an integrated circuit?

a device where transistors, resistors, and diodes exist on a single chip of silicon

What is a byte? How to remember?

a group of 8 bits Imagery: rhymes with "mite" which have 8 legs - Image: a biker mite (nose ring, leather jacket) with the word "bit" tattooed on each leg

What is a flip-flop? What type of device is a flip-flop? What are flip-flops the basis for?

a sequential logic device that performs 2 primary functions: 1) they provide a means for storing data and 2) a means for switching between 2 binary states. A flip-flop is a bistable device. Has the ability to stay in a particular output state until the input signals cause it to change state. Flip-flops form the basis of information storage and processing in digital computers.

What are the Associative Laws of boolean algebra? How to remember?

associative = groupings of same type (all AND or all OR) does not matter as long as order remains the same. How To Remember: like associate Mrs. Ragouzis knows a friend, who knows a friend, who knows Kevin Bacon. - You can associate any of the people in the list with the other people to their right or left. You just can't switch the order or it all falls apart.

What are asynchronous inputs? What are synchronous inputs? Why are they named such? Differentiate between the two.

asynchronous: preset and clear functions on flip-flop that instantaneously override any other inputs. - Name comes from the fact that their effect can be asserted at any time. NOT triggered by a clock. synchronous: clock inputs, "synchronous with the clock"

What is the generic expansion equation for a number for any base system? How do you convert from an arbitrary base to base 10?

b = base n = number of digits Use the equation in the image.

What is the binary number system? Generic equation? Example: show the expanded version of 1101_2. Why is the base 2 numbering system used in digital devices? What symbols are used to represent digits in a binary number system?

base 2 number system Why?: Because the operation of digital devices is based on transistors that switch between 2 states: ON (saturated state) and OFF (cutoff state) Symbols: 1 = ON 2 = OFF

What are edge-triggered flip-flops? Types? What on a schematic indicates an edge-triggered flip-flop?

clocked flip-flops whose outputs can changed based on either a positive or negative edge of a clock pulse. Types: 1) Positive-edge-triggered 2) Negative-edge-triggered Schematic: - The arrow (small angle bracket) on the input edge of the flip-flop indicates it is edge-triggered.

What are the Commutative Laws of boolean algebra? How to remember?

commutative = order doesn't matter How to remember: 1. commutation = exchange / trade 2. Many ways to commute, and all end up in same destination. Imagery: sounds like "commute" - A map that show my mom's commute from Greeley to Fort Collins along Highway 85 and my commute along highway 34 and I-25. I go east first, then north. She goes north first, then east. We both arrive at the same place.

How do you convert btw decimal and binary coded decimal (BCD)? Convert 123 (base 10) to BCD.

create the 4-bit code for each decimal digit Example: 123 = 0001 0010 0011 (BCD)

What is the JK flip-flop? Draw the schematic of one.

flip-flop similar to a RS flip-flop, except it can have a state where both of the inputs are high (1) simultaneously. - Toggling is possible w/ this flip-flop. - J is analogous to S (set) input in the RS flip-flop - K is analogous to R (reset) input in the RS flip-flop

What is boolean algebra?

mathematical system used to define the rules for expressing and simplifying binary logic statements.

How do you denote a logic AND, a logic OR, and a logic NOT using algebra symbols?

plus (+) = logic OR dot (·) = logic AND (X with a line over it/X bar) = logic NOT

What are the asynchronous inputs: preset and clear used to do on a flip-flop?

preset input: used to set or initialize the output Q of the flip-flop to high (1). clear input: used to clear or reset the output Q of the flip-flop to low (0). Note: preset and clear should not be used simultaneously

Flip-flops are usually clocked. What does this mean?

the "clock" is a third input signal to the device that coordinates the changes in the output of the device.

What is a bit? How to remember?

the digits of a binary number. Ex. 1011 - each number is a bit that makes up this 4-bit binary number

What is the fan-out of a digital gate?

the maximum number of similar digital inputs that can be driven by the gate's output. Imagery: has fan in the name - Image: a ceiling fan with combinational logic gates as the blades. The blades are the similar digital inputs. Fan-out is the max number of blades. Output is electricity going into fan.

What does it mean for a flip-flop to toggle?

the output changes value - Ex. a 1 would become 0, and a 0 would become a 1

What are the two families of digital integrated circuits?

1. TTL (transistor-transistor logic) 2. CMOS (complementary metal-oxide semiconductors)

6.1: Introduction

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What does the internal design of an RS flip-flop look like? What components does it consist of? Draw a diagram of it.

Components: - Combinational logic gates with internal feedback from the outputs to the inputs of the NAND gates. - 2 NAND gates - 2 INV gates (inverter)