Logic Gates Part – III : Combinational Logic Gates Combinational Logic Circuits are made up from...

Logic GatesPart – III : Combinational Logic

Gates Combinational Logic Circuits are made up from basic logic NAND,

NOR or NOT gates that are "combined" or connected together to produce more complicated switching circuits.

Ex-OR and Ex-NOR gates are combinational gates.

Course Name: Digital Logic Design Level(UG/PG): UG Author(s) : Phani Swathi ChittaMentor: Aruna Adil, Prof. C. Amarnath

*The contents in this ppt are licensed under Creative Commons Attribution-NonCommercial-ShareAlike 2.5 India license

Learning ObjectivesAfter interacting with this Learning Object, the learner will be able to:• Explain operations and applications of combinational logic gates

Definitions of the components/Keywords:

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1 1. LOGIC: Reasoning conducted or assessed according to strict principles of

validity.

2. LOGIC GATE: The gate is a digital circuit with one or more inputs but only one

output that can be activated by particular combinations of inputs.

3. INPUT: The signal which is given to get the desired output

4. OUTPUT: Anything that comes out

5. XOR GATE:The output is high (1) if its inputs are at opposite states.

6. XNOR GATE: :The output is high (1) if its inputs are at same states.

7. TRUTH TABLE : A truth table is a table that shows all the input-output

possibilities of a logic circuit.

8. LOGICAL EXPRESSION: A logical expression consists of one or more logical

operators and logical, numeric, or relational operands.

9. LOGICAL LAWS: laws which govern the logical circuits.

• The operation of a logic gate can be easily understood with the help of Truth Table.


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1 • Combinational Logic Circuits are made up from basic logic NAND, NOR or NOT gates that are "combined" or connected together to produce more complicated switching circuits. These logic gates are the building blocks of combinational logic circuits.•The Ex-OR and Ex-NOR gates are combinational gates.•The Ex-OR gate is an abbreviation for Exclusive-OR gate. It can also be written as XOR.• Any combinational circuit can be implemented with only NAND and NOR gates.•These gates are particularly useful in arithmetic operations as well as error-detection and correction circuits.

• Boolean algebra is based on a number of properties that govern what manipulations can be performed on variables. The basic properties are shown below. For animator- below table must be shown after this statement• Boolean variables can only take on one of two values, namely True and False.• The values are called by different names. The most common pairs of names for True are {True, T, High, HI, H, 1} and the most common names for False are (False, F, Low, LO, L, 0}. • In this LO, we use 0 and 1


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1


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Symbols and expressions used to represent logic gates:

Ex - OR:

This can be expressed as: X = A AND NOT B OR NOT A AND B

Ex – NOR:

This can be expressed as: X = A AND B OR NOT A AND NOT B

B

A X = A • B = A + B = A B + A B

B

AX = A + B = A B + A B

Input Output

A+B = A∙B

A∙B = A+B

• De Morgan's Theorems are one among the most powerful and useful theorems of Boolean Algebra. These theorems stem directly from the duality of Boolean logic and can be expressed as follows.

Description of the gates:

Ex-OR: The XOR gate is a digital logic gate that produces a HIGH (1) output only when its inputs are at opposite states. A LOW(0) output occurs when inputs are same. The XOR operation is denoted with symbol +


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1

B

A X = A + B

Input 1 Input 2 Output

A B

0 0 0

0 1 1

1 0 1

1 1 0

Description of the gates: Ex-NOR: The XNOR gate is a digital logic gate that implements XOR function with a complemented (inverted) output. A LOW(0) output occurs when inputs are at opposite states. The gate produces a HIGH (1) output only when its inputs are same. Therefore XNOR gate is often referred to as the Equivalence gate.

- It is also referred to as the coincidence operation because it produces the output of 1 when its inputs coincide in value; i.e., they are both 0 or both 1.

- The XNOR operation is denoted with symbol

• A small circle or a bubble at the end of an XOR gate is used to signify the XNOR function.


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1

B

AX = A • B = A + B

Input 1 Input 2 Output

A B

0 0 1

0 1 0

1 0 0

1 1 1

•

Master Layout

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1

• Give START, PAUSE and STOP buttons• Give a slider to control the speed of animation• Give a STEPPER button that allows the user to follow the simulation procedure step by step. After every step the simulation pauses until the STEPPER button is pressed•Give a text area to display the status of the simulation

Fig. A

Simulation Area

GATE Menu XOR XNOR

• Seven radio buttons Basic concept of GATE Electronic Logic using Switches Applications of XOR Implementation of XOR using NAND Implementation of XOR using NOR Implementation of XNOR using NAND Implementation of XNOR using NOR

Input A

Input B

Output

GATE Symbol

Truth Table:

Control Area

10Legend: Toggle Switch

‘1’ - input is HIGH ‘0’- input is LOW

For XOR refer slide 10 – 72 For XNOR refer slide 73 – 124 Toggle Switch: a switch that

has two positions . It has a mechanism to do two things one at a time

**The demo itself is interactivity in this LO**

Step 1: 1

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XOR

Instruction for the animator Text to be displayed in the working area (DT)

• Initially only Gate menu is enabled and all other buttons are disabled (check slide 125 for more details)

• When user selects any Gate from Gate Menu, Show the respective gate symbol and display as it is shown in master layout fig. A

• Once XOR gate is selected, then enable/highlight five radio buttons – basic concept of the gate, electronic logic, application of XOR, implementation of XOR using NAND and implementation of XOR using NOR

• Follow the steps as shown in stepwise process.

• If ‘basic concept of the GATE’ radio button is selected, the process should follow the steps shown from slide 10 – 18

• If ‘electronic logic’ is selected, the process should follow the steps shown from slide 19 – 27

• If ‘application of XOR’ is selected, the process should follow the steps shown from slide 28 – 41

• If ‘implementation of XOR using NAND’ is selected, the process should follow the steps shown from slide 42 – 55

• If ‘implementation of XOR using NOR’ is selected, the process should follow the steps shown from slide 56 – 72

• The text in DT should appear in parallel to the figures

• Select Gate from Gate menu

• The 2-input XOR gate symbol and its truth table.

• 1 represents input HIGH

• 0 represents input LOW

Input A Input B Output

Input A

Input B

Output

10Legend:

Truth Table:


Step 2: 1

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Basic concept of the XOR gate


• If ‘basic concept of gate’ radio button is selected, the process should follow the steps shown from slide 10 - 18

• Initially show the figure in step 1, then highlight small grey squares in the switch and display 0 and 0 near the horizontal lines of the gate as shown

• Also show 0 and 0 in the truth table

• Then display 0 + 0 inside the gate

• The text in DT should appear in parallel to the figure

• Input A is 0

• Input B is 0


0 0Input A

Input B

Output0

0

10Legend:

Step 3: 1

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4Instruction for the animator Text to be displayed in the working area (DT)

• Then show the output in the given square box.

• Change the color of the box and output as well


• The output is 0


0 0 0Input A

Input B

Output

0

0

0

10Legend:

Step 4: 1

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4 Instruction for the animator Text to be displayed in the working area (DT)

• Then highlight grey square in the switch at input A and red square at input B

• Display inputs as 0 and 1 near the gate as shown

• Also show 0 and 1 in the truth table as shown and keep filling the truth table along with old values



• Input A is 0

• Input B is 1


0 0 0

0 1

Input A

Input B

Output

10Legend:

0

1

Step 5: 1

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• The output is 1


0 0 0

0 1 1

Input A

Input B

Output

10Legend:

1

0

1

Step 6: 1

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2


• Then highlight red square in the switch at input A and grey square at input B





• Input A is 1

• Input B is 0

10Legend:


0 0 0

0 1 1

1 0

Input A

Input B

Output1

0

Step 7: 1

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• The output is 1

10Legend:


0 0 0

0 1 1

1 0 1

Input A

Input B

Output

1

1

0

Step 8: 1

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• Then highlight red square in the switch at input A and red square at input B





• Input A is 1

• Input B is 1

10Legend:


0 0 0

0 1 1

1 0 1

1 1

Input A

Input B

Output1

1

Step 9: 1

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• The output is 0

• Select any radio button to view different ways of implementing XOR

10Legend:


0 0 0

0 1 1

1 0 1

1 1 0

Input A

Input B

Output

0

1

1

Step 10: 1

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XOR gate using Switches


• If electronic logic radio button is selected, the process should follow the steps shown from slide 19 - 27

• Initially show the figure in step 10

• Input is given only in the toggle switch. Depending upon the input given, the switch position must change. The movement of switch must be smooth


• Performing XOR logic using electronic switches

• Switch open represents input 0

• Switch close represents input 1

• LED off represents 0

• LED on represents 1

10Legend:

Input A Input B OutputInput A

Input B

Output

Using Electronic switches

LED

Battery

Switch open

Switch closed

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• Then highlight small grey squares in the switch and display 0 and 0 near the horizontal lines of the gate

• Also show 0 and 0 in the truth table as shown

• The black lines at switch A and switch B should move to meet 0 and 0

• The red circles in the switches represent supply from battery


• Switch A is at 0 representing input A as 0

• Switch B is at 0 representing input B as 0

Step 11:

10Legend:


0 0Input A

Input B

Output0

0


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• Show LED in OFF state




• As switch A is at 0 and switch B is at 0, there is no continuous path.

• Therefore there is no supply of power for LED to glow

• So the LED is OFF representing 0

Step 12:

10Legend:


0 0 0Input A

Input B

Output0

0

0


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4Instruction for the animator Text to be displayed in the working

area (DT)




• The black line at switch A should be moved to 0 and the line at switch B should be moved to 1




Step 13: Input A Input B Output

0 0 0

0 1

Input A

Input B

Output

10Legend:

0

1


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• Show LED glowing that represents LED in ON state




• As switch A is at 0 and switch B is at 1, there is a continuous path for the current to flow thereby supplying power for LED to glow

• So the LED is ON representing 1


0 0 0

0 1 1

Input A

Input B

Output

10Legend:

1

0

1Using Electronic switches

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4Instruction for the animator Text to be displayed in the

working area (DT)









0 0 0

0 1 1

1 0

Input A

Input B

Output1

0

10Legend:


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• As switch A is at1 and switch B is at 0, there is a continuous path for the current to flow thereby supplying power for LED to glow



0 0 0

0 1 1

1 0 1

Input A

Input B

Output

1

1

0

10Legend:


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• The black lines at switch A and switch B should move to meet 1 and 1





0 0 0

0 1 1

1 0 1

1 1

Input A

Input B

Output1

1

10Legend:


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• Select any radio button to view different ways of implementing XOR gate


0 0 0

0 1 1

1 0 1

1 1 0

Input A

Input B

Output

0

1

1

10Legend:


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XOR - Application


• If ‘application of XOR’ radio button is selected, the process should follow the steps shown from slide 28 – 41


• Input is given only in the toggle switch. Depending upon the input given, the respective button must be highlighted.(refer next slide for details)

• Show the switch on the switch board is ON

• * You can change the shape or model of power adaptor, vending machine. This image is given for reference to understand how the figure should look like in the LO


• Implementation of XOR logic using vending machine

• Tea button OFF represents input 0

• Tea button ON represents input 1

• Coffee button OFF represents 0

• Coffee button ON represents 1

• Empty cup represents 0

• Filled cup represents 1

Step 19:

Input A

Input B

Output

10Legend:


Tea button

Coffee button

Cup

When the power supply is ON,Tea button : 0 - OFF, 1- ONCoffee button: 0- OFF, 1- ONCup: 0- Empty, 1- Full

Refer slide 29(next slide) for figure

Tea button

Coffee button

Power adaptor

Switch is ON

Switch board

Empty cup

Power button (it should be ON always)

When • A-0 and B-0 are selected, no buttons are selected and no coffee/ tea in the cup• A-0 and B-1 are selected, highlight coffee(orange) button or show a hand pressing coffee button and the cup is filled with coffee(show the flow also)• A-1 and B-0 are selected, highlight tea(green) button or show a hand pressing tea button and the cup is filled with tea(show the flow also)• A-1 and B-1 are selected, highlight both buttons or show a hand pressing both buttons but no coffee/tea in the cup

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• Show that no button on the vending machine is pressed


• Tea button OFF represents input A as 0

• Coffee button OFF represents input B as 0

Step 20:

10Legend:

Input A

Input B

Output0

0


Tea button

Coffee button

Cup

OFF (0) OFF (0)


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• The cup should be shown empty




• As no button is pressed on the machine, no liquid comes out and the cup is empty

• So the empty cup represents 0

Step 21:

10Legend:

Input A

Input B

Output0

0

0


Tea button

Coffee button

Cup

OFF (0) OFF (0) Empty (0)

Refer slide 31(previous slide) for figure

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4Instruction for the animator Text to be displayed in the working area

(DT)




• Show that coffee button is pressed (if you can, show a hand pressing the button)


• Tea button is OFF representing input A as 0

• Coffee button is pressed representing input B as 1

Step 22: Input A

Input B

Output0

1

10Legend:


Tea button

Coffee button

Cup


OFF (0) ON (1)


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• Show that cup is filled with coffee (show coffee falling into the cup)




• As coffee button is pressed on the machine, the cup is filled up with coffee

• So the filled cup represents 1

Step 23:

Input A

Input B

Output

1

0

1

10Legend:


Tea button

Coffee button

Cup


OFF (0) ON (1) FULL (1)

Refer slide 34(previous slide) for figure

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Step 24: Input A

Input B

Output1

0

10Legend:





• Show that tea button is pressed (if you can, show a hand pressing the button)


• Tea button is pressed representing input A as 1

• Coffee button is OFF representing input B as 0


Tea button

Coffee button

Cup


OFF (0) ON (1) FULL (1)

ON (1) OFF (0)


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Step 25:

Input A

Input B

Output1

0

1

10Legend:


• Show that cup is filled with tea (show tea falling into the cup)




• As tea button is pressed on the machine, the cup is filled up with tea

• So the filled cup represents 1


Tea button

Coffee button

Cup


OFF (0) ON (1) FULL (1)

ON (1) OFF (0) FULL (1)Refer slide 37(previous slide) for figure

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• Show that both buttons on the vending machine are pressed


• Tea button ON represents input A as 1

• Coffee button ON represents input B as 1

Step 26:

Input A

Input B

Output1

1

10Legend:


Tea button

Coffee button

Cup


OFF (0) ON (1) FULL (1)

ON (1) OFF (0) FULL (1)

ON (1) ON (1)Refer slide 40(next slide) for figure

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• The cup should be shown empty




• As both buttons are pressed on the machine, no liquid comes out and the cup is empty

• This is because an invalid signal is sent to the machine

• So the empty cup represents 0

• Thus using a vending machine, XOR logic is explained

Step 27: Input A

Input B

Output1

1

0

10Legend:


Tea button

Coffee button

Cup


OFF (0) ON (1) FULL (1)

ON (1) OFF (0) FULL (1)

ON (1) ON (1) Empty (0)Refer slide 40(previous slide) for figure

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Implementation of Ex-OR using NAND


• Initially show figure in step 28

• The order of steps (20 steps) should follow the sequence given in table. Also see slides (44 – 55) for more details

• There must be delay between each step (for delay refer table in next slide)

• All the minimizations should appear below the figure

• When showing minimization of expressions given in next slide, all the steps should come one after the other. (with a delay of 1 second between steps)


• Implementation of XOR using NAND

• Inputs A and B

• Output X

• For the minimization rules, refer Boolean algebra and De Morgan’s Theorems

• Thus using NAND, XOR logic is implemented

Step 28:

Expression Set - 1

Expression Set - 2

Expression Set - 3

Inputs Output

Equation - 1

Sr. No Sequence of steps to be displayed Delay between each step

1 Figure in black -

2 Text ‘Inputs’ After 2 seconds

3 A and B in red After 2 seconds

4 Color change of input lines After 2 seconds

5 Color change of output line After 2 seconds

6 Equation – 1 After 2 seconds



9 Expression set – 1 Show at a time

10 Minimization of expression set – 1

11 Color change of input lines

12 Color change of output line

13 Expression set – 2




17 Text ‘Output’

18 ‘X’ in red



Steps (4 – 20): refer slides (44 – 55)

Expression Set - 1

Expression Set - 1

Expression Set - 1Expression Set - 1

In expression set -1, first display only left hand side of = (equal to) in the figure. Then minimization has to be shown After minimization is done then show the right hand side expression in the figure

Expression Set - 2

Expression Set - 2


Expression Set - 3

Expression Set - 3


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Implementation of Ex-OR using NORStep 29:

Expression Set - 1

Expression Set - 2

Inputs Output








• Implementation of XOR using NOR

• Inputs A and B

• Output X


• Thus using NOR, XOR logic is implemented

Equation - 1

Equation - 2

Equation - 3


1 Figure in black -



















20 ‘X’ in red




Expression Set - 1

Expression Set - 1


Expression Set - 2

Expression Set - 2


Step 1: 1

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XNOR


• Initially only Gate menu is enabled and all other buttons are disabled (check slide 125 for more details)

• When user selects any Gate from Gate Menu, Show the respective gate symbol and display as it is shown in master layout fig. A

• Once the gate is selected, enable/highlight four radio buttons – basic concept of the gate, electronic logic, implementation of XNOR using NAND, implementation of XNOR using NOR

• Follow the steps as shown in stepwise process.

• If ‘basic concept of the GATE’ radio button is selected, the process should follow the steps shown from slide 73 – 81

• If ‘electronic logic’ is selected, the process should follow the steps shown from slide 82 – 90

• If ‘implementation of XNOR using NAND’ is selected, the process should follow the steps shown from slide 91 – 110

• If ‘implementation of XNOR using NOR’ is selected, the process should follow the steps shown from slide 111 – 124

• The text in DT should appear in parallel to the figures

• Select Gate from Gate menu

• The 2-input XNOR gate symbol and its truth table.

• 1 represents input HIGH

• 0 represents input LOW


Input A

Input B

Output

10Legend:

Truth Table:


Step 2: 1

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Basic concept of the XNOR gate


• If ‘basic concept of gate’ radio button is selected, the process should follow the steps shown from slide 73 – 81

• Initially show the figure in step 1, then highlight small grey squares in the switch and display 0 and 0 near the horizontal lines of the gate as shown

• Also show 0 and 0 in the truth table

• Then display 0 . 0 inside the gate


• Input A is 0

• Input B is 0


0 0Input A

Input B

Output0

0

10Legend:

Step 3: 1

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• The output is 1


0 0 1Input A

Input B

Output

1

0

0

10Legend:

Step 4: 1

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2

4 Instruction for the animator Text to be displayed in the working area (DT)






• Input A is 0

• Input B is 1


0 0 1

0 1

Input A

Input B

Output

10Legend:

0

1

Step 5: 1

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2





• The output is 0


0 0 1

0 1 0

Input A

Input B

Output

10Legend:

0

0

1

Step 6: 1

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2







• Input A is 1

• Input B is 0

10Legend:


0 0 1

0 1 0

1 0

Input A

Input B

Output1

0

Step 7: 1

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2





• The output is 0

10Legend:


0 0 1

0 1 0

1 0 0

Input A

Input B

Output

0

1

0

Step 8: 1

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2







• Input A is 1

• Input B is 1

10Legend:


0 0 1

0 1 0

1 0 0

1 1

Input A

Input B

Output1

1

Step 9: 1

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• The output is 1

• Select any radio button to view different ways of implementing XNOR gate

10Legend:


0 0 1

0 1 0

1 0 0

1 1 1

Input A

Input B

Output

1

1

1

Step 10: 1

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4

XNOR gate using Switches


• If electronic logic radio button is selected, the process should follow the steps shown from slide 82 – 90


• Input is given only in the toggle switch. Depending upon the input given, the switch position must change. The movement of switch must be smooth


• Performing XNOR logic using electronic switches

• Switch open represents input 0

• Switch close represents input 1

• LED off represents 0

• LED on represents 1

10Legend:

Input A Input B OutputInput A

Input B

Output


LED

Battery

Switch open

Switch closed

1

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2


area (DT)



• The black lines at switch A and switch B should be moved up to meet 0 and 0




Step 11:

10Legend:


0 0Input A

Input B

Output0

0


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4






• As both the switches are at 0, there is a continuous path for the current to flow thereby supplying power for LED to glow


Step 12:

10Legend:


0 0 1Input A

Input B

Output0

0

1


1

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2


area (DT)









0 0 1

0 1

Input A

Input B

Output

10Legend:

0

1


1

5

3

2

4










0 0 1

0 1 0

Input A

Input B

Output

10Legend:

0

0

1Using Electronic switches

1

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2


area (DT)









0 0 1

0 1 0

1 0

Input A

Input B

Output1

0

10Legend:


1

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4










0 0 1

0 1 0

1 0 0

Input A

Input B

Output

0

1

0

10Legend:


1

5

3

2


area (DT)




• The black lines at switch A and switch B should be moved down to meet 1 and 1





0 0 1

0 1 0

1 0 0

1 1

Input A

Input B

Output1

1

10Legend:


1

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2






• As both the switches are at 1, there is a continuous path for the current to flow thereby supplying power for LED to glow


• Select any radio button to view different ways of implementing XNOR gate


0 0 1

0 1 0

1 0 0

1 1 1

Input A

Input B

Output

1

1

1

10Legend:


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2

4

Implementation of Ex-NOR using NANDStep 19:








• Implementation of XNOR using NAND

• Inputs A and B

• Output X


• Thus using NAND, XNOR logic is implemented


Expression Set - 1

Expression Set - 2

Expression Set - 3

Expression Set - 4

Inputs Output

Equation- 1


1 Figure in black -






















21 Text ‘Output’ After 2 seconds

22 ‘X’ in red After 2 seconds




Expression Set - 1

Expression Set - 1


Expression Set - 2

Expression Set - 2


Expression Set - 3

Expression Set - 3


Refer next slide for expression set - 4

Expression Set - 4

Minimization of expressions for the animation (XNOR using NAND):

Expression Set - 4


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Implementation of Ex-NOR using NORStep 20: Expression Set - 1

Expression Set - 2

Expression Set - 3

Inputs Output



• The order of steps (20 steps) should follow the sequence given in table. Also see slides (113 – 124 ) for more details





• Implementation of XNOR using NOR

• Inputs A and B

• Output X


• Thus using NOR, XNOR logic is implemented

Equation- 1


1 Figure in black -

















18 ‘X’ in red




Expression Set - 1

Expression Set - 1


Expression Set - 2

Expression Set - 2


Expression Set - 3

Expression Set - 3


Introduction

Credits

125

Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…

(Further Reading)

Try it yourself

Interactivity:

Analogy

Slide 1

Slide 3

Slide 125 – 129

Slide 130

Electrical Engineering

•The demo itself is interactivity in this LO• Initially only Gate menu must be enabled, all other buttons should be disabled• Text “ Select any gate from Gate Menu” is to be displayed in DT area •When user selects any Gate from Gate Menu, Show the respective gate symbol, empty truth table as shown in master layout fig. A• After gate is selected, then enable respective radio buttons • Also display the text in DT area “ Select any radio button to view different ways of implementing ____ logic” (the blank is to be filled depending upon the respective gate selected) • After any one radio button is selected, enable START button• After START is pressed, enable PAUSE, Auto-run, Stepper, RESET/STOP buttons • Display the text “ Press Auto-run or Stepper buttons to view the animation” • Then display the text “ Click on red or grey buttons of toggle switch to give input” •Input –

if user clicks red, input should be displayed as 1 if user clicks grey, input should be displayed as 0

10Legend: Toggle switch

Questionnaire1. Which of the figures (a to d) is equivalent to figure A

Answers:

a) c)

b) d)

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3Fig. A

***Answers are given in red

Questionnaire2. What is X (output waveform) if A and B (input waveforms) are given

as shown in figure?

Answers:

a)

b)

c)

d)

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3

Questionnaire

3. Which of the figures (a to d) is the De Morgan’s equivalent of figure A?

Answers:

a) b)

c) d)

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Fig. A

Questionnaire4. Consider a staircase having an overhead light. There are two toggle switches A and B (one each on one level of the staircase) to turn the light ON and OFF. (See fig. A) – next slide Initially both the switches are in ‘0’ state i.e., OFF. The light is switched ON using switch B but it is switched OFF using switch A. Which logic gate can be used to implement the above logic?

Hint: Compute truth table of the logic gate assuming switches A and B as its inputs and light as its output

Answers:

a) NAND

b) NOR

c) XNOR

d) XOR

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Fig. A

Questionnaire5. Which of the following are TRUE about XNOR?

I. It produces a LOW output when its inputs are same

II. It produces a HIGH output when its inputs are same

III. It is also referred to as Equivalence gate

IV. Output of XNOR is same as the inverted output of XOR

Answers:

a) I, III,IV

b) II, III, IV

c) III,IV

d) Only I

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Links for further readingReference websites:http://en.wikipedia.org/wiki/Logic_gate

http://www.wisc-online.com/Objects/ViewObject.aspx?ID=dig1302

http://www.mekanizmalar.com/logic_gates.html

http://www.williamson-labs.com/480_logic.htm

http://isweb.redwoods.cc.ca.us/instruct/calderwoodd/diglogic/

http://www.w3professors.com/Pages/Courses/DCLD/Digital-Circuits-and-Logic-Designs.html

http://homepages.inf.ed.ac.uk/rbf/HIPR2/arthops.htm

Books:Digital Systems: Principles and Applications by Ronald-J.-Tocci

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