# What are the examples of logic gates

## Technical computer Science

Logic gate

Thorsten Thormählen

November 24, 2020

Part 3, Chapter 3

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### notation

Type | font | Examples |
---|---|---|

Variables (scalars) | italic | $ a, b, x, y $ |

Functions | upright | $ \ mathrm {f}, \ mathrm {g} (x), \ mathrm {max} (x) $ |

Vectors | bold, elements line by line | $ \ mathbf {a}, \ mathbf {b} = \ begin {pmatrix} x \ y \ end {pmatrix} = (x, y) ^ \ top, $ $ \ mathbf {B} = (x, y, z) ^ \ top $ |

Matrices | typewriter | $ \ mathtt {A}, \ mathtt {B} = \ begin {bmatrix} a & b \ c & d \ end {bmatrix} $ |

amounts | calligraphic | $ \ mathcal {A}, B = \ {a, b \}, b \ in \ mathcal {B} $ |

Number ranges, coordinate spaces | double crossed | $ \ mathbb {N}, \ mathbb {Z}, \ mathbb {R} ^ 2, \ mathbb {R} ^ 3 $ |

### content

- Gate symbols
- Realization of a Boolean function using logic gates
- Practical example: building a logic circuit with ICs

### Gate symbols according to DIN 40900

### Gate symbols, US ANSI 91

### Gate symbols

- The DIN symbols are used throughout this lecture
- The more recent recommendation of the International Electrotechnical Commission in IEC 60617-12 essentially corresponds to the rectangular DIN symbols
- However, the US standard is very common in practice (especially in English-language literature)

### Representation of switching networks with gates

- In order to build up switching networks from several gates, inputs and outputs of gates can be connected with solid lines running at right angles
- A right-angled branch assumes that there is a connection between the lines
- This can also be clarified with a filled in point
- In order to mark connected intersections, the completed point is absolutely necessary, otherwise the lines are interpreted as not connected.
- The fact that two crossing lines are not connected can also be made clear by a semicircle

### More compact representation

- Both inputs and outputs can be inverted by adding the (unfilled) negation circle.
- Example: $ y = \ lnot (\ lnot a \ land b) $

- AND and OR gates can also have more than two inputs
- An OR gate with $ n $ inputs implements the expression $ y = x_1 \ lor x_2 \ lor \ dots \ lor x_n $
- An AND gate with $ n $ inputs implements the expression $ y = x_1 \ land x_2 \ land \ dots \ land x_n $
- Example: $ y = (a \ lor b \ lor c) \ land \ lnot d \ land \ lnot e $

### Practical example: building a logic circuit with ICs

CD4572UB

- In the following we want to build a real circuit that contains some logic gates
- We use the CMOS IC CD4572UB from Texas Instruments, which provides 4 inverters, 1 NAND and 1 NOR gate

### Practical example: building a logic circuit with ICs

- We will use LEDs ("Light-Emitting Diodes") to visualize the states of the inputs and outputs
- For wired LEDs, the cathode (-) is the shorter leg and the longer leg is the anode (+)
- An LED must always be operated with a series resistor that sets the current through the LED

### Ohm's law

- To calculate the magnitude of the resistance, we need Ohm's law
- Ohm's law makes a statement about voltage and current strength at a resistor:
- Voltage $ U $: force on charge carrier, unit volt $ [\ mathrm {V}] $
- Amperage $ I $: charge carriers flowing through per unit of time, unit amperes $ [\ mathrm {A}] $

- Ohm's law says that the current $ I $ flowing through a resistor $ R $ is proportional to the voltage $ U $ that is dropped across the resistor
$ U = R \ cdot I \ Leftrightarrow I = \ frac {U} {R} \ Leftrightarrow R = \ frac {U} {I} $

- Resistance $ R $: proportionality factor between voltage and current strength, unit ohm $ [\ Omega] $

- How big is the current $ I $ in this circuit?

- Answer: $ I = \ frac {U} {R} = \ frac {4.5 \, \ mathrm {V}} {200 \, \ Omega} = 0.0225 \, \ mathrm {A} = 22.5 \, \ mathrm {mA} $

### Ohm's law

- Voltage divider
- The same current $ I_0 $ flows everywhere in the circuit above
- According to Ohm's law:
$ I_0 = \ frac {U_1} {R_1} $ and $ I_0 = \ frac {U_2} {R_2} $ and $ I_0 = \ frac {U_0} {R_1 + R_2} $

- The conversion results in the ratio of the voltages $ U_1 $ and $ U_2 $:
$ \ frac {U_1} {R_1} = \ frac {U_2} {R_2} \ Leftrightarrow \ frac {U_1} {U_2} = \ frac {R_1} {R_2} $

- Flow divider
- In the lower circuit, the same voltage $ U_0 $ drops across both resistors
- This results in the ratio of the currents $ I_1 $ and $ I_2 $:
$ U_0 = R_1 \ cdot I_1 = R_2 \ cdot I_2 \ Leftrightarrow \ frac {I_1} {I_2} = \ frac {R_2} {R_1} $

### Calculating the series resistance of an LED

- The data sheet of the LED we use shows that it should be operated with a current of $ I_0 = 20 \, \ mathrm {mA} $. According to the data sheet, in this case $ U_2 = 2.25 \, \ mathrm {V} $ drops above the LED.
- With a voltage supply with $ U_0 = 4.5 \, \ mathrm {V} $, this results in $ R_1 $ for the series resistor
$ \ begin {align} I_0 & = \ frac {U_1} {R_1} = \ frac {U_0 - U_2} {R_1} \ \ Leftrightarrow R_1 & = \ frac {U_0 - U_2} {I_0} \ & = \ frac {4.5 \, \ mathrm {V} - 2.25 \, \ mathrm {V}} {20 \, \ mathrm {mA}} = 112.5 \, \ Omega \ end {align} $

### Practical example: building a logic circuit with ICs

- The circuit should be implemented with a breadboard
- In the columns for the power supply ("+" or "-"), the slots are vertically connected to one another
- Otherwise the slots are horizontally connected to each other, respectively ("a" to "e") and ("f" to "j")
- A 4.5 volt flat battery is used as the power supply

### Practical example: NOT gate

- When the button is open, the yellow LED lights up and the red one is off
- When the button is closed, the red LED lights up and the yellow one is off
- Example application alarm system: button determines whether the door is open or closed; red LED indicates alarm is switched on; yellow LED alerts the security service

### Practical example: NAND gate

- The red LEDs indicate the status of the two buttons
- The yellow LED only does not light up when both buttons are closed
- Example application: alarm system for two windows

### Practical example: building a logic circuit with ICs

- This video shows (in rapid succession) the structure of the NOT gate and NAND gate circuit from the previous foils on a breadboard

### Practical example: building a logic circuit with ICs

- In the circuits shown, the inputs of the logic IC were each provided with a so-called pull-down resistor of $ 10 \, \ mathrm {k \ Omega} $
- If the switches are open, the input would otherwise have an undefined potential and the behavior at the output would be random
- When the switch is open, the pull-down resistor pulls the input towards ground
- When the switch is closed, the supply voltage is applied to the input and a small leakage current flows through the pull-down resistor $ I _ {\ tiny \ text {loss}} = \ frac {4,5 \ mathrm {V}} {10 \, \ mathrm {k \ Omega}} = 0.45 \, \ mathrm {mA} $

### Realization with n-channel field effect transistors

- Inverters, NAND and NOR gates, as they are used in the IC CD4572UB, can be implemented e.g. by means of n-channel field effect transistors, which we got to know in chapter 1.2 "History"

NAND

$ x $ | $ y $ | $ z $ |

0 | 0 | 1 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

NOR

$ x $ | $ y $ | $ z $ |

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 0 |

### Realization with CMOS technology

- Due to the resistance, however, a relatively large power loss occurs, which is why today's CMOS technology uses p- and n-channel field effect transistors (the CD4572UB is also a CMOS IC)
- The n-channel FET switches through at logic 1, while the p-channel FET switches through at logic 0

### Any questions?

Suggestions or suggestions for improvement can also be sent to me by e-mail: Contact

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