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

TypefontExamples
Variables (scalars)italic$ a, b, x, y $
Functionsupright$ \ mathrm {f}, \ mathrm {g} (x), \ mathrm {max} (x) $
Vectorsbold, elements line by line$ \ mathbf {a}, \ mathbf {b} = \ begin {pmatrix} x \ y \ end {pmatrix} = (x, y) ^ \ top, $ $ \ mathbf {B} = (x, y, z) ^ \ top $
Matricestypewriter$ \ mathtt {A}, \ mathtt {B} = \ begin {bmatrix} a & b \ c & d \ end {bmatrix} $
amountscalligraphic$ \ mathcal {A}, B = \ {a, b \}, b \ in \ mathcal {B} $
Number ranges, coordinate spacesdouble 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 $
001
011
101
110

NOR

$ x $$ y $$ z $
001
010
100
110

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?

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