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electrical_engineering_1:preparation_properties_proportions [2023/03/16 14:09] mexleadmin optimized the language |
electrical_engineering_1:preparation_properties_proportions [2023/10/11 11:27] mexleadmin [Bearbeiten - Panel] |
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Zeile 1: | Zeile 1: | ||
- | ====== 1. Preparation, | + | # |
+ | |||
+ | ====== 1 Preparation, | ||
===== 1.1 Physical Proportions ===== | ===== 1.1 Physical Proportions ===== | ||
Zeile 7: | Zeile 9: | ||
By the end of this section, you will be able to: | By the end of this section, you will be able to: | ||
- | - know the basic physical quantities and the associated SI units. | + | - know the fundamental |
- | - know the most important prefixes. Be able to assign a power of ten to the respective abbreviation ($\rm{G}$, $\rm{M}$, $\rm{k}$, $\rm{d}$, $\rm{c}$, $\rm{m}$, $\rm{µ}$, $\rm{n}$). | + | - know the most important prefixes. Be able to assign a power of ten to the respective abbreviation (${\rm G}$, ${\rm M}$, ${\rm k}$, ${\rm d}$, ${\rm c}$, ${\rm m}$, ${\rm µ}$, ${\rm n}$). |
- insert given numerical values and units into an existing quantity equation. From this, you should be able to calculate the correct result using a calculator. | - insert given numerical values and units into an existing quantity equation. From this, you should be able to calculate the correct result using a calculator. | ||
- assign the Greek letters. | - assign the Greek letters. | ||
Zeile 31: | Zeile 33: | ||
^ Base quantity | ^ Base quantity | ||
- | | Time | Second | + | | Time | Second |
- | | Length | + | | Length |
- | | el. Current | + | | el. Current |
- | | Mass | Kilogram | + | | Mass | Kilogram |
- | | Temperature | + | | Temperature |
- | | amount of \\ substance | + | | amount of \\ substance |
- | | luminous \\ intensity | + | | luminous \\ intensity |
</ | </ | ||
</ | </ | ||
Zeile 43: | Zeile 45: | ||
* For practical applications of physical laws of nature, **physical quantities** are put into mathematical relationships. | * For practical applications of physical laws of nature, **physical quantities** are put into mathematical relationships. | ||
* There are basic quantities based on the SI system of units (French for Système International d' | * There are basic quantities based on the SI system of units (French for Système International d' | ||
- | * In order to determine the basic quantities quantitatively (quantum = Latin for "how big"), **physical units** are defined, e.g. $\rm{metre}$ for length. | + | * In order to determine the basic quantities quantitatively (quantum = Latin for //how big//), **physical units** are defined, e.g. ${\rm metre}$ for length. |
* In electrical engineering, | * In electrical engineering, | ||
- | * Each physical quantity is indicated by a product of **numerical value** and **unit**: \\ e.g. $I = 2~\rm{A}$ | + | * Each physical quantity is indicated by a product of **numerical value** and **unit**: \\ e.g. $I = 2~{\rm A}$ |
- | * This is the short form of $I = 2\cdot 1~\rm{A}$ | + | * This is the short form of $I = 2\cdot 1~{\rm A}$ |
* $I$ is the physical quantity, here: electric current strength | * $I$ is the physical quantity, here: electric current strength | ||
* $\{I\} = 2 $ is the numerical value | * $\{I\} = 2 $ is the numerical value | ||
- | * $ [I] = 1~\rm{A}$ is the (measurement) unit, here: $\rm{Ampere}$ | + | * $ [I] = 1~{\rm A}$ is the (measurement) unit, here: ${\rm Ampere}$ |
Zeile 55: | Zeile 57: | ||
==== derived quantities, SI units, and prefixes ==== | ==== derived quantities, SI units, and prefixes ==== | ||
- | * Besides the basic quantities, there are also quantities derived from them, e.g. $1~{{\rm{m}}\over{\rm{s}}}$. | + | * Besides the basic quantities, there are also quantities derived from them, e.g. $1~{{{\rm m}}\over{{\rm s}}}$. |
* SI units should be preferred for calculations. These can be derived from the basic quantities **without a numerical factor**. | * SI units should be preferred for calculations. These can be derived from the basic quantities **without a numerical factor**. | ||
- | * The pressure unit bar ($\rm{bar}$) is an SI unit. | + | * The pressure unit bar (${\rm bar}$) is an SI unit. |
- | * BUT: The obsolete pressure unit " | + | * BUT: The obsolete pressure unit " |
* To prevent the numerical value from becoming too large or too small, it is possible to replace a decimal factor with a prefix. These are listed in <tabref tab02>. | * To prevent the numerical value from becoming too large or too small, it is possible to replace a decimal factor with a prefix. These are listed in <tabref tab02>. | ||
Zeile 64: | Zeile 66: | ||
< | < | ||
^ prefix ^ prefix symbol ^ meaning^ | ^ prefix ^ prefix symbol ^ meaning^ | ||
- | | Yotta | $\rm{Y}$ | $10^{24}$ | + | | Yotta | ${\rm Y}$ | $10^{24}$ |
- | | Zetta | $\rm{Z}$ | $10^{21}$ | + | | Zetta | ${\rm Z}$ | $10^{21}$ |
- | | Exa | $\rm{E}$ | $10^{18}$ | + | | Exa | ${\rm E}$ | $10^{18}$ |
- | | Peta | $\rm{P}$ | $10^{15}$ | + | | Peta | ${\rm P}$ | $10^{15}$ |
- | | Tera | $\rm{T}$ | $10^{12}$ | + | | Tera | ${\rm T}$ | $10^{12}$ |
- | | Giga | $\rm{G}$ | $10^{9}$ | + | | Giga | ${\rm G}$ | $10^{9}$ |
- | | Mega | $\rm{M}$ | $10^{6}$ | + | | Mega | ${\rm M}$ | $10^{6}$ |
- | | Kilo | $\rm{k}$ | $10^{3}$ | + | | Kilo | ${\rm k}$ | $10^{3}$ |
- | | Hecto | $\rm{h}$ | $10^{2}$ | + | | Hecto | ${\rm h}$ | $10^{2}$ |
- | | Deka | $\rm{de}$ | $10^{1}$ | + | | Deka | ${\rm de}$ | $10^{1}$ |
</ | </ | ||
< | < | ||
^ prefix ^ prefix symbol ^ meaning^ | ^ prefix ^ prefix symbol ^ meaning^ | ||
- | | Deci | $\rm{d}$ | $10^{-1}$ | + | | Deci | ${\rm d}$ | $10^{-1}$ |
- | | Centi | $\rm{c}$ | $10^{-2}$ | + | | Centi | ${\rm c}$ | $10^{-2}$ |
- | | Milli | $\rm{m}$ | $10^{-3}$ | + | | Milli | ${\rm m}$ | $10^{-3}$ |
- | | Micro | $\rm{u}$, $µ$ | $10^{-6}$ | + | | Micro | ${\rm u}$, $µ$ | $10^{-6}$ |
- | | Nano | $\rm{n}$ | $10^{-9}$ | + | | Nano | ${\rm n}$ | $10^{-9}$ |
- | | Piko | $\rm{p}$ | $10^{-12}$ | + | | Piko | ${\rm p}$ | $10^{-12}$ |
- | | Femto | $\rm{f}$ | $10^{-15}$ | + | | Femto | ${\rm f}$ | $10^{-15}$ |
- | | Atto | $\rm{a}$ | $10^{-18}$ | + | | Atto | ${\rm a}$ | $10^{-18}$ |
- | | Zeppto | $\rm{z}$ | $10^{-21}$ | + | | Zeppto | ${\rm z}$ | $10^{-21}$ |
- | | Yocto | $\rm{y}$ | $10^{-24}$ | + | | Yocto | ${\rm y}$ | $10^{-24}$ |
</ | </ | ||
</ | </ | ||
Zeile 109: | Zeile 111: | ||
=== Quantity equations === | === Quantity equations === | ||
- | The vast majority of physical equations result in a physical unit that is not equal to $1$. | + | The vast majority of physical equations result in a physical unit that does not equal $1$. |
\\ \\ | \\ \\ | ||
- | Example: Force $F = m \cdot a$ with $[\rm{F}] = 1~kg \cdot {{\rm{m}}\over{\rm{s}^2}}$ | + | Example: Force $F = m \cdot a$ with $[{\rm F}] = 1~\rm kg \cdot {{{\rm m}}\over{{\rm s}^2}}$ |
\\ \\ | \\ \\ | ||
Zeile 127: | Zeile 129: | ||
This results in a dimensionless quantity relative to the reference value. | This results in a dimensionless quantity relative to the reference value. | ||
- | Example: | + | Example: |
As a reference the following values are often used: | As a reference the following values are often used: | ||
Zeile 140: | Zeile 142: | ||
<callout title=" | <callout title=" | ||
- | Let a body with the mass $m = 100~\rm{kg}$ be given. The body is lifted by the height $s=2~\rm{m}$. \\ | + | Let a body with the mass $m = 100~{\rm kg}$ be given. The body is lifted by the height $s=2~{\rm m}$. \\ |
What is the value of the needed work? | What is the value of the needed work? | ||
Zeile 148: | Zeile 150: | ||
Work = Force $\cdot$ displacement | Work = Force $\cdot$ displacement | ||
\\ $W = F \cdot s \quad\quad\quad\; | \\ $W = F \cdot s \quad\quad\quad\; | ||
- | \\ $W = m \cdot g \cdot s \quad\quad$ where $m=100~\rm{kg}$, $s=2~m$ and $g=9.81~{{\rm{m}}\over{\rm{s}^2}}$ | + | \\ $W = m \cdot g \cdot s \quad\quad$ where $m=100~{\rm kg}$, $s=2~m$ and $g=9.81~{{{\rm m}}\over{{\rm s}^2}}$ |
- | \\ $W = 100~kg \cdot 9.81 ~{{\rm{m}}\over{\rm{s}^2}} \cdot 2~\rm{m} $ | + | \\ $W = 100~kg \cdot 9.81 ~{{{\rm m}}\over{{\rm s}^2}} \cdot 2~{\rm m} $ |
- | \\ $W = 100 \cdot 9.81 \cdot 2 \;\; \cdot \;\; \rm{kg} \cdot {{\rm{m}}\over{\rm{s}^2}} | + | \\ $W = 100 \cdot 9.81 \cdot 2 \;\; \cdot \;\; {\rm kg} \cdot {{{\rm m}}\over{{\rm s}^2}} |
- | \\ $W = 1962 \quad\quad \cdot \quad\quad\; | + | \\ $W = 1962 \quad\quad \cdot \quad\quad\; |
- | \\ $W = 1962~\rm{Nm} = 1962~\rm{J} $ | + | \\ $W = 1962~{\rm Nm} = 1962~{\rm J} $ |
</ | </ | ||
Zeile 208: | Zeile 210: | ||
==== Exercises ==== | ==== Exercises ==== | ||
- | <panel type=" | + | {{tagtopic>chapter1_1& |
- | {{youtube> | + | |
- | </ | + | |
- | + | ||
- | <panel type=" | + | |
- | Convert the following values step by step: | + | |
- | * A vehicle speed of $80.00~\rm{km/ | + | |
- | * An energy of $60.0~\rm{J}$ in $\rm{kWh}$ ($1~\rm{J} = 1~\rm{Joule} = 1~\rm{Watt}\cdot \rm{second}$) <button size=" | + | |
- | * The number of electrolytically deposited single positively charged copper ions of $1.2~\rm{Coulombs}$ (a copper ion has the charge of about $1.6 \cdot 10^{-19}~\rm{C}$)< | + | |
- | * Absorbed energy of a small IoT consumer, which consumes $1~\rm{µW}$ uniformly in $10 ~\rm{days}$ <button size=" | + | |
- | </ | + | |
- | + | ||
- | <panel type=" | + | |
- | Convert the following values step by step: | + | |
- | How many minutes could an ideal battery with $10~\rm{kWh}$ operate a consumer with $3~\rm{W}$? | + | |
- | </ | + | |
- | + | ||
- | <panel type=" | + | |
- | Convert the following values step by step: | + | |
- | How much energy does an average household consume per day when consuming an average power of $500~\rm{W}$? | + | |
- | </ | + | |
===== 1.2 Introduction to the Structure of Matter ===== | ===== 1.2 Introduction to the Structure of Matter ===== | ||
Zeile 246: | Zeile 228: | ||
</ | </ | ||
- | * Explanation of the charge on the basis of the atomic models according to Bohr and Sommerfeld (see <imgref BildNr0> | + | * Explanation of the charge |
* Atoms consist of | * Atoms consist of | ||
* Atomic nucleus (with protons and neutrons) | * Atomic nucleus (with protons and neutrons) | ||
* Electron shell | * Electron shell | ||
* Electrons are carriers of the elementary charge $|e|$ | * Electrons are carriers of the elementary charge $|e|$ | ||
- | * elementary charge $|e| = 1.6022\cdot 10^{-19}~\rm{C}$ | + | * elementary charge $|e| = 1.6022\cdot 10^{-19}~{\rm C}$ |
* Proton is the antagonist, i.e. has the opposite charge | * Proton is the antagonist, i.e. has the opposite charge | ||
* Sign is arbitrarily chosen: | * Sign is arbitrarily chosen: | ||
Zeile 295: | Zeile 277: | ||
In the insulator, charge carriers are firmly bound to the atomic shells. | In the insulator, charge carriers are firmly bound to the atomic shells. | ||
- | \\ \\ \\ | + | \\ \\ |
Examples: | Examples: | ||
Zeile 304: | Zeile 286: | ||
==== Exercises ==== | ==== Exercises ==== | ||
- | <panel type=" | + | {{tagtopic>chapter1_2& |
- | How many electrons make up the charge of one coulomb? | + | |
- | </ | + | |
- | + | ||
- | <panel type=" | + | |
- | A balloon has a charge of $Q=7~\rm{nC}$ on its surface. How many additional electrons are on the balloon? | + | |
- | </ | + | |
===== 1.3 Effects of electric charges and current ===== | ===== 1.3 Effects of electric charges and current ===== | ||
Zeile 333: | Zeile 309: | ||
* first attempt (see <imgref BildNr1> | * first attempt (see <imgref BildNr1> | ||
* Two charges ($Q_1$ and $Q_2$) are suspended at a distance of $r$. | * Two charges ($Q_1$ and $Q_2$) are suspended at a distance of $r$. | ||
- | * Charges are generated by the high voltage source and transferred to the two test specimens | + | * Charges are generated by the high-voltage source and transferred to the two test specimens |
* Result | * Result | ||
- | * samples with same charges $\rightarrow$ | + | * samples with same charges $\rightarrow$ |
* samples with charges of different sign $\rightarrow$ attraction | * samples with charges of different sign $\rightarrow$ attraction | ||
* Findings | * Findings | ||
Zeile 346: | Zeile 322: | ||
Setup for own experiments \\ | Setup for own experiments \\ | ||
{{url> | {{url> | ||
- | Take a charge ($+1~\rm{nC}$) and position it. Measure the field across a sample charge (a sensor). | + | Take a charge ($+1~{\rm nC}$) and position it. Measure the field across a sample charge (a sensor). |
</ | </ | ||
< | < | ||
- | Experiment | + | Experiment |
{{youtube> | {{youtube> | ||
</ | </ | ||
- | * Qualitative investigation | + | * Qualitative investigation |
* two charges ($Q_1$ and $Q_2$) at distance $r$ | * two charges ($Q_1$ and $Q_2$) at distance $r$ | ||
* additional measurement of the force $F_C$ (e.g. via spring balance) | * additional measurement of the force $F_C$ (e.g. via spring balance) | ||
Zeile 370: | Zeile 346: | ||
<callout icon=" | <callout icon=" | ||
The Coulomb force (in a vacuum) can be calculated via. \\ $\boxed{ F_C = {{1} \over {4\pi\cdot\varepsilon_0}} \cdot {{Q_1 \cdot Q_2} \over {r^2}} }$ \\ | The Coulomb force (in a vacuum) can be calculated via. \\ $\boxed{ F_C = {{1} \over {4\pi\cdot\varepsilon_0}} \cdot {{Q_1 \cdot Q_2} \over {r^2}} }$ \\ | ||
- | where $\varepsilon_0 = 8.85 \cdot 10^{-12} \cdot ~{\rm{C}^2 \over {\rm{m}^2\cdot \rm{N}}} = 8.85 \cdot 10^{-12} \cdot ~{{\rm{As}} \over {\rm{Vm}}}$ | + | where $\varepsilon_0 = 8.85 \cdot 10^{-12} \cdot ~{{\rm C}^2 \over {{\rm m}^2\cdot |
</ | </ | ||
Zeile 411: | Zeile 387: | ||
* the above-mentioned conductor with a cross-section $A$ perpendicular to the conductor | * the above-mentioned conductor with a cross-section $A$ perpendicular to the conductor | ||
* the quantity of charges $\Delta Q = n \cdot e$, which in a certain period of time $\Delta t$, pass through the area $A$ | * the quantity of charges $\Delta Q = n \cdot e$, which in a certain period of time $\Delta t$, pass through the area $A$ | ||
- | * In the case of a uniform charge transport over a longer period | + | * In the case of a uniform charge transport over a longer period, i.e. direct current (DC), the following applies: |
- | * The amount of charges per time flowing through the surface is constant: \\ ${{\Delta Q} \over {\Delta t}} = const. = I$ | + | * The amount of charges per time flowing through the surface is constant: \\ ${{\Delta Q} \over {\Delta t}} = {\rm const.} = I$ |
* $I$ denotes the strength of the direct current. | * $I$ denotes the strength of the direct current. | ||
- | * The unit of $I$ is the SI unit $\rm{Ampere}$: $1~\rm{A} = {{1~\rm{C}}\over{1~\rm{s}}}$ . Thus, for the unit coulomb applies: $1~\rm{C} = 1~\rm{A} \cdot \rm{s}$ | + | * The unit of $I$ is the SI unit ${\rm Ampere}$: $1~{\rm A} = {{1~{\rm C}}\over{1~{\rm s}}}$ . Thus, for the unit coulomb applies: $1~{\rm C} = 1~{\rm A} \cdot {\rm s}$ |
<callout icon=" | <callout icon=" | ||
- | The current of $1~\rm{A}$ flows when an amount of charge of $1~\rm{C}$ is transported in $1~\rm{s}$ through the cross-section of the conductor. | + | The current of $1~{\rm A}$ flows when an amount of charge of $1~{\rm C}$ is transported in $1~{\rm s}$ through the cross-section of the conductor. |
- | Before 2019: The current of $1~\rm{A}$ flows when two parallel conductors, each $1~\rm{m}$ long and $1~\rm{m}$ apart, exert a force of $F_C = 0.2\cdot 10^{-6}~\rm{N}$ on each other. | + | Before 2019: The current of $1~{\rm A}$ flows when two parallel conductors, each $1~{\rm m}$ long and $1~{\rm m}$ apart, exert a force of $F_C = 0.2\cdot 10^{-6}~{\rm N}$ on each other. |
</ | </ | ||
Zeile 471: | Zeile 447: | ||
==== Exercises ==== | ==== Exercises ==== | ||
- | <panel type=" | + | {{tagtopic>chapter1_4& |
- | + | ||
- | < | + | |
- | < | + | |
- | </ | + | |
- | {{drawio> | + | |
- | </ | + | |
- | + | ||
- | Let the charge gain per time on an object be given. | + | |
- | * Determine the current $I$ from the $Q$-$t$-diagram <imgref BildNr3> and plot them into the diagram. | + | |
- | * How could you proceed if the amount of charge on the object changes non-linearly? | + | |
- | + | ||
- | </ | + | |
- | + | ||
- | <panel type=" | + | |
- | + | ||
- | How many electrons pass through a control cross-section of a metallic conductor, when the current of $40~\rm{mA}$ flows for $4.5~\rm{s}$? | + | |
- | + | ||
- | </ | + | |
===== 1.5 Voltage, Potential, and Energy ===== | ===== 1.5 Voltage, Potential, and Energy ===== | ||
Zeile 521: | Zeile 479: | ||
* The energy turnover is proportional to the amount of charge $q$ transported. | * The energy turnover is proportional to the amount of charge $q$ transported. | ||
* In many cases, the " | * In many cases, the " | ||
- | * V for Voltage is in the English literature often used to denote the unit $\rm{V}$ AS WELL AS the quantity $V$ (in German $U$ is used for the quantity): | + | * V for Voltage is in the English literature often used to denote the unit ${\rm V}$ AS WELL AS the quantity $V$ (in German $U$ is used for the quantity): |
* e.g. | * e.g. | ||
- | * $VCC = 5~\rm{V}$ : Voltage supply of an IC (__V__oltage __C__ommon __C__ollector), | + | * $VCC = 5~{\rm V}$ : Voltage supply of an IC (__V__oltage __C__ommon __C__ollector), |
- | * $V_{S+} = 15~\rm{V}$ : Voltage supply of an operational amplifier (__V__oltage __S__upply plus). | + | * $V_{S+} = 15~{\rm V}$ : Voltage supply of an operational amplifier (__V__oltage __S__upply plus). |
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
Zeile 578: | Zeile 536: | ||
<callout icon=" | <callout icon=" | ||
* Voltage is always a potential difference. | * Voltage is always a potential difference. | ||
- | * The unit of voltage is $\rm{Volt}$: $1~\rm{V}$ | + | * The unit of voltage is ${\rm Volt}$: $1~{\rm V}$ |
</ | </ | ||
<callout icon=" | <callout icon=" | ||
- | A voltage of $1~\rm{V}$ is present between two points if a charge of $1~\rm{C}$ undergoes an energy change of $1~\rm{J} = 1~\rm{Nm}$ between these two points. | + | A voltage of $1~{\rm V}$ is present between two points if a charge of $1~{\rm C}$ undergoes an energy change of $1~{\rm J} = 1~{\rm Nm}$ between these two points. |
- | From $W=U \cdot Q$ also the unit results: $1~\rm{Nm} = 1~ \rm{V} \cdot \rm{As} \rightarrow 1~ \rm{V} = 1~{{\rm{Nm}}\over{\rm{As}}}$ | + | From $W=U \cdot Q$ also the unit results: $1~{\rm Nm} = 1~ {\rm V} \cdot {\rm As} \rightarrow 1~ {\rm V} = 1~{{{\rm Nm}}\over{{\rm As}}}$ |
</ | </ | ||
Zeile 601: | Zeile 559: | ||
==== Exercises ==== | ==== Exercises ==== | ||
- | <panel type=" | + | # |
+ | # | ||
< | < | ||
Zeile 609: | Zeile 568: | ||
</ | </ | ||
- | Explain whether the voltages $U_{Batt}$, $U_{12}$ and $U_{21}$ in <imgref BildNr21> | + | Explain whether the voltages $U_{\rm Batt}$, $U_{12}$ and $U_{21}$ in <imgref BildNr21> |
- | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | |
- | </WRAP></WRAP></panel> | + | # |
+ | * Which terminal has the higher potential? | ||
+ | * From where to where does the arrow point? | ||
+ | # | ||
+ | |||
+ | |||
+ | # | ||
+ | * '' | ||
+ | * For $U_{\rm Batt}$: The arrow starts at terminal 1 and ends at terminal 2. So $U_{\rm Batt}=U_{12}>0$ | ||
+ | * $U_{21}<0$ | ||
+ | # | ||
+ | |||
+ | # | ||
Zeile 638: | Zeile 610: | ||
The reason for this conversion is the resistance e.g. of the conductor or other loads. | The reason for this conversion is the resistance e.g. of the conductor or other loads. | ||
- | A resistor is an electrical component with two connections (or terminals). Components with two terminals are called two-terminal | + | A resistor is an electrical component with two connections (or terminals). Components with two terminals are called two-terminal |
Zeile 647: | Zeile 619: | ||
In general, the cause-and-effect relationship is such that an applied voltage across the resistor produces the current flow. However, the reverse relationship also applies: as soon as an electric current flows across a resistor, a voltage drop is produced over the resistor. | In general, the cause-and-effect relationship is such that an applied voltage across the resistor produces the current flow. However, the reverse relationship also applies: as soon as an electric current flows across a resistor, a voltage drop is produced over the resistor. | ||
In electrical engineering, | In electrical engineering, | ||
+ | |||
+ | The values of the resistors are standardized in such a way, that there is a fixed number of different values between $1~\Omega$ and $10~\Omega$ or between $10~\rm k\Omega$ and $100~\rm k\Omega$. These ranges, which cover values up to the tenfold number, are called decades. In general, the resistors are ordered in the so-called {{wp>E series of preferred numbers}}, like 6 values in a decade, which is named E6 (here: $1.0~\rm k\Omega$, $1.5~\rm k\Omega$, $2.2~\rm k\Omega$, $3.3~\rm k\Omega$, $4.7~\rm k\Omega$, $6.8~\rm k\Omega$). As higher the number (e.g. E24) more different values are available in a decade, and as more precise the given value is. | ||
+ | |||
+ | For larger resistors with wires, the value is coded by four to six colored bands (see [[https:// | ||
+ | |||
+ | < | ||
+ | </ | ||
+ | {{drawio> | ||
+ | |||
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
Zeile 657: | Zeile 638: | ||
{{drawio> | {{drawio> | ||
- | * For linear resistors, the resistance value is $R={{U_R}\over{I_R}}=const.$ and thus independent of $U_R$. | + | * For linear resistors, the resistance value is $R={{U_R}\over{I_R}}={\rm const.} $ and thus independent of $U_R$. |
- | * **Ohm' | + | * **Ohm' |
* In <imgref BildNr13> | * In <imgref BildNr13> | ||
- | * The reciprocal value (inverse) of the resistance is called the conductance: | + | * The reciprocal value (inverse) of the resistance is called the conductance: |
</ | </ | ||
Zeile 696: | Zeile 677: | ||
<WRAP > | <WRAP > | ||
< | < | ||
- | ^ Material | + | ^ Material |
| Silver | | Silver | ||
| Copper | | Copper | ||
Zeile 713: | Zeile 694: | ||
<callout icon=" | <callout icon=" | ||
The resistance can be calculated by \\ $\boxed{R = \rho \cdot {{l}\over{A}} } $ | The resistance can be calculated by \\ $\boxed{R = \rho \cdot {{l}\over{A}} } $ | ||
- | * $\rho$ is the material dependent resistivity with the unit: $[\rho]={{[R]\cdot[A]}\over{l}}=1~{{\Omega\cdot \rm{m}^{\not{2}}}\over{\not{\rm{m}}}}=1~\Omega\cdot \rm{m}$ | + | * $\rho$ is the material dependent resistivity with the unit: $[\rho]={{[R]\cdot[A]}\over{l}}=1~{{\Omega\cdot |
- | * Often, instead of $1~\Omega\cdot \rm{m}$, the unit $1~{{\Omega\cdot {\rm{mm}^2}}\over{\rm{m}}}$ is used. It holds that $1~{{\Omega\cdot {\rm{mm}^2}}\over{\rm{m}}}= 10^{-6}~\Omega \rm{m}$ | + | * Often, instead of $1~\Omega\cdot |
</ | </ | ||
Zeile 734: | Zeile 715: | ||
* Chemical environment (chemoresistive effect e.g. chemical analysis of breathing air) | * Chemical environment (chemoresistive effect e.g. chemical analysis of breathing air) | ||
- | In order to summarize these influences in a formula, the mathematical tool of {{wp> | + | To summarize these influences in a formula, the mathematical tool of {{wp> |
This will be shown here practically for the thermoresistive effect. | This will be shown here practically for the thermoresistive effect. | ||
The thermoresistive effect, or the temperature dependence of the resistivity, | The thermoresistive effect, or the temperature dependence of the resistivity, | ||
The starting point for this is again an experiment. The ohmic resistance is to be determined as a function of temperature. | The starting point for this is again an experiment. The ohmic resistance is to be determined as a function of temperature. | ||
- | For this purpose, the resistance is measured | + | For this purpose, the resistance is measured |
<WRAP group>< | <WRAP group>< | ||
Zeile 754: | Zeile 735: | ||
* The constant is replaced by $c = R_0 \cdot \alpha$ | * The constant is replaced by $c = R_0 \cdot \alpha$ | ||
- | * $\alpha$ here is the linear temperature coefficient with unit: $ [\alpha] = {{1}\over{[\vartheta]}} = {{1}\over{\rm{K}}} $ | + | * $\alpha$ here is the linear temperature coefficient with unit: $ [\alpha] = {{1}\over{[\vartheta]}} = {{1}\over{{\rm K}}} $ |
* Besides the linear term, it is also possible to increase the accuracy of the calculation of $R(\vartheta)$ with higher exponents of the temperature influence. This approach will be discussed in more detail in the mathematics section below. | * Besides the linear term, it is also possible to increase the accuracy of the calculation of $R(\vartheta)$ with higher exponents of the temperature influence. This approach will be discussed in more detail in the mathematics section below. | ||
* These temperature coefficients are described with Greek letters: $\alpha$, $\beta$, $\gamma$, ... | * These temperature coefficients are described with Greek letters: $\alpha$, $\beta$, $\gamma$, ... | ||
Zeile 779: | Zeile 760: | ||
Where: | Where: | ||
- | * $\alpha$ the (linear) temperature coefficient with unit: $ [\alpha] = {{1}\over{\rm{K}}} $ | + | * $\alpha$ the (linear) temperature coefficient with unit: $ [\alpha] = {{1}\over{{\rm K}}} $ |
- | * $\beta$ the (quadratic) temperature coefficient with unit: $ [\beta] = {{1}\over{\rm{K}^2}} $ | + | * $\beta$ the (quadratic) temperature coefficient with unit: $ [\beta] = {{1}\over{{\rm K}^2}} $ |
- | * $\gamma$ the temperature coefficient with unit: $ [\gamma] = {{1}\over{\rm{K}^3}} $ | + | * $\gamma$ the temperature coefficient with unit: $ [\gamma] = {{1}\over{{\rm K}^3}} $ |
- | * $\vartheta_0$ is the given reference temperature, | + | * $\vartheta_0$ is the given reference temperature, |
The further the temperature range deviates from the reference temperature, | The further the temperature range deviates from the reference temperature, | ||
Zeile 790: | Zeile 771: | ||
<callout icon=" | <callout icon=" | ||
- | In addition to the specification of the parameters $\alpha$, | + | In addition to the specification of the parameters $\alpha$, |
- | This is a different variant of approximation, | + | This is a different variant of approximation, |
It is based on the {{wp> | It is based on the {{wp> | ||
- | For the temperature dependence of the resistance, the Arrhenius equation links the inhibition of carrier motion by lattice vibrations to the temperature $R(T) \sim e^{{B}\over{T}} $ . | + | For the temperature dependence of the resistance, the Arrhenius equation links the inhibition of carrier motion by lattice vibrations to the temperature $R(T) \sim {\rm e}^{{\rm B}\over{T}} $ . |
- | A series expansion can again be applied: $R(T) \sim e^{A + {{B}\over{T}} + {{C}\over{T^2}} + ...}$. | + | A series expansion can again be applied: $R(T) \sim {\rm e}^{{\rm A} + {{\rm B}\over{T}} + {{\rm C}\over{T^2}} + ...}$. |
- | However, often only $B$ is given. \\ By taking the ratio of any temperature $T$ and $T_{25}=298.15~\rm{K}$ ($\hat{=} 25~°\rm{C}$) we get: | + | However, often only $B$ is given. \\ By taking the ratio of any temperature $T$ and $T_{25}=298.15~{\rm K}$ ($\hat{=} 25~°{\rm C}$) we get: |
- | ${{R(T)}\over{R_{25}}} = {{exp \left({{B}\over{T}}\right)} \over {exp \left({{B}\over{298.15 ~\rm{K}}}\right)}} $ with $R_{25}=R(T_{25})$ | + | ${{R(T)}\over{R_{25}}} = {{{\rm exp} \left({{\rm B}\over{T}}\right)} \over {{\rm exp} \left({{\rm B}\over{298.15 ~{\rm K}}}\right)}} $ with $R_{25}=R(T_{25})$ |
This allows the final formula to be determined: | This allows the final formula to be determined: | ||
- | $R(T) = R_{25} \cdot exp \left( | + | $R(T) = R_{25} \cdot {\rm exp} \left( |
</ | </ | ||
Zeile 807: | Zeile 788: | ||
=== Types of temperature-dependent Resistors === | === Types of temperature-dependent Resistors === | ||
- | Besides the temperature dependence as a disturbing influence, | + | Besides the temperature dependence as a negative, |
- | These are called thermistors (a portmanteau of __therm__ally sensitive res__istor__). Thermistors are basically | + | These are called thermistors (a portmanteau of __therm__ally sensitive res__istor__). Thermistors are divided into hot conductors and cold conductors. |
A special form of thermistors is materials that have been explicitly optimized for minimum temperature dependence (e.g. Constantan or Isaohm). | A special form of thermistors is materials that have been explicitly optimized for minimum temperature dependence (e.g. Constantan or Isaohm). | ||
Zeile 831: | Zeile 812: | ||
* Such a PTC thermistor is also called // | * Such a PTC thermistor is also called // | ||
* Examples are doped semiconductors or metals. | * Examples are doped semiconductors or metals. | ||
- | * Applications are temperature sensors. For this purpose, they often offer a wide temperature range and good linearity (e.g. PT100 in the range of $-100~°\rm{C}$ to $200~°\rm{C}$). | + | * Applications are temperature sensors. For this purpose, they often offer a wide temperature range and good linearity (e.g. PT100 in the range of $-100~°{\rm C}$ to $200~°{\rm C}$). |
* [[https:// | * [[https:// | ||
Zeile 860: | Zeile 841: | ||
<panel type=" | <panel type=" | ||
- | Assume that a soft pencil lead is 100 % graphite. What is the resistance of a $5.0~\rm{cm}$ long and $0.20~\rm{mm}$ wide line if it has a height of $0.20~\rm{µm}$? | + | Assume that a soft pencil lead is 100 % graphite. What is the resistance of a $5.0~{\rm cm}$ long and $0.20~{\rm mm}$ wide line if it has a height of $0.20~{\rm µm}$? |
The resistivity is given by <tabref tab04>. | The resistivity is given by <tabref tab04>. | ||
Zeile 868: | Zeile 849: | ||
<button size=" | <button size=" | ||
\begin{align*} | \begin{align*} | ||
- | R = 10~\rm{k} \Omega | + | R = 10~{\rm k} \Omega |
\end{align*}</ | \end{align*}</ | ||
Zeile 875: | Zeile 856: | ||
<panel type=" | <panel type=" | ||
- | Let a cylindrical coil in the form of a multi-layer winding be given - this could for example occur in windings of a motor. The cylindrical coil has an inner diameter of $d_i=70~\rm{mm}$ and an outer diameter of $d_a = 120~\rm{mm}$. The number of turns is $n_W=1350$ turns, the wire diameter is $d=2.0~\rm{mm}$ and the specific conductivity of the wire is $\kappa_{Cu}=56 \cdot 10^6 ~{{\rm{S}}\over{\rm{m}}}$. | + | Let a cylindrical coil in the form of a multi-layer winding be given - this could for example occur in windings of a motor. The cylindrical coil has an inner diameter of $d_i=70~{\rm mm}$ and an outer diameter of $d_a = 120~{\rm mm}$. The number of turns is $n_W=1350$ turns, the wire diameter is $d=2.0~{\rm mm}$ and the specific conductivity of the wire is $\kappa_{Cu}=56 \cdot 10^6 ~{{{\rm S}}\over{{\rm m}}}$. |
- | First calculate the wound wire length and then the ohmic resistance of the entire coil. | + | First, calculate the wound wire length and then the ohmic resistance of the entire coil. |
</ | </ | ||
Zeile 883: | Zeile 864: | ||
The power supply line to a consumer has to be replaced. Due to the application, | The power supply line to a consumer has to be replaced. Due to the application, | ||
- | * The old aluminium supply cable had a specific conductivity $\kappa_{Al}=33\cdot 10^6 ~\rm{{S}\over{m}}$ and a cross-section $A_{Al}=115~\rm{mm}^2$. | + | * The old aluminium supply cable had a specific conductivity $\kappa_{Al}=33\cdot 10^6 ~{\rm {S}\over{m}}$ and a cross-section $A_{Al}=115~{\rm mm}^2$. |
- | * The new copper supply cable has a specific conductivity $\kappa_{Cu}=56\cdot 10^6 ~\rm{{S}\over{m}}$ | + | * The new copper supply cable has a specific conductivity $\kappa_{Cu}=56\cdot 10^6 ~{\rm {S}\over{m}}$ |
Which wire cross-section $A_{Cu}$ must be selected? | Which wire cross-section $A_{Cu}$ must be selected? | ||
Zeile 925: | Zeile 906: | ||
The energy expenditure per time unit represents the **power**: \\ | The energy expenditure per time unit represents the **power**: \\ | ||
- | $\boxed{P={{\Delta W}\over{\Delta t}}}$ with the unit $[P]={{[W]}\over{[t]}}=1~\rm{{J}\over{s}} = 1~\rm{{Nm}\over{s}} = 1 ~\rm{V\cdot A} = 1~\rm{W}$ | + | $\boxed{P={{\Delta W}\over{\Delta t}}}$ with the unit $[P]={{[W]}\over{[t]}}=1~{\rm {J}\over{s}} = 1~{\rm {Nm}\over{s}} = 1 ~{\rm V\cdot A} = 1~{\rm W}$ |
- | For a constant power $P$ and an initial energy $W(t=0~\rm{s})=0$ holds: \\ | + | For a constant power $P$ and an initial energy $W(t=0~{\rm s})=0$ holds: \\ |
$\boxed{W=P \cdot t}$ \\ | $\boxed{W=P \cdot t}$ \\ | ||
If the above restrictions do not apply, the generated/ | If the above restrictions do not apply, the generated/ | ||
Zeile 942: | Zeile 923: | ||
This gives the power (i.e. energy converted per unit time): \\ | This gives the power (i.e. energy converted per unit time): \\ | ||
- | $\boxed{P=U_{12} \cdot I}$ with the unit $[P]= 1 ~\rm{V\cdot A} = 1~\rm{W} \quad$ ... $\rm{W}$ here stands for the physical unit watts. | + | $\boxed{P=U_{12} \cdot I}$ with the unit $[P]= 1 ~{\rm V\cdot A} = 1~{\rm W} \quad$ ... ${\rm W}$ here stands for the physical unit watts. |
For ohmic resistors: | For ohmic resistors: | ||
Zeile 951: | Zeile 932: | ||
^ Name of the nominal quantity ^ physical quantity ^ description^ | ^ Name of the nominal quantity ^ physical quantity ^ description^ | ||
- | | Nominal power (= rated power) | + | | Nominal power (= rated power) |
- | | Nominal current (= rated current) | + | | Nominal current (= rated current) |
- | | Nominal voltage (= rated voltage) | + | | Nominal voltage (= rated voltage) |
==== Efficiency ==== | ==== Efficiency ==== | ||
- | The usable (= outgoing) $P_O$ power of a real system is always smaller than the supplied (incoming) power $P_I$. | + | The usable (= outgoing) $P_{\rm O}$ power of a real system is always smaller than the supplied (incoming) power $P_{\rm I}$. |
This is due to the fact, that there are additional losses in reality. \\ | This is due to the fact, that there are additional losses in reality. \\ | ||
- | The difference is called power loss $P_L$. It is thus valid: | + | The difference is called power loss $P_{\rm loss}$. It is thus valid: |
- | $P_I = P_O + P_L$ | + | $P_{\rm I} = P_{\rm O} + P_{\rm loss}$ |
- | Instead of the power loss $P_V$, the efficiency $\eta$ is often given: | + | Instead of the power loss $P_{\rm loss}$, the efficiency $\eta$ is often given: |
- | $\boxed{\eta = {{P_{O}}\over{P_{I}}}\overset{!}{< | + | $\boxed{\eta = {{P_{\rm O}}\over{P_{\rm I}}}\overset{!}{< |
For systems connected in series (cf. <imgref BildNr23> | For systems connected in series (cf. <imgref BildNr23> | ||
- | $\boxed{\eta = {{P_{O}}\over{P_{I}}} = {\not{P_{1}}\over{P_{I}}}\cdot {\not{P_{2}}\over \not{P_{1}}}\cdot {{P_{O}}\over \not{P_{2}}} = \eta_1 \cdot \eta_3 \cdot \eta_3}$ | + | $\boxed{\eta = {{P_{\rm O}}\over{P_{\rm I}}} = {\not{P_{1}}\over{P_{\rm I}}}\cdot {\not{P_{2}}\over \not{P_{1}}}\cdot {{P_{\rm O}}\over \not{P_{2}}} = \eta_1 \cdot \eta_2 \cdot \eta_3}$ |
< | < | ||
Zeile 989: | Zeile 970: | ||
</ | </ | ||
- | <panel type=" | + | # |
+ | # | ||
- | An SMD resistor is used on a circuit board for current measurement. The resistance value should be $R=0.2~\Omega$, the maximum power $P_M=250 ~\rm{mW}$. | + | An SMD resistor is used on a circuit board for current measurement. The resistance value should be $R=0.20~\Omega$, |
What is the maximum current that can be measured? | What is the maximum current that can be measured? | ||
- | </ | + | # |
+ | The formulas $R = {{U} \over {I}}$ and $P = {U} \cdot {I}$ can be combined to get: | ||
+ | \begin{align*} | ||
+ | P = R \cdot I^2 | ||
+ | \end{align*} | ||
+ | |||
+ | This can be rearranged into | ||
+ | |||
+ | \begin{align*} | ||
+ | I = + \sqrt{ {{P} \over{R} } } | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | \begin{align*} | ||
+ | I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A} | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | |||
+ | # | ||
<panel type=" | <panel type=" | ||
Zeile 1004: | Zeile 1009: | ||
</ | </ | ||
- | * The battery monitor BQ769x0 measures the charge and discharge currents of a lithium-ion battery | + | * The battery monitor BQ769x0 measures the charge and discharge currents of a lithium-ion battery |
* Draw an equivalent circuit with a voltage source (battery), measuring resistor and load resistor $R_L$. Also, draw the measurement voltage and load voltage. | * Draw an equivalent circuit with a voltage source (battery), measuring resistor and load resistor $R_L$. Also, draw the measurement voltage and load voltage. | ||
- | * The shunt should have a resistance value of $1~\rm{m}\Omega$. What maximum charge/ | + | * The shunt should have a resistance value of $1~{\rm m}\Omega$. What maximum charge/ |
* What power loss is generated at the shunt in the extreme case? | * What power loss is generated at the shunt in the extreme case? | ||
* Now the efficiency is to be calculated | * Now the efficiency is to be calculated | ||
- | * Find the efficiency as a function of $R\_S$ and $R_L$. Note that the same current flows through both resistors. | + | * Find the efficiency as a function of $\rm R\_S$ and $R_\rm L$. Note that the same current flows through both resistors. |
- | * Special task: The battery is to have a nominal voltage of $10~\rm{V}$ (3 cells) and the maximum discharge current is to flow. What efficiency results from the measurement alone? | + | * Special task: The battery is to have a nominal voltage of $10~{\rm V}$ (3 cells) and the maximum discharge current is to flow. What efficiency results from the measurement alone? |
</ | </ | ||
<panel type=" | <panel type=" | ||
- | A water pump ($\eta_P = 60~\%$) has an electric motor drive ($\eta_M=90~\%$). | + | A water pump ($\eta_\rm P = 60~\%$) has an electric motor drive ($\eta_\rm M=90~\%$). |
- | The pump has to pump $500~\rm{l}$ water per minute up to $12~\rm{m}$ difference in height. | + | The pump has to pump $500~{\rm l}$ water per minute up to $12~{\rm m}$ difference in height. |
* What must be the rated power of the motor? | * What must be the rated power of the motor? | ||
- | * What current does the motor draw from the $230~\rm{V}$ mains? (assumption: | + | * What current does the motor draw from the $230~{\rm V}$ mains? (assumption: |
</ | </ | ||
<panel type=" | <panel type=" | ||
- | Often, parts of a circuit have to be protected from over-current, | + | Often, parts of a circuit have to be protected from over-current, |
This is usually done by a fuse or a circuit breaker, which opens up the connection and therefore disables the path for the current. | This is usually done by a fuse or a circuit breaker, which opens up the connection and therefore disables the path for the current. | ||
A problem with the commonly used fuses is, that once the fuse is blown (=it has been tripped) it has to be changed. \\ | A problem with the commonly used fuses is, that once the fuse is blown (=it has been tripped) it has to be changed. \\ | ||
Zeile 1038: | Zeile 1043: | ||
{{drawio> | {{drawio> | ||
- | In the given circuit below, a fuse $F$ shall protect another component shown as $R_L$, which could be a motor or motor driver for example. | + | In the given circuit below, a fuse $F$ shall protect another component shown as $R_\rm L$, which could be a motor or motor driver for example. |
In general, the fuse $F$ can be seen as a (temperature variable) resistance. | In general, the fuse $F$ can be seen as a (temperature variable) resistance. | ||
- | The source voltage $U_S$ is $50~\rm{V}$ and $R_L=250~\Omega$. | + | The source voltage $U_\rm S$ is $50~{\rm V}$ and $R_{\rm L}=250~\Omega$. |
{{drawio> | {{drawio> | ||
For this fuse, the component " | For this fuse, the component " | ||
- | When this fuse trips, it has to carry nearly the full source voltage and dissipates a power of $0.8~\rm{W}$. | + | When this fuse trips, it has to carry nearly the full source voltage and dissipates a power of $0.8~{\rm W}$. |
- | * First assume that the fuse is not blown. The resistance of the fuse at this is $1~\Omega$, which is negligible compared to $R_L$. What is the value of the current flowing through $R_L$? | + | * First assume that the fuse is not blown. The resistance of the fuse at this is $1~\Omega$, which is negligible compared to $R_{\rm L}$. What is the value of the current flowing through $R_{\rm L}$? |
* Assuming for the next questions that the fuse has to carry the full source voltage and the given power is dissipated. | * Assuming for the next questions that the fuse has to carry the full source voltage and the given power is dissipated. | ||
* Which value will the resistance of the fuse have? | * Which value will the resistance of the fuse have? | ||
* What is the current flowing through the fuse, when it is tripped? | * What is the current flowing through the fuse, when it is tripped? | ||
- | * Compare this resistance of the fuse with $R_L$. Is the assumption, that all of the voltage drops on the fuse is feasible? | + | * Compare this resistance of the fuse with $R_{\rm L}$. Is the assumption, that all of the voltage drops on the fuse feasible? |
</ | </ | ||
Zeile 1057: | Zeile 1062: | ||
===== Further Reading ===== | ===== Further Reading ===== | ||
- | - [[http:// | + | - [[http:// |
- [[https:// | - [[https:// | ||
+ | # | ||