Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_1:preparation_properties_proportions [2023/03/10 18:20] – mexleadmin | electrical_engineering_1:preparation_properties_proportions [2024/10/10 15:17] (aktuell) – mexleadmin | ||
|---|---|---|---|
| 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. | ||
| - always calculate with numerical value and unit. | - always calculate with numerical value and unit. | ||
| Zeile 16: | Zeile 18: | ||
| < | < | ||
| - | A nice 10 minute intro into some of the main topics of this chapter | + | A nice 10-minute intro into some of the main topics of this chapter |
| {{youtube> | {{youtube> | ||
| </ | </ | ||
| 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: ${\rm Ampere}$ |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== derived | + | ==== derived |
| - | * 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 98: | Zeile 100: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Physical | + | ==== Physical |
| * Physical equations allow a connection of physical quantities. | * Physical equations allow a connection of physical quantities. | ||
| Zeile 108: | Zeile 110: | ||
| <callout color=" | <callout color=" | ||
| - | === Quantity | + | === Quantity |
| - | 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 122: | Zeile 124: | ||
| < | < | ||
| <callout color=" | <callout color=" | ||
| - | === normalized | + | === normalized |
| In normalized quantity equations, the measured value or calculated value of a quantity equation is divided by a reference value. | In normalized quantity equations, the measured value or calculated value of a quantity equation is divided by a reference value. | ||
| 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 reference | + | As a reference |
| * Nominal values (maximum permissible value in continuous operation) or | * Nominal values (maximum permissible value in continuous operation) or | ||
| * Maximum values (maximum value achievable in the short term) | * Maximum values (maximum value achievable in the short term) | ||
| 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~Nm = 1962~J $ | + | \\ $W = 1962~{\rm Nm} = 1962~{\rm J} $ |
| </ | </ | ||
| </ | </ | ||
| - | ==== Letters for physical | + | ==== Letters for physical |
| In physics and electrical engineering, | In physics and electrical engineering, | ||
| Zeile 163: | Zeile 165: | ||
| Thus explains $C$ for // | Thus explains $C$ for // | ||
| But, maybe you already know that $C$ is used for the thermal capacity as well as for the electrical capacity. | But, maybe you already know that $C$ is used for the thermal capacity as well as for the electrical capacity. | ||
| - | The Latin alphabet | + | The Latin alphabet |
| For this reason, Greek letters are used for various physical quantities (see <tabref tab03>). | For this reason, Greek letters are used for various physical quantities (see <tabref tab03>). | ||
| 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 $1~\mu W$ 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 opposite charge | + | * Proton is the antagonist, i.e. has the opposite charge |
| * Sign is arbitrarily chosen: | * Sign is arbitrarily chosen: | ||
| * Electron charge: $-e$ | * Electron charge: $-e$ | ||
| * proton charge: $+e$ | * proton charge: $+e$ | ||
| - | * all charges on/in bodies can only occur as integer multiples of the elementary charge. | + | * All charges on/in bodies can only occur as integer multiples of the elementary charge. |
| * Due to the small numerical value of $e$, the charge is considered as a continuum when viewed macroscopically. | * Due to the small numerical value of $e$, the charge is considered as a continuum when viewed macroscopically. | ||
| Zeile 266: | Zeile 248: | ||
| - | ==== Conductivity ==== | + | ==== Conductivity |
| <WRAP group>< | <WRAP group>< | ||
| <callout color=" | <callout color=" | ||
| Zeile 282: | Zeile 264: | ||
| === Semiconductor === | === Semiconductor === | ||
| - | In semiconductors, | + | In semiconductors, |
| Examples: | Examples: | ||
| * Silicon | * Silicon | ||
| - | * diamond | + | * Diamond |
| </ | </ | ||
| 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 | + | ===== 1.3 Effects of Electric Charges |
| < | < | ||
| === Learning Objectives === | === Learning Objectives === | ||
| 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 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 ($+1~nC$) 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 $F_{\rm C}$ (e.g. via spring balance) |
| * Experiment results: | * Experiment results: | ||
| - | * Force increases linearly with larger charge $Q_1$ or $Q_2$. \\ $ F_C \sim Q_1$ and $ F_C \sim Q_2$ | + | * Force increases linearly with larger charge $Q_1$ or $Q_2$. \\ $ F_{\rm C} \sim Q_1$ and $ F_{\rm C} \sim Q_2$ |
| - | * Force falls quadratic with greater distance $r$ \\ $ F_C \sim {1 \over {r^2}}$ | + | * Force falls quadratic with greater distance $r$ \\ $ F_{\rm C} \sim {1 \over {r^2}}$ |
| - | * with a proportionality factor $a$: \\ $ F_C = a \cdot {{Q_1 \cdot Q_2} \over {r^2}}$ | + | * with a proportionality factor $a$: \\ $ F_{\rm C} = a \cdot {{Q_1 \cdot Q_2} \over {r^2}}$ |
| * Proportionality factor $a$ | * Proportionality factor $a$ | ||
| - | * The proportionality factor $a$ is defined | + | * The proportionality factor $a$ is defined |
| * $a$ thus becomes: | * $a$ thus becomes: | ||
| * $a = {{1} \over {4\pi\cdot\varepsilon}}$ | * $a = {{1} \over {4\pi\cdot\varepsilon}}$ | ||
| Zeile 369: | Zeile 345: | ||
| <callout icon=" | <callout icon=" | ||
| - | The Coulomb force (in a vacuum) can be calculated via. \\ $\boxed{ | + | The Coulomb force (in a vacuum) can be calculated via. \\ $\boxed{ |
| - | 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 $F_{\rm L} = 0.2\cdot 10^{-6}~{\rm N}$ on each other. |
| </ | </ | ||
| Zeile 454: | Zeile 430: | ||
| <callout icon=" | <callout icon=" | ||
| An electrode is a connection (or pin) of an electrical component. \\ | An electrode is a connection (or pin) of an electrical component. \\ | ||
| - | As a rule, the dimension of an electrode is characterized | + | Looking at a component, the electrode is characterized |
| - | The name of the electrode is given as following: | + | The name of the electrode is given as follows: |
| * **A**node: Electrode at which the current enters the component. | * **A**node: Electrode at which the current enters the component. | ||
| * Cathode: Electrode at which the current exits the component. (in German // | * Cathode: Electrode at which the current exits the component. (in German // | ||
| - | As a mnemonic you can remember the structure, shape and electrodes | + | As a mnemonic, you can remember the diode' |
| </ | </ | ||
| Zeile 471: | Zeile 447: | ||
| ==== Exercises ==== | ==== Exercises ==== | ||
| - | <panel type=" | + | {{tagtopic>chapter1_4& |
| - | < | + | ===== 1.5 Voltage, Potential, and Energy ===== |
| - | < | + | |
| - | </ | + | |
| - | {{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 ===== | + | |
| < | < | ||
| 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 652: | Zeile 633: | ||
| <WRAP group>< | <WRAP group>< | ||
| <callout color=" | <callout color=" | ||
| - | === Linear | + | === Linear |
| < | < | ||
| </ | </ | ||
| {{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 666: | Zeile 647: | ||
| </ | </ | ||
| <callout color=" | <callout color=" | ||
| - | === Non-linear | + | === Non-linear |
| < | < | ||
| </ | </ | ||
| Zeile 673: | Zeile 654: | ||
| * The point in the $U$-$I$ diagram in which a system rests is called the operating point. In the <imgref BildNr14> | * The point in the $U$-$I$ diagram in which a system rests is called the operating point. In the <imgref BildNr14> | ||
| * For nonlinear resistors, the resistance value is $R={{U_R}\over{I_R(U_R)}}=f(U_R)$. This resistance value depends on the operating point. | * For nonlinear resistors, the resistance value is $R={{U_R}\over{I_R(U_R)}}=f(U_R)$. This resistance value depends on the operating point. | ||
| - | * Often small changes around the operating point are of interest (e.g. for small disturbances of load machines). For this purpose, the differential resistance $r$ (also dynamic resistance) is determined: \\ $\boxed{r={{dU_R}\over{dI_R}} \approx{{\Delta U_R}\over{\Delta I_R}} }$ with unit $[R]=1~\Omega$. | + | * Often small changes around the operating point are of interest (e.g. for small disturbances of load machines). For this purpose, the differential resistance $r$ (also dynamic resistance) is determined: \\ $\boxed{r={{{\rm d}U_R}\over{{\rm d}I_R}} \approx{{\Delta U_R}\over{\Delta I_R}} }$ with unit $[R]=1~\Omega$. |
| * As with the resistor $R$, the reciprocal of the differential resistance $r$ is the differential conductance $g$. | * As with the resistor $R$, the reciprocal of the differential resistance $r$ is the differential conductance $g$. | ||
| - | * In <imgref BildNr14> | + | * In <imgref BildNr14> |
| </ | </ | ||
| </ | </ | ||
| - | ==== Resistance as a material | + | ==== Resistance as a Material |
| < | < | ||
| - | Clear explanation of resistivity | + | Good explanation of resistivity |
| {{youtube> | {{youtube> | ||
| </ | </ | ||
| Zeile 696: | Zeile 677: | ||
| <WRAP > | <WRAP > | ||
| < | < | ||
| - | ^ Material | + | ^ Material |
| - | | Silver | + | | Silver |
| - | | Copper | + | | Copper |
| - | | Aluminium | + | | Gold |
| - | | Gold | $2.2\cdot 10^{-2}$ | + | | Aluminium |
| - | | Lead | + | | Lead | $2.1\cdot 10^{-1}$ |
| - | | Graphite | + | | Graphite |
| - | | Battery Acid (Lead-acid Battery) | $1.5\cdot 10^4$ | | + | | Battery Acid (Lead-acid Battery) |
| - | | Blood | $1.6\cdot 10^{6}$ | + | | Blood |
| - | | (Tap) Water | $2 \cdot 10^{7}$ | + | | (Tap) Water |
| - | | Paper | $1\cdot 10^{15} ... 1\cdot 10^{17}$ | + | | Paper |
| </ | </ | ||
| 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 729: | Zeile 710: | ||
| The resistance value is - apart from the influences of geometry and material mentioned so far - also influenced by other effects. These are e.g.: | The resistance value is - apart from the influences of geometry and material mentioned so far - also influenced by other effects. These are e.g.: | ||
| * Heat (thermoresistive effect, e.g. in the resistance thermometer) | * Heat (thermoresistive effect, e.g. in the resistance thermometer) | ||
| - | * Light (photosensitive effect, e.g. in the component | + | * Light (photosensitive effect, e.g. in the component |
| * Magnetic field (magnetoresistive effect, e.g. in hard disks) | * Magnetic field (magnetoresistive effect, e.g. in hard disks) | ||
| * Pressure (piezoresistive effect e.g. tire pressure sensor) | * Pressure (piezoresistive effect e.g. tire pressure sensor) | ||
| * 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 753: | Zeile 734: | ||
| $R(\vartheta) = R_0 + c\cdot (\vartheta - \vartheta_0)$ | $R(\vartheta) = R_0 + c\cdot (\vartheta - \vartheta_0)$ | ||
| - | * 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{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: $\alpha$, $\beta$, $\gamma$, ... |
| + | * Sometimes in the datasheets the value $\alpha$ is named as TCR (" | ||
| <WRAP group>< | <WRAP group>< | ||
| Zeile 779: | Zeile 761: | ||
| 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 772: | ||
| <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, for example {{electrical_engineering_1: |
| - | ${{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( |
| </ | </ | ||
| - | === Types of temperature dependent | + | === Types of temperature-dependent |
| - | 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 | + | A special form of thermistors |
| <WRAP group>< | <WRAP group>< | ||
| Zeile 817: | Zeile 799: | ||
| * As the name suggests, the NTC has a __n__egative __t__emperature __c__oefficient. This leads to lower resistance at higher temperatures. | * As the name suggests, the NTC has a __n__egative __t__emperature __c__oefficient. This leads to lower resistance at higher temperatures. | ||
| - | * Such a NTC thermistor is also called // | + | * Such an NTC thermistor is also called // |
| * Examples are semiconductors | * Examples are semiconductors | ||
| * Applications are inrush current limiters and temperature sensors. For the desired operating point, a strongly non-linear curve is selected there (e.g. fever thermometer). | * Applications are inrush current limiters and temperature sensors. For the desired operating point, a strongly non-linear curve is selected there (e.g. fever thermometer). | ||
| Zeile 831: | Zeile 813: | ||
| * 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 844: | Zeile 826: | ||
| ==== Resistor Packages ==== | ==== Resistor Packages ==== | ||
| - | The packages are not explained in detail here. The video shows the smaller available packages. In the 3rd semester and higher we will use 0603 size resistors. | + | The packages are not explained in detail here. The video shows the smaller available packages. In the 3rd semester and higher we will use 0603-size resistors. |
| < | < | ||
| Zeile 860: | Zeile 842: | ||
| <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 850: | ||
| <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 857: | ||
| <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 $d_{\rm i}=70~{\rm mm}$ and an outer diameter of $d_{\rm a} = 120~{\rm mm}$. | ||
| + | The number of turns is $n_{\rm W}=1350$ turns, the wire diameter is $d=2.0~{\rm mm}$ and the specific conductivity of the wire is $\kappa_{\rm Cu}=56 \cdot 10^6 ~{{{\rm S}}\over{{\rm m}}}$. | ||
| - | 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 867: | ||
| 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 $\kappa_{\rm Al}=33\cdot 10^6 ~{\rm {S}\over{m}}$ and a cross-section $A_{\rm Al}=115~{\rm mm}^2$. |
| - | * 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 $\kappa_{\rm Cu}=56\cdot 10^6 ~{\rm {S}\over{m}}$ |
| - | Which wire cross-section $A_{Cu}$ must be selected ? | + | Which wire cross-section $A_{\rm Cu}$ must be selected? |
| </ | </ | ||
| Zeile 900: | Zeile 884: | ||
| < | < | ||
| - | === Goal === | + | === Learning Objectives |
| After this lesson you should be able to: | After this lesson you should be able to: | ||
| - Be able to calculate the electrical power and energy across a resistor. | - Be able to calculate the electrical power and energy across a resistor. | ||
| Zeile 906: | Zeile 890: | ||
| < | < | ||
| - | A nice 10 minute intro into power and efficiency (a cutout from 2:40 to 12:15 from a full video of EEVblog) | + | A nice 10-minute intro into power and efficiency (a cutout from 2:40 to 12:15 from a full video of EEVblog) |
| {{youtube> | {{youtube> | ||
| </ | </ | ||
| ==== Determining the electrical Power in a DC Circuit ==== | ==== Determining the electrical Power in a DC Circuit ==== | ||
| - | From chapter [[#1.5 Voltage, potential and energy]] it is known that a movement of a charge across a potential difference corresponds to a change in energy. | + | From chapter [[#1.5 Voltage, potential, and energy]] it is known that a movement of a charge across a potential difference corresponds to a change in energy. |
| Charge transport therefore automatically means energy expenditure. Often, however, the energy expenditure per unit of time is of interest. | Charge transport therefore automatically means energy expenditure. Often, however, the energy expenditure per unit of time is of interest. | ||
| Zeile 925: | Zeile 909: | ||
| 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/ | ||
| Besides the current flow from the source to the consumer (and back), also power flows from the source to the consumer. | Besides the current flow from the source to the consumer (and back), also power flows from the source to the consumer. | ||
| - | In the following circuit the color code shows the incoming and outgoing power. | + | In the following circuit, the color code shows the incoming and outgoing power. |
| < | < | ||
| Zeile 942: | Zeile 926: | ||
| 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 935: | ||
| ^ 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 984: | Zeile 968: | ||
| <panel type=" | <panel type=" | ||
| - | The first 5:20 minutes is a recap of the fundamentals of calculation | + | The first 5:20 minutes is a recap of the fundamentals of calculating |
| {{youtube> | {{youtube> | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| </ | </ | ||
| - | <panel type=" | + | # |
| + | # | ||
| - | A 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 1012: | ||
| </ | </ | ||
| - | * 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 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 open up the connection and therefore | + | This is usually done by a fuse or a circuit breaker, which opens up the connection and therefore |
| 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. \\ | ||
| Since opening up electronics and changing the fuse is not reasonable for consumer electronics, | Since opening up electronics and changing the fuse is not reasonable for consumer electronics, | ||
| Zeile 1032: | Zeile 1040: | ||
| This expansion moves the conducting paths apart. | This expansion moves the conducting paths apart. | ||
| The system will stay in a state, where a minimum current is flowing, which maintains just enough heat dissipation for the expansion. | The system will stay in a state, where a minimum current is flowing, which maintains just enough heat dissipation for the expansion. | ||
| - | This process is also reversible: When cooled down, the conducting paths gets re-connected. | + | This process is also reversible: When cooled down, the conducting paths get re-connected. |
| - | This components are also called **polymer positive temperature coefficient** | + | These components are also called **polymer positive temperature coefficient** |
| - | In the diagram below the internal structure and the resistance over the temperature | + | In the diagram below the internal structure and the resistance over the temperature |
| {{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 $50~V$ 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 1065: | ||
| ===== Further Reading ===== | ===== Further Reading ===== | ||
| - | - [[http:// | + | - [[http:// |
| - [[https:// | - [[https:// | ||
| + | # | ||