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 16:26] – [Bearbeiten - Panel] 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 (G, M, k, d, c, m, µ, 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 30: | Zeile 32: | ||
| < | < | ||
| - | ^ 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. $metre$ 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. $I = 2~A$ | + | * 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 $I = 2\cdot 1~A$ | + | * 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~A$ is the (measurement) unit, here: 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. $1~{{m}\over{s}}$. | + | * 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 ($bar$) 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 | Y | + | | Yotta | ${\rm Y}$ |
| - | | Zetta | Z | + | | Zetta | ${\rm Z}$ |
| - | | Exa | E | + | | Exa | ${\rm E}$ |
| - | | Peta | P | + | | Peta |
| - | | Tera | T | + | | Tera |
| - | | Giga | G | + | | Giga |
| - | | Mega | M | + | | Mega |
| - | | Kilo | k | + | | Kilo |
| - | | Hecto | h | + | | Hecto | ${\rm h}$ |
| - | | Deka | de | $10^{1}$ | + | | Deka |
| </ | </ | ||
| < | < | ||
| ^ prefix ^ prefix symbol ^ meaning^ | ^ prefix ^ prefix symbol ^ meaning^ | ||
| - | | Deci | d | + | | Deci |
| - | | Centi | c | + | | Centi | ${\rm c}$ |
| - | | Milli | m | + | | Milli | ${\rm m}$ |
| - | | Micro | u, $\mu$ | $10^{-6}$ | + | | Micro | ${\rm u}$, $µ$ | $10^{-6}$ |
| - | | Nano | n | + | | Nano |
| - | | Piko | p | + | | Piko |
| - | | Femto | f | + | | Femto | ${\rm f}$ |
| - | | Atto | a | + | | Atto |
| - | | Zeppto | z | + | | Zeppto | ${\rm z}$ |
| - | | Yocto | y | + | | Yocto | ${\rm y}$ |
| </ | </ | ||
| </ | </ | ||
| < | < | ||
| + | \\ | ||
| Importance of orders of magnitude in engineering (when the given examples in the video are unclear: we will get into this.) | Importance of orders of magnitude in engineering (when the given examples in the video are unclear: we will get into this.) | ||
| {{youtube> | {{youtube> | ||
| Zeile 97: | Zeile 100: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Physical | + | ==== Physical |
| * Physical equations allow a connection of physical quantities. | * Physical equations allow a connection of physical quantities. | ||
| * There are two types of physical equations to distinguish (at least in German): | * There are two types of physical equations to distinguish (at least in German): | ||
| - | * Quantity equations (Größengleichungen ) | + | * Quantity equations (in German: //Größengleichungen// ) |
| - | * Normalized quantity equations (also called related quantity equations, normierte Größengleichungen) | + | * Normalized quantity equations (also called related quantity equations, |
| < | < | ||
| <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 $[F] = 1~kg \cdot {{m}\over{s^2}}$ | + | Example: Force $F = m \cdot a$ with $[{\rm F}] = 1~\rm kg \cdot {{{\rm m}}\over{{\rm s}^2}}$ |
| \\ \\ | \\ \\ | ||
| Zeile 121: | 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 139: | Zeile 142: | ||
| <callout title=" | <callout title=" | ||
| - | Let a body with the mass $m = 100~kg$ be given. The body is lifted by the height $s=2~m$. \\ | + | 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 147: | 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 $m=100~kg$, $s=2~m$ and $g=9.81~{{m}\over{s^2}}$ | + | \\ $W = m \cdot g \cdot s \quad\quad$ where $m=100~{\rm kg}$, $s=2~m$ and $g=9.81~{{{\rm m}}\over{{\rm s}^2}}$ |
| - | \\ $W = 100~kg \cdot 9.81{{m}\over{s^2}} \cdot 2m $ | + | \\ $W = 100~kg \cdot 9.81 ~{{{\rm m}}\over{{\rm s}^2}} \cdot 2~{\rm m} $ |
| - | \\ $W = 100\cdot 9.81 \cdot 2 \;\; \cdot \;\; kg \cdot {{m}\over{s^2}} \cdot m$ | + | \\ $W = 100 |
| - | \\ $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 162: | 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 207: | 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~km/ | + | |
| - | * An energy of $60.0~J$ in $kWh$ ($1~J = 1~Joule = 1~Watt*second$) <button size=" | + | |
| - | * The number of electrolytically deposited single positively charged copper ions of $1.2~Coulombs$ (a copper ion has the charge of about $1.6 \cdot 10^{-19}~C$)< | + | |
| - | * Absorbed energy of a small IoT consumer, which consumes $1~\mu W$ uniformly in $10 ~days$ <button size=" | + | |
| - | </ | + | |
| - | + | ||
| - | <panel type=" | + | |
| - | Convert the following values step by step: | + | |
| - | How many minutes could an ideal battery with $10~kWh$ operate a consumer with $3~W$? | + | |
| - | </ | + | |
| - | + | ||
| - | <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 $500~W$? How many chocolate bars ($2' | + | |
| - | </ | + | |
| - | ===== 1.2 Introduction to the structure | + | ===== 1.2 Introduction to the Structure |
| < | < | ||
| Zeile 245: | 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 $|e| = 1.6022\cdot 10^{-19}~C$ | + | * 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 265: | Zeile 248: | ||
| - | ==== Conductivity ==== | + | ==== Conductivity |
| <WRAP group>< | <WRAP group>< | ||
| <callout color=" | <callout color=" | ||
| Zeile 281: | Zeile 264: | ||
| === Semiconductor === | === Semiconductor === | ||
| - | In semiconductors, | + | In semiconductors, |
| Examples: | Examples: | ||
| * Silicon | * Silicon | ||
| - | * diamond | + | * Diamond |
| </ | </ | ||
| Zeile 294: | 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 303: | 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 $Q=7~nC$ 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 332: | 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 345: | 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 368: | 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 $\varepsilon_0 = 8.85 \cdot 10^{-12} \cdot ~{C^2 \over {m^2\cdot N}} = 8.85 \cdot 10^{-12} \cdot ~{{As} \over {Vm}}$ | + | where $\varepsilon_0 = 8.85 \cdot 10^{-12} \cdot ~{{\rm C}^2 \over {{\rm m}^2\cdot |
| </ | </ | ||
| Zeile 410: | 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 ampere: $1~A = {{1~C}\over{1~s}}$ . Thus, for the unit coulomb applies: $1~C = 1~A\cdot s$ | + | * 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 $1~A$ flows when an amount of charge of $1~C$ is transported in $1~s$ 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 $1~A$ flows when two parallel conductors, each $1~m$ long and $1~m$ apart, exert a force of $F_C = 0.2\cdot 10^{-6}~N$ 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 453: | 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 **K**athode) | + | * 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 470: | 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 $40~mA$ flows for $4.5~s$? | + | |
| - | + | ||
| - | </ | + | |
| - | + | ||
| - | ===== 1.5 Voltage, Potential and Energy ===== | + | |
| < | < | ||
| Zeile 520: | 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 often used to denote the unit AND the quantity (in German $U$ is used for the quantity): | + | * V for Voltage is in the English literature |
| * e.g. | * e.g. | ||
| - | * $VCC = 5~V$ : 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), |
| - | * $V_{S+} = 15~V$ : 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 577: | Zeile 536: | ||
| <callout icon=" | <callout icon=" | ||
| * Voltage is always a potential difference. | * Voltage is always a potential difference. | ||
| - | * The unit of voltage is Volt: $1~V$ | + | * The unit of voltage is ${\rm Volt}$: $1~{\rm V}$ |
| </ | </ | ||
| <callout icon=" | <callout icon=" | ||
| - | A voltage of $1~V$ is present between two points if a charge of $1~C$ undergoes an energy change of $1~J = 1~Nm$ 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: $1~Nm = 1~V\cdot As \rightarrow 1~V = 1~{{Nm}\over{As}}$ | + | 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 600: | Zeile 559: | ||
| ==== Exercises ==== | ==== Exercises ==== | ||
| - | <panel type=" | + | # |
| + | # | ||
| < | < | ||
| Zeile 608: | 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 637: | 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 646: | 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 651: | 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 665: | Zeile 647: | ||
| </ | </ | ||
| <callout color=" | <callout color=" | ||
| - | === Non-linear | + | === Non-linear |
| < | < | ||
| </ | </ | ||
| Zeile 672: | 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 695: | 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 712: | 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: $[\rho]={{[R]\cdot[A]}\over{l}}=1~{{\Omega\cdot m^{\not{2}}}\over{\not{m}}}=1~\Omega\cdot m$ | + | * $\rho$ is the material dependent resistivity with the unit: $[\rho]={{[R]\cdot[A]}\over{l}}=1~{{\Omega\cdot |
| - | * Often, instead of $1~\Omega\cdot m$, the unit $1~{{\Omega\cdot {mm^2}}\over{m}}$ is used. It holds that $1~{{\Omega\cdot {mm^2}}\over{m}}= 10^{-6}~\Omega m$ | + | * Often, instead of $1~\Omega\cdot |
| </ | </ | ||
| Zeile 728: | 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. | ||
| + | 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. For this purpose, the resistance is measured | + | 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 | ||
| <WRAP group>< | <WRAP group>< | ||
| Zeile 749: | 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 775: | Zeile 761: | ||
| Where: | Where: | ||
| - | * $\alpha$ the (linear) temperature coefficient with unit: $ [\alpha] = {{1}\over{K}} $ | + | * $\alpha$ the (linear) temperature coefficient with unit: $ [\alpha] = {{1}\over{{\rm K}}} $ |
| - | * $\beta$ the (quadratic) temperature coefficient with unit: $ [\beta] = {{1}\over{K^2}} $ | + | * $\beta$ the (quadratic) temperature coefficient with unit: $ [\beta] = {{1}\over{{\rm K}^2}} $ |
| - | * $\gamma$ the temperature coefficient with unit: $ [\gamma] = {{1}\over{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 786: | 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 {\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 $T_{25}=298.15~K$ ($\hat{=} 25~°C$) 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 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 |
| 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>< | ||
| + | |||
| === NTC Thermistor === | === NTC Thermistor === | ||
| * 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 Heißleiter in German ("hot conductor" | + | * 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 823: | Zeile 811: | ||
| * As the name suggests, the PTC has a __p__ositive __t__emperature __c__oefficient. This leads to lower resistance at lower temperatures. | * As the name suggests, the PTC has a __p__ositive __t__emperature __c__oefficient. This leads to lower resistance at lower temperatures. | ||
| - | * Such a PTC thermistor is also called Kaltleiter in German ("cold conductor" | + | * 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 $-100~°C$ to $200~°C$). | + | * 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 838: | 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 854: | Zeile 842: | ||
| <panel type=" | <panel type=" | ||
| - | Assume that a soft pencil lead is 100 % graphite. What is the resistance of a $5.0~cm$ long and $0.20~mm$ wide line if it has a height of $0.20~\mu m$? | + | 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 862: | Zeile 850: | ||
| <button size=" | <button size=" | ||
| \begin{align*} | \begin{align*} | ||
| - | R = 10~k \Omega | + | R = 10~{\rm k} \Omega |
| \end{align*}</ | \end{align*}</ | ||
| Zeile 869: | 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 $d_i=70mm$ and an outer diameter of $d_a = 120~mm$. The number of turns is $n_W=1350$ turns, the wire diameter is $d=2.0~mm$ and the specific conductivity of the wire is $\kappa_{Cu}=56 \cdot 10^6 ~{{S}\over{m}}$. | + | 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 877: | 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 $\kappa_{Al}=33\cdot 10^6 ~{{S}\over{m}}$ and a cross-section $A_{Al}=115~mm^2$. | + | * 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 $\kappa_{Cu}=56\cdot 10^6 ~{{S}\over{m}}$ | + | * 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 894: | 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 900: | 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. Charge transport therefore automatically means energy expenditure. Often, however, the energy expenditure per unit of time is of interest. | + | 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. | ||
| < | < | ||
| Zeile 918: | 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 $[P]={{[W]}\over{[t]}}=1~{{J}\over{s}} = 1~{{Nm}\over{s}} = 1 ~V\cdot A = 1~W$ | + | $\boxed{P={{\Delta W}\over{\Delta t}}}$ 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 $W(t=0~s)=0$ 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. In the following circuit the color code shows the incoming and outgoing power. | + | 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. | ||
| < | < | ||
| Zeile 934: | 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 $[P]= 1 ~V\cdot A = 1~W \quad$ ... $W$ 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 943: | 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 $P_I$. This is due to the fact, that there are additional losses in reality. \\ | + | The usable (= outgoing) $P_{\rm O}$ power of a real system is always smaller than the supplied (incoming) power $P_{\rm I}$. |
| - | The difference is called power loss $P_L$. It is thus valid: | + | This is due to the fact, that there are additional losses in reality. \\ |
| + | 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 975: | 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 $P_M=250 ~mW$. | + | 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 995: | 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 $1~m\Omega$. 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 $10~V$ (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 $500~l$ water per minute up to $12~m$ 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 $230~V$ 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, |
| - | Since opening up electronics and changing the fuse is not reasonable for consumer electronics, | + | This is usually done by a fuse or a circuit breaker, which opens up the connection and therefore |
| - | In the diagram below the internal structure and the resistance over the temperature | + | 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, | ||
| + | These consist of a polymer (=" | ||
| + | When more and more current is flowing, more and more heat is generated. | ||
| + | At one distinct temperature, | ||
| + | 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. | ||
| + | This process is also reversible: When cooled down, the conducting paths get re-connected. | ||
| + | These components are also called **polymer positive temperature coefficient** | ||
| + | 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 general, the fuse $F$ can be seen as a (temperature variable) resistance. | + | 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. |
| - | The source voltage $U_S$ is $50~V$ and $R_L=250~\Omega$. | + | In general, the fuse $F$ can be seen as a (temperature variable) resistance. |
| + | 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 $0.8~W$. | + | 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 1038: | Zeile 1065: | ||
| ===== Further Reading ===== | ===== Further Reading ===== | ||
| - | - [[http:// | + | - [[http:// |
| - [[https:// | - [[https:// | ||
| + | # | ||