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electrical_engineering_1:preparation_properties_proportions [2023/03/28 17:26] mexleadmin [Bearbeiten - Panel] |
electrical_engineering_1:preparation_properties_proportions [2023/10/12 03:48] (aktuell) mexleadmin [Bearbeiten - Panel] |
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| Zeile 1: | Zeile 1: | ||
| - | ====== 1. Preparation, | + | # |
| + | |||
| + | ====== 1 Preparation, | ||
| ===== 1.1 Physical Proportions ===== | ===== 1.1 Physical Proportions ===== | ||
| Zeile 112: | Zeile 114: | ||
| \\ \\ | \\ \\ | ||
| - | Example: Force $F = m \cdot a$ with $[{\rm F}] = 1~\ rm kg \cdot {{{\rm m}}\over{{\rm s}^2}}$ | + | Example: Force $F = m \cdot a$ with $[{\rm F}] = 1~\rm kg \cdot {{{\rm m}}\over{{\rm s}^2}}$ |
| \\ \\ | \\ \\ | ||
| Zeile 208: | Zeile 210: | ||
| ==== Exercises ==== | ==== Exercises ==== | ||
| - | <panel type=" | + | {{tagtopic>chapter1_1& |
| - | {{youtube> | + | |
| - | </ | + | |
| - | + | ||
| - | <panel type=" | + | |
| - | Convert the following values step by step: | + | |
| - | * A vehicle speed of $80.00~{\rm km/h}$ in $m/s$ <button size=" | + | |
| - | * An energy of $60.0~{\rm J}$ in ${\rm kWh}$ ($1~{\rm J} = 1~{\rm Joule} = 1~{\rm Watt}\cdot {\rm second}$) <button size=" | + | |
| - | * The number of electrolytically deposited single positively charged copper ions of $1.2~{\rm Coulombs}$ (a copper ion has the charge of about $1.6 \cdot 10^{-19}~{\rm C}$)< | + | |
| - | * Absorbed energy of a small IoT consumer, which consumes $1~{\rm µW}$ uniformly in $10 ~{\rm days}$ <button size=" | + | |
| - | </ | + | |
| - | + | ||
| - | <panel type=" | + | |
| - | Convert the following values step by step: | + | |
| - | How many minutes could an ideal battery with $10~{\rm kWh}$ operate a consumer with $3~{\rm W}$? | + | |
| - | </ | + | |
| - | + | ||
| - | <panel type=" | + | |
| - | Convert the following values step by step: | + | |
| - | How much energy does an average household consume per day when consuming an average power of $500~{\rm W}$? How many chocolate bars ($2' | + | |
| - | </ | + | |
| ===== 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) | ||
| Zeile 304: | Zeile 286: | ||
| ==== Exercises ==== | ==== Exercises ==== | ||
| - | <panel type=" | + | {{tagtopic>chapter1_2& |
| - | How many electrons make up the charge of one coulomb? | + | |
| - | </ | + | |
| - | + | ||
| - | <panel type=" | + | |
| - | A balloon has a charge of $Q=7~{\rm nC}$ on its surface. How many additional electrons are on the balloon? | + | |
| - | </ | + | |
| ===== 1.3 Effects of electric charges and current ===== | ===== 1.3 Effects of electric charges and current ===== | ||
| Zeile 350: | Zeile 326: | ||
| < | < | ||
| - | Experiment | + | Experiment |
| {{youtube> | {{youtube> | ||
| </ | </ | ||
| - | * Qualitative investigation | + | * Qualitative investigation |
| * two charges ($Q_1$ and $Q_2$) at distance $r$ | * two charges ($Q_1$ and $Q_2$) at distance $r$ | ||
| * additional measurement of the force $F_C$ (e.g. via spring balance) | * additional measurement of the force $F_C$ (e.g. via spring balance) | ||
| Zeile 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: \\ ${{\Delta Q} \over {\Delta t}} = {\rm const.} = I$ | * 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. | ||
| Zeile 471: | Zeile 447: | ||
| ==== Exercises ==== | ==== Exercises ==== | ||
| - | <panel type=" | + | {{tagtopic>chapter1_4& |
| - | + | ||
| - | < | + | |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | + | ||
| - | Let the charge gain per time on an object be given. | + | |
| - | * Determine the current $I$ from the $Q$-$t$-diagram <imgref BildNr3> and plot them into the diagram. | + | |
| - | * How could you proceed if the amount of charge on the object changes non-linearly? | + | |
| - | + | ||
| - | </ | + | |
| - | + | ||
| - | <panel type=" | + | |
| - | + | ||
| - | How many electrons pass through a control cross-section of a metallic conductor, when the current of $40~{\rm mA}$ flows for $4.5~{\rm s}$? | + | |
| - | + | ||
| - | </ | + | |
| ===== 1.5 Voltage, Potential, and Energy ===== | ===== 1.5 Voltage, Potential, and Energy ===== | ||
| Zeile 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 647: | Zeile 619: | ||
| In general, the cause-and-effect relationship is such that an applied voltage across the resistor produces the current flow. However, the reverse relationship also applies: as soon as an electric current flows across a resistor, a voltage drop is produced over the resistor. | In general, the cause-and-effect relationship is such that an applied voltage across the resistor produces the current flow. However, the reverse relationship also applies: as soon as an electric current flows across a resistor, a voltage drop is produced over the resistor. | ||
| In electrical engineering, | In electrical engineering, | ||
| + | |||
| + | The values of the resistors are standardized in such a way, that there is a fixed number of different values between $1~\Omega$ and $10~\Omega$ or between $10~\rm k\Omega$ and $100~\rm k\Omega$. These ranges, which cover values up to the tenfold number, are called decades. In general, the resistors are ordered in the so-called {{wp>E series of preferred numbers}}, like 6 values in a decade, which is named E6 (here: $1.0~\rm k\Omega$, $1.5~\rm k\Omega$, $2.2~\rm k\Omega$, $3.3~\rm k\Omega$, $4.7~\rm k\Omega$, $6.8~\rm k\Omega$). As higher the number (e.g. E24) more different values are available in a decade, and as more precise the given value is. | ||
| + | |||
| + | For larger resistors with wires, the value is coded by four to six colored bands (see [[https:// | ||
| + | |||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | |||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 657: | Zeile 638: | ||
| {{drawio> | {{drawio> | ||
| - | * For linear resistors, the resistance value is $R={{U_R}\over{I_R}}=const.$ and thus independent of $U_R$. | + | * For linear resistors, the resistance value is $R={{U_R}\over{I_R}}={\rm const.} $ and thus independent of $U_R$. |
| - | * **Ohm' | + | * **Ohm' |
| * In <imgref BildNr13> | * In <imgref BildNr13> | ||
| * The reciprocal value (inverse) of the resistance is called the conductance: | * The reciprocal value (inverse) of the resistance is called the conductance: | ||
| Zeile 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> |
| </ | </ | ||
| Zeile 734: | Zeile 715: | ||
| * Chemical environment (chemoresistive effect e.g. chemical analysis of breathing air) | * Chemical environment (chemoresistive effect e.g. chemical analysis of breathing air) | ||
| - | In order to summarize these influences in a formula, the mathematical tool of {{wp> | + | To summarize these influences in a formula, the mathematical tool of {{wp> |
| This will be shown here practically for the thermoresistive effect. | This will be shown here practically for the thermoresistive effect. | ||
| The thermoresistive effect, or the temperature dependence of the resistivity, | The thermoresistive effect, or the temperature dependence of the resistivity, | ||
| The starting point for this is again an experiment. The ohmic resistance is to be determined as a function of temperature. | The starting point for this is again an experiment. The ohmic resistance is to be determined as a function of temperature. | ||
| - | For this purpose, the resistance is measured | + | For this purpose, the resistance is measured |
| <WRAP group>< | <WRAP group>< | ||
| Zeile 790: | Zeile 771: | ||
| <callout icon=" | <callout icon=" | ||
| - | In addition to the specification of the parameters $\alpha$, | + | In addition to the specification of the parameters $\alpha$, |
| This is a different variant of approximation, | This is a different variant of approximation, | ||
| It is based on the {{wp> | It is based on the {{wp> | ||
| - | For the temperature dependence of the resistance, the Arrhenius equation links the inhibition of carrier motion by lattice vibrations to the temperature $R(T) \sim e^{{B}\over{T}} $ . | + | For the temperature dependence of the resistance, the Arrhenius equation links the inhibition of carrier motion by lattice vibrations to the temperature $R(T) \sim {\rm e}^{{\rm B}\over{T}} $ . |
| - | A series expansion can again be applied: $R(T) \sim e^{A + {{B}\over{T}} + {{C}\over{T^2}} + ...}$. | + | A series expansion can again be applied: $R(T) \sim {\rm e}^{{\rm A} + {{\rm B}\over{T}} + {{\rm C}\over{T^2}} + ...}$. |
| However, often only $B$ is given. \\ By taking the ratio of any temperature $T$ and $T_{25}=298.15~{\rm K}$ ($\hat{=} 25~°{\rm C}$) we get: | However, often only $B$ is given. \\ By taking the ratio of any temperature $T$ and $T_{25}=298.15~{\rm K}$ ($\hat{=} 25~°{\rm C}$) we get: | ||
| - | ${{R(T)}\over{R_{25}}} = {{exp \left({{B}\over{T}}\right)} \over {exp \left({{B}\over{298.15 ~{\rm K}}}\right)}} $ with $R_{25}=R(T_{25})$ | + | ${{R(T)}\over{R_{25}}} = {{{\rm exp} \left({{\rm B}\over{T}}\right)} \over {{\rm exp} \left({{\rm B}\over{298.15 ~{\rm K}}}\right)}} $ with $R_{25}=R(T_{25})$ |
| This allows the final formula to be determined: | This allows the final formula to be determined: | ||
| - | $R(T) = R_{25} \cdot exp \left( | + | $R(T) = R_{25} \cdot {\rm exp} \left( |
| </ | </ | ||
| Zeile 807: | Zeile 788: | ||
| === Types of temperature-dependent Resistors === | === Types of temperature-dependent Resistors === | ||
| - | Besides the temperature dependence as a disturbing influence, | + | Besides the temperature dependence as a negative, |
| - | These are called thermistors (a portmanteau of __therm__ally sensitive res__istor__). Thermistors are basically | + | These are called thermistors (a portmanteau of __therm__ally sensitive res__istor__). Thermistors are divided into hot conductors and cold conductors. |
| A special form of thermistors is materials that have been explicitly optimized for minimum temperature dependence (e.g. Constantan or Isaohm). | A special form of thermistors is materials that have been explicitly optimized for minimum temperature dependence (e.g. Constantan or Isaohm). | ||
| Zeile 875: | Zeile 856: | ||
| <panel type=" | <panel type=" | ||
| - | Let a cylindrical coil in the form of a multi-layer winding be given - this could for example occur in windings of a motor. The cylindrical coil has an inner diameter of $d_i=70~{\rm mm}$ and an outer diameter of $d_a = 120~{\rm mm}$. The number of turns is $n_W=1350$ turns, the wire diameter is $d=2.0~{\rm mm}$ and the specific conductivity of the wire is $\kappa_{Cu}=56 \cdot 10^6 ~{{{\rm S}}\over{{\rm m}}}$. | + | Let a cylindrical coil in the form of a multi-layer winding be given - this could for example occur in windings of a motor. |
| + | The cylindrical coil has an inner diameter of $d_{\rm i}=70~{\rm mm}$ 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 866: | ||
| 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 989: | Zeile 972: | ||
| </ | </ | ||
| - | <panel type=" | + | # |
| + | # | ||
| - | An SMD resistor is used on a circuit board for current measurement. The resistance value should be $R=0.2~\Omega$, and 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 $R=0.20~\Omega$, and the maximum power $P_M=250 ~\rm mW $. |
| 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 1011: | ||
| </ | </ | ||
| - | * The battery monitor BQ769x0 measures the charge and discharge currents of a lithium-ion battery | + | * The battery monitor BQ769x0 measures the charge and discharge currents of a lithium-ion battery |
| * Draw an equivalent circuit with a voltage source (battery), measuring resistor and load resistor $R_L$. Also, draw the measurement voltage and load voltage. | * Draw an equivalent circuit with a voltage source (battery), measuring resistor and load resistor $R_L$. Also, draw the measurement voltage and load voltage. | ||
| * The shunt should have a resistance value of $1~{\rm m}\Omega$. What maximum charge/ | * The shunt should have a resistance value of $1~{\rm m}\Omega$. What maximum charge/ | ||
| Zeile 1040: | Zeile 1047: | ||
| 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 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 $U_\rm S$ is $50~{\rm V}$ and $R_L=250~\Omega$. | + | The source voltage $U_\rm S$ is $50~{\rm V}$ and $R_{\rm L}=250~\Omega$. |
| {{drawio> | {{drawio> | ||
| Zeile 1046: | Zeile 1053: | ||
| 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 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 1064: | ||
| ===== Further Reading ===== | ===== Further Reading ===== | ||
| - | - [[http:// | + | - [[http:// |
| - [[https:// | - [[https:// | ||
| + | # | ||