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electrical_engineering_1:preparation_properties_proportions [2023/03/31 14:45]
mexleadmin 		electrical_engineering_1:preparation_properties_proportions [2023/10/12 03:48] (aktuell)
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Zeile 1: Zeile 1:
-====== 1. Preparation, Properties, and Proportions ======+#@DefLvlBegin_HTML~1,1.~@#  
 + 
 +====== 1 Preparation, Properties, and Proportions ======
  
 ===== 1.1 Physical Proportions ===== ===== 1.1 Physical Proportions =====
Zeile 208: Zeile 210:
 ==== Exercises ==== ==== Exercises ====
  
-<panel type="info" title="Exercise 1.1.1 Conversions I - precalculated example for the conversion of units"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +{{tagtopic>chapter1_1&nodate&nouser&noheader&nofooter&order=custom}}
-{{youtube>DLjHyd0pFos}} +
-</WRAP></WRAP></panel> +
- +
-<panel type="info" title="Exercise 1.1.2 Conversions II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +
-Convert the following values step by step: +
-  * A vehicle speed of $80.00~{\rm km/h}$ in $m/s$ <button size="xs" type="link" collapse="Loesung_1_1_2_1_Endergebnis">{{icon>eye}} Final result</button><collapse id="Loesung_1_1_2_1_Endergebnis" collapsed="true"> $ 22.22~{\rm {m}\over{s}}$</collapse> +
-  * An energy of $60.0~{\rm J}$ in ${\rm kWh}$ ($1~{\rm J} = 1~{\rm Joule} = 1~{\rm Watt}\cdot {\rm second}$) <button size="xs" type="link" collapse="Loesung_1_1_2_2_Endergebnis">{{icon>eye}} Final result</button><collapse id="Loesung_1_1_2_2_Endergebnis" collapsed="true"> $ 1.67 \cdot 10^{-5}~{\rm kWh}$</collapse> +
-  * The number of electrolytically deposited single positively charged copper ions of $1.2~{\rm Coulombs}$ (a copper ion has the charge of about $1.6 \cdot 10^{-19}~{\rm C}$)<button size="xs" type="link" collapse="Loesung_1_1_2_3_Endergebnis">{{icon>eye}} Final result</button><collapse id="Loesung_1_1_2_3_Endergebnis" collapsed="true"> $7.5 \cdot 10^{18} ~{\rm ions}$</collapse> +
-  * Absorbed energy of a small IoT consumer, which consumes $1~{\rm µW}$ uniformly in $10 ~{\rm days}$ <button size="xs" type="link" collapse="Loesung_1_1_2_4_Endergebnis">{{icon>eye}} Final result</button><collapse id="Loesung_1_1_2_4_Endergebnis" collapsed="true"> $0.864~{\rm Ws} = 0.864~{\rm J}$</collapse> +
-</WRAP></WRAP></panel> +
- +
-<panel type="info" title="Exercise 1.1.3 Conversions III"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +
-Convert the following values step by step: +
-How many minutes could an ideal battery with $10~{\rm kWh}$ operate a consumer with $3~{\rm W}$? +
-</WRAP></WRAP></panel> +
- +
-<panel type="info" title="Exercise 1.1.4 Conversions IV"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +
-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 ($2'000~{\rm kJ}$ each) does this correspond to? +
-</WRAP></WRAP></panel>+
  
 ===== 1.2 Introduction to the Structure of Matter ===== ===== 1.2 Introduction to the Structure of Matter =====
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 ==== Exercises ==== ==== Exercises ====
  
-<panel type="info" title="Exercise 1.2.1 Charges I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +{{tagtopic>chapter1_2&nodate&nouser&noheader&nofooter&order=custom}}
-How many electrons make up the charge of one coulomb? +
-</WRAP></WRAP></panel> +
- +
-<panel type="info" title="Exercise  1.2.2 Charges II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +
-A balloon has a charge of $Q=7~{\rm nC}$ on its surface. How many additional electrons are on the balloon? +
-</WRAP></WRAP></panel>+
  
 ===== 1.3 Effects of electric charges and current ===== ===== 1.3 Effects of electric charges and current =====
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 ==== Exercises ==== ==== Exercises ====
  
-<panel type="info" title="Exercise 1.4.1 Determining the current from the charge per time"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +{{tagtopic>chapter1_4&nodate&nouser&noheader&nofooter&order=custom}}
- +
-<WRAP> +
-<imgcaption BildNr3 | Time course of the charge> +
-</imgcaption> +
-{{drawio>Zeitverlauf_Ladung.svg}} +
-</WRAP> +
- +
-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? +
- +
-</WRAP></WRAP></panel> +
- +
-<panel type="info" title="Exercise 1.4.2 Electron flow"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +
- +
-How many electrons pass through a control cross-section of a metallic conductor, when the current of $40~{\rm mA}$ flows for $4.5~{\rm s}$? +
- +
-</WRAP></WRAP></panel>+
  
 ===== 1.5 Voltage, Potential, and Energy ===== ===== 1.5 Voltage, Potential, and Energy =====
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 ==== Exercises ==== ==== Exercises ====
  
-<panel type="info" title="Exercise 1.5.1 Direction of the voltage"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +#@TaskTitle_HTML@#1.5.1 Direction of the voltage  
 +#@TaskText_HTML@#
  
 <WRAP> <WRAP>
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 </WRAP> </WRAP>
  
-Explain whether the voltages $U_{Batt}$, $U_{12}$ and $U_{21}$ in <imgref BildNr21> are positive or negative according to the voltage definition. +Explain whether the voltages $U_{\rm Batt}$, $U_{12}$ and $U_{21}$ in <imgref BildNr21> are positive or negative according to the voltage definition. 
-~~PAGEBREAK~~ ~~CLEARFIX~~ + 
-</WRAP></WRAP></panel>+#@HiddenBegin_HTML~1,Hints~@# 
 +  * Which terminal has the higher potential?  
 +  * From where to where does the arrow point?  
 +#@HiddenEnd_HTML~1,Hints~@# 
 + 
 + 
 +#@HiddenBegin_HTML~2,Result~@# 
 +  * ''+'' is the higher potential. Terminal 1 has the higher potential. $\varphi_1 > \varphi_2$ 
 +  * For $U_{\rm Batt}$: The arrow starts at terminal 1 and ends at terminal 2. So $U_{\rm Batt}=U_{12}>0$ 
 +  * $U_{21}<0$ 
 +#@HiddenEnd_HTML~1l2,Result~@# 
 + 
 +#@TaskEnd_HTML@# 
  
  
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, circuit diagrams use idealized components in a {{wp>Lumped-element model}}. The resistance of the wires is either neglected - if it is very small compared to all other resistance values - or drawn as a separate "lumped" resistor. In electrical engineering, circuit diagrams use idealized components in a {{wp>Lumped-element model}}. The resistance of the wires is either neglected - if it is very small compared to all other resistance values - or drawn as a separate "lumped" resistor.
 +
 +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://www.digikey.com/en/resources/conversion-calculators/conversion-calculator-resistor-color-code|conversion tool]]). For smaller resistors without wires, often numbers are printed onto the components ([[https://www.digikey.com/en/resources/conversion-calculators/conversion-calculator-smd-resistor-code|conversion tool]])
 +
 +<imgcaption BildNr13 | examples for a real 15kOhm resistor>
 +</imgcaption>
 +{{drawio>examplesForResistors.svg}}
 +
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
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 {{drawio>linearer_Widerstand_UI.svg}} {{drawio>linearer_Widerstand_UI.svg}}
  
-  * 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's law** results: \\ $\boxed{R={{U_R}\over{I_R}}}$ with unit $[R]={{[U_R]}\over{[I_R]}}= 1{{V}\over{A}}= 1~\Omega$+  * **Ohm's law** results: \\ $\boxed{R={{U_R}\over{I_R}}}$ with unit $[R]={{[U_R]}\over{[I_R]}}= 1{\rm {V}\over{A}}= 1~\Omega$
   * In <imgref BildNr13> the value $R$ can be read from the course of the straight line $R={{{\Delta U_R}}\over{\Delta I_R}}$   * In <imgref BildNr13> the value $R$ can be read from the course of the straight line $R={{{\Delta U_R}}\over{\Delta I_R}}$
   * The reciprocal value (inverse) of the resistance is called the conductance: $G={{1}\over{R}}$ with unit $1~{\rm S}$ (${\rm Siemens}$). This value can be seen as a slope in the $U$-$I$ diagram.   * The reciprocal value (inverse) of the resistance is called the conductance: $G={{1}\over{R}}$ with unit $1~{\rm S}$ (${\rm Siemens}$). This value can be seen as a slope in the $U$-$I$ diagram.
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> an operating point is marked with a circle in the left diagram.   * The point in the $U$-$I$ diagram in which a system rests is called the operating point. In the <imgref BildNr14> an operating point is marked with a circle in the left diagram.
   * 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> the differential conductance $g$ can be read from the slope of the straight line at each point $g={{dI_R}\over{dU_R}}$+  * In <imgref BildNr14> the differential conductance $g$ can be read from the slope of the straight line at each point $g={{{\rm d}I_R}\over{{\rm d}U_R}}$
 </callout> </callout>
  
Zeile 790: Zeile 771:
 <callout icon="fa fa-info" color="grey" title="Outlook"> <callout icon="fa fa-info" color="grey" title="Outlook">
  
-In addition to the specification of the parameters $\alpha$,$\beta$, ..., the specification of $R_{25}$ and $B_{25}$ can occasionally be found. +In addition to the specification of the parameters $\alpha$,$\beta$, ..., the specification of $R_{25}$ and $\rm B_{25}$ can occasionally be found. 
 This is a different variant of approximation, which refers to the temperature of $25~°{\rm C}$.  This is a different variant of approximation, which refers to the temperature of $25~°{\rm C}$. 
 It is based on the {{wp>Arrhenius equation}}, which links reaction kinetics to temperature in chemistry.  It is based on the {{wp>Arrhenius equation}}, which links reaction kinetics to temperature in chemistry. 
-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( B_{25} \cdot \left({{1}\over{T}} - {{1}\over{298.15~{\rm K}}} \right) \right)  $+$R(T) = R_{25} \cdot {\rm exp} \left( {\rm B}_{25} \cdot \left({{1}\over{T}} - {{1}\over{298.15~{\rm K}}} \right) \right)  $
  
 </callout> </callout>
Zeile 875: Zeile 856:
 <panel type="info" title="Exercise 1.6.3 Resistance of a cylindrical coil"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 1.6.3 Resistance of a cylindrical coil"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-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 conductor resistance must remain the same. The power supply line to a consumer has to be replaced. Due to the application, the conductor resistance must remain the same.
-  * 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?
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
Zeile 989: Zeile 972:
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
-<panel type="info" title="Exercise 1.7.2 Power"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> +#@TaskTitle_HTML@#1.7.2 Power 
 +#@TaskText_HTML@#
  
-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?
  
-</WRAP></WRAP></panel>+#@HiddenBegin_HTML~pow1,Solution~@# 
 +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*} 
 + 
 +#@HiddenEnd_HTML~pow1,Solution ~@# 
 + 
 +#@HiddenBegin_HTML~pow2,Result~@# 
 +\begin{align*} 
 +I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A}   
 +\end{align*} 
 + 
 +#@HiddenEnd_HTML~pow2,Result ~@# 
 + 
 + 
 +#@TaskEnd_HTML@# 
  
 <panel type="info" title="Exercise 1.7.3 Power loss and efficiency I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>  <panel type="info" title="Exercise 1.7.3 Power loss and efficiency I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> 
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>PPTCfusecircuit.svg}} {{drawio>PPTCfusecircuit.svg}}
Zeile 1046: Zeile 1053:
 For this fuse, the component "[[https://www.mouser.de/datasheet/2/643/ds_CP_0zcg_series-1960332.pdf|0ZCG0020AF2C]]"((the datasheet is not needed for this exercise)) is used.  For this fuse, the component "[[https://www.mouser.de/datasheet/2/643/ds_CP_0zcg_series-1960332.pdf|0ZCG0020AF2C]]"((the datasheet is not needed for this exercise)) is used. 
 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?
  
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
Zeile 1060: Zeile 1067:
   - [[https://www.youtube.com/watch?v=KGJqykotjog&ab_channel=AtomsandSporks|How electric flow really works]]: No, there are no free electrons in the wire, and the electrons are not colliding with the atoms or atomic cores...   - [[https://www.youtube.com/watch?v=KGJqykotjog&ab_channel=AtomsandSporks|How electric flow really works]]: No, there are no free electrons in the wire, and the electrons are not colliding with the atoms or atomic cores...
  
 +#@DefLvlEnd_HTML~1,1.~@#