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electrical_engineering_1:preparation_properties_proportions [2023/07/21 13:59]
mexleadmin 		electrical_engineering_1:preparation_properties_proportions [2023/10/12 03:48] (aktuell)
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 #@DefLvlBegin_HTML~1,1.~@#  #@DefLvlBegin_HTML~1,1.~@# 
  
-====== 1. Preparation, Properties, and Proportions ======+====== 1 Preparation, Properties, and Proportions ======
  
 ===== 1.1 Physical Proportions ===== ===== 1.1 Physical Proportions =====
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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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 Explain whether the voltages $U_{\rm Batt}$, $U_{12}$ 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@# 
  
  
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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 $[R]={{[U_R]}\over{[I_R]}}= 1{\rm {V}\over{A}}= 1~\Omega$   * **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 $R={{{\Delta U_R}}\over{\Delta I_R}}$
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   * 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>
  
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 <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>
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 <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.
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 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?
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 </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%> 
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 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}}
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 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?
  
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