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electrical_engineering_1:preparation_properties_proportions [2023/07/21 23:18] mexleadminelectrical_engineering_1:preparation_properties_proportions [2023/10/11 11:29] mexleadmin
Zeile 1: Zeile 1:
 #@DefLvlBegin_HTML~1,1.~@#  #@DefLvlBegin_HTML~1,1.~@# 
  
-====== 1Preparation, Properties, and Proportions ======+====== 1 Preparation, Properties, and Proportions ======
  
 ===== 1.1 Physical Proportions ===== ===== 1.1 Physical Proportions =====
Zeile 638: Zeile 638:
 {{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 $R={{{\Delta U_R}}\over{\Delta I_R}}$
Zeile 654: 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 779: Zeile 779:
  
 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({{\rm B}\over{T}}\right)} \over {{\rm exp} \left({{\rm 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:
Zeile 976: Zeile 976:
 What is the maximum current that can be measured? What is the maximum current that can be measured?
  
-#@HiddenBegin_HTML~1,Solution~@#+#@HiddenBegin_HTML~pow1,Solution~@#
 The formulas $R = {{U} \over {I}}$ and $P = {U} \cdot {I}$ can be combined to get: The formulas $R = {{U} \over {I}}$ and $P = {U} \cdot {I}$ can be combined to get:
 \begin{align*} \begin{align*}
Zeile 988: Zeile 988:
 \end{align*} \end{align*}
  
-#@HiddenEnd_HTML~1,Solution ~@#+#@HiddenEnd_HTML~pow1,Solution ~@#
  
-#@HiddenBegin_HTML~2,Result~@#+#@HiddenBegin_HTML~pow2,Result~@#
 \begin{align*} \begin{align*}
 I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A}   I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A}  
 \end{align*} \end{align*}
  
-#@HiddenEnd_HTML~2,Result ~@#+#@HiddenEnd_HTML~pow2,Result ~@#
  
  
Zeile 1045: Zeile 1045:
 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 1051: Zeile 1051:
 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>