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electrical_engineering_1:preparation_properties_proportions [2023/04/03 13:24] – mexleadminelectrical_engineering_1:preparation_properties_proportions [2023/10/04 09:17] – mexleadmin
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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 1.2.1 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@# 
  
  
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 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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   * 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 $R={{U_R}\over{I_R}}=const.$ 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.
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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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 </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%>