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ee1:preparation_properties_proportions [2023/05/31 09:27] – mexleadminee1:preparation_properties_proportions [2023/10/03 20:39] (aktuell) – 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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 By the end of this section, you will be able to: By the end of this section, you will be able to:
   - know the fundamental physical quantities and the associated SI units.   - know the fundamental physical quantities and the associated SI units.
-  - know the most important prefixes. Be able to assign a power of ten to the respective abbreviation (${\rm G}$, ${\rm M}$, ${\rm k}$, ${\rm d}$, ${\rm c}$, ${\rm m}$, ${\rm µ}$, ${\rm n}$).+  - know the most important prefixes. Be able to assign a power of ten to the respective abbreviation (${\rm G}$, ${\rm M}$, ${\rm k}$, ${\rm d}$, ${\rm c}$, ${\rm m}$, ${\rm Âµ}$, ${\rm n}$).
   - insert given numerical values and units into an existing quantity equation. From this, you should be able to calculate the correct result using a calculator.   - insert given numerical values and units into an existing quantity equation. From this, you should be able to calculate the correct result using a calculator.
   - assign the Greek letters.   - assign the Greek letters.
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   * For practical applications of physical laws of nature, **physical quantities** are put into mathematical relationships.   * For practical applications of physical laws of nature, **physical quantities** are put into mathematical relationships.
-  * There are basic quantities based on the SI system of units (French for Système International d'Unités), see below.+  * There are basic quantities based on the SI system of units (French for Système International d'Unités), see below.
   * In order to determine the basic quantities quantitatively (quantum = Latin for //how big//), **physical units** are defined, e.g. ${\rm metre}$ for length.   * In order to determine the basic quantities quantitatively (quantum = Latin for //how big//), **physical units** are defined, e.g. ${\rm metre}$ for length.
   * In electrical engineering, the first three basic quantities (cf. <tabref tab01> ) are particularly important. \\ Mass is important for the representation of energy and power.   * In electrical engineering, the first three basic quantities (cf. <tabref tab01> ) are particularly important. \\ Mass is important for the representation of energy and power.
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 | Centi  | ${\rm c}$      | $10^{-2}$   |  | Centi  | ${\rm c}$      | $10^{-2}$   | 
 | Milli  | ${\rm m}$      | $10^{-3}$   |  | Milli  | ${\rm m}$      | $10^{-3}$   | 
-| Micro  | ${\rm u}$, $µ$  | $10^{-6}$   | +| Micro  | ${\rm u}$, $µ$  | $10^{-6}$   | 
 | Nano   | ${\rm n}$      | $10^{-9}$   |  | Nano   | ${\rm n}$      | $10^{-9}$   | 
 | Piko   | ${\rm p}$      | $10^{-12}$  |  | Piko   | ${\rm p}$      | $10^{-12}$  | 
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   * Physical equations allow a connection of physical quantities.   * Physical equations allow a connection of physical quantities.
   * There are two types of physical equations to distinguish (at least in German):   * There are two types of physical equations to distinguish (at least in German):
-    * Quantity equations (in German: //Größengleichungen// ) +    * Quantity equations (in German: //Größengleichungen// ) 
-    * Normalized quantity equations (also called related quantity equations, in German //normierte Größengleichungen//)+    * Normalized quantity equations (also called related quantity equations, in German //normierte Größengleichungen//)
  
 <WRAP> <WRAP>
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   * Similar to the transport of a mass in the gravitational field, energy is needed to transport the charge in the "voltage field"   * Similar to the transport of a mass in the gravitational field, energy is needed to transport the charge in the "voltage field"
   * We will look at the specific electric field in the next semester   * We will look at the specific electric field in the next semester
-  * A point charge $q$ is moved from electrode ① to electrode ②. \\ The charge resembles a moving point of mass in the gravitational field.+  * A point charge $q$ is moved from electrode â‘  to electrode â‘¡. \\ The charge resembles a moving point of mass in the gravitational field.
   * $\rightarrow$ there is a turnover of energy.   * $\rightarrow$ there is a turnover of energy.
   * The energy turnover is proportional to the amount of charge $q$ transported.   * The energy turnover is proportional to the amount of charge $q$ transported.
-  * In many cases, the "energetic path" from ① to ② has to be characterized in charge-independent terms: \\ $\boxed{{W_{1,2}\over{q}} = U_{1,2}}$+  * In many cases, the "energetic path" from â‘  to â‘¡ has to be characterized in charge-independent terms: \\ $\boxed{{W_{1,2}\over{q}} = U_{1,2}}$
   * V for Voltage is in the English literature often used to denote the unit ${\rm V}$ AS WELL AS the quantity $V$ (in German $U$ is used for the quantity):   * V for Voltage is in the English literature often used to denote the unit ${\rm V}$ AS WELL AS the quantity $V$ (in German $U$ is used for the quantity):
   * e.g.   * e.g.
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 <callout icon="fa fa-comment" color="blue" title="Definition of the conventional direction of the voltage (according to DIN5489)"> <callout icon="fa fa-comment" color="blue" title="Definition of the conventional direction of the voltage (according to DIN5489)">
-The voltage of $U_{12}$ along a path from point ① to ② becomes positive when the potential in ① is greater than the potential in ②.+The voltage of $U_{12}$ along a path from point â‘  to â‘¡ becomes positive when the potential in â‘  is greater than the potential in â‘¡.
 </callout> </callout>
  
 ==== 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~2,Result~@# 
 + 
 +#@TaskEnd_HTML@# 
  
  
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 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.+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]]) 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]])
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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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   * $\beta$ the (quadratic) temperature coefficient with unit: $ [\beta] = {{1}\over{{\rm K}^2}} $   * $\beta$ the (quadratic) temperature coefficient with unit: $ [\beta] = {{1}\over{{\rm K}^2}} $
   * $\gamma$ the temperature coefficient with unit: $ [\gamma] = {{1}\over{{\rm K}^3}} $   * $\gamma$ the temperature coefficient with unit: $ [\gamma] = {{1}\over{{\rm K}^3}} $
-  * $\vartheta_0$ is the given reference temperature, usually $0~°{\rm C}$ or $25~°{\rm C}$.+  * $\vartheta_0$ is the given reference temperature, usually $0~°{\rm C}$ or $25~°{\rm C}$.
  
 The further the temperature range deviates from the reference temperature, the more temperature coefficients are required to reproduce the actual curve (<imgref BildNr22>). The further the temperature range deviates from the reference temperature, the more temperature coefficients are required to reproduce the actual curve (<imgref BildNr22>).
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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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   * As the name suggests, the NTC has a __n__egative __t__emperature __c__oefficient. This leads to lower resistance at higher temperatures.   * As the name suggests, the NTC has a __n__egative __t__emperature __c__oefficient. This leads to lower resistance at higher temperatures.
-  * Such an NTC thermistor is also called //Heißleiter// in German ("hot conductor").+  * Such an NTC thermistor is also called //Heißleiter// in German ("hot conductor").
   * Examples are semiconductors   * Examples are semiconductors
   * Applications are inrush current limiters and temperature sensors. For the desired operating point, a strongly non-linear curve is selected there (e.g. fever thermometer).   * Applications are inrush current limiters and temperature sensors. For the desired operating point, a strongly non-linear curve is selected there (e.g. fever thermometer).
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   * Such a PTC thermistor is also called //Kaltleiter// in German ("cold conductor").   * Such a PTC thermistor is also called //Kaltleiter// in German ("cold conductor").
   * Examples are doped semiconductors or metals.   * Examples are doped semiconductors or metals.
-  * Applications are temperature sensors. For this purpose, they often offer a wide temperature range and good linearity (e.g. PT100 in the range of $-100~°{\rm C}$ to $200~°{\rm C}$).+  * Applications are temperature sensors. For this purpose, they often offer a wide temperature range and good linearity (e.g. PT100 in the range of $-100~°{\rm C}$ to $200~°{\rm C}$).
   * [[https://www.geogebra.org/m/VVA2YUJQ#material/EQQm5kbT|Interactive example]] for PTC thermistors   * [[https://www.geogebra.org/m/VVA2YUJQ#material/EQQm5kbT|Interactive example]] for PTC thermistors
  
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 <panel type="info" title="Exercise 1.6.2 Resistance of a pencil stroke"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 1.6.2 Resistance of a pencil stroke"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-Assume that a soft pencil lead is 100 % graphite. What is the resistance of a $5.0~{\rm cm}$ long and $0.20~{\rm mm}$ wide line if it has a height of $0.20~{\rm µm}$?+Assume that a soft pencil lead is 100 % graphite. What is the resistance of a $5.0~{\rm cm}$ long and $0.20~{\rm mm}$ wide line if it has a height of $0.20~{\rm Âµm}$?
  
 The resistivity is given by <tabref tab04>. The resistivity is given by <tabref tab04>.
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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 flow diagram 
 +#@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~poflodi1,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~poflodi1,Solution ~@# 
 + 
 +#@HiddenBegin_HTML~poflodi2,Result~@# 
 +\begin{align*} 
 +I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A}   
 +\end{align*} 
 + 
 +#@HiddenEnd_HTML~poflodi2,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%>