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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
electrical_engineering_1:preparation_properties_proportions [2023/07/21 15:05] mexleadmin [Bearbeiten - Panel] |
electrical_engineering_1:preparation_properties_proportions [2023/10/12 03:48] (aktuell) mexleadmin [Bearbeiten - Panel] |
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- | ====== 1. Preparation, | + | ====== 1 Preparation, |
===== 1.1 Physical Proportions ===== | ===== 1.1 Physical Proportions ===== | ||
Zeile 638: | Zeile 638: | ||
{{drawio> | {{drawio> | ||
- | * 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' | * **Ohm' | ||
* In <imgref BildNr13> | * In <imgref BildNr13> | ||
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> | * The point in the $U$-$I$ diagram in which a system rests is called the operating point. In the <imgref BildNr14> | ||
* 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> | + | * In <imgref BildNr14> |
</ | </ | ||
Zeile 771: | Zeile 771: | ||
<callout icon=" | <callout icon=" | ||
- | In addition to the specification of the parameters $\alpha$, | + | In addition to the specification of the parameters $\alpha$, |
This is a different variant of approximation, | This is a different variant of approximation, | ||
It is based on the {{wp> | It is based on the {{wp> | ||
- | 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( | + | $R(T) = R_{25} \cdot {\rm exp} \left( |
</ | </ | ||
Zeile 856: | Zeile 856: | ||
<panel type=" | <panel type=" | ||
- | 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 864: | Zeile 866: | ||
The power supply line to a consumer has to be replaced. Due to the application, | The power supply line to a consumer has to be replaced. Due to the application, | ||
- | * 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? |
</ | </ | ||
Zeile 970: | Zeile 972: | ||
</ | </ | ||
- | <panel type=" | + | # |
+ | # | ||
- | 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? | ||
- | </ | + | # |
+ | 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*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | \begin{align*} | ||
+ | I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A} | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | |||
+ | # | ||
<panel type=" | <panel type=" | ||
Zeile 1021: | 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> | {{drawio> | ||
Zeile 1027: | Zeile 1053: | ||
For this fuse, the component " | For this fuse, the component " | ||
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? |
</ | </ |