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electrical_engineering_1:preparation_properties_proportions [2023/05/31 09:26] mexleadmin |
electrical_engineering_1:preparation_properties_proportions [2023/10/11 11:29] mexleadmin |
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- | ====== 1. Preparation, | + | ====== 1 Preparation, |
===== 1.1 Physical Proportions ===== | ===== 1.1 Physical Proportions ===== | ||
Zeile 559: | Zeile 559: | ||
==== Exercises ==== | ==== Exercises ==== | ||
- | <panel type=" | + | # |
+ | # | ||
< | < | ||
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Explain whether the voltages $U_{\rm Batt}$, $U_{12}$ and $U_{21}$ in <imgref BildNr21> | Explain whether the voltages $U_{\rm Batt}$, $U_{12}$ and $U_{21}$ in <imgref BildNr21> | ||
- | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | |
- | </WRAP></WRAP></panel> | + | # |
+ | * Which terminal has the higher potential? | ||
+ | * From where to where does the arrow point? | ||
+ | # | ||
+ | |||
+ | |||
+ | # | ||
+ | * '' | ||
+ | * For $U_{\rm Batt}$: The arrow starts at terminal 1 and ends at terminal 2. So $U_{\rm Batt}=U_{12}>0$ | ||
+ | * $U_{21}<0$ | ||
+ | # | ||
+ | |||
+ | # | ||
Zeile 606: | Zeile 620: | ||
In electrical engineering, | In electrical engineering, | ||
- | 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: $1.0~\rm k\Omega$, | + | 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:// | For larger resistors with wires, the value is coded by four to six colored bands (see [[https:// | ||
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{{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> | ||
* The reciprocal value (inverse) of the resistance is called the conductance: | * The reciprocal value (inverse) of the resistance is called the conductance: | ||
Zeile 640: | 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 757: | 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 956: | Zeile 970: | ||
</ | </ | ||
- | <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 1007: | 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> | {{drawio> | ||
Zeile 1013: | Zeile 1051: | ||
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? |
</ | </ |