Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung Nächste Überarbeitung Beide Seiten der Revision | ||
electrical_engineering_2:inductances_in_circuits [2023/03/17 12:52] mexleadmin |
electrical_engineering_2:inductances_in_circuits [2023/10/03 19:13] mexleadmin |
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Zeile 1: | Zeile 1: | ||
- | ====== 6. Inductances in Circuits ====== | + | ====== 6 Inductances in Circuits ====== |
===== 6.1 Basic Circuits ===== | ===== 6.1 Basic Circuits ===== | ||
Zeile 7: | Zeile 7: | ||
==== 6.1.1 Series Circuits ==== | ==== 6.1.1 Series Circuits ==== | ||
- | Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff' | + | Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff' |
\begin{align*} L_{eq} &= {{\sum_i \Psi_i}\over{I}} = \sum_i L_i \end{align*} | \begin{align*} L_{eq} &= {{\sum_i \Psi_i}\over{I}} = \sum_i L_i \end{align*} | ||
Zeile 22: | Zeile 22: | ||
==== 6.1.2 Parallel Circuits ==== | ==== 6.1.2 Parallel Circuits ==== | ||
- | For parallel circuits one can also start with the principles based on Kirchhoff' | + | For parallel circuits, one can also start with the principles based on Kirchhoff' |
\begin{align*} u_{\rm eq}= u_1 = u_2 = ... \\ \end{align*} | \begin{align*} u_{\rm eq}= u_1 = u_2 = ... \\ \end{align*} | ||
Zeile 34: | Zeile 34: | ||
\begin{align*} | \begin{align*} | ||
| | ||
- | \int u_{\rm ind} {\rm d}t &= L \cdot I \\ | + | \int u_{\rm ind} {\rm d}t &= L \cdot i \\ |
i &= {{1}\over{L}} \cdot \int u_{\rm ind} {\rm d}t \\ | i &= {{1}\over{L}} \cdot \int u_{\rm ind} {\rm d}t \\ | ||
\end{align*} | \end{align*} | ||
Zeile 51: | Zeile 51: | ||
==== 6.1.3 in AC Circuits ==== | ==== 6.1.3 in AC Circuits ==== | ||
- | For AC circuits (i.e. with sinusoidal signals) the impedance $Z$ based on the real part $R$ and imaginary part $X$ has to be considered. | + | For AC circuits (i.e. with sinusoidal signals) the impedance $Z$ based on the real part $R$ and imaginary part $X$ has to be considered. |
\begin{align*} \underline{Z} = {{\underline{u}}\over{\underline{i}}} = {{1}\over{\underline{i}}} \cdot \underline{u} | \begin{align*} \underline{Z} = {{\underline{u}}\over{\underline{i}}} = {{1}\over{\underline{i}}} \cdot \underline{u} | ||
Zeile 61: | Zeile 61: | ||
\end{align*} | \end{align*} | ||
- | Without limiting the generality, one can assume the current $i$ to be: $i = I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}$.\\ | + | Without limiting the generality, one can assume the current $i$ to be: $i = I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}$.\\ |
Once inserted, the formula gets: | Once inserted, the formula gets: | ||
\begin{align*} | \begin{align*} | ||
- | \underline{Z} &= {{1} \over {I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}}} \cdot L {{ {\rm d}} \over {{\rm d}t} } \left( I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0} \right) | + | \underline{Z} &= {{1} \over {I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}}} \cdot L {{ {\rm d}} \over {{\rm d}t} } \left( I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0} \right) |
- | &= {{1} \over {I \cdot \sqrt{2} \cdot e^{j \cdot \omega t + \varphi_0}}} \cdot L I \cdot \sqrt{2} \cdot {{ {\rm d}} \over {{\rm d}t} } \left( e^{j \cdot \omega t + \varphi_0} \right) | + | &= {{1} \over {I \cdot \sqrt{2} \cdot {\rm e}^{{\rm j} \cdot \omega t + \varphi_0}}} \cdot L I \cdot \sqrt{2} \cdot {{ {\rm d}} \over {{\rm d}t} } \left( |
- | &= {{1} \over {\qquad\quad\; | + | &= {{1} \over {\qquad\quad\; |
- | &= {{1} \over {\qquad\quad\; | + | &= {{1} \over {\qquad\quad\; |
- | \underline{Z} &= L \cdot j\omega | + | \underline{Z} &= L \cdot {\rm j}\omega |
\end{align*} | \end{align*} | ||
Zeile 82: | Zeile 82: | ||
===== 6.3 Resonance Phenomena ===== | ===== 6.3 Resonance Phenomena ===== | ||
- | Similar to the approach | + | Similar to last semester's approach, |
==== 6.3.1 RLC - Series Resonant Circuit ==== | ==== 6.3.1 RLC - Series Resonant Circuit ==== | ||
Zeile 99: | Zeile 99: | ||
\begin{align*} | \begin{align*} | ||
- | \underline{U}_I | + | \underline{U}_I |
- | \underline{U}_I | + | \underline{U}_I |
- | \underline{Z}_{\rm eq} & | + | \underline{Z}_{\rm eq} & |
\end{align*} | \end{align*} | ||
Zeile 118: | Zeile 118: | ||
\begin{align*} | \begin{align*} | ||
\varphi_u = \varphi_Z | \varphi_u = \varphi_Z | ||
- | & | + | & |
\end{align*} | \end{align*} | ||
Zeile 143: | Zeile 143: | ||
|voltage $U_C$ \\ at the capacitor | | $\boldsymbol{\LARGE{U}}$ \\ because $\frac{1}{\omega C}$ becomes very large | | $\boldsymbol{\frac{1}{\omega_0 C} \cdot I = \frac{1}{\omega_0 C} \cdot \frac{U}{R} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}\cdot U}}$ | | $\boldsymbol{\small{0}}$ \\ because $\frac{1}{\omega C}$ becomes very small | | |voltage $U_C$ \\ at the capacitor | | $\boldsymbol{\LARGE{U}}$ \\ because $\frac{1}{\omega C}$ becomes very large | | $\boldsymbol{\frac{1}{\omega_0 C} \cdot I = \frac{1}{\omega_0 C} \cdot \frac{U}{R} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}\cdot U}}$ | | $\boldsymbol{\small{0}}$ \\ because $\frac{1}{\omega C}$ becomes very small | | ||
- | The calculation in the table shows that in the resonance case, the voltage across the capacitor or inductor deviates from the input voltage by a factor $\color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}}$. This quantity is called **quality or Q-factor** | + | The calculation in the table shows that in the resonance case, the voltage across the capacitor or inductor deviates from the input voltage by a factor $\color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}}$. This quantity is called **quality or Q-factor** |
\begin{align*} | \begin{align*} | ||
- | \boxed{ | + | \boxed{ |
- | = \left.\frac{U_L}{U} \right\vert_{\omega = \omega_0} | + | = \left.\frac{U_L}{U} \right\vert_{\omega = \omega_0} |
- | = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}} } | + | = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}} } |
\end{align*} | \end{align*} | ||
- | The quality can be greater than, less than, or equal to 1. The quality $Q$ does not have a unit and should not be confused with the charge $Q$. | + | The quality can be greater than, less than, or equal to 1. The quality $Q_{\rm S}$ does not have a unit and should not be confused with the charge $Q$. |
* If the quality is very high, the overshoot of the voltages at the impedances becomes very large in the resonance case. This is useful and necessary in various applications, | * If the quality is very high, the overshoot of the voltages at the impedances becomes very large in the resonance case. This is useful and necessary in various applications, | ||
Zeile 158: | Zeile 158: | ||
The reciprocal of the $Q$ is called **attenuation** | The reciprocal of the $Q$ is called **attenuation** | ||
- | \begin{align*} \boxed{ d_{\rm S} = \frac{1}{Q_S} = R \sqrt{\frac{C}{L}} } \end{align*} | + | \begin{align*} \boxed{ d_{\rm S} = \frac{1}{Q_{\rm S}} = R \sqrt{\frac{C}{L}} } \end{align*} |
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
Zeile 216: | Zeile 216: | ||
<panel type=" | <panel type=" | ||
- | A $R$-$L$-$C$ series circuit uses a capacity of $C=1 ~\rm µF$. The circuit is fed by a voltage source with $U_I$ at $f_1 = 50~\rm Hz$. | + | A $R$-$L$-$C$ series circuit uses a capacity of $C=1 ~\rm µF$. A voltage source with $U_I$ feeds the circuit |
- Which values does $R$ and $L$ need to have, when the resonance voltage $|\underline{U}_L|$ and $|\underline{U}_C|$ at $f_1$ shall show the double value of the input voltage $U_I$? | - Which values does $R$ and $L$ need to have, when the resonance voltage $|\underline{U}_L|$ and $|\underline{U}_C|$ at $f_1$ shall show the double value of the input voltage $U_I$? | ||
Zeile 225: | Zeile 225: | ||
<panel type=" | <panel type=" | ||
- | A given $R$-$L$-$C$ series circuit is fed with a frequency, | + | A given $R$-$L$-$C$ series circuit is fed with a frequency, $20~\%$ larger than the resonance frequency keeping the amplitude of the input voltage constant. In this situation, the circuit shows a current that is $30~\%$ lower than the maximum current value. |
- Calculate the Quality $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$. | - Calculate the Quality $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$. |