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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 [2023/03/24 14:52] mexleadmin |
electrical_engineering_2:inductances_in_circuits [2023/10/03 19:13] mexleadmin |
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- | ====== 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_{\rm eq} &= {{\sum_i \Psi_i}\over{I}} = \sum_i L_i \end{align*} |
A similar result can be derived from the induced voltage $u_{ind}= L {{{\rm d}i}\over{{\rm d}t}}$, when taking the situation of a series circuit (i.e. $i_1 = i_2 = i_1 = ... = i_{\rm eq}$ and $u_{\rm eq}= u_1 + u_2 + ...$): | A similar result can be derived from the induced voltage $u_{ind}= L {{{\rm d}i}\over{{\rm d}t}}$, when taking the situation of a series circuit (i.e. $i_1 = i_2 = i_1 = ... = i_{\rm eq}$ and $u_{\rm eq}= u_1 + u_2 + ...$): | ||
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==== 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 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 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 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 $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$. |