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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 =====
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 ==== 6.1.1 Series Circuits ==== ==== 6.1.1 Series Circuits ====
  
-Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff's mesh law ($i=const$) the series circuit of inductions can be interpreted as a single current $i$ which generates multiple linked fluxes $\Psi$. Since the current must stay constant in the series circuit, the following applies for the equivalent inductor of a series connection of single ones:+Based on $L = {{ \Psi(t)}\over{i}}$ and Kirchhoff's mesh law ($i=\rm const$) the series circuit of inductions can be interpreted as a single current $i$ which generates multiple linked fluxes $\Psi$. Since the current must stay constant in the series circuit, the following applies for the equivalent inductor of a series connection of single ones:
  
 \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*}
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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's mesh law:+For parallel circuits, one can also start with the principles based on Kirchhoff's mesh law:
  
 \begin{align*} u_{\rm eq}= u_1 = u_2 = ... \\ \end{align*} \begin{align*} u_{\rm eq}= u_1 = u_2 = ... \\ \end{align*}
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 ==== 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. In order to do so, one has to solve:+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. To do so, one has to solve:
  
 \begin{align*} \underline{Z} = {{\underline{u}}\over{\underline{i}}} = {{1}\over{\underline{i}}} \cdot \underline{u}  \end{align*} \begin{align*} \underline{Z} = {{\underline{u}}\over{\underline{i}}} = {{1}\over{\underline{i}}} \cdot \underline{u}  \end{align*}
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 ===== 6.3 Resonance Phenomena ===== ===== 6.3 Resonance Phenomena =====
  
-Similar to the approach last semester we now focus on circuits with inductors $L$. For preparation, please recap the chapter [[:electrical_engineering_1:circuits_under_different_frequencies|Circuits under different Frequencies]] from last semester.+Similar to last semester's approach, we now focus on circuits with inductors $L$. For preparation, please recap the chapter [[:electrical_engineering_1:circuits_under_different_frequencies|Circuits under different Frequencies]] from last semester.
  
 ==== 6.3.1 RLC - Series Resonant Circuit ==== ==== 6.3.1 RLC - Series Resonant Circuit ====
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 <panel type="info" title="Exercise 6.3.1 Series Resonant Circuit I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 6.3.1 Series Resonant Circuit I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-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 at $f_1 = 50~\rm Hz$.
  
   - 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 $U_I$?
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 <panel type="info" title="Exercise 6.3.2 Series Resonant Circuit II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 6.3.2 Series Resonant Circuit II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-A given $R$-$L$-$C$ series circuit is fed with a frequency, which is $20~\%$ 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.+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 $Q = {{1}\over{R}}\sqrt{{{L}\over{C}}}$.