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electrical_engineering_and_electronics_1:block20 [2025/12/02 18:44] – mexleadminelectrical_engineering_and_electronics_1:block20 [2026/01/20 15:39] (aktuell) – [20.3 Exercises] mexleadmin
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-====== Block 20 — Electromagnetic Induction and Energy ======+====== Block 20 — Inductance and Energy ======
  
-===== Learning objectives =====+===== 20.0 Intro ===== 
 + 
 +==== 20.0.1 Learning objectives ====
 <callout> <callout>
 After this 90-minute block, you can After this 90-minute block, you can
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 </callout> </callout>
  
-===== Preparation at Home =====+==== 20.0.2 Preparation at Home ====
  
 Well, again  Well, again 
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   * ...   * ...
  
-===== 90-minute plan =====+==== 20.0.3 90-minute plan ====
   - Warm-up (x min):    - Warm-up (x min): 
     - ....      - .... 
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   - Wrap-up (x min): Summary box; common pitfalls checklist.   - Wrap-up (x min): Summary box; common pitfalls checklist.
  
-===== Conceptual overview =====+==== 20.0.4 Conceptual overview ====
 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
   - ...   - ...
 </callout> </callout>
  
-===== Core content =====+===== 20.1 Core content =====
  
-==== Self-Induction ====+==== 20.1.1 Self-Induction ====
  
 Up to now, we investigated the induction of electric voltages and currents based on the change of an external flux ${\rm d}\Psi / {\rm d}t$.  Up to now, we investigated the induction of electric voltages and currents based on the change of an external flux ${\rm d}\Psi / {\rm d}t$. 
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 <WRAP> <imgcaption ImgNr15 | Self-Induction of a Coil> </imgcaption> {{drawio>SelfInductionCoil.svg}} </WRAP> <WRAP> <imgcaption ImgNr15 | Self-Induction of a Coil> </imgcaption> {{drawio>SelfInductionCoil.svg}} </WRAP>
  
-The created field density of the coil can be derived from Ampere's Circuital Law +Given by the [[block16#Recap of the fieldline images]] in Block16, we know that the $H$-field is given by magnetic voltage $\theta(t) = N \cdot i$ as:
 \begin{align*}  \begin{align*} 
-\theta(t) &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} \\  +   {H}(t) &=                         {{N \cdot i }\over {l}} \\ 
-          &= \int & \vec{H}_{\rm inner}(t) \cdot {\rm d}\vec{s} & + & \int \vec{H}_{\rm outer}(t) \cdot {\rm d} \vec{s} \\  +
-          &= \int & \vec{H}(t) \cdot {\rm d}\vec{s}             & + &   0 \\  +
-          &=      & {H}(t) \cdot l \\ +
 \end{align*} \end{align*}
  
-With magnetic voltage $\theta(t) = N \cdot i$ this lead to the magnetic flux density $B(t)$+This lead to the magnetic flux density $B(t)$
  
 \begin{align*}  \begin{align*} 
-N \cdot i &= {H}(t) \cdot l \\  
-   {H}(t) &=                         {{N \cdot i }\over {l}} \\  
    {B}(t) &= \mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \\     {B}(t) &= \mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \\ 
 \end{align*} \end{align*}
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 \end{align*} \end{align*}
  
-The changing flux $\Phi$ is now creating an induced electric voltage and current, which counteracts the initial change of the current. +The changing flux $\Phi$ is now creating an induced electric voltage and current, which counteracts the initial change of the current. \\
 This effect is called **Self Induction**. The induced electric voltage $u_{\rm ind}$ is given by: This effect is called **Self Induction**. The induced electric voltage $u_{\rm ind}$ is given by:
  
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 \end{align*} \end{align*}
  
-The result means that the induced electric voltage $u_{\rm ind}$ is proportional to the change of the current ${{\rm d}\over{{\rm d}t}}i$. +The result means that the induced electric voltage $u_{\rm ind}$ is proportional to the change of the current ${{\rm d}\over{{\rm d}t}}i$. \\
 The proportionality factor is also called **Self-inductance**  $L$ (or often simply called inductance). The proportionality factor is also called **Self-inductance**  $L$ (or often simply called inductance).
  
-===== 4.5 Inductance =====+==== 20.1.2 Inductance ====
  
 The inductance is another passive basic component of the electric circuit.  The inductance is another passive basic component of the electric circuit. 
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 \end{align*} \end{align*}
  
-==== Inductance of different Components ====+==== 20.1.3 Inductance of different Components ====
  
 === Long Coil === === Long Coil ===
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 +==== 20.1.4  Inductances in Circuits ====
  
 +Focus here: uncoupled inductors!
  
 +=== Series Circuits ===
  
-===== Common pitfalls =====+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_{\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 $u_{\rm eq}= u_1 + u_2 + ...$): 
 + 
 +\begin{align*}  
 +& u_{\rm eq}                                       & = &u_1                                    & + &u_2                        &+ ... \\  
 +& L_{\rm eq} {{{\rm d}i_{\rm eq} }\over{{\rm d}t}} & = &L_{1} {{{\rm d}i_{1} }\over{{\rm d}t}} & + &L_{2} {{di_{2} }\over{dt}} &+ ... \\  
 +& L_{\rm eq} {{{\rm d}i }\over{{\rm d}t}}          & = &L_{1} {{{\rm d}i     }\over{{\rm d}t}} & + &L_{2} {{di     }\over{dt}} &+ ... \\  
 +& L_{\rm eq}                                       & = &L_{1}                                  & + &L_{2}                      &+ ... \\  
 +\end{align*} 
 + 
 +===Parallel Circuits === 
 + 
 +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*} 
 + 
 +and Kirchhoff's nodal law: 
 + 
 +\begin{align*} i_{\rm eq}= i_1 + i_2 + ... \\ \end{align*} 
 + 
 +Here, the formula for the induced voltage has to be rearranged: 
 + 
 +\begin{align*}  
 +     u_{\rm ind}          &= L {{{\rm d}i}\over{{\rm d}t}} \quad \quad \quad \quad \bigg| \int(){\rm d}t \\  
 +\int u_{\rm ind} {\rm d}t &= L \cdot i \\  
 +                        i &= {{1}\over{L}} \cdot \int u_{\rm ind} {\rm d}t \\  
 +\end{align*} 
 + 
 +By this, we get: 
 + 
 +\begin{align*}  
 +                                  i_{\rm eq}          &=& i_1                                       &+& i_2                                       &+& ... \\  
 +{{1}\over{L_{\rm eq}}} \cdot \int u_{\rm eq} {\rm d}t &=& {{1}\over{L_1}} \cdot \int u_{1} {\rm d}t &+& {{1}\over{L_2}} \cdot \int u_{2} {\rm d}t &+& ... \\  
 +{{1}\over{L_{\rm eq}}} \cdot \int u          {\rm d}t &=& {{1}\over{L_1}} \cdot \int u     {\rm d}t &+& {{1}\over{L_2}} \cdot \int u     {\rm d}t &+& ... \\  
 +{{1}\over{L_{\rm eq}}}                                &=& {{1}\over{L_1}}                           &+& {{1}\over{L_2}}                           &+& ... \\  
 +\end{align*} 
 + 
 +<callout icon="fa fa-exclamation" color="red" title="Notice:"> The inductor behaves in the parallel and series circuit similar to the resistor. </callout> 
 + 
 +==== 20.1.5 Energy of the magnetic Field ==== 
 + 
 +not covered  
 +===== 20.2 Common pitfalls =====
   * ...   * ...
  
-===== Exercises =====+===== 20.3 Exercises ===== 
  
 {{page>electrical_engineering_and_electronics:task_ljxf80q7vxywehqf_with_calculation&nofooter}} {{page>electrical_engineering_and_electronics:task_ljxf80q7vxywehqf_with_calculation&nofooter}}
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 \begin{align*}  \begin{align*} 
 L_1 &= \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}} \\ L_1 &= \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}} \\
-    &= 4\pi \cdot 10^{-7} {\rm {{H}\over{m}}} \cdot 1 \cdot (390)^2 \cdot {{\pi \cdot (0.03~\rm m)^2 }\over {0.18 ~\rm m}}+    &= 4\pi \cdot 10^{-7} {\rm {{H}\over{m}}} \cdot 1 \cdot (390)^2 \cdot {{\pi \cdot ({{0.03~\rm m}\over{2}})^2 }\over {0.18 ~\rm m}}
 \end{align*} \end{align*}
  
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 #@HiddenBegin_HTML~4511R,Result~@# #@HiddenBegin_HTML~4511R,Result~@#
 \begin{align*}  \begin{align*} 
-L_1 &= 3.0 ~\rm mH+L_1 &= 0.75 ~\rm mH
 \end{align*} \end{align*}
 #@HiddenEnd_HTML~4511R,Result~@# #@HiddenEnd_HTML~4511R,Result~@#
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 #@HiddenBegin_HTML~4512R,Result~@# #@HiddenBegin_HTML~4512R,Result~@#
 \begin{align*}  \begin{align*} 
-L_1 &= 12 ~\rm mH+L_1 &= 4 ~\rm mH
 \end{align*} \end{align*}
 #@HiddenEnd_HTML~4512R,Result~@# #@HiddenEnd_HTML~4512R,Result~@#
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 #@HiddenBegin_HTML~4513R,Result~@# #@HiddenBegin_HTML~4513R,Result~@#
 \begin{align*}  \begin{align*} 
-L_1 &= 6.0 ~\rm mH+L_1 &= 1.5 ~\rm mH
 \end{align*} \end{align*}
 #@HiddenEnd_HTML~4513R,Result~@# #@HiddenEnd_HTML~4513R,Result~@#
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 #@HiddenBegin_HTML~4514R,Result~@# #@HiddenBegin_HTML~4514R,Result~@#
 \begin{align*}  \begin{align*} 
-L_4 &= 3.0 ~\rm H+L_4 &= 0.75 ~\rm H
 \end{align*} \end{align*}
 #@HiddenEnd_HTML~4514R,Result~@# #@HiddenEnd_HTML~4514R,Result~@#
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 <panel type="info" title="Exercise 4.5.2 Self Induction II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 4.5.2 Self Induction II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-A cylindrical air coil (length $l=40 ~\rm cm$, diameter $d=5.0 ~\rm cm$, and a number of windings $N=300$) passes a current of $30 ~\rm A$. The current shall be reduced linearly in $2.0 ~\rm ms$ down to $0.0 ~\rm A$.+A cylindrical air coil (length $l=40 ~\rm cm$, radius $r=5.0 ~\rm cm$, and a number of windings $N=300$) passes a current of $30 ~\rm A$. The current shall be reduced linearly in $2.0 ~\rm ms$ down to $0.0 ~\rm A$.
  
 What is the amount of the induced voltage $u_{\rm ind}$?  What is the amount of the induced voltage $u_{\rm ind}$?