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--- electrical_engineering_and_electronics_1:block18 [2025/12/02 18:34] – mexleadmin
+++ electrical_engineering_and_electronics_1:block18 [2025/12/02 18:52] (aktuell) – mexleadmin
@@ Zeile 237: / Zeile 237: @@
 which is identical to the potential difference between the ends of the rod that we determined earlier.
+==== Linked Flux ====
+When looking at the magnetic field in a coil multiple windings capture the passing flux, see <imgref ImgNr14> (a).
+It can also be interpreted in such a way that the flux is going through the closed surface of the circuit multiple times (in picture (b)).
+<WRAP> <imgcaption ImgNr14 | Example for a linked Flux> </imgcaption> {{drawio>LinkedFlux.svg}} </WRAP>
+The **linked flux** $\Psi$ is defined as the resulting flux given by the sum of the partial fluxes of the closed circuit.
+\begin{align*}
+\boxed{ \Psi = \sum_{i=1}^n \Phi_{i} }
+\end{align*}
+For the case of a coil with $N$ windings, this leads to:
+\begin{align*}
+\boxed{ \Psi = N \cdot \Phi }
+\end{align*}
+<callout icon="fa fa-exclamation" color="red" title="Notice:">
+When calculating the force, use the flux within the material ($\Phi$). \\
+When investigating effects in the coil, use the linked flux ($\Psi$). \\
+</callout>
 ===== Common pitfalls =====
@@ Zeile 242: / Zeile 269: @@
 ===== Exercises =====
+{{page>electrical_engineering_and_electronics:task_rdz03rspbwusy7wk_with_calculation&nofooter}}
+{{page>electrical_engineering_and_electronics:task_ludzwiuhjxitz85b_with_calculation&nofooter}}
+<panel type="info" title="Exercise 4.1.4 Effects of induction I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A change of magnetic flux is passing a coil with a single winding. The following pictures <imgref ImgNrEx01> show different flux-time-diagrams as examples.
+  * Create for each $\Phi(t)$-diagram the corresponding $u_{\rm ind}(t)$-diagram!
+  * Write down each maximum value of $u_{\rm ind}(t)$
+<WRAP> <imgcaption ImgNrEx01| Flux-Time-Diagrams> </imgcaption> <WRAP> {{drawio>FluxTimeDia1.svg}} \\ </WRAP></WRAP>
+<button size="xs" type="link" collapse="Solution_4_1_4_1_Solution">{{icon>eye}} Solution for (a)</button><collapse id="Solution_4_1_4_1_Solution" collapsed="true">
+For partwise linear $u_{\rm ind}$ one can derive:
+\begin{align*}
+u_{\rm ind} &= -{{{\rm d}\Phi}\over{{\rm d}t}} \\
+            &= -{{\Delta \Phi}\over{\Delta t}}
+\end{align*}
+For diagram (a):
+  * $t= 0.0 ... 0.6 ~\rm s$: $u_{\rm ind} = -{{0 ~\rm Vs}\over{0.6 ~\rm s}}= 0$
+  * $t= 0.6 ... 1.5 ~\rm s$: $u_{\rm ind} = -{{-3.75\cdot 10^{-3} ~\rm Vs}\over{0.9 ~\rm s}}= +4.17 ~\rm mV$
+  * $t= 1.5 ... 2.1 ~\rm s$: $u_{\rm ind} = -{{0 ~\rm Vs}\over{0.6 ~\rm s}}= 0$
+</collapse>
+<button size="xs" type="link" collapse="Solution_4_1_4_1_Finalresult">
+{{icon>eye}} Final result for (a)</button><collapse id="Solution_4_1_4_1_Finalresult" collapsed="true">
+<WRAP> <imgcaption ImgNrEx01| Flux-Time-Diagrams Solution> </imgcaption> <WRAP> {{drawio>FluxTimeDia1Solution.svg}} \\
+</WRAP></WRAP> \\
+</collapse>
+</WRAP></WRAP></panel>
+<panel type="info" title="Exercise 4.1.5 Effects of induction II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A changing of magnetic flux is passing a coil with a single winding and induces the voltage $u_{\rm ind}(t)$.
+The following pictures <imgref ImgNrEx02> show different voltage-time diagrams as examples.
+  * Create for each $u_{\rm ind}(t)$-diagram the corresponding $\Phi(t)$-diagram!
+  * Write down each maximum value of $\Phi(t)$
+Note the given start value $\Phi_0$ for each flux.
+<WRAP> <imgcaption ImgNrEx02| Voltage-Time-Diagrams> </imgcaption> <WRAP> {{drawio>FluxTimeDia2.svg}} \\ </WRAP></WRAP>
+#@HiddenBegin_HTML~415_1S,Solution for (a)~@#
+For partwise linear $u_{\rm ind}$ one can derive:
+\begin{align*}
+u_{\rm ind}        &= -{{{\rm d}\Phi}\over{{\rm d}t}} \\
+\rightarrow  \Phi  &= -\int_0^t{ u_{\rm ind} \;{\rm d}t} \\
+\Phi               &= \Phi_0 -\sum_k {u_{{\rm ind},~k} \; \Delta t} \\
+\end{align*}
+For diagram (a):
+  * $t= 0.00 ... 0.04 ~\rm s\quad$: $\quad \Phi = \Phi_0 - {0 \cdot \; \Delta t} \quad\quad\quad\quad\quad\quad\quad= 0 ~\rm Wb$
+  * $t= 0.04 ... 0.10 ~\rm s\quad$: $\quad \Phi =   0 {~\rm Wb} - {{30 ~\rm mV} \cdot \; (t - 0.04 ~\rm s)} = \quad {1.2 ~\rm mWb} - t \cdot 30 ~\rm mV$
+  * $t= 0.10 ... 0.14 ~\rm s\quad$: $\quad \Phi =   {1.2 ~\rm mWb} - {0.10 ~\rm s} \cdot 30 ~\rm mV \quad = - {1.8 ~\rm mWb}$
+#@HiddenEnd_HTML~415_1S,Solution ~@#
+#@HiddenBegin_HTML~415_1R,Result for (a)~@#
+{{drawio>FluxTimeDia2Res.svg}}
+#@HiddenEnd_HTML~415_1R,Result~@#
+</WRAP></WRAP></panel>
 <panel type="info" title="Exercise 4.1.1 Magnetic Field Strength around a horizontal straight Conductor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
@@ Zeile 440: / Zeile 542: @@
 </collapse> </WRAP></WRAP></panel>
+<panel type="info" title="Exercise 4.1.6 Coil in magnetic Field I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A single winding is located in a homogenous magnetic field ($B = 0.5 ~\rm T$) between the pole pieces.
+The winding has a length of $150 ~\rm mm$ and a distance between the conductors of $50 ~\rm mm$ (see <imgref ImgNrEx03>).
+  * Determine the function $u_{\rm ind}(t)$, when the coil is rotating with $3000 ~\rm min^{-1}$.
+  * Given a current of $20 ~\rm A$ through the winding: What is the torque $M(\varphi)$ depending on the angle between the surface vector of the winding and the magnetic field?
+<WRAP> <imgcaption ImgNrEx03| Winding between Pole Pieces> </imgcaption> <WRAP> {{drawio>WindingPolePieces.svg}} \\ </WRAP></WRAP>
+<button size="xs" type="link" collapse="Solution_4_1_6_1_Solution">{{icon>eye}} Solution</button><collapse id="Solution_4_1_6_1_Solution" collapsed="true">
+\begin{align*}
+u_{\rm ind} &= -   {{{\rm d}\Phi}\over{{\rm d}t}} \\
+            &= -         {{\rm d}\over{{\rm d}t}} B \cdot A \\
+            &= - B \cdot {{\rm d}\over{{\rm d}t}} A\\
+            &= - B \cdot {{\rm d}\over{{\rm d}t}} \left(l \cdot d \cdot \cos(\omega t) \right)\\
+            &= + B \cdot l \cdot d \cdot \omega \cdot \sin(\omega t)\\
+\end{align*}
+</collapse> </WRAP></WRAP></panel>
+<panel type="info" title="Exercise 4.1.7 Coil in magnetic Field II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A rectangular coil is given by the sizes $a=10 ~\rm cm$, $b=4 ~\rm cm$, and the number of windings $N=200$.
+This coil moves with a constant speed of $v=2 ~\rm m/s$ perpendicular to a homogeneous magnetic field ($B=1.3 ~\rm T$ on a length of $l=5 ~\rm cm$).
+The area of the coil is tilted with regard to the field in $\alpha=60°$ and enters the field from the left side (see <imgref ImgNrEx04>).
+  * Determine the function $u_{\rm ind}(t)$ on the coil along the given path. Sketch of the $u_{\rm ind}(t)$ diagram.
+  * What is the maximum induced voltage $u_{\rm ind,Max}$?
+<WRAP> <imgcaption ImgNrEx04| Winding between Pole Pieces> </imgcaption> <WRAP> {{drawio>WindingPolePieces2.svg}} \\ </WRAP></WRAP>
+<button size="xs" type="link" collapse="Solution_4_1_7_1_Solution">{{icon>eye}} Solution</button><collapse id="Solution_4_1_7_1_Solution" collapsed="true">
+Let assume, that $l$ is in the $x$-axis, $\vec{B}$ in the $y$-axis and $a$.
+\\ \\
+**Step 1**: Calculate the effective area, perpendicular to the $\vec{B}$-field (independent from whether the area is in the $\vec{B}$-field or not).
+For this $b$ has to be projected onto the plane perpendicular to the $\vec{B}$-field: $b_{\rm eff}= b \cdot \cos \alpha$
+\begin{align*}
+A_{\rm eff} &= a \cdot b \cdot \cos \alpha
+\end{align*}
+**Step 2**: Focus on entering and exiting the $\vec{B}$-field. \\
+Induction only occurs for ${{\rm d}\over{{\rm d}t}}(A\cdot B)\neq 0$, so here: when the area $A_{\rm eff}$ enters and leave the constant $\vec{B}$-field.
+When entering the $\vec{B}$-field the area $A$ with $0<A<A_{eff}$ is in the field.
+The area moves with $v$. Therefore, after $\Delta t = b_{\rm eff} \cdot v$ the full $\vec{B}$-field is provided onto the area $A_{\rm eff}$:
+\begin{align*}
+u_{\rm ind} &= -        {{{\rm d}\Psi}\over{{\rm d}t}} \\
+            &= -N \cdot       {{\rm d}\over{{\rm d}t}} B \cdot A \\
+            &= -N \cdot           {{1}\over{\Delta t}} B \cdot A_{\rm eff} \\
+            &= -N \cdot {{1}\over{b \cdot \cos \alpha \cdot v}} B \cdot a \cdot b \cdot \cos \alpha \\
+            &= -N \cdot B \cdot {{a}\over{v}}\\
+\end{align*}
+The following diagram shows ...
+  * ... how one can derive the effective width $b_{\rm eff}$, which is projected onto the plane perpendicular to the $\vec{B}$-field: $b_{\rm eff}= b \cdot \cos \alpha$
+  * ... what happens on the effective area $A_{\rm eff}$: there is a change of the field lines in the area only for entering and leaving the $\vec{B}$-field.
+  * ... how the $u_{\rm ind}(t)$ looks as a graph: the part of $A_{\rm eff}$, where the $\vec{B}$-field passes through increase linearly due to the constant speed $v$
+Be aware, that the task did not provide a clue for the direction of windings and therefore it provides no clue for the polarization of the induced voltage. \\
+So, the course of the voltage when entering or exiting is not uniquely given.
+<WRAP> <imgcaption ImgNrEx04s| Solution> </imgcaption> <WRAP>{{drawio>WindingPolePieces2solution.svg}}  \\ </WRAP></WRAP>
+</collapse> </WRAP></WRAP></panel>
 ===== Embedded resources =====