Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- electrical_engineering_and_electronics_1:block20 [2025/12/02 17:12] – angelegt mexleadmin
+++ electrical_engineering_and_electronics_1:block20 [2025/12/02 18:50] (aktuell) – mexleadmin
@@ Zeile 31: / Zeile 31: @@
 ===== Core content =====
-...
+==== Self-Induction ====
+Up to now, we investigated the induction of electric voltages and currents based on the change of an external flux ${\rm d}\Psi / {\rm d}t$.
+For the induced current $i_{\rm ind}$, we found that it counteracts the change of the external flux (Lenz law).
+But what happens, when there is no external field - only a coil which creates the flux change itself (see <imgref ImgNr46>)?
+<WRAP> <imgcaption ImgNr46 | Induction Phenomenons> </imgcaption> {{drawio>InductionPhenomenons.svg}} </WRAP>
+To understand this, we will investigate the situation for a long coil (<imgref ImgNr15>).
+<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
+\begin{align*}
+\theta(t) &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} \\
+          &= \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*}
+With magnetic voltage $\theta(t) = N \cdot i$ this lead to the magnetic flux density $B(t)$
+\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}} \\
+\end{align*}
+Based on the magnetic flux density $B(t)$ it is possible to calculate the flux $\Phi(t)$:
+\begin{align*}
+\Phi(t) &= \iint_A \vec{B}(t)                                      \cdot {\rm d}\vec{A} \\
+        &= \iint_A \mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \cdot {\rm d}A \\
+        &=         \mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \cdot A \\
+\end{align*}
+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:
+\begin{align*}
+u_{\rm ind} &= - N \cdot {{{\rm d}                                                   \Phi(t)}\over{{\rm d}t}} \\
+            &= - N \cdot {{{\rm d} (\mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \cdot A)}\over{{\rm d}t}} \\
+            &= - N \cdot            \mu_0 \mu_{\rm r} \cdot {{N \cdot A }\over {l}} \cdot {{{\rm d}i}\over{{\rm d}t}} \\
+\end{align*}
+\begin{align*}
+\boxed{ u_{\rm ind} = - \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}} \cdot {{{\rm d}i}\over{{\rm d}t}} \\ } \\
+\text{for a long coil}
+\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 proportionality factor is also called **Self-inductance**  $L$ (or often simply called inductance).
+===== 4.5 Inductance =====
+The inductance is another passive basic component of the electric circuit.
+Besides the ohmic resistor $R$ and the capacitor $C$, the inductor $L$ is the lump component entailing the inductance.
+Generally, the inductance is defined by:
+\begin{align*}
+\boxed{ L = \left|{{u_{\rm ind}}\over{{\rm d}i / {\rm d}t}}\right| \\ }
+\end{align*}
+The inductance $L$ can also be described differently based on Lenz law $u_{\rm ind} = - {{\rm d}\over{{\rm d}t}}\Psi(t)$ :
+\begin{align*}
+L &= \left|{{u_{\rm ind}}\over{{\rm d}i / {\rm d}t}}\right| \\
+  &=  {{{d \Psi(t)}/{dt}}\over{{\rm d}i / {\rm d}t}} \\
+\end{align*}
+\begin{align*} \boxed{ L = {{ \Psi(t)}\over{i}} } \end{align*}
+One can also consider an inductor a "conservative person": it does not like to see abrupt changes in the passing current.
+It reacts to any change in the current with a counteracting voltage since the current change leads to a changing flux and - therefore - an induced voltage.
+The <imgref BildNr5> shows an inductor in series with a resistor and a switch (any real switch also behaves as a capacitor, when open).
+Once the simulation is started, the inductor directly counteracts the current, which is why the current only slowly increases.
+The unit of the inductance is $\rm 1 ~Henry = 1 ~H = 1 {{Vs}\over{A}} = 1{{Wb}\over{A}} $
+<WRAP> <imgcaption BildNr5 | Example of a Circuit with an Inductor> </imgcaption> \\ {{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCBMCmC0CcIAsA6ADAdgBwFYMYGYcd4CyCsQc0rlIq4wwAoANyiRoLSQ5qfo0anWkJBoUOZgBsQZAGzh5XAosiReQ9GghDmAJzmqoGo4u6bwaNMwIYVigWaX9x45gHc+cns4vuADyUCKDRKMAw1JmQOEABXZiCIykgwRywUsEReMEobIKRUqAxeJDJijBjckAAlROQ0NXlFJCyoeWzYhIL1UJCy+khGqsoAGXrOSqycOn54Sl5TbuQwGbAy2fAkFtiAS09vJwVwMEF6oZpUlKGTCFKaQGTCers1NBmCeFeFhpAnoLtFh05PBFhh6PdfswAPbgEDySwBLBgHTQEBYLS6KBYsSXdwwkLwtyI5FgAAmonQayxgjktIkaDOQA noborder}} </WRAP>
+Mathematically the voltages can be described in the following way:
+\begin{align*}
+u_0 &= u_R &+ &u_L \\
+    &= i \cdot R & + &     {{{\rm d}\Psi}\over{{\rm d}t}} \\
+    &= i \cdot R & + &L \cdot {{{\rm d}i}\over{{\rm d}t}} \\
+\end{align*}
+==== Inductance of different Components ====
+=== Long Coil ===
+In the last sub-chapter, the formula of a long coil was already investigated.
+By these, the inductance of a long coil is
+\begin{align*}
+\boxed{L_{\rm long \; coil} = \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}}}
+\end{align*}
+=== Toroidal Coil ===
+The toroidal coil was analyzed in the last chapter(see [[:electrical_engineering_2:the_magnetostatic_field#magnetic_field_strength_part_1toroidal_coil|magnetic Field Strength Part 1: Toroidal Coil]]).
+Here, a rectangular intersection a assumed (see <imgref ImgNr16>).
+<WRAP> <imgcaption ImgNr16 | Self-Induction of a toroidal Coil> </imgcaption> {{drawio>SelfInductionToroCoil.svg}} </WRAP>
+This leads to
+\begin{align*} H(t) = {{N \cdot i}\over {l}} \end{align*}
+with the mean magnetic path length (= length of the average field line) $l = \pi(r_{\rm o} + r_{\rm i})$:
+\begin{align*} H(t) = {{N \cdot i}\over { \pi(r_{\rm o} + r_{\rm i})}} \end{align*}
+The inductance $L$ can be calculated by
+\begin{align*}
+L_{\rm toroidal \; coil} &=         {{ \Psi(t)}\over{i}} \\
+                         &= {{ N \cdot \Phi(t)}\over{i}} \\
+\end{align*}
+With the magnetic flux density $B(t) = \mu_0 \mu_{\rm r} H(t) = \mu_0 \mu_{\rm r} {{i \cdot N }\over {l}}$ and the cross section $A = h (r_{\rm o} - r_{\rm i})$, we get:
+\begin{align*}
+\quad \quad L_{\rm toroidal \; coil} &= {{   N \cdot \mu_0 \mu_{\rm r} {{i \cdot N } \over { \pi(r_{\rm o} + r_{\rm i})}} \cdot h(r_{\rm o} - r_{\rm i})}\over{i}} \\
+                                     &= {{ N^2 \cdot \mu_0 \mu_{\rm r}                                                    \cdot h(r_{\rm o} - r_{\rm i})}\over{ \pi(r_{\rm o} + r_{\rm i})}} \\
+\end{align*}
+\begin{align*}
+\boxed{ L_{\rm toroidal \; coil} = \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{ h(r_{\rm o} - r_{\rm i})}\over{ \pi(r_{\rm o} + r_{\rm i})}} }
+\end{align*}
+==== 6 Inductances in Circuits ====
+Focus here: uncoupled inductors!
+=== Series Circuits ===
+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>
 ===== Common pitfalls =====
@@ Zeile 37: / Zeile 222: @@
 ===== Exercises =====
 {{page>electrical_engineering_and_electronics:task_ljxf80q7vxywehqf_with_calculation&nofooter}}
 {{page>electrical_engineering_and_electronics:task_unkkahm3u0v9azny_with_calculation&nofooter}}
-{{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 4.5.1 Self Induction I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+Calculate the inductance for the following settings
+. Cylindrical long air coil with $N=390$, winding diameter $d=3.0 ~\rm cm$ and length $l=18 ~\rm cm$
+#@HiddenBegin_HTML~4511S,Solution~@#
+\begin{align*}
+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}}
+\end{align*}
+#@HiddenEnd_HTML~4511S,Solution ~@#
+#@HiddenBegin_HTML~4511R,Result~@#
+\begin{align*}
+L_1 &= 3.0 ~\rm mH
+\end{align*}
+#@HiddenEnd_HTML~4511R,Result~@#
+. Similar coil geometry as explained in 1. , but with double the number of windings
+#@HiddenBegin_HTML~4512S,Solution~@#
+\begin{align*}
+L_2 &= \mu_0 \mu_{\rm r} \cdot N_2^2 \cdot {{A }\over {l}} \\
+    &= \mu_0 \mu_{\rm r} \cdot (2\cdot N)^2 \cdot {{A }\over {l}} \\
+    &= \mu_0 \mu_{\rm r} \cdot 4\cdot N^2 \cdot {{A }\over {l}} \\
+    &= 4\cdot L_1 \\
+\end{align*}
+#@HiddenEnd_HTML~4512S,Solution ~@#
+#@HiddenBegin_HTML~4512R,Result~@#
+\begin{align*}
+L_1 &= 12 ~\rm mH
+\end{align*}
+#@HiddenEnd_HTML~4512R,Result~@#
+. Two coils as explained in 1. in series
+#@HiddenBegin_HTML~4513S,Solution~@#
+multiple inductances in series just add up. One can think of adding more windings to the first coil in the formula..
+#@HiddenEnd_HTML~4513S,Solution ~@#
+#@HiddenBegin_HTML~4513R,Result~@#
+\begin{align*}
+L_1 &= 6.0 ~\rm mH
+\end{align*}
+#@HiddenEnd_HTML~4513R,Result~@#
+. Similar coil geometry and number of windings as explained in 1. , but with an iron core ($\mu_{\rm r}=1000$)
+#@HiddenBegin_HTML~4514S,Solution~@#
+\begin{align*}
+L_4 &= \mu_0 \mu_{\rm r,4} \cdot N^2 \cdot {{A }\over {l}} \\
+    &= \mu_0 \cdot 1000 \cdot N^2 \cdot {{A }\over {l}} \\
+    &= 1000 \cdot L_4 \\
+\end{align*}
+#@HiddenEnd_HTML~4514S,Solution ~@#
+#@HiddenBegin_HTML~4514R,Result~@#
+\begin{align*}
+L_4 &= 3.0 ~\rm H
+\end{align*}
+#@HiddenEnd_HTML~4514R,Result~@#
+</WRAP></WRAP></panel>
+<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 $0.0 ~\rm A$.
+What is the amount of the induced voltage $u_{\rm ind}$?
+#@HiddenBegin_HTML~4521S,Solution~@#
+The requested induced voltage can be derived by:
+\begin{align*}
+L                                      &= \left|{{u_{\rm ind}}\over{{\rm d}i / {\rm d}t}}\right| \\
+\rightarrow  \left|u_{\rm ind}\right|  &= L \cdot \left|{{{\rm d}i}\over{{\rm d}t}}\right| \\
+                                       &= L \cdot \left|{{\Delta i}\over{\Delta t}}\right| \\
+\end{align*}
+Therefore, we just need the inductance $L$, since ${{\Delta i}\over{\Delta t}}$ is defined as $30 ~\rm A$ per $2 ~\rm ms$:
+\begin{align*}
+L &= \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}} \\
+\end{align*}
+So, the result can be derived as:
+\begin{align*}
+\left|u_{\rm ind}\right| &= \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}} \cdot \left|{{\Delta i}\over{\Delta t}}\right| \\
+                         &= 4\pi \cdot 10^{-7} {\rm {{H}\over{m}}} \cdot 1 \cdot (300)^2 \cdot {{\pi \cdot (0.05~\rm m)^2 }\over {0.40 ~\rm m}} \cdot {{30 ~\rm A}\over{2 ~\rm ms}}
+\end{align*}
+#@HiddenEnd_HTML~4521S,Solution ~@#
+#@HiddenBegin_HTML~1,Result~@#
+\begin{align*}
+\left|u_{\rm ind}\right| &= 33 ~\rm V\end{align*}
+#@HiddenEnd_HTML~1,Result~@#
+</WRAP></WRAP></panel>
+<panel type="info" title="Exercise 4.5.3 Self Induction III"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A coil with the inductance $L=20 ~\rm µH$ passes a current of $40 ~\rm A$. The current shall be reduced linearly in $5 ~\rm µs$ down to $0 ~\rm A$ (see <imgref ImgNrEx05>).
+  * What is the amount of the induced voltage $u_{\rm ind}$?
+  * Sketch the course of $u_{\rm ind}(t)$!
+<WRAP> <imgcaption ImgNrEx05| Circuit and timing Diagram> </imgcaption> <WRAP> {{drawio>CircuitAndTiming.svg}} \\ </WRAP></WRAP>
+</WRAP></WRAP></panel>
 ===== Embedded resources =====