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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
electrical_engineering_2:magnetic_circuits [2023/05/17 10:23] mexleadmin [Effects in the electric Circuits] |
electrical_engineering_2:magnetic_circuits [2024/05/03 16:02] (aktuell) mexleadmin [Magnetic Circuit with two Sources] |
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Zeile 1: | Zeile 1: | ||
- | ====== 5. Magnetic Circuits ====== | + | ====== 5 Magnetic Circuits ====== |
< | < | ||
Zeile 9: | Zeile 9: | ||
In this chapter, we will investigate, | In this chapter, we will investigate, | ||
- | ===== 5.1 Linear | + | ===== 5.1 Linear |
For the upcoming calculations, | For the upcoming calculations, | ||
Zeile 168: | Zeile 168: | ||
Calculate the magnetic resistances of cylindrical coreless (=ironless) coils with the following dimensions: | Calculate the magnetic resistances of cylindrical coreless (=ironless) coils with the following dimensions: | ||
- | - $l=35.8~\rm cm$, $d=1.9~\rm cm$ | + | - $l=35.8~\rm cm$, $d=1.90~\rm cm$ |
- | - $l=22.5~\rm cm$, $d=1.5~\rm cm$ | + | - $l=11.1~\rm cm$, $d=1.50~\rm cm$ |
- | <button size=" | + | # |
- | - $1.5\cdot 10^5 ~\rm {{1}\over{H}}$ | + | The magnetic resistance is given by: |
- | - $3.0\cdot 10^5 ~\rm {{1}\over{H}}$ | + | \begin{align*} |
+ | \ R_{\rm m} & | ||
+ | \end{align*} | ||
- | </collapse> | + | With |
+ | * the area $ A = \left({{d}\over{2}}\right)^2 \cdot \pi $ | ||
+ | * the vacuum magnetic permeability $\mu_{0}=4\pi\cdot 10^{-7} ~\rm H/m$, and | ||
+ | * the relative permeability $\mu_{\rm r}=1$. | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | - $1.00\cdot 10^9 ~\rm {{1}\over{H}}$ | ||
+ | - $0.50\cdot 10^9 ~\rm {{1}\over{H}}$ | ||
+ | # | ||
</ | </ | ||
Zeile 393: | Zeile 405: | ||
$k_{21}$ describes how much of the flux from coil $1$ is acting on coil $2$ (similar for $k_{12}$): | $k_{21}$ describes how much of the flux from coil $1$ is acting on coil $2$ (similar for $k_{12}$): | ||
- | \begin{align*} k_{21} = {{\Phi_{21}}\over{\Phi_{11}}} \\ \end{align*} | + | \begin{align*} k_{21} = \pm {{\Phi_{21}}\over{\Phi_{11}}} \\ \end{align*} |
- | When $k_{21}=100~\%$, | + | The sign of $k_{21}$ depends on the direction of $\Phi_{21}$ relative to $\Phi_{22}$! If the directions are the same, the positive sign applies, if the directions are oposite, the minus sign applies. |
+ | |||
+ | When $k_{21}=+100~\%$, there is no flux in the middle leg but only in the second coil and in the same direction as the flux that originates from second coil. \\ | ||
+ | When $k_{21}=-100~\%$, | ||
For $k_{21}=0~\%$ all the flux is in the middle leg circumventing the second coil, i.e. there is no coupling. | For $k_{21}=0~\%$ all the flux is in the middle leg circumventing the second coil, i.e. there is no coupling. | ||
Zeile 409: | Zeile 424: | ||
& | & | ||
\end{align*} | \end{align*} | ||
+ | |||
+ | Note, that also $M_{21}$ and $M_{12}$ can be either positiv or negative, depending on the sign of the coupling coefficients. | ||
The formula is finally: | The formula is finally: | ||
Zeile 473: | Zeile 490: | ||
* the mutual inductances are equal: $M_{12} = M_{21} = M$ | * the mutual inductances are equal: $M_{12} = M_{21} = M$ | ||
* the mutual inductance $M$ is: $M = \sqrt{M_{12}\cdot M_{21}} = k \cdot \sqrt {L_{11}\cdot L_{22}}$ | * the mutual inductance $M$ is: $M = \sqrt{M_{12}\cdot M_{21}} = k \cdot \sqrt {L_{11}\cdot L_{22}}$ | ||
- | * The resulting *total coupling* $k$ is given as \begin{align*} k = \sqrt{k_{12}\cdot k_{21}} \end{align*} | + | * The resulting |
==== Effects in the electric Circuits ==== | ==== Effects in the electric Circuits ==== | ||
Zeile 499: | Zeile 516: | ||
< | < | ||
- | In this case, the **mutual induction | + | In this case, the **mutual induction |
The formula of the shown circuitry is then: | The formula of the shown circuitry is then: | ||
Zeile 513: | Zeile 530: | ||
< | < | ||
- | In this case, the **mutual induction | + | In this case, the **mutual induction |
The formula of the shown circuitry is then: | The formula of the shown circuitry is then: | ||
\begin{align*} | \begin{align*} | ||
- | u_1 &= R_1 \cdot i_1 &+ L_{11} \cdot {{{\rm d}i_1}\over{{\rm d}t}} &- M \cdot {{{\rm d}i_2}\over{{\rm d}t}} & \\ | + | u_1 &= R_1 \cdot i_1 &+ L_{11} \cdot {{{\rm d}i_1}\over{{\rm d}t}} & + M \cdot {{{\rm d}i_2}\over{{\rm d}t}} & \\ |
- | u_2 &= R_2 \cdot i_2 &+ L_{22} \cdot {{{\rm d}i_2}\over{{\rm d}t}} &- M \cdot {{{\rm d}i_1}\over{{\rm d}t}} & \\ | + | u_2 &= R_2 \cdot i_2 &+ L_{22} \cdot {{{\rm d}i_2}\over{{\rm d}t}} & + M \cdot {{{\rm d}i_1}\over{{\rm d}t}} & \\ |
\end{align*} | \end{align*} | ||
Zeile 599: | Zeile 616: | ||
\boxed{W_m = {{1}\over{2}}L\cdot I^2 } | \boxed{W_m = {{1}\over{2}}L\cdot I^2 } | ||
\end{align*} | \end{align*} | ||
+ | |||
+ | |||
+ | |||
+ | |||
==== magnetic Energy of a magnetic Circuit ==== | ==== magnetic Energy of a magnetic Circuit ==== | ||
- | With this formula also the stored energy in a magnetic circuit can be calculated. | + | With this formula also the stored energy in a magnetic circuit can be calculated. For this, the formula be rewritten by the properties linked flux $\Psi = N \cdot \Phi = L \cdot I$ and magnetic voltage $\theta=N \cdot I = \Phi \cdot R_{\rm m}$ of the magnetic circuit: \begin{align*} \boxed{W_{\rm m} = {{1}\over{2}} \Psi \cdot I = {{1}\over{2}} {{\Psi^2}\over{L}}= {{1}\over{2}}{{\Phi^2 }\over{N^2 \cdot L}} = {{1}\over{2}} \Phi^2 \cdot R_{\rm m} = {{1}\over{2}}{{\theta^2 }\over{R_{\rm m}}}} \end{align*} |
- | For this, the formula be rewritten by the properties linked flux $\Psi = N \cdot \Phi = L \cdot I$ and magnetic voltage $\theta=N \cdot I = \Phi \cdot R_{\rm m}$ of the magnetic circuit: | + | |
- | \begin{align*} | + | |
- | \boxed{W_{\rm m} = {{1}\over{2}} | + | |
- | = {{1}\over{2}}{{\Psi^2 }\over{L}}} | + | |
- | \end{align*} | + | |
==== magnetic Energy of a toroid Coil ==== | ==== magnetic Energy of a toroid Coil ==== | ||
Zeile 758: | Zeile 774: | ||
The big difference there is, that there the magnetic flux $\Phi$ is not interpreted as an analogy to the electric current $I$ but to the electric charge $Q$. | The big difference there is, that there the magnetic flux $\Phi$ is not interpreted as an analogy to the electric current $I$ but to the electric charge $Q$. | ||
This model can solve more questions, however, is a bit less intuitive based on this course and less commonly used compared to the {{https:// | This model can solve more questions, however, is a bit less intuitive based on this course and less commonly used compared to the {{https:// | ||
+ | |||
+ | ==== Moving a Plate into an Air Gap ==== | ||
+ | |||
+ | < | ||
==== Switch Reluctance Motor ==== | ==== Switch Reluctance Motor ==== |