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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung Nächste Überarbeitung Beide Seiten der Revision | ||
electrical_engineering_2:magnetic_circuits [2023/05/23 08:49] ott [magnetic Energy of a magnetic Circuit] |
electrical_engineering_2:magnetic_circuits [2024/05/10 16:02] mexleadmin [5.3 Mutual Induction and Coupling] |
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- | ====== 5. Magnetic Circuits ====== | + | ====== 5 Magnetic Circuits ====== |
< | < | ||
Zeile 308: | Zeile 308: | ||
===== 5.3 Mutual Induction and Coupling ===== | ===== 5.3 Mutual Induction and Coupling ===== | ||
- | Situation: Two coils $1$ and $2$ near each other. \\ Questions: | + | Imagine charging your phone wirelessly by simply placing it on a charging pad. |
+ | This seamless experience is made possible by the fascinating phenomenon of mutual induction and coupling between two coils. | ||
+ | |||
+ | This situation is depicted in <imgref ImgNr09>: | ||
+ | When an alternating current flows through one coil (Coil $1$), it creates a time-varying magnetic field that induces a voltage in the nearby coil (Coil $2$), even though they are not physically connected. | ||
+ | This mutual influence is governed by the principle of electromagnetic induction. | ||
+ | |||
+ | <WRAP center 35%> < | ||
+ | |||
+ | The key factor determining the strength of mutual induction is the mutual inductance ($M$) between the coils. | ||
+ | It quantifies the magnetic flux linkage and depends on factors like the number of turns, current, and relative orientation of the coils. | ||
+ | |||
+ | While geometric properties play a role, the fundamental principle can be described using electric properties alone, making mutual induction a versatile concept with numerous applications, | ||
+ | |||
+ | * Wireless power transfer | ||
+ | * Transformers | ||
+ | * Inductive coupling in communication systems | ||
+ | * Inductive sensors | ||
+ | |||
+ | As we explore this chapter, we'll delve into the mathematical models, equations, and practical considerations of mutual induction and coupling, unlocking a world of innovative technologies that shape our modern lives. | ||
+ | We explicitly try to answer the following questions: | ||
* Which effect do the coils have on each other? | * Which effect do the coils have on each other? | ||
* Can we describe the effects with mainly electric properties (i.e. no geometric properties) | * Can we describe the effects with mainly electric properties (i.e. no geometric properties) | ||
- | <WRAP center 35%> < | ||
==== Effect of Coils on each other ==== | ==== Effect of Coils on each other ==== | ||
Zeile 405: | Zeile 424: | ||
$k_{21}$ 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 421: | Zeile 443: | ||
& | & | ||
\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 485: | Zeile 509: | ||
* 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 511: | Zeile 535: | ||
< | < | ||
- | 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 525: | Zeile 549: | ||
< | < | ||
- | 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 613: | Zeile 637: | ||
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- | ==== magnetic Energy of a magnetic Circuit ==== | ||
- | |||
- | |||
- | ==== magnetic Energy of a magnetic Circuit ==== | ||
- | ==== magnetic Energy of a magnetic Circuit ==== | ||
==== magnetic Energy of a magnetic Circuit ==== | ==== magnetic Energy of a magnetic Circuit ==== |