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electrical_engineering_2:magnetic_circuits [2023/05/22 10:55] 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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| Zeile 1: | Zeile 1: | ||
| - | ====== 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 612: | Zeile 636: | ||
| \end{align*} | \end{align*} | ||
| - | ==== magnetic Energy of a magnetic Circuit ==== | + | |
| + | |||
| ==== 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. For this, the formula be rewritten by the properties linked flux $\Psi = N \cdot \Phi = L \c= {{1}\over{2}}{{\Psi^2 }\over{L}}} dot I$ 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}}{{\Phi^2 }\over{N^2 \dot L}}} = {{1}\over{2}}{{\theta^2 }\over{R_m^2 \dot N^2 \dot L}}} = {{1}\over{2}}{{\theta^2 }\over{R_m}}} \end{align*} | + | 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*} |
| ==== magnetic Energy of a toroid Coil ==== | ==== magnetic Energy of a toroid Coil ==== | ||