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electrical_engineering_2:magnetic_circuits [2023/05/23 08:49] – [magnetic Energy of a magnetic Circuit] ottelectrical_engineering_2:magnetic_circuits [2024/05/10 16:02] – [5.3 Mutual Induction and Coupling] mexleadmin
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-====== 5. Magnetic Circuits ======+====== 5 Magnetic Circuits ======
  
 <callout> For this and the following chapter the online Book 'DC Electrical Circuit Analysis - A Practical Approach' is strongly recommended as a reference. In detail this is chapter [[https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/DC_Electrical_Circuit_Analysis_-_A_Practical_Approach_(Fiore)/10%3A_Magnetic_Circuits_and_Transformers/10.3%3A_Magnetic_Circuits|10.3 Magnetic Circuits]] </callout> <callout> For this and the following chapter the online Book 'DC Electrical Circuit Analysis - A Practical Approach' is strongly recommended as a reference. In detail this is chapter [[https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/DC_Electrical_Circuit_Analysis_-_A_Practical_Approach_(Fiore)/10%3A_Magnetic_Circuits_and_Transformers/10.3%3A_Magnetic_Circuits|10.3 Magnetic Circuits]] </callout>
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 ===== 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%> <imgcaption ImgNr09 | Mutual Induction of two Coils> </imgcaption> {{drawio>MutualInductionTwoCoils1.svg}} </WRAP> 
 + 
 +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, including: 
 + 
 +  * 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%> <imgcaption ImgNr09 | Mutual Induction of two Coils> </imgcaption> {{drawio>MutualInductionTwoCoils1.svg}} </WRAP> 
  
 ==== Effect of Coils on each other ==== ==== Effect of Coils on each other ====
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 $k_{21}$ 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 $k_{21}=100~\%$, there is no flux in the middle leg but only in the second coil. \\ +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 $k_{21}=-100~\%$, there is no flux in the middle leg but only in the second coil and in opposite direction as the flux that originates from the second coil. \\ 
 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.
  
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        &= k_{21}                  \cdot {{N_1 \cdot N_2 }\over {R_{\rm m1}}} \\         &= k_{21}                  \cdot {{N_1 \cdot N_2 }\over {R_{\rm m1}}} \\ 
 \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: 
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   * 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 **total coupling** $k$ is given as \begin{align*} k = \rm{sgn}(k_{12}) \sqrt{k_{12}\cdot k_{21}} \end{align*}
  
 ==== Effects in the electric Circuits ==== ==== Effects in the electric Circuits ====
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 <imgcaption ImgNr12 | Example Circuits with positive Polarity> </imgcaption> {{drawio>posCoupling.svg}} <imgcaption ImgNr12 | Example Circuits with positive Polarity> </imgcaption> {{drawio>posCoupling.svg}}
  
-In this case, the **mutual induction added positively**.+In this case, the **mutual induction is positiv $(M>0)$**.
  
 The formula of the shown circuitry is then:  The formula of the shown circuitry is then: 
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 <WRAP> <imgcaption ImgNr13 | Example Circuits with negative Polarity> </imgcaption> {{drawio>negCoupling.svg}} </WRAP> <WRAP> <imgcaption ImgNr13 | Example Circuits with negative Polarity> </imgcaption> {{drawio>negCoupling.svg}} </WRAP>
  
-In this case, the **mutual induction added negatively**.+In this case, the **mutual induction is negativ $(M<0)$***.
  
 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*}
  
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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 ====