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electrical_engineering_2:magnetic_circuits [2023/05/17 10:45]
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mexleadmin [Magnetic Circuit with two Sources]
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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>
Zeile 9: Zeile 9:
 In this chapter, we will investigate, how far we come with such an analogy and where it can be practically applied. In this chapter, we will investigate, how far we come with such an analogy and where it can be practically applied.
  
-===== 5.1 Linear magnetic Circuits =====+===== 5.1 Linear Magnetic Circuits =====
  
 For the upcoming calculations, the following assumptions are made For the upcoming calculations, the following assumptions are made
Zeile 405: 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 $k_{21}=100~\%$, 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.
  
Zeile 421: Zeile 424:
        &= 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: 
Zeile 485: 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 **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: 
Zeile 525: Zeile 530:
 <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*}
  
Zeile 611: 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}}  \Psi\cdot I  +
-                 = {{1}\over{2}}{{\Psi^2 }\over{L}}}  +
-\end{align*}+
  
 ==== magnetic Energy of a toroid Coil ==== ==== magnetic Energy of a toroid Coil ====
Zeile 770: 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://en.wikipedia.org/wiki/Magnetic_circuit#Resistance–reluctance_model|Magnetic_circuit}}, which was also presented in this chapter. This model can solve more questions, however, is a bit less intuitive based on this course and less commonly used compared to the {{https://en.wikipedia.org/wiki/Magnetic_circuit#Resistance–reluctance_model|Magnetic_circuit}}, which was also presented in this chapter.
 +
 +==== Moving a Plate into an Air Gap ====
 +
 +<WRAP> <imgcaption ImgNr81 | a magnetic circuit with a moving plate> {{electrical_engineering_2:plate_in_airgap_50_.gif}} {{electrical_engineering_2:stiftt_in_luftspalt_2.3.gif}} </imgcaption> </WRAP>
  
 ==== Switch Reluctance Motor ==== ==== Switch Reluctance Motor ====