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| electrical_engineering_2:magnetic_circuits [2024/06/02 16:03] – [Bearbeiten - Panel] mexleadmin | electrical_engineering_2:magnetic_circuits [2025/05/27 07:56] (aktuell) – mexleadmin | ||
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| Zeile 7: | Zeile 7: | ||
| < | < | ||
| - | 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 have come with such an analogy and where it can be practically applied. |
| ===== 5.1 Linear Magnetic Circuits ===== | ===== 5.1 Linear Magnetic Circuits ===== | ||
| Zeile 17: | Zeile 17: | ||
| - The fields inside of airgaps are homogeneous. This is true for small air gaps. | - The fields inside of airgaps are homogeneous. This is true for small air gaps. | ||
| - | One can calculate a lot of simple magnetic circuits when these assumptions and focusing on the average field line are applied. | + | One can calculate a lot of simple magnetic circuits when these assumptions |
| < | < | ||
| Zeile 140: | Zeile 140: | ||
| |Simplifications |The simplifications often work for good results \\ (small wire diameter, relatively constant resistivity) |The simplification is often too simple \\ (widespread beyond the mean magnetic path length, non-linearity of the permeability) | | |Simplifications |The simplifications often work for good results \\ (small wire diameter, relatively constant resistivity) |The simplification is often too simple \\ (widespread beyond the mean magnetic path length, non-linearity of the permeability) | | ||
| - | <panel type=" | + | <panel type=" |
| A coil is set up onto a toroidal plastic ring ($\mu_{\rm r}=1$) with an average circumference of $l_R = 300 ~\rm mm$. | A coil is set up onto a toroidal plastic ring ($\mu_{\rm r}=1$) with an average circumference of $l_R = 300 ~\rm mm$. | ||
| Zeile 164: | Zeile 164: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| 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: | ||
| Zeile 192: | Zeile 192: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| Calculate the magnetic resistances of an airgap with the following dimensions: | Calculate the magnetic resistances of an airgap with the following dimensions: | ||
| Zeile 208: | Zeile 208: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| Calculate the magnetic voltage necessary to create a flux of $\Phi=0.5 ~\rm mVs$ in an airgap with the following dimensions: | Calculate the magnetic voltage necessary to create a flux of $\Phi=0.5 ~\rm mVs$ in an airgap with the following dimensions: | ||
| Zeile 224: | Zeile 224: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| Calculate the magnetic flux created on a magnetic resistance of $R_m = 2.5 \cdot 10^6 ~\rm {{1}\over{H}}$ with the following magnetic voltages: | Calculate the magnetic flux created on a magnetic resistance of $R_m = 2.5 \cdot 10^6 ~\rm {{1}\over{H}}$ with the following magnetic voltages: | ||
| Zeile 242: | Zeile 242: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | A core shall consist of two parts as seen in <imgref ImgExNr08> | + | A core shall consist of two parts, as seen in <imgref ImgExNr08> |
| In the coil, with $600$ windings shall pass the current $I=1.30 ~\rm A$. | In the coil, with $600$ windings shall pass the current $I=1.30 ~\rm A$. | ||
| Zeile 250: | Zeile 250: | ||
| The mean magnetic path lengths are $l_1 = 200 ~\rm mm$ and $l_2 = 130 ~\rm mm$. | The mean magnetic path lengths are $l_1 = 200 ~\rm mm$ and $l_2 = 130 ~\rm mm$. | ||
| - | The air gaps on the coupling joint between both parts have the length $\delta=0.23 ~\rm mm$ each. | + | The air gaps on the coupling joint between both parts have the length $\delta = 0.23 ~\rm mm$ each. |
| The permeability of the ferrite is $\mu_r = 3000$. | The permeability of the ferrite is $\mu_r = 3000$. | ||
| The cross-section area $A_{\delta}$ of the airgap can be considered the same as $A_2$ | The cross-section area $A_{\delta}$ of the airgap can be considered the same as $A_2$ | ||
| Zeile 270: | Zeile 270: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| The magnetic circuit in <imgref ImgExNr09> | The magnetic circuit in <imgref ImgExNr09> | ||
| Zeile 289: | Zeile 289: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| The choke coil shown in <imgref ImgExNr10> | The choke coil shown in <imgref ImgExNr10> | ||
| Zeile 458: | Zeile 458: | ||
| - | <panel type=" | + | <panel type=" |
| The magnetical configuration in <imgref ExImgNr01> | The magnetical configuration in <imgref ExImgNr01> | ||
| Zeile 489: | Zeile 489: | ||
| \end{align*} | \end{align*} | ||
| - | With the given geometry this leads to | + | With the given geometry, this leads to |
| \begin{align*} | \begin{align*} | ||
| R_{\rm m1} &= {{1}\over{\mu_0 \mu_{\rm r}}}{{l}\over{A}}\cdot \left(3 + {{1\cdot 2}\over{1 + 2}}\right) \\ | R_{\rm m1} &= {{1}\over{\mu_0 \mu_{\rm r}}}{{l}\over{A}}\cdot \left(3 + {{1\cdot 2}\over{1 + 2}}\right) \\ | ||
| Zeile 543: | Zeile 543: | ||
| # | # | ||
| - | For Electric vehicles sometimes wireless charging systems are employed. These use the principle of mutual inductance to transfer power from a charging pad on the ground to the vehicle' | + | For Electric vehicles, sometimes wireless charging systems are employed. These use the principle of mutual inductance to transfer power from a charging pad on the ground to the vehicle' |
| This system consists of two coils: a transmitter coil embedded in the charging pad and a receiver coil mounted on the underside of the vehicle. | This system consists of two coils: a transmitter coil embedded in the charging pad and a receiver coil mounted on the underside of the vehicle. | ||
| Zeile 588: | Zeile 588: | ||
| * the direction of the windings, and | * the direction of the windings, and | ||
| - | * the orientation/ | + | * The orientation/ |
| <WRAP center 50%> < | <WRAP center 50%> < | ||
| Zeile 598: | Zeile 598: | ||
| < | < | ||
| - | In this case, the **mutual induction is positiv | + | In this case, the **mutual induction is positive |
| The formula of the shown circuitry is then: | The formula of the shown circuitry is then: | ||
| Zeile 606: | Zeile 606: | ||
| \end{align*} | \end{align*} | ||
| - | === negative | + | === Negative |
| - | The polarity is negative when only one current | + | The polarity is negative when only one current flows into the dotted pin and the other one out of the dotted pin (see <imgref ImgNr13> |
| < | < | ||
| - | In this case, the **mutual induction is negativ | + | In this case, the **mutual induction is negative |
| The formula of the shown circuitry is then: | The formula of the shown circuitry is then: | ||
| Zeile 620: | Zeile 620: | ||
| \end{align*} | \end{align*} | ||
| - | <panel type=" | + | <panel type=" |
| A toroidal core (ferrite, $\mu_{\rm r} = 900$) has a cross-sectional area of $A = 500 ~\rm mm^2$ and an average circumference of $l=280 ~\rm mm$. | A toroidal core (ferrite, $\mu_{\rm r} = 900$) has a cross-sectional area of $A = 500 ~\rm mm^2$ and an average circumference of $l=280 ~\rm mm$. | ||
| - | On the core, there are two coils $N_1=500$ and $N_2=250$ wound. The currents on the coils are $I_1 = 250 ~\rm mA$ and $I_2=300 ~\rm mA$. | + | At the core, there are two coils $N_1=500$ and $N_2=250$ wound. The currents on the coils are $I_1 = 250 ~\rm mA$ and $I_2=300 ~\rm mA$. |
| - The coils shall pass the currents with positive polarity (see the image **A** in <imgref ImgEx14> | - The coils shall pass the currents with positive polarity (see the image **A** in <imgref ImgEx14> | ||
| Zeile 636: | Zeile 636: | ||
| **Step 1 - Draw an equivalent magnetic circuit** | **Step 1 - Draw an equivalent magnetic circuit** | ||
| - | Since there are no branches all of the core can be lumped | + | Since there are no branches, all of the core can be lumped |
| < | < | ||
| Zeile 642: | Zeile 642: | ||
| Hopkinson' | Hopkinson' | ||
| - | It connects the magnetic flux $\Phi$ and the magnetic voltage $\theta$ on the single magnetic resistor $R_m$. \\ | + | It connects the magnetic flux $\Phi$ and the magnetic voltage $\theta$ on the single magnetic resistor $R_\rm m$. \\ |
| It also connects the single magnetic fluxes $\Phi_x$ (with $x = {1,2}$) and the single magnetic voltages $\theta_x$. \\ | It also connects the single magnetic fluxes $\Phi_x$ (with $x = {1,2}$) and the single magnetic voltages $\theta_x$. \\ | ||
| Zeile 674: | Zeile 674: | ||
| # | # | ||
| - | - $0.10 ~\rm mVs$ | + | - $0.10 ~\rm mVs$ |
| - $0.40 ~\rm mVs$ | - $0.40 ~\rm mVs$ | ||
| # | # | ||
| Zeile 683: | Zeile 683: | ||
| The magnetic field of a coil stores magnetic energy. | The magnetic field of a coil stores magnetic energy. | ||
| - | The energy transfer from the electric circuit to the magnetic field is also the cause of the " | + | The energy transfer from the electric circuit to the magnetic field is also the cause of the " |
| The energetic turnover for charging an conductor from $i(t_0=0)=0$ to $i(t_1)=I$ is given by: | The energetic turnover for charging an conductor from $i(t_0=0)=0$ to $i(t_1)=I$ is given by: | ||
| Zeile 709: | Zeile 709: | ||
| ==== magnetic Energy of a toroid Coil ==== | ==== magnetic Energy of a toroid Coil ==== | ||
| - | The formula can also be used for calculating the stored energy of a toroid coil with $N$ windings, the cross-section $A$, and an average length $l$ of a field line. | + | The formula can also be used for calculating the stored energy of a toroid coil with $N$ windings, the cross-section $A$, and an average length $l$ of a field line. \\ |
| By this, the following formulas can be used: | By this, the following formulas can be used: | ||
| - | \begin{align*} | + | - For the magnetic voltage: $\theta = H \cdot l = N \cdot I $ \\ |
| - | \theta = H \cdot l = N \cdot I \\ | + | - For the magnetic flux: $\Phi = B \cdot A $ |
| - | \Phi = B \cdot A | + | |
| - | \end{align*} | + | |
| With the above-mentioned formulas of the magnetic circuit, we get: | With the above-mentioned formulas of the magnetic circuit, we get: | ||
| Zeile 735: | Zeile 733: | ||
| ==== generalized magnetic Energy ==== | ==== generalized magnetic Energy ==== | ||
| - | The general term to find the magnetic energy (e.g. for inhomogeneous magnetic fields) is given by | + | The general term to find the magnetic energy (e.g., for inhomogeneous magnetic fields) is given by |
| \begin{align*} | \begin{align*} | ||
| W_{\rm m} &= \iiint_V{w_{\rm m} {\rm d}V} \\ | W_{\rm m} &= \iiint_V{w_{\rm m} {\rm d}V} \\ | ||
| Zeile 755: | Zeile 753: | ||
| \end{align*} | \end{align*} | ||
| - | Multiplying with $i$ and with $dt$ we get the principle of conservation of energy $dw = u \cdot i \cdot {\rm d}t$ for each small time step. | + | Multiplying with $i$ and with $dt$, we get the principle of conservation of energy $dw = u \cdot i \cdot {\rm d}t$ for each small time step. |
| \begin{align*} | \begin{align*} | ||
| Zeile 765: | Zeile 763: | ||
| \begin{align*} | \begin{align*} | ||
| dW_{\rm m} &= N {{{\rm d}\Phi}\over{{\rm d}t}} \cdot i \cdot {\rm d}t \\ | dW_{\rm m} &= N {{{\rm d}\Phi}\over{{\rm d}t}} \cdot i \cdot {\rm d}t \\ | ||
| - | | + | |
| & | & | ||
| & | & | ||
| Zeile 787: | Zeile 785: | ||
| In <imgref ImgNr15> the situation for a magnetic material with a linear relationship between $B$ and $H$ is shown. | In <imgref ImgNr15> the situation for a magnetic material with a linear relationship between $B$ and $H$ is shown. | ||
| - | Given by the maximum current $I_{\rm max}$ the maximum field strength $H_{\rm max}$ can be derived. | + | Given the maximum current $I_{\rm max}$ the maximum field strength $H_{\rm max}$ can be derived. |
| In the circuit in <imgref ImgNr14>, | In the circuit in <imgref ImgNr14>, | ||
| Therefore, the $B$-$H$-curve gets passed through positive and negative values of $H$ and $H$ along the line of $B=\mu H$. | Therefore, the $B$-$H$-curve gets passed through positive and negative values of $H$ and $H$ along the line of $B=\mu H$. | ||
| Zeile 794: | Zeile 792: | ||
| The situation for integrating the area in the graph is also shown: | The situation for integrating the area in the graph is also shown: | ||
| - | For each step ${\rm d}B$ the corresponding value of the field strength $H$ has to be integrated. | + | For each step ${\rm d}B$, the corresponding value of the field strength $H$ has to be integrated. |
| For $B_0=0$ to $B=B_{\rm max}$ the magnetic energy is | For $B_0=0$ to $B=B_{\rm max}$ the magnetic energy is | ||
| Zeile 817: | Zeile 815: | ||
| As an example, the situation of the field strength $H(t_1)=H_1$ is shown. | As an example, the situation of the field strength $H(t_1)=H_1$ is shown. | ||
| This shall be the field strength after magnetizing the ferrite material to $H_{\rm max}$ (yellow arrows) and then partly demagnetizing the material again (blue arrow). | This shall be the field strength after magnetizing the ferrite material to $H_{\rm max}$ (yellow arrows) and then partly demagnetizing the material again (blue arrow). | ||
| - | The magnetization corresponds to an energy intake | + | The magnetization corresponds to an energy intake |
| - | Moving along the $H$-$B$-curve, | + | Moving along the $H$-$B$-curve, |
| This means that the magnetization and demagnetization take place lossless in this example. | This means that the magnetization and demagnetization take place lossless in this example. | ||
| - | This is a good approximation for magnetically soft materials, however, does not work for magnetically hard materials like a permanent magnet. | + | This is a good approximation for magnetically soft materials; however, |
| Here, hysteresis also has to be considered. | Here, hysteresis also has to be considered. | ||
| Zeile 828: | Zeile 826: | ||
| < | < | ||
| - | ===== Tasks ===== | + | ===== Exercise |
| - | <panel type=" | + | <panel type=" |
| The <imgref ImgTask01> | The <imgref ImgTask01> | ||
| Zeile 845: | Zeile 843: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | The book [[https:// | + | The book [[https:// |
| </ | </ | ||
| Zeile 854: | Zeile 852: | ||
| An alternative interpretation of the magnetic circuits is the {{https:// | An alternative interpretation of the magnetic circuits is the {{https:// | ||
| - | 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 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, |
| ==== Moving a Plate into an Air Gap ==== | ==== Moving a Plate into an Air Gap ==== | ||