Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
electrical_engineering_and_electronics_2:block09 [2026/05/17 00:23] – mexleadminelectrical_engineering_and_electronics_2:block09 [2026/05/17 01:09] (current) – mexleadmin
Line 112: Line 112:
 This is the starting point for the transformer. This is the starting point for the transformer.
 </panel> </panel>
 +
 +==== Polarity and the dot convention ====
 +
 +Before we start with the transformer, we have to look on a common convention for the orientation of two coils to each other. \\
 +
 +<WRAP>
 +<panel type="default">
 +<imgcaption fig_direction_of_coupling|Dot convention: the dots indicate corresponding winding ends.></imgcaption>
 +{{drawio>electrical_engineering_and_electronics_2:directionofcoupling.svg}}
 +</panel>
 +</WRAP>
 +
 +<callout>
 +**Rule of thumb**
 +
 +  * If both currents enter dotted terminals, the fluxes support each other.
 +  * If one current enters a dotted terminal and the other current leaves a dotted terminal, the fluxes oppose each other.
 +</callout>
 +
  
 ==== Ideal single-phase transformer ==== ==== Ideal single-phase transformer ====
Line 293: Line 312:
   * \(\Phi_{11}\): total flux created by coil \(1\).   * \(\Phi_{11}\): total flux created by coil \(1\).
   * \({\color{blue}{\Phi_{21}}}\): part of this flux that also links coil \(2\).   * \({\color{blue}{\Phi_{21}}}\): part of this flux that also links coil \(2\).
-  * \({\color{orange}{\Phi_{\rm S1}}}\): stray or leakage flux that does **not** link coil \(2\).+  * \({\color{orange}{\Phi_{\rm \sigma 1}}}\): stray or leakage flux that does **not** link coil \(2\). The greek sigma $\sigma$ is used to depict the term "stray"
  
-The instantaneous voltage induced in coil \(2\) is+For an example, we will have a look onto the instantaneous voltage induced in coil \(2\):
  
 \[ \[
 \begin{align*} \begin{align*}
 u_{{\rm ind},2}(t) u_{{\rm ind},2}(t)
 +=
 +\frac{{\rm d}{\color{blue}{\Psi_{21}}}}{{\rm d}t}
 = =
 N_2\frac{{\rm d}{\color{blue}{\Phi_{21}}}}{{\rm d}t}. N_2\frac{{\rm d}{\color{blue}{\Phi_{21}}}}{{\rm d}t}.
Line 309: Line 330:
 \[ \[
 \begin{align*} \begin{align*}
 +\boxed{
 \underline{U}_{{\rm ind},2}(t) \underline{U}_{{\rm ind},2}(t)
 = =
-j\omega N_2 {\color{blue}{ \underline{\Phi}_{21} }}.+j\omega {\color{blue}{ \underline{\Psi}_{21} }} 
 += 
 +j\omega N_2 {\color{blue}{ \underline{\Phi}_{21} }} 
 +}.
 \end{align*} \end{align*}
 \] \]
 +
 +We need these complex represenations for the next steps into the transformer. \\
  
 <panel type="info" title="Analogies"> <panel type="info" title="Analogies">
Line 443: Line 470:
   * A large \(M\) means strong interaction.     * A large \(M\) means strong interaction.  
   * A small \(M\) means weak interaction.   * A small \(M\) means weak interaction.
 +
 +\\
  
 <panel type="info" title="Engineering example: wireless charging"> <panel type="info" title="Engineering example: wireless charging">
Line 452: Line 481:
 </panel> </panel>
  
-==== Polarity and the dot convention ====+==== voltages by mutual inductances and resitances ====
  
-The sign of the mutual term depends on the winding direction and on the chosen current reference arrows.+For positive coupling, we get the following complex representation (since $u(t) = L\frac{{\rm d}i}{{\rm d}t} \: \longrightarrow  \:\underline{U}=j\omega L$):
  
-<WRAP> +\[ 
-<panel type="default"> +\begin{align*} 
-<imgcaption fig_direction_of_coupling|Dot convention: the dots indicate corresponding winding ends.></imgcaption> +\underline{U}_1 &= R_1 \underline{I}_1 + {\color{green} {j\omega L_{11} \underline{I}_1 }} + {\color{blue}{j\omega M       \underline{I}_2 }}, 
-{{drawio>electrical_engineering_and_electronics_2:directionofcoupling.svg}} +\\[4pt] 
-</panel> +\underline{U}_2 &= R_2 \underline{I}_2 + {\color{blue}  {j\omega M      \underline{I}_1 }} + {\color{green}{j\omega L_{22} \underline{I}_2 }}. 
-</WRAP>+\end{align*} 
 +\]
  
-<callout> +For negative coupling, the sign of the \(M\)-term changes in the chosen equation system, see <imgref fig_positive_coupling> and <imgref fig_negative_coupling>.
-**Rule of thumb** +
- +
-  * If both currents enter dotted terminals, the mutual fluxes support each other. +
-  * If one current enters a dotted terminal and the other current leaves a dotted terminal, the mutual fluxes oppose each other. +
-</callout>+
  
 <WRAP group> <WRAP group>
Line 474: Line 499:
 <panel type="default"> <panel type="default">
 <imgcaption fig_positive_coupling|Positive coupling: currents enter corresponding dotted terminals.></imgcaption> <imgcaption fig_positive_coupling|Positive coupling: currents enter corresponding dotted terminals.></imgcaption>
-{{:electrical_engineering_2:poscoupling.svg?500}}+{{drawio>electrical_engineering_and_electronics_2:poscoupling.svg}}
 </panel> </panel>
 </WRAP> </WRAP>
Line 481: Line 506:
 <panel type="default"> <panel type="default">
 <imgcaption fig_negative_coupling|Negative coupling: only one current enters a dotted terminal.></imgcaption> <imgcaption fig_negative_coupling|Negative coupling: only one current enters a dotted terminal.></imgcaption>
-{{:electrical_engineering_2:negcoupling.svg?500}}+{{drawio>electrical_engineering_and_electronics_2:negcoupling.svg}}
 </panel> </panel>
 </WRAP> </WRAP>
 </WRAP> </WRAP>
  
-For positive coupling: +<panel type="info" title="Tunnel Analogy for AC circuits">
- +
-\[ +
-\begin{align*} +
-u_1 +
-&= +
-{\color{green}{L_{11}\frac{{\rm d}i_1}{{\rm d}t}}} +
-+ +
-{\color{blue}{M\frac{{\rm d}i_2}{{\rm d}t}}}, +
-\\[4pt] +
-u_2 +
-&= +
-{\color{blue}{M\frac{{\rm d}i_1}{{\rm d}t}}} +
-+ +
-{\color{green}{L_{22}\frac{{\rm d}i_2}{{\rm d}t}}}. +
-\end{align*} +
-\] +
- +
-For negative coupling, the sign of the \(M\)-term changes in the chosen equation system. +
- +
- +
- +
-<panel type="info" title="Tunnel Analogy">+
 The dots are like matching openings for magnetic action. \\ The dots are like matching openings for magnetic action. \\
-A positive current (e.g. $i_1$) entering the dotted terminal of one winding produces a positive induced voltage (e.g. aligned with $u_2$) at the dotted terminal of the other winding. \\ +A positive current (e.g. $\underline{I}_1$) entering the dotted terminal of one winding produces a positive induced voltage (e.g. aligned with $\underline{U}_2$) at the dotted terminal of the other winding. \\ 
-With only a load $R_2$ connected to the secondary side, this voltage tends to drive current out of the dotted terminal into the load ($i_2$ has to be inverted, since the transformer is a source then).+With only a load $R_2$ connected to the secondary side, this voltage tends to drive current out of the dotted terminal into the load ($\underline{I}_2$ has to be inverted, since the transformer is a source then).
 </panel> </panel>