Real transformer

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The real transformer is usually a linearized model of a transformer , which extends the ideal transformer to include stray fields, ohmic losses, hysteresis losses and possibly capacitive effects.

In a real transformer, not all of the magnetic flux produced by one of the coils also flows through the other coil. This phenomenon is called scattering . In many applications, scattering is undesirable, in others (e.g. resonant converters) it is an important component of the topology, since additional coils can be saved with specifically selected scattering factors.

Equivalent circuit diagram for low frequencies

A real transformer with windings and magnetic fluxes

Electric and magnetic circuit

The adjacent figure is clearly shown in the following circuit diagram:

This circuit diagram includes two electrical circuits and one magnetic. The coil windings N ₁ and N ₂ act as a coupling between the electrical and magnetic circuit.

The leakage fluxes are shown in the equivalent circuit diagram as magnetic resistances R _{m, σ1} and R _{m, σ2} . A current I ₁ creates a flow in the magnetic circuit at N ₁ , so that . The magnetic resistance R _{m, h} that the main flux Φ _h experiences is divided here for reasons of symmetry. In a very good approximation, R _{m, h is} the magnetic resistance of the iron yoke. The resistors R ₁ and R ₂ represent the resistance of the coil windings. ${\displaystyle \Theta _{1}=N_{1}\cdot I_{1}}$ ${\textstyle \Phi _{\sigma 1}={\frac {\Theta _{1}}{R_{m,\sigma 1}}}}$

Symmetrical equivalent circuit

In the next step, the magnetic circuit is transformed into electrical circuits:

The coil windings N ₁ and N ₂ merge to form an ideal transformer with a transformation ratio . In the transformation by the gyratorischen couplings coils connected in parallel, magnetic resistances connected in series inductances are: , , and . ${\textstyle {\ddot {u}}={\frac {N_{1}}{N_{2}}}}$ ${\textstyle L_{\sigma 1}={\frac {N_{1}^{2}}{R_{m,\sigma 1}}}}$ ${\textstyle L_{\sigma 2}={\frac {N_{2}^{2}}{R_{m,\sigma 2}}}}$ ${\textstyle L_{h,1}={\frac {N_{1}^{2}}{R_{m,h}/2}}}$ ${\textstyle L_{h,2}={\frac {N_{2}^{2}}{R_{m,h}/2}}}$

Note : Often there is also talk of primary and secondary main inductance, which however mean the transformation of both coil halves on one side and must therefore never be drawn in the same equivalent circuit diagram! This often leads to confusion.

T equivalent circuit diagram

Every linear two-port can be represented as a T equivalent circuit diagram. The steps shown above give the individual components of this circuit diagram a direct, real reference. The ideal transformer is extracted from the circuit diagram:

A new addition here is the resistance R _Fe , which symbolizes the hysteresis losses in the core that have not been taken into account so far .

During the transformation through the transformer coupling [ü], parallel / series connections are retained and the following applies: and The two inductances L _{h, 1} and L _{h, 2} merge to form the (primary) main inductance . ${\textstyle L'_{\sigma 2}={\ddot {u}}^{2}\cdot L_{\sigma 2}}$ ${\textstyle R'_{2}={\ddot {u}}^{2}\cdot R_{2}}$ ${\textstyle L_{h}={\frac {L_{h,1}\cdot L'_{h,2}}{L_{h,1}+L'_{h,2}}}={\frac {N_{1}^{2}}{R_{m,h}}}}$

The T equivalent circuit shown above expands the ideal transformer [ü] by the real effects. The transmission behavior of the real transformer results from the concatenation of the ideal and the real transformer in chain parameters (current direction I ₂ away from the two-port!) As follows:

${\displaystyle {\begin{pmatrix}U_{1}\\I_{1}\end{pmatrix}}=T_{real}\cdot {\begin{pmatrix}U_{2}\\I_{2}\end{pmatrix}}}$ wherein   with  ${\displaystyle T_{real}={\begin{pmatrix}1+{\frac {Z_{1}}{Z_{h}}}&Z_{1}+Z_{2}+{\frac {Z_{1}\cdot Z_{2}}{Z_{h}}}\\{\frac {1}{Z_{h}}}&1+{\frac {Z_{2}}{Z_{h}}}\end{pmatrix}}\cdot {\begin{pmatrix}{\ddot {u}}&0\\0&{\frac {1}{\ddot {u}}}\end{pmatrix}}}$ ${\displaystyle {\begin{aligned}&Z_{1}=R_{1}+i\omega L_{\sigma 1}\\&Z_{2}=R'_{2}+i\omega L'_{\sigma 2}\\&Z_{h}={\frac {R_{\text{Fe}}\cdot i\omega L_{h}}{R_{\text{Fe}}+i\omega L_{h}}}\end{aligned}}}$

The T equivalent circuit is usually used for circuit analysis.

Parameters

Self inductance

The self-inductance of a transformer is calculated as follows:

${\displaystyle L_{1}={\frac {N_{1}\cdot \Phi _{11}}{I_{1}}}={\frac {N_{1}\cdot (\Lambda _{h}+\Lambda _{\sigma 1})\cdot \Theta _{1}}{I_{1}}}=N_{1}^{2}\cdot (\Lambda _{h}+\Lambda _{\sigma 1})=L_{h}+L_{\sigma 1}}$

${\displaystyle L_{2}={\frac {N_{2}\cdot \Phi _{22}}{I_{2}}}={\frac {L_{h}}{{\ddot {u}}^{2}}}+L_{\sigma 2}}$

where the magnetic conductance and Φ _{11 is} the portion of the magnetic flux that is caused by the current I ₁ ( superposition ) and runs through the coil windings N ₁ (Φ ₂₂ equivalent). ${\displaystyle \Lambda ={\frac {1}{R_{m}}}}$

The self-inductances can be measured at the transformer terminals when the other side of the transformer is open (idle). Note: In the magnetic circuit from the first circuit diagram, N ₂ acts like a short circuit when the terminals in the electrical circuit are open ( gyratory coupling ).

Mutual inductance

The mutual inductances are defined as:

${\displaystyle M_{12}={\frac {N_{1}\cdot \Phi _{12}}{I_{2}}}={\frac {N_{1}\cdot k_{2}\cdot \Phi _{22}}{I_{2}}}={\frac {N_{1}}{I_{2}}}\cdot {\frac {\Lambda _{h}}{\Lambda _{h}+\Lambda _{\sigma 2}}}\cdot (\Lambda _{h}+\Lambda _{\sigma 2})\cdot \Theta _{2}=N_{1}\cdot N_{2}\cdot \Lambda _{h}}$

${\displaystyle M_{21}={\frac {N_{2}\cdot \Phi _{21}}{I_{1}}}={\frac {N_{2}\cdot k_{1}\cdot \Phi _{11}}{I_{1}}}=N_{1}\cdot N_{2}\cdot \Lambda _{h}}$

As shown above, the two mutual inductances are the same in the case of linear materials (as previously assumed). Thus the size is referred to as mutual inductance . ${\displaystyle M_{12}=M_{21}:=M}$ ${\displaystyle M}$

Coupling factor

The so-called flux coupling factors are defined as follows:

${\displaystyle k_{1}={\frac {\Phi _{21}}{\Phi _{11}}}={\frac {\Lambda _{h}}{\Lambda _{h}+\Lambda _{\sigma 1}}}}$

${\displaystyle k_{2}={\frac {\Phi _{12}}{\Phi _{22}}}={\frac {\Lambda _{h}}{\Lambda _{h}+\Lambda _{\sigma 2}}}}$

${\displaystyle \Phi _{12}}$is the portion of the magnetic flux through which is caused by electricity . Accordingly, the following applies: . ${\displaystyle N_{1}}$ ${\displaystyle I_{2}}$ ${\displaystyle \Phi _{h}=\Phi _{12}+\Phi _{21}}$

You can also write:

${\displaystyle k_{1}\cdot L_{1}=N_{1}^{2}\cdot \Lambda _{h}}$

→  ${\displaystyle \Lambda _{h}={\frac {k_{1}\cdot L_{1}}{N_{1}^{2}}}={\frac {k_{2}\cdot L_{2}}{N_{2}^{2}}}}$

→  ${\displaystyle \Lambda _{h}^{2}=\Lambda _{h}\cdot \Lambda _{h}=\left({\frac {k_{1}\cdot L_{1}}{N_{1}^{2}}}\right)\cdot \left({\frac {k_{2}\cdot L_{2}}{N_{2}^{2}}}\right)}$

→  ${\displaystyle M=N_{1}\cdot N_{2}\cdot \Lambda _{h}=N_{1}\cdot N_{2}\cdot {\sqrt {\left({\frac {k_{1}\cdot L_{1}}{N_{1}^{2}}}\right)\cdot \left({\frac {k_{2}\cdot L_{2}}{N_{2}^{2}}}\right)}}={\sqrt {k_{1}\cdot k_{2}\cdot L_{1}\cdot L_{2}}}:=k\cdot {\sqrt {L_{1}\cdot L_{2}}}}$

with a common coupling factor where is. For perfect coupling is . ${\displaystyle k={\sqrt {k_{1}\cdot k_{2}}}={\frac {M}{\sqrt {L_{1}\cdot L_{2}}}}}$ ${\displaystyle 0<k<1}$ ${\displaystyle k=1}$

Spreading factor

Measurement of the short circuit inductance

Equivalent to the coupling factor, scattering factors can be defined as follows:

${\displaystyle \sigma _{1}=1-k_{1}={\frac {\Lambda _{\sigma 1}}{\Lambda _{h}+\Lambda _{\sigma 1}}}={\frac {L_{\sigma 1}}{L_{1}}}}$

${\displaystyle \sigma _{2}=1-k_{2}={\frac {\Lambda _{\sigma 2}}{\Lambda _{h}+\Lambda _{\sigma 2}}}={\frac {L_{\sigma 2}}{L_{2}}}}$

The factor is called the Blondel coefficient or simply the scatter factor. ${\displaystyle \sigma =1-k^{2}=\sigma _{1}+\sigma _{2}-\sigma _{1}\cdot \sigma _{2}}$

In the case of a symmetrical, capacitance-free transformer ( ), the Blondel coefficient can e.g. B. easily determine with an LCR meter by inductance measurement: ${\textstyle \Lambda _{\sigma 1}=\Lambda _{\sigma 2}\leftrightarrow L_{\sigma 1}=L'_{\sigma 2}\leftrightarrow \sigma _{1}=\sigma _{2}}$

${\displaystyle \sigma =2\sigma _{1}-\sigma _{1}^{2}={\frac {L_{\mathrm {kurz} }}{L_{\mathrm {offen} }}}}$

where L _open  =  L _{1 is} the measurable inductance at the input with an open output and L _{short is} the measurable short-circuit inductance at the input with a short-circuit of the output (input and output are interchangeable here).

Individual evidence

1. ↑ cf. Dieter Zastrow: Electrical engineering: a basic textbook . 17th edition. Springer-Verlag, 2010, ISBN 978-3-8348-0562-1 , section: Realer Transformator , p. 330 .
2. ↑ Manfred Albach: Inductivities in power electronics . Springer Fachmedien Wiesbaden, Wiesbaden 2017, ISBN 978-3-658-15080-8 , doi : 10.1007 / 978-3-658-15081-5 .
3. ↑ Practical Basics of Electrical Engineering Experiment: Magnetic Circle. HTW Dresden, August 2011, accessed on September 5, 2015, p. 9.
4. ↑ Wolfgang Demtröder: Experimentalphysik 2: Electricity and optics . 5th edition. Springer, Berlin / Heidelberg 2009, ISBN 978-3-540-68219-6 , pp. 150-154 .
5. ↑ Ing: GdE: Models of the transformer - Wikibooks, collection of free textbooks, non-fiction and specialist books. In: de.wikibooks.org. Retrieved November 4, 2016 .