Transformer and gyratory coupling

The terms transformer coupling and gyratory coupling come from the four-pole theory . They describe the mutual effect between input and output variables of special network elements that exchange energy between the same or different physical domains (e.g. electrical, magnetic, mechanical, thermodynamic) in compliance with the principle of energy conservation .

General

As input and output variables of the two ports used, special pairs of flow variables and difference variables are used, which can be clearly calculated from the Massieu-Gibbs function.

• Current and voltage (electrical systems)
• magnetic flux change and magnetic tension (magnetic systems)
• Force and speed (translational mechanical systems)
• Torque and angular speed (rotary mechanical systems)
• Mass flow and gravitational potential (mechanical systems of heavy mass)
• Volume flow and pressure ( acoustics , fluid mechanics )
• Entropy flow and temperature difference (thermodynamic systems)

Transformer coupling

The essential characteristic of a transformer coupling is that, in compliance with the principle of energy conservation, flow quantities are coupled with flow quantities and difference quantities with difference quantities.

From a network theoretical point of view, several matrix forms are possible, which can also be converted into one another under certain mathematical conditions. A matrix form that generally always exists is the chain matrix .

If the size is a flow coordinate and the size is a difference coordinate and the entry gate is labeled with the index 1 and the exit gate with the index 2, a transformational coupling can generally be achieved via a chain matrix of the form ${\displaystyle \mu }$ ${\displaystyle \lambda }$

${\displaystyle {\begin{pmatrix}{\lambda _{1}}\\{\mu _{1}}\end{pmatrix}}={\begin{pmatrix}{A_{11}}&{0}\\{0}&{A_{22}}\end{pmatrix}}{\begin{pmatrix}{\lambda _{2}}\\{\mu _{2}}\end{pmatrix}}}$

describe.

Gyratory coupling

In the gyratory coupling, while adhering to the principle of energy conservation, flow quantities with difference quantities and difference quantities with flow quantities are generated via a chain matrix of the form

${\displaystyle {\begin{pmatrix}{\lambda _{1}}\\{\mu _{1}}\end{pmatrix}}={\begin{pmatrix}{0}&{A_{12}}\\{A_{21}}&{0}\end{pmatrix}}{\begin{pmatrix}{\lambda _{2}}\\{\mu _{2}}\end{pmatrix}}}$

coupled. With regard to the signs , it is assumed for both matrices that the chain matrix is arrowed on the input side as a consumer and on the output side as a generator .

The principle of energy conservation is expressed in the fact that the determinant of the chain matrix is ​​always 1 ( for the transformer and for the gyrator). ${\displaystyle A_{22}={\frac {1}{A_{11}}}}$ ${\displaystyle A_{21}=-{\frac {1}{A_{12}}}}$

Examples from physics and technology

A piezoelectric element represents a gyratorischen converter from network theoretical point of view. The matrix elements of the switching matrix are determined by the geometry of the piezoelectric ceramic and the piezoelectric constant force formed.

An electrostatic converter ( condenser microphone ) is also based on the principle of gyratory conversion. The matrix elements of the coupling matrix are formed by the geometry of the capacitor and the phantom voltage.   

The transformer principle can be found in electrothermal converters like the Peltier elements . Here the matrix elements of the coupling matrix are determined by the Seebeck coefficient .   

literature

• Arno Lenk, Günther Pfeifer and Roland Werthschützky: Electromechanical systems, mechanical and acoustic networks, their interactions and applications . Springer, Berlin 2000, ISBN 978-3-540-67941-7 .
• Jörg Grabow: Generalized networks in mechatronics. 1st edition. Oldenbourg Verlag, Munich 2013, ISBN 978-3-486-71261-2 .
• Gottfried Falk: Physics: Number and Reality: The conceptual and mathematical foundations of a universal quantitative description of nature. Birkhäuser Verlag, Basel 1990, ISBN 978-3-7643-2550-3 .
• Klaus Janschek: System design of mechatronic systems: methods - models - concepts. Springer, 2010, ISBN 978-3-540-78876-8 .