Mutual induction

The mutual induction or inductive coupling is mutual magnetic influence between two or more spatially adjacent electrical circuits by the electromagnetic induction due to a change of the magnetic flux Φ . Mutual induction is the basis of many technical devices, the most important of which is the transformer . In other cases it can be an undesirable effect, such as in the area of electromagnetic compatibility .

Related coupling types in this context are capacitive coupling , galvanic coupling and radiation coupling .

principle

Flux linkage of two conductor loops

A current-carrying (first) conductor loop causes, depending on its geometry, the generation of a magnetic flux density B in its spatial environment. Due to the Biot-Savart law, this is directly proportional to the instantaneous value of the current :

${\displaystyle {\vec {B}}({\vec {r}})\propto i_{1}}$

The magnetic flux Φ ₂ , which flows through the loop 2 with the area , is calculated as ${\displaystyle A}$

${\displaystyle \Phi _{2}=\int _{A}{\vec {B}}({\vec {r}})\cdot \mathrm {d} {\vec {A}}}$,

which corresponds to the sum of all scalar products of all infinitesimally small area vectors with the magnetic flux density vectors that penetrate these areas . ${\displaystyle \mathrm {d} {\vec {A}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle \mathrm {d} {\vec {A}}}$

But the actual realization of Faraday in the law of induction was that it is not the flow itself, but the change in the flow over time, that is responsible for the induction. The following equation can thus be derived for the mutual inductance (also referred to as " coupling inductance ") analogously to self-induction : ${\displaystyle \Phi }$ ${\displaystyle {\dot {\Phi }}}$ ${\displaystyle M}$ ${\displaystyle L}$

${\displaystyle {\frac {\mathrm {d} \Phi _{2}}{\mathrm {d} t}}=M_{21}\cdot {\frac {\mathrm {d} i_{1}}{\mathrm {d} t}}}$

Based on this definition, mutual inductance can be viewed as a generalization of self-inductance. Like this one, it is given in the SI unit Henry [H] .

symmetry

An essential property is the symmetry of the flux linkages: The mutual inductance from system 1 to system 2 is the same as for the opposite case:

${\displaystyle M_{21}=M_{12}\quad }$

In many cases, this relationship facilitates the practical calculation of flux linkages. For example, it is easy to calculate an expression for the flux linkage of a long coil with a smaller, concentrically mounted receiver coil. The reverse case, namely the concatenation of the flux of the small with the large coil, would presumably encounter considerable analytical difficulties without knowledge of the above relation. The described symmetry, which is also referred to as the magnetic reciprocity theorem, can be proven with the mathematical means of vector analysis with the aid of Maxwell's equations .

Proof of magnetic reciprocity

The magnetic field B can be expressed as the rotation of a vector potential :

${\displaystyle {\vec {B}}={\vec {\nabla }}\times {\vec {A}}}$

The magnetic flux through the second conductor loop is then ( denotes an infinitesimal surface element) ${\displaystyle \mathrm {d} {\vec {a}}}$

${\displaystyle \Phi _{2}=\int _{L_{2}}{\vec {B}}\cdot \mathrm {d} {\vec {a}}=\int _{L_{2}}{\big (}{\vec {\nabla }}\times {\vec {A}}{\big )}\cdot \mathrm {d} {\vec {a}}=\int _{L_{2}}{\vec {A}}\cdot \mathrm {d} {\vec {s}}_{2}}$

But now the vector potential can be traced back to the line integral of the current in the first conductor loop (this is another way of writing the Biot-Savart law): ${\displaystyle A}$ ${\displaystyle i}$

${\displaystyle {\vec {A}}({\vec {r}})={\frac {\mu _{0}i}{4\pi }}\int _{L_{1}}{\frac {\mathrm {d} {\vec {s}}_{1}}{\vert {\vec {r}}-{\vec {x}}_{1}\vert }}}$

Inserting this into the penultimate equation gives:

${\displaystyle \Phi _{2}={\frac {\mu _{0}i}{4\pi }}\int _{L_{1}}\int _{L_{2}}{\frac {\mathrm {d} {\vec {s}}_{1}\cdot \mathrm {d} {\vec {s}}_{2}}{\vert {\vec {x}}_{2}-{\vec {x}}_{1}\vert }}}$

${\displaystyle M}$ will therefore

${\displaystyle M_{12}={\frac {\mu _{0}}{4\pi }}\int _{L_{1}}\int _{L_{2}}{\frac {\mathrm {d} {\vec {s}}_{1}\cdot \mathrm {d} {\vec {s}}_{2}}{\vert {\vec {x}}_{2}-{\vec {x}}_{1}\vert }}={\frac {\mu _{0}}{4\pi }}\int _{L_{2}}\int _{L_{1}}{\frac {\mathrm {d} {\vec {s}}_{2}\cdot \mathrm {d} {\vec {s}}_{1}}{\vert {\vec {x}}_{1}-{\vec {x}}_{2}\vert }}=M_{21}}$

application

Principle of inductive coupling, with field description (A) and as a network model (B)

In the field of electromagnetic compatibility (EMC), mutual inductance is also referred to as magnetic coupling or inductive coupling and describes the usually undesired magnetic coupling of neighboring electrical circuits. The magnetic flux caused by the current in a circuit, such as the circuit consisting of the AC voltage source U ₁ in the adjacent circuit diagram , caused by magnetic coupling in the second circuit, shown with the AC voltage source U ₂ , an additional induced source voltage, which in this circuit as unwanted interference can occur.

The modeling can be carried out as a field model (A) with the variable magnetic field, or equivalent to this in the area of ​​network theory with the help of the mutual inductance M _s , as shown in the right figure in case (B). The counter-induced voltage in the second conductor loop, which is caused by the current from the first conductor loop, is: ${\displaystyle Ug_{2}}$ ${\displaystyle i_{1}}$

${\displaystyle Ug_{2}=M_{s}\cdot {\frac {\mathrm {d} i_{1}}{\mathrm {d} t}}}$

Due to the symmetry, a mutual inductance M _{s is} a reciprocal quadrupole .

Due to the higher energy density of the magnetic field compared to the electric field, a relatively high power transmission at medium frequencies can also be achieved by means of inductive coupling. This fact is exploited in transformers or electrical drives such as the gap motor .

In the field of message transmission, inductive coupling is used in the context of inductive transmission , for example in the case of contactless signal transmission of a sensor signal between sensor and display device or contactless chip cards, the so-called RFID .

Specialist literature

• Pascal Leuchtmann: Introduction to electromagnetic field theory . Pearson Studies, 2005, ISBN 3-8273-7144-9 .
• Horst Stöcker: Pocket book of physics . 6th edition. Publisher Harri Deutsch, Frankfurt am Main 2010, ISBN 978-3-8171-1860-1 .
• Günter Springer: Expertise in electrical engineering . 18th edition. Europa-Lehrmittel, Wuppertal 1989, ISBN 3-8085-3018-9 .