Impedance inverter

An impedance inverter , in the German literature as dual booster is designated, an electronic circuit which the impedance of a real existing dipole by inversion in a to the original dual impedance transforms. It is the "counterpart" to the impedance converter . ${\displaystyle {\underline {Z}}_{L}}$ ${\displaystyle {\underline {Z}}_{E}}$

General

The ideal model of an impedance inverter is a linear two-port in whose chain matrix only the secondary diagonal is occupied ( complex quantities are underlined):

${\displaystyle \left({\underline {A}}\right)={\begin{pmatrix}0&{\underline {a}}_{12}\\{\underline {a}}_{21}&0\end{pmatrix}}}$

If the (load) impedance is connected to its output port L, the following (input) impedance is set at input port E in accordance with the calculation methods of the two-port theory : ${\displaystyle {\underline {Z}}_{L}}$ ${\displaystyle {\underline {Z}}_{E}}$

${\displaystyle {\underline {Z}}_{E}={\frac {{\underline {a}}_{12}}{{\underline {a}}_{21}\cdot {\underline {Z}}_{L}}}}$

The inversion factor is a selectable factor that determines the type of inversion. The chain matrix shows that an impedance inverter is generally a non-reversible active two-port, because apart from special cases, both the determinant and the forward and backward power ratios are not equal to 1. ${\displaystyle {\frac {{\underline {a}}_{12}}{{\underline {a}}_{21}}}}$ ${\displaystyle \det \left({\underline {A}}\right)}$

Therefore real impedance inverters have to be built as electronic, active circuits. In practice, one or more operational amplifiers and other passive components are used for this.

variants

Depending on the choice of the inversion factor, a distinction is made between two typical types of impedance inverters:

• When positive impedance inverter ( english positive impedance inverter PII) of the inversion factor is real and positive.
• When negative impedance inverter ( English negative impedance inverter , NII) of the inversion factor is real and negative, d. H. In addition to the inversion, the sign is also rotated.

Special case

A positive impedance inverter with the secondary condition

${\displaystyle {\underline {a}}_{12}={\frac {1}{{\underline {a}}_{21}}}=r_{G}}$

called gyrator with the Gyrationswiderstand r _G . On the one hand applies to the determinant of its chain matrix . This is why a gyrator is an irreversible (sometimes referred to as anti-reciprocal ) two-port. On the other hand, the power ratio in both directions is 1. It is therefore (ideally) losslessly passive and represents the counterpart to the ideal transformer . For example, it can convert a capacitance C into an inductance , because the following applies: ${\displaystyle \det \left({\underline {A}}\right)=-1}$ ${\displaystyle L=C\cdot r_{G}^{2}}$

${\displaystyle {\underline {Z}}_{E}={\frac {r_{G}^{2}}{\frac {1}{j\omega C}}}=j\omega C\cdot r_{G}^{2}}$

As a result, the gyrator has achieved great importance both in network theory and in real circuit practice.

literature

• Reinhold Paul: Basic electrical engineering textbook - Volume 2: Networks . 3. Edition. Springer, 1996, ISBN 978-3-540-55866-8 .
• FH Lange: Signals and Systems - Volume 2: Controlled Electronic Systems . Verlag Technik, Berlin 1968, DNB  366518496 .