Impedance converter

An impedance converter (also referred to as a proportional converter in the literature ) is an electronic circuit which multiplies the impedance of a real two-terminal network by a generally complex factor and thus converts it into a desired impedance . It is the "counterpart" to the impedance inverter . ${\displaystyle {\underline {Z}}_{L}}$ ${\displaystyle {\underline {Z}}_{E}}$

The impedance converter is used in particular in the context of the optimized circuit technology of analog filters to convert capacitive reactances into inductive reactances or also ohmic resistances into negative resistances.

General

An ideal impedance converter is a linear two-port in whose chain matrix only the main diagonal is occupied ( complex quantities are underlined):

${\displaystyle \left({\underline {A}}\right)={\begin{pmatrix}{\underline {a}}_{11}&0\\0&{\underline {a}}_{22}\end{pmatrix}}}$

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

${\displaystyle {\underline {Z}}_{E}={\frac {{\underline {a}}_{11}}{{\underline {a}}_{22}}}\cdot {\underline {Z}}_{L}={\underline {k}}\cdot {\underline {Z}}_{L}}$

The complex conversion factor is a selectable, usually constant factor that determines the type of conversion. The chain matrix shows that an impedance converter is generally a non-reversible active two-port, because apart from special cases, both the determinant and the power ratio are not equal to 1. ${\displaystyle {\underline {k}}={\frac {{\underline {a}}_{11}}{{\underline {a}}_{22}}}}$ ${\displaystyle \det \left({\underline {A}}\right)}$

This is why impedance converters are built as electronic, active circuits. One or more operational amplifiers and passive components such as resistors and capacitors are used for this.

If, in a special case, both the determinant and the (forward and backward) power transmission are equal to 1, then this (positive) impedance converter degenerates into the ideal transformer with the transformation ratio u . ${\displaystyle u={\underline {a}}_{22}={\frac {1}{{\underline {a}}_{11}}}}$

species

Depending on the choice of conversion factor , a distinction is made between the following typical types of impedance converters: ${\displaystyle {\underline {k}}}$

• Positive Impedance Converter (PIC): The factor is positive, real and practically greater than 1. An example is the capacity multiplier . ${\displaystyle {\underline {k}}}$
• Negative Impedance Converter (NIC): In this important case, the factor is negative and real. It is used to invert the sign of the impedance. With a NIC, a negative resistance can be formed from an ohmic resistance, which always has a positive value . With a negative resistance, the current decreases as the voltage increases. Due to this property, oscillators can be constructed or resonant circuits can be undamped with the NIC . ${\displaystyle {\underline {k}}}$
• Generalized impedance converter (GIC): The factor is complex and usually depends on the angular frequency ω: k = k (jω). It is used, for example , to convert capacitive impedances, such as capacitors, into inductive impedances, such as coils represent in direct form . In this way, coils that are complex to produce in electronic circuits such as analog filters can be replaced by capacitors that are simpler and more cost-effective to produce. ${\displaystyle {\underline {k}}}$

The “ super capacitances ” and “ super inductances ” obtained within the framework of the Bruton transformation also represent special applications of the general impedance converter and are used in particular in the circuit technology of analog filters. In this transformation, also known as FDNR technology for Frequency Dependent Negative Resistance , the frequency dependency of k (jω) is used and frequency-independent ohmic resistances are replaced by frequency-dependent capacitors. As part of the Bruton transformation, existing capacitors are transformed into “super capacitors ” with low-pass filters , the real impedance of which depends on the square of the frequency. With high-pass filters , so-called "superinductivities" occur, an inductance whose real impedance depends on the square of the frequency.

literature

• Lutz v. Wangenheim: Active Filters and Oscillators . 1st edition. Springer, 2008, ISBN 978-3-540-71737-9 .
• Theodore Deliyannis, J. Kel Fidler, Yichuang Sun: Continuous-Time Active Filter Design . 1st edition. Crc Press, 1999, ISBN 978-0-8493-2573-1 .
• Reinhold Paul: Electrical engineering basic textbook Volume 2: Networks . 3. Edition. Springer, 1996, ISBN 978-3-540-55866-8 .

Web links

• http://www.krucker.ch/DiverseDok/Impedanzkonverter.pdf Examples for impedance converters