page_title: Network synthesis
section_title: Cauer II form
hierarchical_section_title: Network synthesis / Synthesis techniques / Cauer synthesis / Cauer II form
context_page_description: Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. The technique is to be compared to network analysis in which the response of a given circuit is calculated. Network synthesis was a great leap forward in circuit design. Prior to network synthesis, only network analysis was available, but this requires that one already knows what form of circuit is to be analysed. There is no guarantee that the chosen circuit will be the closest possible match to the desired response, nor that the circuit is the simplest possible. Network synthesis directly addresses both these issues. Network synthesis has historically been concerned with synthesising passive networks, but is not limited to such circuits.
  The field was founded by Wilhelm Cauer after reading Ronald M. Foster's 1924 paper A reactance theorem. Foster's theorem provided a method of synthesising LC circuits with arbitrary number of elements by a partial fraction expansion of the impedance function.
context_section_description: Cauer II form expands  in exactly the same way as Cauer I form except that lowest degree term is extracted first in the continued fraction expansion rather than the highest degree term as is done in Cauer I form.  The example used for the Cauer I form and the Foster forms when expanded as a Cauer II form results in some elements having negative values.  This particular PRF, therefore, cannot be realised in passive components as a Cauer II form without the inclusion of transformers or mutual inductances.
  The essential reason that the example  cannot be realised as a Cauer II form is that this form has a high-pass topology.  The first element extracted in the continued fraction is a series capacitor.  This makes it impossible for the zero of  at the origin to be realised.  The Cauer I form, on the other hand, has a low-pass topology and naturally has a zero at the origin.  However, the  of this function can be realised as a Cauer II form since the first element extracted is a shunt inductor.  This gives a pole at the origin for , but that translates to the necessary zero at the origin for .  The continued fraction expansion is,
  and the realised network is shown in the figure.
caption: Example of a Cauer II form realisation