# On the Realization of Rational Admittance Matrices Using Voltage-Controlled Voltage Sources and RC One-Ports

About: This article is published in IEEE Transactions on Circuit Theory.The article was published on 1969-11-01. It has received 8 citations till now. The article focuses on the topics: Voltage divider & Admittance.

##### Citations

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TL;DR: In this article, a synthesis procedure is presented whereby the network function T (s) can be realized as an active RC multiport network with grounded ports, based on V(s) = T (S) U (s), where T is a q × p matrix of real rational functions of the complex variable s, the realization requires a minimum number of grounded capacitors and no more than 2 (p+n) inverting, grounded voltage amplifiers or p+n differential output.

Abstract: A synthesis procedure–easily implemented as a digital computer program– is presented whereby the network function T (s) can be realized as an active RC multiport network with grounded ports. Based on V (s) = T (s) U (s), where T (s) is a q × p matrix of real rational functions of the complex variable s, the realization requires a minimum number of grounded capacitors–n = degree { T (s)}–and no more than 2 (p+n) inverting, grounded voltage amplifiers or p+n differential output, grounded voltage amplifiers. Note: These properties of the realization are desirable if the network is to be fabricated as an integrated circuit.

15 citations

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TL;DR: In this paper, a synthesis procedure, easily implemented as a digital computer program, is presented whereby any p \times p matrix Y(s) of real rational functions of the complex frequency variable s can be realized as the short-circuit admittance matrix of a p -port active RC network.

Abstract: A synthesis procedure, easily implemented as a digital computer program, is presented whereby any p \times p matrix Y(s) of real rational functions of the complex frequency variable s can be realized as the short-circuit admittance matrix of a p -port active RC network. The realization requires a minimum number of capacitors- n = degree \{Y(s)\} -and no more than 2(p+n) inverting common ground voltage-controlled voltage sources. All the capacitors and ports are grounded.

12 citations

### Cites methods from "On the Realization of Rational Admi..."

...A synthesis technique developed by Goldman and Ghausi [ 4 ] requires no more than 2p common-ground voltage-controlled voltage sources, of which p have differential outputs and p have positive gains....

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.... [ 4 ] E. S. Kuh and R. A. Rohrer, Theory of Linear Active Networks....

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TL;DR: In this paper, a new and practical synthesis procedure is presented for the realization of an arbitrary n × n matrix of real rational functions of the complex frequency variable as the short-circuit admittance matrix of a transformerless active RC n-port network.

Abstract: A new and practical synthesis procedure is presented for the realization of an arbitrary n × n matrix of real rational functions of the complex frequency variable as the short-circuit admittance matrix of a transformerless active RC n-port network. The realization requires a theoretically minimum number of capacitors p, where p is the degree of the matrix, and no more than (n + p + 1) grounded finite-gain phase-inverting voltage-controlled voltage sources. All the capacitors and ports are grounded.
The freedom implicit in the synthesis procedure allows the inclusion of constraints on the passive element values. Furthermore, in special cases the realization is achieved with a reduced number of conductances and voltage-controlled voltage sources. The synthesis procedure is simple to apply and can readily be implemented on a digital computer. Several examples are given.

10 citations

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TL;DR: In this article, it is proved that an arbitrary n × n matrix of real rational functions of the complex frequency variable can be realized as the short-circuit admittance matrix of a grounded transformerless active RC n-port network containing (n+1) grounded finite-gain phase-inverting voltage-controlled voltage sources (VCVSs).

Abstract: A new procedure for the synthesis of active RC networks when grounded finite-gain phase-inverting voltage-controlled voltage sources serve as active elements is developed. It is proved that an arbitrary n × n matrix of real rational functions of the complex frequency variable can be realized as the short-circuit admittance matrix of a grounded transformerless active RC n-port network containing (n+1) grounded finite-gain phase-inverting voltage-controlled voltage sources (VCVSs). In general all the (n+1) grounded VCVSs are necessary.
The structure proposed to prove a general theorem is later simplified for the realization of a restricted but important class of real rational matrices to obtain considerable savings in the computation volume and in the number of passive components used for the realization of the network.
Examples are given to illustrate presented synthesis procedures.

5 citations

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TL;DR: In this article, a new and practical technique for synthesizing arbitrary rational admittance matrices is presented, which includes the inclusion of the RC constraints and the ability to include the frequency-dependent nonideal controlled-source models in the synthesis procedure.

Abstract: A new and practical technique for synthesizing arbitrary rational admittance matrices is presented. Of special interest is the inclusion of the RC constraints and the ability to include the frequency-dependent nonideal controlled-source models in the synthesis procedure.

5 citations

##### References

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TL;DR: The following basic theorem concerning active RC networks is proved in this paper : an arbitrary N × N matrix of real rational functions in the complex-frequency variable can be realized as the short-circuit admittance matrix of a transformerless active RC N-port network containing N real-coefficient controlled sources.

Abstract: The following basic theorem concerning active RC networks is proved: Theorem: An arbitrary N × N matrix of real rational functions in the complex-frequency variable (a) can be realized as the short-circuit admittance matrix of a transformerless active RC N-port network containing N real-coefficient controlled sources, and (b) cannot, in general, be realized as the short-circuit admittance matrix of an active RC network containing less than N controlled sources.

26 citations

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TL;DR: In this paper, it was shown that an arbitrary symmetric N × N matrix of real rational functions in the complex-frequency variable (a) can be realized as the immittance matrix of an N-port network containing only resistors, capacitors, inductors, ideal transformers, and negative-RC impedances if M < N.

Abstract: Theorem: An arbitrary symmetric N × N matrix of real rational functions in the complex-frequency variable (a) can be realized as the immittance matrix of an N-port network containing only resistors, capacitors, and N negative-RC impedances, and (b) cannot, in general, be realized as the immittance matrix of an N-port network containing resistors, capacitors, inductors, ideal transformers, and M negative-RC impedances if M < N. The necessary and sufficient conditions for the immittance-matrix realization of transformerless networks of capacitors, self-inductors, resistors, and negative resistors follow as a special case of the theorem. In addition, an earlier result is extended by presenting a procedure for the realization of an arbitrary N × N short-circuit admittance matrix as an unbalanced transformerless active RC network requiring no more than N controlled sources. The passive RC structure has the interesting property that it can always be realized as a (3 N + 1)-terminal network of two-terminal impedances with common reference node and no internal nodes. The active subnetwork can always be realized with N negative-impedance converters.

24 citations

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14 citations

01 Jan 1968

TL;DR: In this article, Martixelli et al. used negative impedance converters or nullors in place of controlled sources to achieve the theoretically minimum number of capacitors, at the expense of an increase in the number of controlled source over that required by Sandberg's method.

Abstract: matrix of real rational fpnctioas of complex freqency. The method achieves realization with the minimlm number of capacitors, m, this beii the “degree” of the matrix. The network is completed by positive resistors ad (n + m) real-co&cimt coatrolled sources Sandberg[’] has shown that an arbitrary n x n matrix Y(s) of real rational functions of complex frequency may be realized by an RC n-port with a minimum of n controlled sources. Other realizations employing negative impedance converters or nullors in place of controlled sources have been described by SandbergL2] and Martix~elli,[~] respectively. The realization described here employs the theoretically minimum number of capacitors, at the expense of an increase in the number of controlled sources over that required by Sandberg’s method.“] The number of capacitors employed here is equal to the “degree” of the matrix Y(s). The concept of “degree” was formalized by Dutfin and Haz~ny.’~] Kalmanf5] showed that this “degree” was also the minimal dimension of the state-space of a linear dynamical system whose transfer matrix is Y(s). He obtained a realization of the dynamical by finding a quadruple of real constant matrices {J, F, G, H} such that Y(s) = J + H(s 1 - F)-’G

9 citations

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01 Jun 1968TL;DR: In this article, an admittance realization of an arbitrary n × n matrix of real rational functions of complex frequency is outlined, which achieves realization with the minimum number of capacitors, m, this being the "degree" of the matrix.

Abstract: An admittance realization is outlined of an arbitrary n × n matrix of real rational functions of complex frequency The method achieves realization with the minimum number of capacitors, m, this being the "degree" of the matrix The network is completed by positive resistors and (n + m) real-coefficient controlled sources

9 citations