### Abstract

It is a classical result, known as Darlington's theorem, that every rational positive-real function z(p) is realizable as the input impedance of a lumped reciprocal reactance two-port tuner N_{t} closed at the far end on 1 Ω. The theorem is evidently false if the 1 Ω termination is replaced by some prescribed non-constant positive-real impedance z_{l} (p). Any z(p) synthesizable in this more restrictive manner is said to be compatible with z_{l}(p) and we write z∼z_{l} to indicate the correspondence. The determination of necessary and sufficient conditions for the validity of z∼z_{l} is the problem of compatible impedances. Of the four better-known network treatments, only that of Schoeffler (IRE Trans. Circuit Theory, CT-8, 131-137 (1961) is completely correct, although severely restricted in scope. In particular, the remaining three contain a common error which appears to have propagated because a constraint on the ratio of even parts z_{e}(p)/z_{le}(p) derived by Schoeffler is unnecessary if z(p) is not minimum-reactance. The main theorems of Wohlers (IEEE Trans. Circuit Theory, CT-12, 528-535 (1965)) and Satyanaryana and Chen (J. Franklin Inst., 309, 267-280 (1980)) are very similar in structure to our Theorem 3 but considerably more complex and do not provide a sufficiently explicit description of the associated tuner N_{t}. In fact (Theorem I), with the exception of a physically irrelevant degeneracy (which is easily detected), N_{t}, when it exists, must possess an impedance matrix Z(p). Moreover, the latter can be effectively parametrized in terms of z(p), z_{l}(p) and a regular-allpass b(p) found as the solution of a standard interpolation problem of the Nevalinna-Pick type. Three fully worked examples clarify the theory and also illustrate many of the numerical steps.

Original language | English (US) |
---|---|

Pages (from-to) | 541-560 |

Number of pages | 20 |

Journal | International Journal of Circuit Theory and Applications |

Volume | 25 |

Issue number | 6 |

State | Published - Nov 1997 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

*International Journal of Circuit Theory and Applications*,

*25*(6), 541-560.

**A new study of the problem of compatible impedances.** / Youla, D. C.; Winter, F.; Pillai, Unnikrishna.

Research output: Contribution to journal › Article

*International Journal of Circuit Theory and Applications*, vol. 25, no. 6, pp. 541-560.

}

TY - JOUR

T1 - A new study of the problem of compatible impedances

AU - Youla, D. C.

AU - Winter, F.

AU - Pillai, Unnikrishna

PY - 1997/11

Y1 - 1997/11

N2 - It is a classical result, known as Darlington's theorem, that every rational positive-real function z(p) is realizable as the input impedance of a lumped reciprocal reactance two-port tuner Nt closed at the far end on 1 Ω. The theorem is evidently false if the 1 Ω termination is replaced by some prescribed non-constant positive-real impedance zl (p). Any z(p) synthesizable in this more restrictive manner is said to be compatible with zl(p) and we write z∼zl to indicate the correspondence. The determination of necessary and sufficient conditions for the validity of z∼zl is the problem of compatible impedances. Of the four better-known network treatments, only that of Schoeffler (IRE Trans. Circuit Theory, CT-8, 131-137 (1961) is completely correct, although severely restricted in scope. In particular, the remaining three contain a common error which appears to have propagated because a constraint on the ratio of even parts ze(p)/zle(p) derived by Schoeffler is unnecessary if z(p) is not minimum-reactance. The main theorems of Wohlers (IEEE Trans. Circuit Theory, CT-12, 528-535 (1965)) and Satyanaryana and Chen (J. Franklin Inst., 309, 267-280 (1980)) are very similar in structure to our Theorem 3 but considerably more complex and do not provide a sufficiently explicit description of the associated tuner Nt. In fact (Theorem I), with the exception of a physically irrelevant degeneracy (which is easily detected), Nt, when it exists, must possess an impedance matrix Z(p). Moreover, the latter can be effectively parametrized in terms of z(p), zl(p) and a regular-allpass b(p) found as the solution of a standard interpolation problem of the Nevalinna-Pick type. Three fully worked examples clarify the theory and also illustrate many of the numerical steps.

AB - It is a classical result, known as Darlington's theorem, that every rational positive-real function z(p) is realizable as the input impedance of a lumped reciprocal reactance two-port tuner Nt closed at the far end on 1 Ω. The theorem is evidently false if the 1 Ω termination is replaced by some prescribed non-constant positive-real impedance zl (p). Any z(p) synthesizable in this more restrictive manner is said to be compatible with zl(p) and we write z∼zl to indicate the correspondence. The determination of necessary and sufficient conditions for the validity of z∼zl is the problem of compatible impedances. Of the four better-known network treatments, only that of Schoeffler (IRE Trans. Circuit Theory, CT-8, 131-137 (1961) is completely correct, although severely restricted in scope. In particular, the remaining three contain a common error which appears to have propagated because a constraint on the ratio of even parts ze(p)/zle(p) derived by Schoeffler is unnecessary if z(p) is not minimum-reactance. The main theorems of Wohlers (IEEE Trans. Circuit Theory, CT-12, 528-535 (1965)) and Satyanaryana and Chen (J. Franklin Inst., 309, 267-280 (1980)) are very similar in structure to our Theorem 3 but considerably more complex and do not provide a sufficiently explicit description of the associated tuner Nt. In fact (Theorem I), with the exception of a physically irrelevant degeneracy (which is easily detected), Nt, when it exists, must possess an impedance matrix Z(p). Moreover, the latter can be effectively parametrized in terms of z(p), zl(p) and a regular-allpass b(p) found as the solution of a standard interpolation problem of the Nevalinna-Pick type. Three fully worked examples clarify the theory and also illustrate many of the numerical steps.

UR - http://www.scopus.com/inward/record.url?scp=0031275574&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031275574&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031275574

VL - 25

SP - 541

EP - 560

JO - International Journal of Circuit Theory and Applications

JF - International Journal of Circuit Theory and Applications

SN - 0098-9886

IS - 6

ER -