#### Document Type

Dissertation

#### Date of Award

Summer 8-31-2003

#### Degree Name

Doctor of Philosophy in Electrical Engineering - (Ph.D.)

#### Department

Electrical and Computer Engineering

#### First Advisor

Richard V. Snyder

#### Second Advisor

Gerald Martin Whitman

#### Third Advisor

Haim Grebel

#### Fourth Advisor

Edip Niver

#### Fifth Advisor

Sridhar Kanamaluru

#### Sixth Advisor

Aly Fathy

#### Abstract

This dissertation presents a theoretical investigation of a practical method to determine *quantitatively* the locations and loci of complex *transmission zeros* (TZ's) of positively and negatively cross-coupled RF or microwave bandpass filter networks.

Bandpass filters can be effectively designed by adjusting the locations of TZ's in the complex s-domain. To locate TZ's, this practical method uses chain matrices for subsystems (discrete parts of the network) of the filter network, and can be extended to other types of filters with cross-coupled sections.

An important result is that a complex doublet, triplet and/or quadruplet, (one-, two-, or four-pairs) of TZ's are shown to result solely from the cross-coupled portion of the circuit.

The several closed-forms of expressions called the *TZ characteristic equation* (TZCE) are obtained in terms of element values of the filter network. The locations and loci of TZ's are obtained by solving the relevant equations. These TZCE's are derived by taking advantage of the bridged-T structure for the cross-coupled part.

The reason for this dissertation is to locate TZ's without having to evaluate the entire transfer function, with all the infinite and DC TZ's as well as the *transmission poles* (TP's).

In the first chapter, definitions of voltage transfer function and chain (ABCD) matrix are discussed to investigate terminated two-port system. The relation between cascaded chain matrices and voltage transfer function is shown.

In the second chapter, a practical bandpass filter network with cross-coupled element is discussed in great detail. The derivations of TZ characteristic equations, the solutions of the equations, and the locations and loci of the TZ's are discussed so that this approach can be extended to generalized networks, including those consisting of combinations of lumped and distributed elements. The transfer function results from a concatenation of chain matrices, and it is expressed as a ratio of rational polynomials, with PR and Hurwitz properties. The reduction of the transfer function into factored polynomials allows for location and identification of TZ's.

In the third and fourth chapters, the application of the theory is discussed. The denominator *characteristic equation* (CE) is solved to locate *reflection zeros* (RZ's), referred to here in as transmission poles (TP's). Note that this identity (*TP's* == *RZ's*) pertains only to the lossless cases. Further examination of lossy networks is part of the work planned in the future.

Several examples of networks are introduced to find out location and locus of the transmission zeros, by directly considering the cancellation of the common terms in the numerator and denominator polynomials to obtain the canonical expressions of characteristic equations.

#### Recommended Citation

Um, Keehong, "Method for theoretically determining the locus and location of the transmission zeros in microwave filter networks" (2003). *Dissertations*. 597.

https://digitalcommons.njit.edu/dissertations/597