Basic Concepts and Theories of Filters This chapter describes basic concepts and theories that form the foundation for design of general RF/microwave filters, including microstrip filters. The topics will cover filter transfer functions, lowpass prototype filters and elements, frequency and element transformations, immittance inverters, Richards’ transformation, and Kuroda identities for distributed elements. Dissipation and unloaded quality factor of filter elements will also be discussed. | Microstrip Filters for RF Microwave Applications. Jia-Sheng Hong M. J. Lancaster Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-38877-7 Hardback 0-471-22161-9 Electronic CHAPTER 3 Basic Concepts and Theories of Filters This chapter describes basic concepts and theories that form the foundation for design of general RF microwave filters including microstrip filters. The topics will cover filter transfer functions lowpass prototype filters and elements frequency and element transformations immittance inverters Richards transformation and Kuroda identities for distributed elements. Dissipation and unloaded quality factor of filter elements will also be discussed. TRANSFER FUNCTIONS General Definitions The transfer function of a two-port filter network is a mathematical description of network response characteristics namely a mathematical expression of S21- On many occasions an amplitude-squared transfer function for a lossless passive filter network is defined as where e is a ripple constant F O represents a filtering or characteristic function and it is a frequency variable. For our discussion here it is convenient to let il represent a radian frequency variable of a lowpass prototype filter that has a cutoff frequency at il il for il 1 rad s . Frequency transformations to the usual radian frequency for practical lowpass highpass bandpass and bandstop filters will be discussed later on. 29 30 BASIC CONCEPTS AND THEORIES OF FILTERS For linear time-invariant networks the transfer function may be defined as a rational function that is N p 21 P D p where N p and D p are polynomials in a complex frequency variable p a jil. For a lossless passive network the neper frequency a 0 and p jil. To find a realizable rational transfer function that produces response characteristics approximating the required response is the so-called approximation problem and in many cases the rational transfer function of can be constructed from the amplitude-squared .
