CHAPTER THREE

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.

3.1 TRANSFER FUNCTIONS

3.1.1 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

(3.1) Numbered Display Equation

where ε is a ripple constant, Fn(Ω) represents a filtering or characteristic function, and Ω is a frequency variable. For our discussion here, it is convenient to let Ω represent a radian frequency variable of a lowpass prototype filter that has a cutoff frequency at Ω= Ωc for Ωc = 1 (rad/s). Frequency transformations to the usual radian frequency for practical lowpass, highpass, bandpass, and bandstop filters will be discussed later.

For linear time-invariant networks, the transfer function may be defined as a rational function, that is

(3.2)

where N(p) and ...

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