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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 de-sign 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
|S21(jV)|2 = 1 + «2F2(V) (3.1)
where « is a ripple constant, Fn(V) represents a filtering or characteristic function, and V is a frequency variable. For our discussion here, it is convenient to let V rep-resent a radian frequency variable of a lowpass prototype filter that has a cutoff fre-quency at V = Vc for Vc = 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 ra-tional function, that is
S21(p) = N(p) (3.2)
where N(p) and D(p) are polynomials in a complex frequency variable p = s + jV. For a lossless passive network, the neper frequency s = 0 and p = jV. To find a real-izable rational transfer function that produces response characteristics approximat-ing the required response is the so-called approximation problem, and in many cases, the rational transfer function of (3.2) can be constructed from the amplitude-squared transfer function of (3.1) [1–2].
For a given transfer function of (3.1), the insertion loss response of the filter, fol-lowing the conventional definition in (2.9), can be computed by
LA(V) = 10 log|S21(jV)|2 dB (3.3)
Since |S11|2 + |S21|2 = 1 for a lossless, passive two-port network, the return loss re-sponse of the filter can be found using (2.9):
LR(V) = 10 log[1 – |S21(jV)|2] dB (3.4)
If a rational transfer function is available, the phase response of the filter can be found as
f21 = Arg S21(jV) (3.5)
Then the group delay response of this network can be calculated by
td(V) = dfd(V) seconds (3.6)
where f21(V) is in radians and V is in radians per second.
3.1.2 The Poles and Zeros on the Complex Plane
The (s, V) plane, where a rational transfer function is defined, is called the complex plane or the p-plane. The horizontal axis of this plane is called the real or s-axis, and the vertical axis is called the imaginary or jV-axis. The values of p at which the function becomes zero are the zeros of the function, and the values of p at which the function becomes infinite are the singularities (usually the poles) of the function. Therefore, the zeros of S21(p) are the roots of the numerator N(p) and the poles of S21(p) are the roots of denominator D(p).
These poles will be the natural frequencies of the filter whose response is de-
3.1 TRANSFER FUNCTIONS 31
scribed by S21(p). For the filter to be stable, these natural frequencies must lie in the left half of the p-plane, or on the imaginary axis. If this were not so, the oscillations would be of exponentially increasing magnitude with respect to time, a condition that is impossible in a passive network. Hence, D(p) is a Hurwitz polynomial [3]; i.e., its roots (or zeros) are in the inside of the left half-plane, or on the jV-axis, whereas the roots (or zeros) of N(p) may occur anywhere on the entire complex plane. The zeros of N(p) are called finite-frequency transmission zeros of the filter.
The poles and zeros of a rational transfer function may be depicted on the p-plane. We will see in the following that different types of transfer functions will be distinguished from their pole-zero patterns of the diagram.
3.1.3 Butterworth (Maximally Flat) Response
The amplitude-squared transfer function for Butterworth filters that have an inser-tion loss LAr = 3.01 dB at the cutoff frequency Vc = 1 is given by
|S21(jV)|2 = 1 + V2n (3.7)
where n is the degree or the order of filter, which corresponds to the number of re-active elements required in the lowpass prototype filter. This type of response is also referred to as maximally flat because its amplitude-squared transfer function defined in (3.7) has the maximum number of (2n – 1) zero derivatives at V = 0. Therefore, the maximally flat approximation to the ideal lowpass filter in the pass-band is best at V = 0, but deteriorates as V approaches the cutoff frequency Vc. Fig-ure 3.1 shows a typical maximally flat response.
FIGURE 3.1 Butterworth (maximally flat) lowpass response.
32 BASIC CONCEPTS AND THEORIES OF FILTERS
A rational transfer function constructed from (3.7) is [1–2]
1
S21(p) = n (p – p) (3.8)
i=1
with
pi = j exp3(2i – 1)p 4
There is no finite-frequency transmission zero [all the zeros of S21(p) are at infini-ty], and the poles pi lie on the unit circle in the left half-plane at equal angular spac-ings, since |pi| = 1 and Arg pi = (2i – 1)p/2n. This is illustrated in Figure 3.2.
3.1.4 Chebyshev Response
The Chebyshev response that exhibits the equal-ripple passband and maximally flat stopband is depicted in Figure 3.3. The amplitude-squared transfer function that de-scribes this type of response is
|S21(jV)|2 = 1 + «2T2(V) (3.9)
where the ripple constant « is related to a given passband ripple LAr in dB by
« = !10 LAr – 1 (3.10)
FIGURE 3.2 Pole distribution for Butterworth (maximally flat) response.
3.1 TRANSFER FUNCTIONS 33
FIGURE 3.3 Chebyshev lowpass response.
Tn(V) is a Chebyshev function of the first kind of order n, which is defined as
cos(n cos–1 V) |V| # 1
n cosh(n cosh–1 V) |V| $ 1
(3.11)
Hence, the filters realized from (3.9) are commonly known as Chebyshev filters. Rhodes [2] has derived a general formula of the rational transfer function from
(3.9) for the Chebyshev filter, that is
n [h2 + sin2(ip/n)]1/2
S21(p) = } (p + p) (3.12) i=1
with
pi = j cos3sin–1 jh + (2i – 1)p 4
h = sinh1}sinh–1}2
Similar to the maximally flat case, all the transmission zeros of S21(p) are located at infinity. Therefore, the Butterworth and Chebyshev filters dealt with so far are sometimes referred to as all-pole filters. However, the pole locations for the Cheby-shev case are different, and lie on an ellipse in the left half-plane. The major axis of the ellipse is on the jV-axis and its size is Ïww hw; the minor axis is on the s-axis and is of size h. The pole distribution is shown, for n = 5, in Figure 3.4.
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