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 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 |S21( j
)|2 =  2 +  F n (
) (3.1) where  is a ripple constant, Fn(
) represents a filtering or characteristic function, and
is a frequency variable For our discussion here, it is convenient to let
represent a radian frequency variable of a lowpass prototype filter that has a cutoff frequency at
=
c for
c = (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) S21(p) =  D(p) (3.2) where N(p) and D(p) are polynomials in a complex frequency variable p =  + j
For a lossless passive network, the neper frequency  = and p = j
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 (3.2) can be constructed from the amplitudesquared transfer function of (3.1) [1–2] For a given transfer function of (3.1), the insertion loss response of the filter, following the conventional definition in (2.9), can be computed by LA(
) = 10 log 2 dB |S21( j
)| (3.3) Since |S11|2 + |S21|2 = for a lossless, passive two-port network, the return loss response of the filter can be found using (2.9): LR(
) = 10 log[1 – |S21( j
)|2] dB (3.4) If a rational transfer function is available, the phase response of the filter can be found as 21 = Arg S21( j
) (3.5) Then the group delay response of this network can be calculated by d21(
) d(
) =  seconds –d
(3.6) where 21(
) is in radians and
is in radians per second 3.1.2 The Poles and Zeros on the Complex Plane The (,
) 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 -axis, and the vertical axis is called the imaginary or j
-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 j
-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 pplane 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 insertion loss LAr = 3.01 dB at the cutoff frequency
c = is given by |S21( j
)|2 =  +
2n (3.7) where n is the degree or the order of filter, which corresponds to the number of reactive 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
= Therefore, the maximally flat approximation to the ideal lowpass filter in the passband is best at
= 0, but deteriorates as
approaches the cutoff frequency
c Figure 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] n S21(p) =   (p – pi) (3.8) i=1 with (2i – 1) pi = j exp  2n
 There is no finite-frequency transmission zero [all the zeros of S21(p) are at infinity], and the poles pi lie on the unit circle in the left half-plane at equal angular spacings, since |pi| = and Arg pi = (2i – 1)/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 describes this type of response is |S21( j
)|2 =  2 +  T n (
) (3.9) where the ripple constant  is related to a given passband ripple LAr in dB by   10 –1 LAr = 10 FIGURE 3.2 Pole distribution for Butterworth (maximally flat) response (3.10) 33 3.1 TRANSFER FUNCTIONS FIGURE 3.3 Chebyshev lowpass response Tn(
) is a Chebyshev function of the first kind of order n, which is defined as Tn(
) = |
| |
| cos(n cos–1
) –1
)  cosh(n cosh (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 [ 2 + sin2(i/n)]1/2  i=1 n S21(p) =   (p + pi) (3.12) i=1 with (2i – 1) pi = j cos sin–1 j +  2n
 1 = sinh  sinh–1  n    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 Chebyshev case are different, and lie on an ellipse in the left half-plane The major axis of the ellipse is on the j
-axis and its size is  1 + 2 ; the minor axis is on the -axis and is of size The pole distribution is shown, for n = 5, in Figure 3.4 34 BASIC CONCEPTS AND THEORIES OF FILTERS FIGURE 3.4 Pole distribution for Chebyshev response 3.1.5 Elliptic Function Response The response that is equal-ripple in both the passband and stopband is the elliptic function response, as illustrated in Figure 3.5 The transfer function for this type of response is |S21( j
)|2 =  + 2F n2(
) FIGURE 3.5 Elliptic function lowpass response (3.13a) 3.1 TRANSFER FUNCTIONS 35 with Fn(
) =  n/2 (
2i –
2)  i=1 M  n/2  (
2s/
2i –
2) for n even i=1 (3.13b) (n–1)/2
 (
2i –
2) i=1 N  (n–1)/2  (
2s/
2i –
2) for n(3) odd i=1 where
i (0 <
i < 1) and
s > represent some critical frequencies; M and N are constants to be defined [4–5] Fn(
) will oscillate between ±1 for |
| 1, and |Fn(
= ±1)| = Figure 3.6 plots the two typical oscillating curves for n = and n = Inspection of Fn(
) in (3.13b) shows that its zeros and poles are inversely proportional, the constant of proportionality being
s An important property of this is that if
i can be found such that Fn(
) has equal ripples in the passband, it will automatically have equal ripples in the stopband The parameter
s is the frequency at which the equal-ripple stopband starts For n even Fn(
s) = M is required, which can be used to define the minimum in the stopband for a specified passband ripple constant  The transfer function given in (3.13) can lead to expressions containing elliptic functions; for this reason, filters that display such a response are called elliptic function filters, or simply elliptic filters They may also occasionally be referred to as Cauer filters, after the person who first introduced the function of this type [6] FIGURE 3.6 Plot of elliptic rational function 36 BASIC CONCEPTS AND THEORIES OF FILTERS 3.1.6 Gaussian (Maximally Flat Group-Delay) Response The Gaussian response is approximated by a rational transfer function [4] a0  n S21(p) = ak pk (3.14) k=0 where p =  + j
is the normalized complex frequency variable, and the coefficients (2n – k)! ak =  n–k k!(n – k)! (3.15) This transfer function posses a group delay that has maximum possible number of zero derivatives with respect to
at
= 0, which is why it is said to have maximally flat group delay around
= and is in a sense complementary to the Butterworth response, which has a maximally flat amplitude The above maximally flat group delay approximation was originally derived by W E Thomson [7] The resulting polynomials in (3.14) with coefficients given in (3.15) are related to the Bessel functions For these reasons, the filters of this type are also called Bessel and/or Thomson filters Figure 3.7 shows two typical Gaussian responses for n = and n = 5, which are obtained from (3.14) In general, the Gaussian filters have a poor selectivity, as can be seen from the amplitude responses in Figure 3.7(a) With increasing filter order FIGURE 3.7 Gaussian (maximally flat group-delay) response: (a) amplitude, (b) group delay 3.1 TRANSFER FUNCTIONS 37 n, the selectivity improves little and the insertion loss in decibels approaches the Gaussian form [1]
2 LA(
) = 10 log e  (2n–1) dB (3.16) Use of this equation gives the dB bandwidth as
3 dB (2 n – 1)l n (3.17) which approximation is good for n Hence, unlike the Butterworth response, the dB bandwidth of a Gaussian filter is a function of the filter order; the higher the filter order, the wider the dB bandwidth However, the Gaussian filters have a quite flat group delay in the passband, as indicated in Figure 3.7(b), where the group delay is normalized by 0, which is the delay at the zero frequency and is inversely proportional to the bandwidth of the passband If we let
=
c = radian per second be a reference bandwidth, then 0 = second With increasing filter order n, the group delay is flat over a wider frequency range Therefore, a high-order Gaussian filter is usually used for achieving a flat group delay over a large passband 3.1.7 All-Pass Response External group delay equalizers, which are realized using all-pass networks, are widely used in communications systems The transfer function of an all-pass network is defined by D(–p) S21(p) =  D(p) (3.18) where p =  + j
is the complex frequency variable and D(p) is a strict Hurwitz polynomial At real frequencies (p = j
), |S21( j
)|2 = S21(p)S21(–p) = so that the amplitude response is unity at all frequencies, which is why it is called the all-pass network However, there will be phase shift and group delay produced by the allpass network We may express (3.18) at real frequencies as S21( j
) = e j21(
), the phase shift of an all-pass network is then 21(
) = –j ln S21( j
) (3.19) and the group delay is given by d21(
) d(ln S21( j
)) d (
) = –  = j  d
d
 (3.20)  dD(–p) dD(p) dp =j   –    D(–p) dp D(p) dp d
p=j
... i.e., a 7-pole (n = 7) Butterworth prototype should be chosen 3.2.2 Chebyshev Lowpass Prototype Filters For Chebyshev lowpass prototype filters having a transfer function given in (3.9) with a passband... group-delay) response: (a) amplitude, (b) group delay 3.1 TRANSFER FUNCTIONS 37 n, the selectivity improves little and the insertion loss in decibels approaches the Gaussian form [1]
2 LA(
)... flat over a wider frequency range Therefore, a high-order Gaussian filter is usually used for achieving a flat group delay over a large passband 3.1.7 All-Pass Response External group delay equalizers,