Microstrip bộ lọc cho các ứng dụng lò vi sóng RF (P3)

3.1.3 Butterworth Maximally Flat Response The amplitude-squared transfer function for Butterworth filters that have an inser-tion loss L Ar= 3.01 dB at the cutoff frequency c= 1 is given

de-of filter elements will also be discussed.

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 Onmany occasions, an amplitude-squared transfer function for a lossless passive filternetwork is defined as

where is a ripple constant, F n(
) represents a filtering or characteristic function,and
is a frequency variable For our discussion here, it is convenient to let
rep-resent a radian frequency variable of a lowpass prototype filter that has a cutoff fre-quency at
=
cfor
c= 1 (rad/s) Frequency transformations to the usual radianfrequency for practical lowpass, highpass, bandpass, and bandstop filters will bediscussed later on

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

where N(p) and D(p) are polynomials in a complex frequency variable p = + j
.For a lossless passive network, the neper frequency = 0 and p = j
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 manycases, 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, lowing the conventional definition in (2.9), can be computed by

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

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 p-

plane We will see in the following that different types of transfer functions will bedistinguished 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 L Ar= 3.01 dB at the cutoff frequency
c= 1 is given by

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 isalso 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
= 0.

Therefore, the maximally flat approximation to the ideal lowpass filter in the band is best at
= 0, but deteriorates as
approaches the cutoff frequency
c Fig-ure 3.1 shows a typical maximally flat response

A rational transfer function constructed from (3.7) is [1–2]

T n(
) is a Chebyshev function of the first kind of order n, which is defined as

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

with

p i = j cos
sin–1j + 

= sinh sinh–1 

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 aresometimes 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 j
-axis and its size is 1 + 2 ; the minor axis is on the -axisand is of size The pole distribution is shown, for n = 5, in Figure 3.4.

3.1.5 Elliptic Function Response

The response that is equal-ripple in both the passband and stopband is the ellipticfunction response, as illustrated in Figure 3.5 The transfer function for this type ofresponse is

1 + 2F n(
)

FIGURE 3.4 Pole distribution for Chebyshev response.

FIGURE 3.5 Elliptic function lowpass response.

Figure 3.6 plots the two typical oscillating curves for n = 4 and n = 5 Inspection

of F n(
) in (3.13b) shows that its zeros and poles are inversely proportional, theconstant of proportionality being
s An important property of this is that if
ican

be found such that F n(
) has equal ripples in the passband, it will automaticallyhave equal ripples in the stopband The parameter
sis the frequency at which the

equal-ripple stopband starts For n even F n(
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 ellipticfunctions; for this reason, filters that display such a response are called ellipticfunction 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]

3.1.6 Gaussian (Maximally Flat Group-Delay) Response

The Gaussian response is approximated by a rational transfer function [4]

Figure 3.7 shows two typical Gaussian responses for n = 3 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

n, the selectivity improves little and the insertion loss in decibels approaches the

Gaussian form [1]

Use of this equation gives the 3 dB bandwidth as

3 dB(2 n – 1)l n 2 (3.17)

which approximation is good for n 3 Hence, unlike the Butterworth response, the

3 dB bandwidth of a Gaussian filter is a function of the filter order; the higher thefilter order, the wider the 3 dB bandwidth

However, the Gaussian filters have a quite flat group delay in the passband, as

in-dicated in Figure 3.7(b), where the group delay is normalized by 0, which is the lay at the zero frequency and is inversely proportional to the bandwidth of the pass-band If we let
=
c= 1 radian per second be a reference bandwidth, then 0= 1

de-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 flatgroup delay over a large passband

3.1.7 All-Pass Response

External group delay equalizers, which are realized using all-pass networks, arewidely used in communications systems The transfer function of an all-pass net-work is defined by

pass network We may express (3.18) at real frequencies as S21( j
) = e j21(
), the

phase shift of an all-pass network is then

An expression for a strict Hurwitz polynomial D(p) is

D(p) =  n



k=1 [p – (–k)]m



k=1 [p – (–i+ j
i )]·[p – (–i– j
i)] (3.21)where –kfor k> 0 are the real left-hand roots, and –i± j
ifor i> 0 and
i> 0

are the complex left-hand roots of D(p), respectively If all poles and zeros of an

all-pass network are located along the -axis, such a network is said to consist of type sections and therefore referred to as C-type all-pass network On the otherhand, if the poles and zeros of the transfer function in (3.18) are all complex withquadrantal symmetry about the origin of the complex plane, the resultant network isreferred to as D-type all-pass network consisting of D-type sections only In prac-tice, a desired all-pass network may be constructed by a cascade connection of indi-vidual C-type and D-type sections Therefore, it is interesting to discuss their char-acteristics separately

C-For a single section C-type all-pass network, the transfer function is

Filter syntheses for realizing the transfer functions, such as those discussed in theprevious section, usually result in the so-called lowpass prototype filters [8–10] A

FIGURE 3.8 Characteristics of single-section C-type all-pass network: (a) pole-zero diagram, (b) group delay response.

FIGURE 3.9 Characteristics of single-section, D-type, all-pass network: (a) pole-zero diagram, (b) group delay response.

(a) (b)

(a) (b)

lowpass prototype filter is in general defined as the lowpass filter whose elementvalues are normalized to make the source resistance or conductance equal to one,

denoted by g0 = 1, and the cutoff angular frequency to be unity, denoted by
c=

1(rad/s) For example, Figure 3.10 demonstrates two possible forms of an n-pole

lowpass prototype for realizing an all-pole filter response, including Butterworth,Chebyshev, and Gaussian responses Either form may be used because both aredual from each other and give the same response It should be noted that in Figure

3.10, g i for i = 1 to n represent either the inductance of a series inductor or the pacitance of a shunt capacitor; therefore, n is also the number of reactive ele- ments If g1is the shunt capacitance or the series inductance, then g0is defined as

ca-the source resistance or ca-the source conductance Similarly, if g n is the shunt

ca-pacitance or the series inductance, g n+1 becomes the load resistance or the loadconductance Unless otherwise specified these g-values are supposed to be the in-ductance in henries, capacitance in farads, resistance in ohms, and conductance inmhos

This type of lowpass filter can serve as a prototype for designing many practicalfilters with frequency and element transformations This will be addressed in thenext section The main objective of this section is to present equations and tables forobtaining element values of some commonly used lowpass prototype filters withoutdetailing filter synthesis procedures In addition, the determination of the degree ofthe prototype filter will be discussed

3.2.1 Butterworth Lowpass Prototype Filters

For Butterworth or maximally flat lowpass prototype filters having a transfer

func-tion given in (3.7) with an inserfunc-tion loss L Ar= 3.01 dB at the cutoff
c= 1, the ment values as referring to Figure 3.10 may be computed by

ele-g0= 1.0

g n+1= 1.0

For convenience, Table 3.1 gives element values for such filters having n = 1 to 9.

As can be seen, the two-port Butterworth filters considered here are always

sym-metrical in network structure, namely, g0= g n+1 , g1= g nand so on

To determine the degree of a Butterworth lowpass prototype, a specification that

is usually the minimum stopband attenuation L AsdB at
=
sfor
s> 1 is given.Hence

For example, if L As= 40 dB and
s = 2, n 6.644, 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 ripple L ArdB and the cutoff frequency
c= 1, the element valuesfor the two-port networks shown in Figure 3.10 may be computed using the follow-ing formulas:

Some typical element values for such filters are tabulated in Table 3.2 for various

passband ripples L Ar , and for the filter degree of n = 1 to 9.

For the required passband ripple L Ar dB, the minimum stopband attenuation L As

dB at
=
s, the degree of a Chebyshev lowpass prototype, which will meet this

specification, can be found by

Using the same example as given above for the Butterworth prototype, i.e., L As 40

dB at
s = 2, but a passband ripple L Ar= 0.1 dB for the Chebyshev response, we

have n 5.45, i.e., n = 6 for the Chebyshev prototype to meet this specification.

This also demonstrates the superiority of the Chebyshev design over the

Butter-worth design for this type of specification

Sometimes, the minimum return loss L Ror the maximum voltage standing wave

ratio VSWR in the passband is specified instead of the passband ripple L Ar If the

re-turn loss is defined by (3.4) and the minimum passband rere-turn loss is L R dB (L R<

0), the corresponding passband ripple is

cosh–1

11

00

0 0

1 1

L L

A A s r

––

11

for n odd for n even

1.0coth2

For example if L R = –16.426 dB, L Ar = 0.1 dB Similarly, since the definition of

TABLE 3.2 Element values for Chebyshev lowpass prototype filters (g0= 1.0,
c= 1)

For passband ripple L Ar= 0.01 dB

3.2.3 Elliptic Function Lowpass Prototype Filters

Figure 3.11 illustrates two commonly used network structures for elliptic function

lowpass prototype filters In Figure 3.11(a), the series branches of

parallel-reso-nant circuits are introduced for realizing the finite-frequency transmission zeros,since they block transmission by having infinite series impedance (open-circuit) atresonance For this form of the elliptic function lowpass prototype [Figure

3.11(a)], g i for odd i(i = 1, 3, · · ·) represent the capacitance of a shunt capacitor,

g i for even i(i = 2, 4, · · ·) represent the inductance of an inductor, and the primed

gi for even i(i = 2, 4, · · ·) are the capacitance of a capacitor in a series branch of parallel-resonant circuit For the dual realization form in Figure 3.11(b), the shunt

branches of series-resonant circuits are used for implementing the

finite-frequen-cy transmission zeros, since they short out transmission at resonance In this case,

referring to Figure 3.11(b), g i for odd i(i = 1, 3, · · ·) are the inductance of a series inductor, g i for even i(i = 2, 4, · · ·) are the capacitance of a capacitor, and primed

gi for even i(i = 2, 4, · · ·) indicate the inductance of an inductor in a shunt branch

of series-resonant circuit Again, either form may be used, because both give thesame response

FIGURE 3.11 Lowpass prototype filters for elliptic function filters with (a) series parallel-resonant branches, (b) its dual with shunt series-resonant branches.

Unlike the Butterworth and Chebyshev lowpass prototype filters, there is no ple formula available for determining element values of the elliptic function low-pass prototype filters Table 3.3 tabulates some useful design data for equally termi-

sim-nated (g0= g n+1= 1) two-port elliptic function lowpass prototype filters shown in

Figure 3.11 These element values are given for a passband ripple L Ar= 0.1 dB, acutoff
c= 1, and various
s, which is the equal-ripple stopband starting frequency,referring to Figure 3.5 Also, listed beside this frequency parameter is the minimum

TABLE 3.3 Element values for elliptic function lowpass prototype filters (g0= g n+1= 1.0,
c= 1,

stopband insertion loss L Asin dB A smaller
simplies a higher selectivity of thefilter at the cost of reducing stopband rejection, as can be seen from Table 3.3 Moreextensive tables of elliptic function filters are available in literature such as [9] and[11].

The degree for an elliptic function lowpass prototype to meet a given tion may be found from the transfer function or design tables such as Table 3.3 Forinstance, considering the same example as used above for the Butterworth and

specifica-Chebyshev prototype, i.e., L As 40 dB at
s = 2 and the passband ripple L Ar= 0.1

dB, we can determine immediately n = 5 by inspecting the design data, i.e.,
sand

L Aslisted in Table 3.3 This also shows that the elliptic function design is superior toboth the Butterworth and Chebyshev designs for this type of specification

3.2.4 Gaussian Lowpass Prototype Filters

The filter networks shown in Figure 3.10 can also serve as the Gaussian lowpassprototype filters, since the Gaussian filters are all-pole filters, as the Butterworth orChebyshev filters are The element values of the Gaussian prototype filters are nor-mally obtained by network synthesis [3–4] For convenience, some element values,which are most commonly used for design of this type filter, are listed in Table 3.4,together with two useful design parameters The first one is the value of
, denoted

by
1%, for which the group delay has fallen off by 1% from its value at
= 0.Along with this parameter is the insertion loss at
1%, denoted by L
1%in dB Not

listed in the table is that for the n = 1 Gaussian lowpass prototype, which is actually

identical to the first-order Butterworth lowpass prototype given in Table 3.1

It can be observed from the tabulated element values that even with the equal

ter-minations (g0 = g n+1 = 1), the Gaussian filters (n 2) are asymmetrical in their

structures It is noteworthy that the higher order (n 5) Gaussian filters extend theflat group delay property into the frequency range where the insertion loss has ex-ceeded 3 dB If we define a 3 dB bandwidth as the passband and require that the

group delay is flat within 1% over the passband, the 5 pole (n = 5) Gaussian

proto-type would be the best choice for the design, with the minimum number of ments This is because the 4 pole Gaussian prototype filter only covers 91% of the 3

ele-dB bandwidth within 1% group delay flatness

TABLE 3.4 Element values for Gaussian lowpass prototype filters (g0= g n+1= 1.0,
s= 1)

3.2.5 All-Pass, Lowpass Prototype Filters

The basic network unit for realizing all-pass, lowpass prototype filters is a lattice

structure, as shown in Figure 3.12(a), where there is a conventional abbreviated

rep-resentation on the right This lattice is not only symmetric with respect to the twoports, but also balanced with respect to ground By inspection, the normalized two-

port Z-parameters of the network are

FIGURE 3.12 Lowpass prototype filters for all-pass filters: (a) basic network unit in a lattice structure; (b) the network elements for C-type, all-pass, lowpass prototype; (c) the network elements for D-type, all-pass, lowpass prototype.

For a single-section, C-type, all-pass, lowpass prototype, the network elements,

as indicated in Figure 3.12(b), are

The network elements for a single section D-type, all-pass, lowpass prototype, as

shown in Figure 3.12(c), are given by



i i

2



where i> 0 and
i> 0 are the two design parameters that will shape the group lay response, as illustrated in Figure 3.9 Since a D-type section is the second-order

de-all-pass network, there are actually two lowpass prototype elements, namely g1and

g2, which will represent both the inductance of an inductor and the capacitance of acapacitor, depending on the locations of these reactive elements, as indicated in Fig-

ure 3.12(c).

Higher-order all-pass prototype filters can be constructed by a chain connection

of several C-type and D-type sections The composite delay curves are then built up

by adding their individual delay contributions to obtain the overall delay istics

So far, we have only considered the lowpass prototype filters, which have a

normal-ized source resistance/conductance g0= 1 and a cutoff frequency
c= 1 To obtainfrequency characteristics and element values for practical filters based on the low-pass prototype, one may apply frequency and element transformations, which will

be addressed in this section

The frequency transformation, which is also referred to as frequency mapping, is

required to map a response such as Chebyshev response in the lowpass prototypefrequency domain
to that in the frequency domain in which a practical filter re-sponse such as lowpass, highpass, bandpass, and bandstop are expressed The fre-quency transformation will have an effect on all the reactive elements accordingly,but no effect on the resistive elements.

In addition to the frequency mapping, impedance scaling is also required to

ac-complish the element transformation The impedance scaling will remove the g0= 1normalization and adjust the filter to work for any value of the source impedance

denoted by Z0 For our formulation, it is convenient to define an impedance scalingfactor 0as

has no effect on the response shape

Let g be the generic term for the lowpass prototype elements in the element

transformation to be discussed Because it is independent of the frequency mation, the following resistive element transformation holds for any type of filter:

transfor-R = 0g for g representing the resistance

for g0being the resistance

for g0being the conductance

Z0/g0

g0 /Y0

 for g representing the capacitance

which is shown in Figure 3.13(a) To demonstrate the use of the element mation, let us consider design of a practical lowpass filter with a cutoff frequency f c

transfor-= 2 GHz and a source impedance Z0= 50 ohms A 3-pole Butterworth lowpass

pro-totype with the structure of Figure 3.10(b) is chosen for this example, which gives

g0 = g4= 1.0 mhos, g1= g3= 1.0 H, and g2= 2.0 F for
c= 1.0 rad/s, from Table3.1 The impedance scaling factor is 0= 50, according to (3.34) The angular cutofffrequency c= 2× 2 × 109rad/s Applying (3.38), we find L1= L3= 3.979 nH and

C2= 3.183 pF The resultant lowpass filter is illustrated in Figure 3.13(b).

FIGURE 3.13 Lowpass prototype to lowpass transformation: (a) basic element transformation, (b) a practical lowpass filter based on the transformation.

It is then obvious that an inductive/capacitive element in the lowpass prototype will

be inversely transformed to a capacitive/inductive element in the highpass filter.With impedance scaling, the element transformation is given by

 for g representing the capacitance

This type of element transformation is shown in Figure 3.14(a) Figure 3.14(b)

demonstrates a practical highpass filter with a cutoff frequency at 2GHz and ohms terminals, which is obtained from the transformation of the 3 pole Butter-worth lowpass prototype given above

50-3.3.3 Bandpass Transformation

Assume that a lowpass prototype response is to be transformed to a bandpass sponse having a passband 2– 1, where 1and 2indicate the passband-edge an-gular frequency The required frequency transformation is

lowpass prototype, we have

j
g 씮 j +

which implies that an inductive/capacitive element g in the lowpass prototype will transform to a series/parallel LC resonant circuit in the bandpass filter The ele- ments for the series LC resonator in the bandpass filter are

ments for the parallel LC resonator in the bandpass filter are