FILTER DESJGN

Filters occur so frequently in the instrumentation and communications industries that no book covering the field of rf circuit dt::sign could be complete without at least one chapter devoted to the subject. Indeed. entire books have been written on the art of filter de­sign alone, so this single chapter cannot possibly cover nil aspects of all types of filters. But it will bmiliarize yOU with the characteristics of four of the most com­~onl)' used filters and will enable you to design ver)' quickl)' and easily a filter that will meet, or exceed. most of the common filter requirements that you will encounter.

\Ve will cover Butterworth, Chebyshev, and Bes­sel filters in all of their common configuratior...:: lou;­pass, high-pass, bandpass, and bandstop. \Ve will Jearn how to take advantage of the attenu~.tion char­acteristics unique to each type of filter. Finally. we will Jearn how to design some very powerful filters in as little as 5 minutes by merely looking through a catalog to choose a design to suit your needs.

BACKGROUND

In Chapter 2, the concept of resonance was ex­plored and we determined the effects that component value chanees had on resonant circuit operation. 'You should nov.: be somewhat familiar with the methods that are used in analyzing passive resonant circuits to nnd quantities, such as loaded Q, insertion loss, and b:r:d,.·.·idth..'Yau .should alsa be c~pa~le of designing

~ ~.>ne- or two-resonator circuits for any loaded Q desiredI (or, at least, determine why you cannot). Quite a few I of the .filter applications that you will encounter, how­I ever, cannot be satis6ed with the simple bandpass j ~rrang:ment given in Chapter ~. There are occasi~ns

when, lOstead of passing a certalO band of frequencIes while rejecting frequencies above and below (band­pass). we would like to a ttenuate a small band of fre· ~uencies while passing all others. This type of filter IS called, appropriately enough, a bandstop filter. Still other requirements call for a low-pass or high-pass response. The characteristic curves for these responses are ~hown in Fig. 3.1. The low-pass filter will allow aH .slgnals below a certain cutoH frequency to pass \\'h1le attenuating all others. A high-pass filter's re­SPOnse is the mirror-image of the low-pass response

and attenuates all signals below a certain cutoff fre­quency while allowing those above cu toff to pass. These types of response simply cannot be handled very well with the two-resonator bandpass dcsi~ns of Chapter 2. ­

In this chapter, we will use the low-pass filter as our wor~orse, as all other responses will be derived from it. So let's take a quick look at a simple low-pass filter and examine its characteristics. F: g. 3·2 is an example of a very simple two-polc, or second-order low-pass filter. The order of a Blter is determined by the slope of the attenuation curve it presents in the stopband. A second-order filter ~s one whose rolloff is a function of the fr~quency squared, or 12 dB per octave. A third-order filter causes a rolloff thnt is pro. portional to frequenc)' cubed, or 18 dB per octave. Thus, the order of n filter can be equated with the number of significant reactive elements that it pre­sents to the source as the signal deviates from the passband.

The circuit of Fig. 3·2 can be anal~'zed in much the same manner as was done in Chapter 2. For instance, an examination of the effects of loaded Q on the reo

.~

sponse would yield the family of curves shown in Fig. 3-3. Surprisingly, even this circuit configuration can cause a peak in the response. This is due to the fact that at some frequenc)', the inductor and capacitor will become resonant and, thus, peak the response if the loaded Q is high enough. The resonant frequency can be determined from

1 ( Eg. 3-1 ) Fr=2~

For low values of loaded Q, however, no response peak will be noticed.

The lortded Q of this filter is dependent upon the individual Q's of the series leg and the shunt leg where, assuming perfect components,

XQl = ---.!: (Eg.3-2)

R.

and,

(Eg.3-3)

44

I

I

45 FtLTD\ DE:sICN

:lnd the total Q is:

_ Q1Q2QlOUl - Ql + Q2 (Eq. 3-4 )

II the total Q of the circuit is gre::aler than a.bout 0.5. thcn for optimum transfer of power from the source to the load, Ql should equal Q2. In this case, at the

./

Frequency

(1\) Low-pa.u.

Frequency

(B) High-paJJ..

II

I Frcqutncy

(C) Bandstop.

Fig. 3-1. Typical filler response curves.

Fig. :)-2. A simple low-pass filter.

peak 'frequency, the response will approach O-d13 in­sertion loss. II the total Q of the nchyork is less th::m ab9ut 0.5, there will be no peak in the response Jncl. for optimum transfer of power, R. should equal Rl.. The peaking of the filter's response is commonly called ripple (defined in Chapter 2) and C:ln vary consider.

$O'----:...--.:-...:...--J..........----~-"---'----'--'--.>--' 0.1 0.1 O.S 1.0 z.o S.O 10

rr~u~1"<T (III,'

Fig. J-J. Typical two-pole filter response C\JrYe-s.

fill. ~~, Threl! ..~:Iement low.p::u.. Blter.

F~ufnC)"

Fig. J-5. Typical response of a. thr~lcm4nt low-pus filter.

0

10

= 20~ c

~ )0 ::I

E < oW

SO

O.l 0.5 1.0 2.0 ~.O

FnquenC'Y (fll.)

. f S VI loadc~ QFIS. J~. Curves showmg requency respon e . for three-element low-pass fiJten.

I

;1b)\, from one filter design to the next depending on r::e' application. As shown, the two-element filter ex­.:bits only one response peak at the edge of the pass­Jndo

It CJn be shown th:Jt the number of peaks within the Jssband is directl), related to the number of ele­.~:lts in the filter by:

Number of Peaks = N - 1

here, S == the number of elements.

,:us. the three-element low-pass filter of Fig. 3-4 auld exhibit two response peaks as shown in Fig. -0 This is true only if the loaded Q is greater than eo Typical response curves for various values of ded Q for the circuit given in Fig. 34 are shown Fig. 3-6. For all odd-order networks, the response dc and at the upper edge of the passband ap­aches 0 dB with dips in the response between the

frequencies. All even-order networks will pro­e 20 insertion loss at dc equal to the amount of band ripple in dB. Keep in mind, however, that er of these two networks, if designed for low values olided Q, can be made to exhibit little or no pass­d ripple. But, as you can see from Figs. 3-3 and the elimination of passband ripple can be ':lade

.. at the expense of bandwicith. '::he smaller the Ie that is allowed, the wider the bandwidth be­es and, therefore, selectivity suffers. Optimum ess in the passband occurs when the loaded Q

'ne three-element circuit is equal to one (1). Any e of loaded Q that is less than one will cause the onse to roll. off noticeably even at very lew uencies, within the defined passband. Thus, not . is the s~lectivity poorer but the passballd inser­loss is too. In an application where there is not

,b signal to begin with, an even further decrease ignal strength could be disastrous. ,ow that we have taken a quick look at two repre­. tive low-pass filters and their associated respomes, ,discuss filters in general:'

igh _Q filters tend to exhibit a far greater initial pe toward the stopband than their low-Q coun­rparts with the same number of elements. Thus, .an)' frequency in the stopband, the attenuationI!I be greater for a high.Q filter than for one wittt lo~'er Q. The penalty for this improvement is e Ulcrease in passband ripple that must occur a result. .. -. o. "

n

w-Q filters tend to have the flattest passband .~nse but their initial attenuation siope at the

edge is small. Thus, the penalty for the re­c~d passband ripple is a decrease in the initial P .and attenuation. ~'Ith the resonant circuits discussed in Chapter

e Source and load resistors loading a filter will\'e t profound effect on the Q of the filter and, re are, On the passband ripple and shape factor

RF CmcuIT DESICl'

of the filter. If a £iltp.~ is inserted between h\l0 I e­

sistance values for which it was not designed, the performance will suHer to an extent dependin n

upon the degree of error in the te~inating im~ pedance values.

4. The final attenuation slope of the response is de­pendent upon thc order of the network. The order of the network is equal to the number of reactive elements in thc low-pass filtcr. Thus, a second-order network (2 elements) falls off at a final attenuation slope of 12 dB per octave, a third-order network (3 eleme~ts) at the rate of 18 dB per octave, and so on, wlth the addition of 6 dB per octave per ele­ment.

~10DERN FILTER DESIGN

Modern ~lter design has evolved through the years from a subject known only to specialists in the field (because of the advanced mathematics involved) to a ~ractic~l well-organized c~talog of ready-to-use cir­CUits available to anyone With a knowledge of eighth grade level math. In fact, an average individual with absolutely no prior practical filter design experience sh.o~~d, be a.ble to sit down, read this chapter, and WithIn 30 mmutes be able to design a practical high­pass, low-pass, bandpass, or bandstop £Iter to his speciBcations. It sounds simple and it is-once a few basic rules are memorized.

The approach we will take in all of the 0t;;signs in this chapter will be to make use of the myriad of normalized low-pass prototypcs that are now avail ­able to 1le designer. The actual design procedure is, th~refore, nothing more than determining your re­qUlrements and, then, finding a filter in a catalog. which satisfies these requirements. Each normalized element value is then scaled to the frequency and im­pedance you desire and, then, transformed to the type of response (bandpass, high-pass, bandstop) tha t you wish. \Vith practice, the procedure becomes very sim­ple and soon you will be defining and designing B1ters.

The concept of normalization may at first seem foreign fo the person who is a newcomer to L'-1e field of Slter design, and the idea of transfonning a low-pass fllter into one that will give one of the other three ~pes of responses might seem absurd. The best advice I can give (to anyone not familiar with these prac­tices and who might feel a bit skeptical at this point) is to press on. The only way to truly realize the beauty and simplicity of this approach is to try a few actual designs. Once you try a few, you wili be hooked, and any other approach to filter design will suddenly seem tedious and unnecessarily complicated.

NORl\1ALlZATION AND TfIE LO\\'-PASS PROTOTYPE

In order to offer a catalog of useful filter circuits to the electronic filter designer, it became necessary to

II ;

I

47 fJLTl1\ DEStC~

t clarclize the presentation of the material. Obvi­s an I'n practice, it would be extremely difficult to1vous , d I hpare the performance an eva uate t e usefulness c~~wo filter networks if tbey were operating under ~wo totaJly different sets of circumstances. Similarly,:h presentation of any comparative design inIorma­'. c for filters, if not standardized, would be totally tIDn f d d' . Z·useless, This concept 0 stan ar I~tlon or norma lZil­

'ion then: is merely a tool used by filter experts to . cs'ent all 6lter design and perfon:nance information pr fl' . d' N l' t' . 3 manner use u to Circuit eSlgners. orma lza Ion ~surcs the designer of the capability of comptlring ~I;e pcrformu.nce of any two filter types when given the same operating conditions.

All of the catalogued filters in this chapter are low­pass filters normalized for a cutoff frequency of one r:JOi3 n per second (0.159 Hz) and for source and load resistors of one ohm. A characteristic response of such ~ filter is shown in Fig. 3-7. The circuit used to gen­erate this response is c:llled the low-pass prototlJPe<

odB

freq u ftK1' (.."

Fig. J-7. Normalized low-pus response.

. Obviously, the design of a filter with s~lch a low cutoff frequency would require component values much larger than those we are ~ccustomed to working with; capacitor values would be in farads rather than microfarads and picofarads, and the inductor vnlues would be in henries rather than in microhenries and 'nanohenries. But onc~ we choose a suitable low-pnss prototype from the c:ltalog, we can cl1ange the im­pcd:lnce level and cutoff frequency of the filter to any

Ivalue we wish through a simple proces.s called scali~. (The net result of this process is a practical filter deSIgn ·. ·;th realizable component values.

r FILTER TYPES

.\t:lnv of the filters used today bear the names of the en who developed them. In this section, we will take look at three such filters and examine their attenua­

iOn characteristics. Their relative merits will be dis­usscd and their low-pass prototypes presented. The hrce filter types discussed will include the Butter­orth. Chebyshev, and Bessel responses.

Ie Duttcrworth Response

The Butterworth BIter is a medium-Q BIter that is 'sed in designs which require the amplitude respqnse

I i

of the' filter to be as flat 'as possible. The Butterworth response is the flattest passband response available and contains no ripple. Tbe typical response of such a filter might look like that of Fig, 3-8.

Since the Butterworth response is only a medium-Q filter, its initial attenuation steepness is not as good as some filters but it u better than others. This char­acteristic often causes the Butterworth response to be called a middle-of-the-road design.

The attenuation of a Butterworth BIter is given by

(Eq. 3·5)

where, w = the frequency at which the attenuation is de·

sired, We =the cutoff frequency (W3 d8) of the filter, n =the number of elements in the filter.

If Equation 3-5 is evaluated at various frequencies for various numbers of elements, a familv of curves is generated which will give a very good graphiC31 representation of the attenuation provided by any order of BIter ~t any frequency. This information is illustrated in Fig. 3-9. Thus, from Fig. 3·9, a 5-element (Bfth'order) Butterwort~ filter will provide an attenu· ation of approximately 30 dB at a freq,uency equal to

odS...----------. -}dB --------------. -- ­

C ~

~ :2 "I

~ <

.. --I

I.: - 1,:

I

•I.

frtquen<y ("I'

Fig. J-8. The Butterworth response.

0

1.2

24

36

~=48 c

~ 60

;:t

n <

84

96

108

1.5 2.0 2.' 3.0 3.5 .&.0

fr~ut"CY R.arioJ(/'.1 \;

Fig. ~.9. Attenuation characteristiC! tor }:: Butterworth fillen, ,/--UJJ'

r(

I

. e the cutoff frequency of the Blter. Notice here tV/Ie . • l' d d thtil t the frequency aXIS 15 norma lze to w/wr; an e

a h begins at the cutoff (-3 dB) point. This graph ~r~remely useful as it provides you with a method f determining, at a glance, the order of a £lter needed

o meet a given attenuation specification. A brief fO mple should illustrate this point (Example 3-1).ex3

EXAMPLE J.l How many elements are required to design a Butter.

worth filter with a cu tolJ frequency of 5.0 hi Hz, if the filter must provide at least 50 dB of attenualion at 150 MH%?

SoJulicm

The first step in the solu tion is to .lind the ra tio 01 IIJI w. :: fjf ••

f 150 MHz 1:~ SO MHz

.= 3

Th\U. at 3 times the cutoff frequency, the rcspomc must be down by at least 50 dB. Referring to Fig. 3-9, it is seen very Quickl)' that a minimum of 6 elements is required to meet \his oesign ~oal. At an IIE, of 3, a 6-e1ement design would provide approximately 5? dB of attenuation. whil~ a ~-e1e­ment design would provJde only about 47 dB, which 1$ not quite Bood enough.

The element values for a normalized Butterworth ow.pass filter operating between equal l-ohm tenni­2tions (source and load) can be found by

I . (2k-l)1T (Eq.3-6). A.. =2 SID • 2n ' k =1,.2, .•• n

t,:e~eilie number of elements, . A, is the k-th reactance in the ladder and may be,

either an inductor or capacitor.

Be i~rUl (2k - 1 )-rr/2n ~s ill i'lioians. ~tVe can use .uation 3-6 to generate our first entry into the cata­g of iow-pass prototypes shown in Table 3-1. The acement of each component of the filter is shown mediately above and below the table.

t,The rules for interpreting Butterworth tables are pIe. The schematic shown above the table is used

. en~ver the ratio Rs/RL is calculated as the design lena. The table is read from the lOp down. Alter­tely, when RL/Ra is calculated, the schematic below

table is used. Then, the element designators in the Ie are read from,:the buttom up. Thus, a four-ele­nt low-pass prototype coul.d appear as shown in ·3-10. Note here that the element values not given Table 3-1 are simply left out of the prototype 1add~r

ark. TIle I-ohm load resistor is then placed di­d}" across the output of the filter. ~member that the cutoff frequency of each filter is dian per second, or 0.159 Hz. Each capacitor value en is in farads, and each inductor value is in ben-

RF Cmevrr DESICN

ries. The network will later be scaled to the impedance and frequenC)1 that is·desired through a simple multi, plication and division process. The component values will then appear much more realistic.

Occasionall)', we have the need to design a filter that will operate between two unequal terrrunations as shown in Fig. 3-11. In this case, the circuit u normal.

1.H8 0.16$

'\, 0.165 I Fig. 3-10. A lour-element Butterworth low-Pius

prototype circuit.

Table 3-1. Butterworth Equal Termination Low-Pass Prototype Element Values (Ra = RL )

tl CJ L2 c~

II

L.

l.

···11 C, L. C7

2 1.414 1..(1.( 3 1.000 2.000 1.000 of 0.765 1.848 1.848 0.765 5 -0.618 1.618 2.000 1.618 0.618 6 0.518 1.414 1.932 1.932 1.414 0.518 7 0.445 1.247 1.802 2.000 1.802 1.247 0.445

n LJ C2 La C. Ls ell L 7

Fig. 3·11. ~neQual tennin,aUoru.

• . Fig. 3-12.. NormalIud unequal tcnninations.

I

ized for a load resistance of 1. ohm, while taking what we get for the source resistance..Dividing both the load and source resistor by 10 will yield a load re­sistance of 1 ohm and a source resistance of 5 ohms JS shown in Fig. 3-12. \Vc can use the normalized terminating resistors to help us find a low.pass proto­type circuit.

Table 3-2 is a list of Butterworth low-pass proto­type values for various r:lUos of sourc~ to load im­pedance (Rs/Rd. The sc~ematic shown above the table is used when Rs/R L IS calculated, and the ele­ment values are read down from the top of the table.

T2ble 3·2A. Butterworth Low-Pass Prototype Element Values

I

I n R3 /Rc. C1 L, C3

2 1.111 1.035 1.835 1.250 0.849 2.121 1.4~9 0.697 2.439 1.661 0.566 2.828 2.000 0.448 3.346 2.500 0.342 4.095 3.333 0.245 5.313 5.000 0.156 1.101

10.000 0.014 14.814 ::0 1.·114 0.707

J ').900 0.808 1.633 1.599 0.800 0.844 1.384 1.926 0.700 0.915 1.165 2.2.17 0.600 1.023 0.965 2.702

~ 0.500 1.18L 0.179 3.2.6 L 0,400 1.425 0.604 4.064 0.300 1.838 0.440 5.363 0.200 2.669 0.284 7.910 0.100 5.161 0.138 15.455

00 1.500 1.333 0.500

4 1.111 0.466 1.502 1.744 1.469 1.250 0.388 . 1.695 1.511 1.811 1.429 0.325 1.862 1.291 2..115 1.661 0.269 2.103 1.082 2.613

",,"2.000 0.218·, 2.452 0.88:J 3.187 . 2.500 0.169 2.986 0.691 4.009 ::3.333 0.124 3.883 0.507 5.338 5.000 0.080 5.684 0.331 1.940

10.000 0.039 11.094 0.162 15.642 co L.S31 1.577 1.082 0.383

n R(,/R~ L, C~

L)

'{9

Alternately, when RL/R g is calculated, the schemanc below the table is used while reading up from the bottom of the table to get the element values (Ex­ample 3-2).

EXAMPLE 3·2 Find the low-pus prototype value for an n = ~ Butler.

worth Biter with unequal terminations: R. = 50 ohms. Rio = 100 ohms.

Solution

Normalizing the two tenninations for Rio = 1 ohm will yield a value of R. = 0.5. Reading down (rom the: top oC Table 3-2, for an n =4 low-pus prototype value. we see that there is no R./R Io =0.5 ratio listed. <AIr second choice then, is to take the vaJue of RJR. = 2, :lnd read up Cro~ the bottom of the table while using the schematic below the table :u the (onn for the low-pus prototype values. This approach results in the )ow-pau prototype cil"C'lit oC Fig. 3-13.

\....\ L"\. 0.%18 0.8U

fig. 3-13. Low-pus prototype circuit (,..r Example 3·2,

Obviously, all possible ratios of source to load reo sistance could not possibly fit on a chart of this size. This, of course, leaves the potential problem of not being able to find the ratio that you need for ::I p:H­ti-:ular design task. The solution to this dilemma is to simply choose a ratio which most closely matches the ratio you need to complete the design. FOl ratios of 100: 1 or so, the best resulu are obtained if you assume this value to be 30 high for practical purposes as to be infinite. Since, in these instances, you arc only approximating the ratio of source to load resis· tance, the filter derived will only approximate the reo sponse that was originally intended. This i" usually not too much of a problem.

The Chebyshev Response

The Chebyshev Slter is a high-Q filter that is usecJ when: (1) a steeper initial descent into the stop,band is required, and (2) the passband response LS. no longer required to be flat. \Vith this type of reqUIre­ment, ripple C:ln be allowed in the pass band. As. m?re ripple is introduced, the initial slope at the beginnIng of the stopband is incre:lsed and produces a mo~e rectangular :lttenuation curve when compared. to t .e rounded Butterworth response. This compa(1s6~n LS

made in Fig. 3-14. Both curves are [or n:::.3 bterd The Chebyshev response shown ha..s 3 dB of passb :1nU ripple and produces a 10 dB improvem~t in stop ~ attenuation over the Butterworth Slter. \..,) r

I

RF CIR~ DESICN

Table 3-2B. Butterworth Low-Pass Protot)'pc Element Values

n Rs/RL C 1 L:! C, L4 C~ L II C; 5 0.900 0.·442 1.027 1.910 1.756 1.389

0.800 0.470 0.8B6 2.061 1.544 1.738 0.700 0.517 0.731 2.285 1.333 2.108 0.600 0.586 0.609 2.600 1.126 2.552 0.500 0.686 0.496 3.051 0.924 3.133 00400 0.838 0.388 3.736 0.727 3.965 0.300 1.094 0.285 4.884 0.537 5.307 0.200 1.608 0.186 7.185 0.352 7.935 0.100 3.512 0.091 14.095 0.173 15.710

GO 1.545 1.694 1..382 0.894 0.309 6 1.111 0.289 1.040 1.322 2.054 1.744 1.335

1.250 0.245 1.116 1.126 2.239 1.550 1.688 1.429 0.207 1.236 0.957 2.499 1.346 2.062 1.667 0.173 1.407 0.801 2.858 1.143 2.509 2.000 0.141 1.653 0.654 3.'359 0.942 3.094 2.500 0.111 . 2.028 0.514 4;141 0.745 3.931 ::1.333 0.082 2.656 0.379 5.433 0.552 5.280 5.000 0.054 3.917 0.248 8.020 0.363 7.922

10.00<i 0.026 7.705 0.122 15.786 0.179 15.738 CIQ 1.553 1.759 1.553 1.202 0.758 0.259

1 0.900 0.299 0'.711 1.404 .1..489 2.125 1.727 1.296 0.800 0.322 0.606 1.517 1.278 2.334 1.546 1.652­0.700 0.357 0.515 1.688 1.091 2.618 1.350 2.028 0.600 0.408 0.432 1.928 0.917 3.005 1.150 2.4;7 0.500 OABO 0.354 2.273 0.751 3.553 0.9S1 3.004 00400 0.590 0.278 2.795 0.592 4.380 0.754 3.904 0.300 0.775 0.206 3.671 0.437 5.761 0.560 5.2.58 0.200 1.145 0.135 5.427 0.287 8.526 0.369 7.908

.---0.100 2.257 0.007 10.700 0.142 16.822 0.182 15.748 GO ' l.sse i,799 1.659 1.397 1.055 0.656 0.223

n Rz/R, L 1 C2 L 3 C. L~ C. L l

~

II l) ~ 4

1",. attenuation of a Chebyshev filter can be found making a few simple but tiresome calculations, and

nbe expressed as:

~. =10 log [1 + (2C.2 (:) '] (Eq. 3-7)

ere,

C.2 (:J' is the' Chebyshev polynomial to the order

n evaluated at (:J', ~ Chebyshev polyno~ials for the first seven orders

arc given in Ta.ble 3-3. The parameter f is given by:

where, Rsa is the passband ripple in decibels.

Note that (:c)' is not the same as (:c). The quan­'0.

.... tity (:J' can be found by defining another parameter:

B =*cosh -I (~) ( Eq. 3-9)

I

..0

Butterworth Rnpo~

Fig. J·14. Comparison of three-element Chebyshev and Butterworth responses.

Table J·3. Chebyshev Polynomials to the Order n

Chebyshev Polynomial

w "

2 2C::)2 ­ 1

3 4 (c::) 3_~ ~_C:)

4 8(:) 4 _ 8(,:)2 + 1 5 16C::)~ - ~!O(~:)3 + 5C::) 6 32(~"J8 - 48(:.r + 18(:,Y~ - 1 7 64(:.) T - 112C:)' + 56(:J 3

- 7(:J

where, n == the order of the filter, ( = the parameter defined in Equation 3-8, cosh -I = the inverse hyperbolic cosine of the quan­

tity in parentheses.

Finally, we have:

(Eq.3·10)

where,

(:J =tho ratio of tho frequenc}J of interest to the

cutoff frequency. cosh =the hyperbolic cosine. ., If vour calculator does not have hyperboltc and m-

verse' hyperbolic functions. they can be manually de­termined from the following relations:

acosh x = O.5(~" + e- )

and cosh -IX =In(x::~)

The preceding equations yield families of attenua­tion curves, each classified according to the amount of

51

.ripple -aHowed in the passband. Several of these fami. lies of curves are shown in Figs. 3-15 through J.1R, and include O.Ol-dB, O.l-dB, O.S-dB. and 1.0-dB ripple. Each curve begins at wI We = 1, which is the normal. ized cutoff, or 3·dB frequency. The passband rippie is. therefore, not shown,

If other families of attenuation curves are needed with different values of passband ripple, the preceding Chebyshev equations can be used to derive them. The problem in Example 3-3 illustrates this.

Obviously, performing the calculations of E:CJmple 3-3 for various values of wI We. ripple. and BIter order is a very time-consuming chore unless a programmable calculator or computer is available to do most of the work for you.

The low·pass prototype element values correspond­ing to the Chebyshev responses of Figs. 3-15 through

.. 3·18 are given in Tables 3-4 through 3-7. Note that the Chebyshev prototype values could not be separated into two distinct sets of tables covering the equal and

0

11

14

J6 iii ~ 43

·ic

~ ::t

~ n ~

3<4

96

108

1.0 2. .5 J.D J.5 ,(.0 S 7 3 9 10

f,~ue"cy lUtio ((/ reI

Fig. 3-15. Attenuation characteristics for a Chebyshe-; 6lter with O.Ol-dD ripple.

0

12

24

36 iii ~ 43 c: .~ 60

::II

E 72 :c

&4

96

108

1.S 1.0 1.5 J.O J ..5 4.0

"t'qu• ...". 'ht'o ('lfe'

. . r ChebyshevFig. :J-16. Attenuation charactenstics or a .

filter with O.l·dB ripple.

-'

52

1.S 2.0 2.5 3.0 3.5 (.0 5 6 7 8 9 10

frequfncy Ratlo (lIf,1

Fia. 3-17. Attenuation charactcrUtics for a Chebyshev ii1ter with O.S-dB ripple.

nequal terminat;ion cases, as was done Eor the But­prototypes. This is because the even order

2, 4, 6, ... ) Chebyshev BIters .cannot have equal The source al1d load must always be

ilferent for proper operation as shown in the tables.

n=

[."\AMPLE 3-3 Find the attenuation of • 4-element, Z.5-dB ripple, low­

pus Chebyshev BIter at CAI/"', = 2.::;. ~ /,0 "

Soliltion 1., -:;. ij)

First evaluate the parameter:

( = " 10~·~/10 - 1

== 0.882

~ut. hnd B.

1 'l/Cv'B= V. [ cosh- J (0.882J.::r" i = 0.1279 ~/uJ L.-

Ith... (..I ..,)' is: /' /" ~ t (w/~.)' =2.5 cosh .1279f. ~ = 2.5204

fina11 Y• we evaluate the fourth order (n == ") Chebyshev POlynomial .t (w/w.)' = 2.52.I c.(:J =8 (:.r - 8 (:.)2 + 1

= 8( 2.5204)· - 8( 2.5204)2 + 1

=273.05 -' .~ .

We can now ev&1uate the nnal equation. ) I

2A.. c 10 loglo [ 1 + (2c. (:.)'] /

=10 log" {l + (0.882)2(273.05)2)

=47.63 dB

'Thus, at an CfI/W. of 2.5. you can expect 47.63 dB of arten­Ilition for this filter.

RF Cmcurr DESIGN

0

12

2(

36 = :!. 48 c .i 1 60 c .! 72 <

&4

90

108

1.5 2.0 2.5 3.0 3.5 4.0 6 B 9 10

Fii. 3-18. Attenuation chuacterbtics for a Chebyshev li.Iter wJth l-dD ripple.

The rules used for interpreting the Butterworth ta­bles apply here also. The schematic sho\1, n ibove the table is used, and the element designators are read down from the top, when the ratio Rs/RL is calculated as a design criteria. Alternatel)', with RL/Rs calcula­tions, use the schematic given bel:"lw L1e table and .cad the element .designatcrs upwards from the bottom of the table. Example 34 is a practice problem for use in understanding the procedure.

EXAl\fPLE 3-4 Find the low-pus prototype ~alues f,,( an n = 5. O.I-d.B

ripple, Chebyshev Siter II the source resistance you are de­si~ln~ for is 50 ohms and the load resist;l.ncc Is 250 ohms.

Solution

Nonnaliution 01 the source and load resiston yields an R./RL = 0.2. A look at Table 3-5. for a O.I-dB ripple Biter with an n = 5 tUld an RalRI. = 0.2, yield.s the' circuit values shown in Fig. 3-19.

f,~"ency R..tio (11f t )

Fig. 3-19. Low-pass pro~otype dre:uit for Example 3-4..

It should be mentioned here that equations could have been presented in this section for deriving the element values for the Chebyshev low-pass prototypes. The equations are extremely long and tediOUS, how­ever, and there would be little to be gained from their presentation.

53 FILTER DESIGN

T:lble 3-4A. Chebyshev Low-P:l.sS Element Values foc O~O I-dD Ripple.

n R!f/Rc. C, L"! 1.101 1.347 1.483 1.111 1.247 1.595 1.250 0.943 1.997 1.-129 0.i59 2.344 1.667 0.609 2.750 :!.OOO 0,479 3.277 2.500 0.303 -1.033 3.333 0.259 5.255 5.000 0.164 7.6:50

to.OOO 0.078 1·1.749 :c 1.412 0.742

J 1.000 1.181 1.8:21 1.181 0.900 1.092 1.660 1.480 0.800 1.097 1.443 1.806 0.700 1.160 1.228 2.165 0.600 1.27.j 1.024 2.598 0.500 1.452 0.829 3.164 0.400 1.734 0.645 3.974 0.300 2.216 0.470 5.280 0.200 3.193 0.305 7.834 0.100 6.141 0.148 15.390

:JO 1.501 1.433 0.591 1.100 0.950 1.938 1.761 1.016 1.111 0.854 1.946 1.744 1.165 1.250 0.618 2.075 1.542 1.617 1.429 0.49S 2.279 1.331 2.008 1.667 0.398 2.571 1.128 2.461 2.000 0.316 2.994 0.926 3.045 2.500 0.242 3.61'1 0.129 3.875 3.333 0.174 4.727 0.538 5.209 5.000 0.112- 6.910 0.352 1.813

10.000 0.054 13.469 0.173 15.510 00 1.529 1.691 1.312 0.523

n Rc.lR, L, C2

L, L]

The Bessel Filter

The initial stopband attenuation of the Bessel filter is very poor and can be approximated by:

AdD =3(:.J2

(Eq. 3-11) ihis B'Cpreuion, however, is not very o.ccuro.te o.bove al\ wI W

f that is cqual to about 2. For valuc! of wI We

g:eater than 2, a straight-line approximation of 6 dB

per octave per element can be made. This yields the family of curves shown in _Fig. 3-20.

Rut why would anyone deliberately design a filter with very poor initial stopband attenuation charJcter. istics? The Bessel filter was originally optimiz~d to obtain a maximally flat group delay or linear phase characteristic in the filter's passband. Thus, selectivity or stopband attenuation is not a primary concern when dealing with the Bessel filter. In high- and medium.Q filters, such as the Chebyshev and Butterworth filters, the phase response is extremely nonlinear over the filter's passband. This phase nonlinearity results in distortion of wideband signals due to the Widely varying time delays associated with the different spectral components of the signal. Bessel filters, on the other hand, with their maximally Aat (constant) group delay are able to pass wideband signals with a minimum of distortion, while still providing some se lectivi ty.

The low-pass prototype element values for the Bes­sel filter are given in Table 3-8. Table 3-8 tabulates the prototi'pe element values for various rntios of source to load resistance,

FREQUENCY AND IlvlPEDANCE SCALING

Once you specify the BIter, choose the appropriate . attenuation response, and writ~ down the low-pass prototype values, the ne:ct step is to transform the prototype circuit into a usable filter. Remember, the cutoff frequency of the prototype circuit is 0.159 Hz (w =1 radl sec), and it operates bet-....een a source and load resistance that are normalized so that Rr.. = 1 ohm.

20 I------+-~~-.d--..::_..,.:::::-_+_-t--t_.....,

eol------4----+--t--t----f--\

~60I-------+---_t_-_+-'\t""~~

O,...------r-----r--"""-T---,-.--r-~

~ 2­c o

:2 II: .!

<

100 ~----+,..--+-__r-+--t--I~

1201L-------lL--JJ:--.J.,,--:-~--~_:10

Fr..qu~"",' (Ilrc'

Fig. J-20. Attenuation characteristic, o( Bessel filten.

"-:

RF Cmcurr DESIC

Table 3·4.B. Chebyshev Low·Pass Element Values for O.Ol.dB Ripple

n

5

6

7

n

Rl'/R L

1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

00

1.101 1.111 1.250 1.4~H)

1.667 2.000 2.500 3.333 5.000

10.000 ~

1.000. 0.900 0.800 0.700 0.600 0.500 00400 0.300 0.200 0.100

00

C1

0.977 0.880 0.877 0.926 1.019 1.166 1.398 1.797 2.604 5.041 1.5-47 0.851 0.760 0.545 0.~.:l6 .. 0.351 0.279 0.21-4 0.155 0.100 0.0:8 1.551 0.913 0.816 0.811 0.857 0.943 1.080 1.297 1.669 2.242 4.701 1.559

L)

~ 1.685 1,456 1.235 1.040 0.863 0.699 0.544 0.398 0.259 0.127 1.795 1.796 1.782 1.884 2.038 2.298 2.678 3.261 4.245 6.22.3

12.171 1.847 1.595 1.362, 1.150 0.967 0.803 0.650 0.507 0.372 0.242 0.119 1.B67

C:!

II

II ~

C3

2.037 2.174 2.379 2.658 3.041 . 3.584 4.403 ~.772

8.~U

16.741 1.645 1.841 1.775 1,489 1.288 i.061 0.86; 0.682 0.S03 0.330 0.162 1.790 2.002 2.089 2.262 2.516 2.872 3.382 4.1SS 5AS4 8.05;

15.B72 1.B66

L3

l)

t.

L. 1.685 1.641 1,499 1.32.3 1.135 0.942 0.749 0.557 0.368 0.182 1.2.37 2.021 2.09<4 2.403 2.735 3.167 3.768 4.687 6.163 9.151

18.1OS 1.598 1.870 1.722 1.525 1.323 1.124 0.928 0.73: 0.546 0.360 0.178 1.765

C,

l~

Rl

C~

0.977 1.27<4 1.607 1.977 2.4z..c 3.009 3.845 5.193 7.828

15.613 0.488 1.631 1.638 1.507 1.332 1.145 0.S.:i4 0.761 0.568 0.376 0."187 1.190 2.002 ~.202

2.465 2.802 3.250 3.875 4.812 6.370 9.484

18.818 1.563

L,

l,

I.,. ,

0.937 1.053 1.504 1.899 ~.357

2.948 3.790 5.143 ;.785

15.595 0.469 1.595 '.581 1.464 1.307 1.131 0.947 0.758 0.568 0.378 0.188 1.161

. C.

Cr

0.913 1.206 , ­1.538 1.910 2.359 2.948 3.790 5.148 7.802

15.652 0..456

LT

~

RLI

The transformation is affected through the following formulas: .

(Eq.3-12)

and

(Eq. 3-13)

Where, C = the final capacitor value,

L = the final inductor value, C. =a low-pass prototype element value, L. = a low-pass prototype element valu~, R = the final load resistor value, f t = the £inal cutoff frequency_

The nonnalized low-pass prototype source resistor must also be transformed to its fin~l value by multi­plying it by the £ina] value of the load resistor (Ex· ample 3-5). Thus, the ratio of the two always remains the same.

55 fILTER DESIGN

Table 3-SA. Chebyshev Low-Pass Proto~pe Element Values for O.I·dB Ripple

n R.,/Rc. Ct L'l

J

2

4

1.355 1.429 1.667 2.000 2.500 J.JJJ 5.000

10.000 ~

1.000 0.900 0.800 O.iOO 0.600 0.500 0.400 0.300 0.200 0.100

CIO

1.355 1.429 1.667 2.000 2.500 3.333 5.000

10.000 CIO

1.209 0.977 ·0.733 0.560 0.417 0.293 0.184 0.087 1.391 1..j33 1.426 1.451 1.521 1.648 1.853 2.18d 2.763 3.942 7.512 1.513

0.992 0.779 0.5i6 0,440 0.329 0.233 0.148 0.070 1.511

I

1.638 1.982 2.489 3.054 3.827 5.050 7.426

14.433 ,0.BI9

1.594 1.494 1.356 1.193 1.017 0.838 0.660 0,486 0.317 0.155 1.510

2.148 2.348 2.730 3.227 3.961 5.178 7.607

14.887­1.768

1.433 1.622 1.871 2.190 2.603 3.159 3.968 5.279 7.850

15.':166 0.716

1.585 1.429 1.185 0.967 0.760 0.560 0.367 0.180 1.455

1.341 l.iOO 2.243 2.856 3.698 5.030 7.614

15.230 0.6":3

n RdRs L t C~ Ls c~

l

L, L)

The process for designing a low~-p·a:3s filter is n very simple one which involves the following procedure:

1. D~fine the response you need by specifying the re­quired attenuation characteristics at selected fre­quencies.

2. Normalize the frequencies of interest by dividing them by the cutoff frequency of the filter. This step forces your data to be in the same form as that of the 3.ttenuation curves of this chapter, where the

.3·dB point on the curve is:

f-=1 f~

EXAMPLE 3·5

S~·aJe the low-pass prototype ~alues of Fi't. 3-19 (F.r:lm. pIe J-t) tu :l cutoff frequency of 50 MHz .:lnd a load resis­t:lnce of 250 ohms.

Solution

Use Equacions 3·12 wd 3-13 to scale each componencas follows:

~ ~",-'

C _ 3.546 (1/ .

'-21r(50X108)(2S0)~\ \,..r.:-,=45 e!-, ~ ~ e.

r _ 9.127 I

..... - 27T( 50 X 10 11 )( 2SO)

=116 pF 7.889

C. = 2"( 50 X 1011 )( 250) =100 pF

L. _ (250) (0.295) - 27T( 50 X 108 )

=235 nH L _ (250)(0.366) 1- 27T(5O X 10") =291 nf!

. Th~ source resistance Is scaled by multiplyinl its normal ized value by the I1nal ~aluc of the load resistor.--- - ~'

RI," •• " =O.2( 250) -' ,

=50 ohms

The fin31 circuit appears in Fig. 3·21.

Fig. 3·21. Low-pus Slter circuit for Example 3-5.

3. Determine the maximum amount of ripple that you can allow in the passband. Remember, the greater the amount of ripple allowed, the mor.c selective the filter is: Higher value~ of ripple may a110w you to elimin~te a few components.

4, Match the normalized attenuation characteristics ( Steps 1 and 2) with the attenuation curves piG­vided in this chapter. Allow yourself a small "fudge­factor" for good measure. This step reveals the mini· mum number of circuit elements that you can get away with-given a certain filter type.

5. Find the lo~v-pass prototype values in the tables.

O. Scale all clements to the frequency arid impedance of the Bnal design.

E:<ample 3-6 diagrams the process of designing a low­pass filter using the preceding steps.

I

55 RF Cmcurr D£SICN

"Table J.SB. ~ow'PassPrototype Eltment Values for O.I·dB' Ripple

L1 l. 4

RL

n Rs/RL •C1 L 2 C3 L. C$ La C, 5 1.000 1.301 1.556 2.241 1.556 1.301

0.900 1.285 1,433 2.380 10488 1.'188 0.800 1.300 1.282 2.582 1:382 ' 1.7380.700 1.358 1.117 2.888 1.24-4 2.062 0.600 1.470 0.947 3.269 1.085 2,48-(0.500 1.654 0.778 3.845 0.913 3.055 0..400 1.954 0.612 4.720 0.733 3.886 0.300 2.477 0.451 6.196 0.550 5.237

~ 0.200 ----3-.546 0.295 9.127 0.386 7.889 0.100 6.787 0.115 17.957 0.182 15.745

CIO 1.561 1.807 1.766 1.417 0.651' 6 1.355 0.942 2.080 1.659 2.2-47 1.53-4 1.277

1.429 0.735 2.z.49 1.454 2.54. 1.405 1.629 1.667 0.542 2.600 1.183 3.06-4 1.185" 2.17t 2.000 0.414 3.068 0.958 3.712 0.979 ~.79"2.S(j() 0.310 3.765 0.749 4.651 0.778 3.645 3.a33 0.220 4.927 0.551 6.195 0.580 4.996 ..5.000 0.139 7.250 0.3ft 9.261 0.384. 7.618

10.000 0.067 H.2ZC 0.178 18.427 0.190 15.350 CIO . 1.53-4 1.884 1.831 1.749 1.39. 0.638

7 1.000 1.262 1.520 2.239 1.680 1.5202.~9 1.2.62 0.900 1.242 1.395 2.361 1.578 2.397 1.459 1.447 0.800 1.255 1.2-45 2.54R 1.443 2.624 1.362 1.697 0.700 1.310 1.083 2.819 1.283 2.942 1.233 2.021 0.600 1.417 0.917 3.205 1.200 3.38. 1.081 2.H4 0.500 1.595 0.753 3.764 u.928 4.015 0.914 3.018 00400 1.885 0.593 4.618 0.742 4.970 0.738 3.855 0.300 2.392 0.431 6.054 0.556 6.569 0.557 5.211 0.200 3..428 '0.286 8.937 0.369 9.770 0.372 7.890 0.100 6.570 0.141 17.603 0.1a.. 19.376 0.186 15.813

co 1.575 1.858 1.921 1.827 1.734 1.379 0.631

n L] C2 L 3 C 4 L$ Ce L,

L, l) ls 4fL L,/,t1 ,

that at an I!f~ of 1/3 (or. fe/f =3) a 5·element, 0.1-_ ~ dB-ripple Chebyshev high-pass filter will also produceHIGH-PASS FILTER DESIGN an attenuation of GO dB .. This is obviously more con·

nee you have learned the mechanics of low-pass venient than baving to refer to more than one set of rdesign. high-pass design becomes a snap. You can curves. .

,',all of the attenuation response curves presented. After finding the response which satisfies all of the far. for the low-pass filters by simply inverting re'quirements. the next step. is to simply refer to the

(lfe axis. For instance, a 5·element, O.l-dB-ripple tables of low-pass prototype values and COP)' do".'n ebyshev low--pass filter will produce an attenuation the prototype values that are called for. High.pass val. ~bout60 dB at an [He of 3 (Fig. 3·16). If yOll were ues for the elements are then obtained directly from king instead with a high.pass filter of the same the low~pass prototype values as follows (refer to and type, you could still ,use Fig. 3-16 to tell you Fig. 3-24): .

•

q:'., (

fa:n:I\ DESIC~ 57

EXAMPLE :J-6 /' Design a ~-pas.5 filter to meet the follQwing spedfica­

tions: f. = 35 MHz. Res~onse greater than 60 dB d.own at 105 M~·z, Ma.:JtImaJly Bat p3..5sband-no ripple, ~ R. = 50 ohms. ; Rio = 500 ohms.

Solution

The need for a maximally Rat passband automatically in­dicates that the design must be a Butterworth response. The nrst step in the design process is to normalize every­thing. Thus.

~=~ " ....~. Rt. 500 q ~'"

=0.1 ~el(t. normalize the frequencies of Interest so thst they may be found in the graph of Fig. 3-9. Thus. we have:

f..... _ 105 MHz _l. _. _f, •• - J5 MHz -- J__

= J 5e We n~xt look at Fig. 3-9 and find a response that is down

at least 60 dB at a frequencyrallo of fIf. =3. Fig. :3-9 Sn­dicates that it will take a minimum of 7 eISriients--to-provide the attenuation specified. Rete: ing to the catalog of Butter­worth low-pa.ss p,-ototype vaJues given in Table 3·2 yields the prototype circuit of Fi~. 3-22.

0.067 0.142

L! ~

\.000

Fig. 3-22~'; Low-pass prototype circuit lor Example 3-6.

We then scale these value3 using Equation, 3-12 and 3-13. The first two value3 are worked out lor you.

e 2.257 , = 211'( 35 X 10~ )500 = 21 pF

J _ (500)(0.067) .A -2"(35 X 10e)

= 152 nH SimiJarly,

Co = 97 pF, e. = 1SJ pF. e =143 pr, t, = 323 nH. 1... =414 nH. R. =50 ohms, Ih. = 500 ohms.

The 6nal circuit is shown 'i~ Fig. 3-23.

''''-..

Table' :J·6A. Chebyshev- Low-PaH Prototype Element

Values for O.S-dB Ripple

'

n R.,/R[, C, L~ C3 L i

2 1.984 0.9&3 1.950 2.000 0.909 2.103 2.500 0.554 3.165 :3.333 0.375 4.411 5.000 0.228 6.700

10.000 0.105 13.322 00 1.307 0.975

:3 1.000 1.864 1.280 1.83-1 0.900 1.918 1.209 2.026 0.800 l.GG7 1.1~0 '2,237 0.700 2.114 1.015 2.511 0.500 2.557 0.759 3.436 0.400 2.985 0.615 4.242 0.300 3.729 0.463 5.575 0.200 5.254 0.309 8.225 0.100 9.890 0.153 16.118

00 1.572 1.518 0.932

/.f'"1 1.98~ 0.920 2.586 1.304 1.828l/ 2.000 0.845 2.720 1.238 1.985

_- 2.500 0.516 3.766 0.869 3.121 3.333 0.:344 5.120 0.621 4.480 5.000 0.210 7.708 OAOO 6.987

10.000 0.098 15.352 0.194 14.2.6'2 00 1.436 1.889 1.521 0.913

n Rc./R., L~ C~ ~ C4

Lt L,

Rl

Simply replace each filter element with an element of the opposite type and with a re­ciprocal value. Thus, L t of Fig. 3-24B is equal to lIe l of Fig. 3-2dA. Likewise, C'1 =l/I.::, and La = lIes.

Stated another way. if the low-pass prototype indi­cates a capacitor of 1.181 farads, then, use an inductor with a value of .1/1.181 =0.847 henry, in.>tc::d, for 3­

high-pass design. However, the source and load re­sistors should not be altered.

The tTansfonnation process results in an attenuation characteristic for the high-pass 5lter that is an eX:l.c, mirror image of the low-pass a.ttenuAtion charact-::ns­tic. The ripple, if there is any, remains the same and the magnitude of the slope of the ,topbnnd (or pass­

58 nF CIRCUIT DESICr-:

Table 3-6B. Chebyshev Low-Pass Prototype Element Values for O.S-dB Ripple

Lz L. 4

• RL

n Rs/Rz, C1 ~ C3 L~ C:. L6 C~ 5 1.000 1.807 1.303 2.691 1.3Q3 l.807

0.900 1.85~ 1.222 2.849 1.238 1.970 0.800 1.926 1.126 3.060 1.157 2.185 0.700 2.035 1.015 3.353 1.058 2..470 0.600 2.200 0.890 3.765 0.942 2.861 0.500 2.-{57 0.754 4.367 0.810 3.-{1-4 0..400 2.870 0.609 5.296 0.66-4 4.245 0.300 3.588 0..459 6.871 0.508 5.625 0.200 5.064 0.306 10.054 0.343 8.367 o.roo 9.556 0.153 19.647 0.173 16.57-{

00 1.630 1.7-40 1.922 1.51-4 0.903 . 1.984 0.9056 2.577 1.368 2.713 1.299 1.796

2.000 0.830 2.704 1.291 2.872 1.237 1.956 2.500 0.506 3.722 0.890 0.881".J09 3.103 3.333 0.337 5.055 0.632 5.699 0.635 -4.481 5.000 0.206 7.615 0..406 8.732 0.412 7.031

10.000 0.096 15.186 0.197 17.681 0.202 1-4..433 7 1.000 1.790 1.296 2.718 1.385 1.2962.718 1.790

O.~OO 1.['.35 ., L~15 2.869 1.308 2.883 1.234 1.953 0.800 1.905 1.118 3.076 1.215 3.l07 1.155 2.168 0.700 2.011 1.007 3.364 1.105 3.416 1.058 2.455 0.600 2.174 0.882 3.i72 0.979 3.852 0.944 2.848 0.500 2.428 0.747 4.370 0.838 2.~a9 0.814 3.405 0.400 2.835 0.60~ 5.295 0.685 5.470 0.609 4.243 0.300 3.546 0.455 6.86i 0.522 7.13-4 0.513 5.635 0.200 5.007 0.303 10.049 0.352 10.496 0.348 8.404 0.100 9.456 0.151 19.649 0.178 20.631 0.176 16.665

co 1.646 1.777 2.03l 1.789 1.924 1.503 0.895

n R,k/R$ L1 C2 L3 C.. La Ce L j

L) l) L~ l7

2nd) skirts remains the same. Example 3-7 illustrates e design of high-pass filters. A closer look at the filter designed in Example 3-7

.t·-eals that it is symmetric. Indeed, all filters given Or the equal tennination class are symmetric. The qual tennination class of filter thus vields a circuit

!hat is easier to design (fewer calcul~tions) and, in most cases, cheaper to build for a high-volume product, due to the number of equal valued components.

THE DUAL NET\VORK

Thus far, we have been referring to the grDup of Iow·p.ass prototype element value tables presented ~lld. then, we choose the schematic that is located

either above or below the tables for the form of the filter that we are designing. depending on the value of RLlRs. Either fonn of the filter will produce exactly the same nttenuation, phase. and group-delay charac­teristics, and each form is called the dual of the other.

An)' filter network in a ladder arrangement. such as the ones presented in this chapter, can be changed into its dual form by application of the following rules:

1. Change all inductors to capacitors, and vice-versa, without changing element values. Thus, 3 henries becomes 3 farads. .

2. Change all resistances into conductances, and vice­versa, with the value unchanged. Thus. 3 ohms be­comes 3 mhos, or 1h ohm.

•

59

l.a21 Table; 3-7A. Cheby"hev Low-P2.3s Prototype Element

II . Values for l.O-dB Ripple

( A) Low-pou prototype circuit.

( B) Equiual6nC high-pa.u prototype circuit.

Fig. 3-24. Low-pa!s to high-pass filter trans(onni.tfon.

3. ~bange all shuot branches to series branches, and vice-versa.

4. Change all elements in series with each other into _ elements that are in parallel with each other. \). ~hange all voltage sources into current sources, and

Vl~e-versa.

fig. 3-26 shows a ladder network and its dual repre­sentation.

Dual networks are convenient, in the case of equal .tcrmin?tions: jf you desire to change the topology of the filter Wlthout changing the response. It is most often used, as shown in Example 3-7, to eliminate an unnecessary inductor which might have crept into the design through some other transformation' process. Inductors are typically' more lower-Q devices than C:1p:1citors and, therefore, exhibit high~r losses. These losses tcnd to cause insertion loss, in addition tb gen­er:111y degrading the overall performance of the filter. The n1;Jmber of inductors in any network should, therefore, be reduced whenever possible.

A littl~ e:cperimentation with dual networks having unequal tenninations will reveal that you can quickly ~et yourself into trouble if you are not careful. This IS. especially true if the toad and source resistance a.re a design criteria and cannot be changed to suit tIle needs of your Stter. Remember, when the dual of a network with unequal terminations is taken, then, the terminations must, by' deSnition, change value as shown in Fig. 3-26.

DANDPASS FILTER DESIGN

The low-pass prototype circuits and response curves gIven In this chapter cnn also be used In the design or bandpass 6Iters. This is done through 3 simple

n R!/Rc. C\ L 2

2 3.000 0.572 3.132 ".000 0.36.5 4.600 8.000 0.157 9.658

0:) 1.213 1.109 J 1.000 2.216 1.088 2.216

___0.500 - ­ 4."3i­ 0.817 2~216 0.333 8.647 0.726 2.216 0.250 8.862 0.680 2.216 .0.12.5 17.725 0.612 2.216

eo 1.652 1.460 1.108 .. ~.OOO· 0.653 4.411 0.814 2.~ 4.000 0.452 7.083 0.812 2.8-t8 8.000 0.209 ' 17.164 0.428 3.2.91

co 1.:}50 2.0]0 1.488 1.1 or,

n RL /R 3 L, Co: L 3 C.

transfonnation process similar to what was done in the high-pass case.

The most difficult task awaiting the designer of a bandpass Slter, if the design is to be derived from the low-pass prototype, is in specifying the bandpass at­tenuation characteristics in tenns of the low-pass re­sponse curves. A method for doing this is shown by the curves in Fig. 3-27, As you can see, when :L low­pass design is transformed into a bandpass design, the attenuation bandwidth ratios remain the same. This means that' a low-pass BIter with a 3-dB cutoff frequency, or a balldwidth of 2 kHz, would transform into a bandpass filter with a 3-dB bandwidth of 2 kHz. If the response of the low-pass network were down '30 dB at a frequency or bandwidth of 4 kHz (fHf: = 2), then the response of the bandpa.s~ network would be down 30 dB at a bandwidth of 4 kHz. Thus. the normalized flIe :lXis of the low-pass attenuation curves becomes a ratio of bandwidths rather than fre· quencies, such that:

where, BW := the bnndwidth nt the required vtUue oE at­

tenuation, . B\V = the 3-dB bandwidth of the bandpass 6lter.

e

60 RF Cmcurr DESICN

Table 3·7B. Chebyshev Low-Pass Prototype Element Values for l.O-dB Ripple

1.1 1.. t.

RL

n R[l/RL CJ ~ C3 L. es Lfj C, 5 1.000 2.207 1.128 3.103 1.128 2.207

0.500 4.4104 0.565 4.653 1.128 2.207 0.333 6.622 0.376 '6.205 1.128 2.207 0.250 8.829 0.282 7.756 1.128 2.207

. 0.125 17.657 O.Hl 13.961 1.128 2.207 co 1.721 1.645 2.061 1.493 1.103

6 3.000 0.579 3.873 0.771 4.711 0.969 2.406 .(.000 0.-'&81 5.&H 0,476 7.351 0.8.(9 2.582 8.000 0..2.27 12.310 0.198 16.740 0.726 2.800

co 1.378 . 2.097 1.690 2.074 1.494 . 1.102 7 1.000 2.204 1.131 3.1.(7 1.19~ 3.147 1.131 2.20.(

0.500 4.408 0.566 6.293 0.895 3.147 1.131 2.20-1 0.333 6.812 0.377 9..4'(} 0.796 3.1.(7 1.131 2.204 0.250 8.815 0.283 12.588 0.747 3.147 1.131 2.204 0.125 17.631 0.141 25.175 0.671 3.147 1.131 2.20-1

co 1.7"" 1.677 2.155 1.703 2.079 1.494 1.]02 . n R L/R3 L, C:z L 3 C. Ull C. L,

1.) 1.) It

Often a bandpass response is not specified, as in Example 3-8. Instead, the requirements are often given as attenuation values at specified frequencies as sho~vn by the curve in Fig.. 3-28. In this case, you must transform tbe stated requirements into informa­lion that takes the form of Equation 3-14. As an ex­ample, consider Fig. 3-28. How do we convert the data that is given into the bandwidth ratios we need?

. Before we can answer that, we have to nnd I3 • Use e following method. . The frequency response of a bandpass filter exhibits

~eometric symmetry. That is, it is only symmetIic when lotted on a logarithmic scale. The center frequency f a geometricall}' symmetric filter is given by the ormula:

Eo =Vf.t (Eq.3-15)

here f. and fb are an)' t\~o frequencies (one above .nd One below the passband) having equal attenua· IOn. Therefore, the center frequency of the response

rve shown in Fig. 3-28 must be

£0=V(45)(75) MHT.

=58.1 MHz '\ ~1

t' \-U:C \ \ ' ]-v

We can use Equation 3·15 again to nnd E,. ~ ....~ ~\}j ~ ~.-f\

58.1 =Vf~( 125) 7/ "\ V \...vJ7G~

or, " fa =27 MHz

Now that f., is known, the data of Fig. 3-28 can be put into the form of Equation 3-1~c,

BW.. OdB _ 125 MHz - 21MHz 'BWUB - 15 J'1Hz - 45 MHz

=3.2rl !-"2­To find a low-pass prototype curve that will satisfy these requirements. simpl)' refer to any of the pertinent graphs presented" in this chapter and Snd a response which will provide 40 dB of attenuation at an flfr of 3.27. (A fourth-ord'er or better But~er'Worth filter will do quite nicely. ) ~ ~ ~~ .

The actual transformation from th~ low-pass to the b:mdpass connguration is accomplished b)' resonating each low~pass element with an element of the opposite type and of the same value. All shunt elements of ~he low-pass 'prototype circuit become para)}el-resonapt

•

...---­L.\~ ~ y

./, ,'F r ;,.; I

fo..T£R DESIGN _, I

((>1 ' "~~61

EXAMPLE 3-7 _--_______. Desilpl an LC high-pass filter wIth an(c-. of 6~Hz and

J minimum attenuation of 40 dB at 30 ~The source 1l1d load resistance are equal at ~ Assume that a a.s-dB passband ripple is tolerab~-

I')

\}.. ~ . Solution

First, nonnalize the attenuation reqUirem{na so that the low-pass attenuation curves may be used. "'

f _ 30 MHz r: - 60 MHz

J"Invertin~. we get: =0.5

f. " l' =­Now, select a normalized low-pass filter that olIers at leut ~O-dB attenuation at a ratio of f./f =2. Reference to Fig. 3.17 (attenuation response of O.5-<fB-ripple Chebyshev fil­lers) indicates that a nonnalize~c;hebyshev will pro­vide the needed attenuation. Table 3-1> contains the ele­ment values for the corresponding network; The normalized low-pass prototype circuit is shown in Fig. 3-25A. Note that the schematic below Table 3-68 was chosen as the low-pass prototype circuit rather than the schematic above the table. The reason for doing this will become obvious after the next Slept Keep in mind. however, that the ratio of R,/Rt. is the same as the ratio of Rt.lR., and is unity. Therefore, it do.-s not matlt:r which [onn is used for the protot;'pe circuit. .

Next. transfonn t~e low-pass circuit to a high-pas3 net­work by replacing each inductor with a cap:lcitor, and vice­versa. using reciprocl1l element values :1$ shown in Fig. 3-25B. Note here that had we begun with the low-pass pro­totype circuit shown above Table 3o$B. this tTansfonnation would have yielded a filter containing three inductors rather lha.n the two shown in Fig. 3-25D. The object in any of these filter desigN is to reduce the number of inducton in the 6nal de5ign. More on this later.

The final step in the design proc~s is to scale the net­worle in both impedance and frequency using Equations 3·[2 and 3-13. The first two calculations are done for you.

1 1.807

C, = 217'( 00 X 101S )( 300)

=4.9 pF

EXAMPLE J·8 finJ the Dutterworth low-pass prototype circuit which.

when trans(onned. would satisfy the following bandpass filter requirements:

BW.. ~ == 2 MHz

B\V... ~. = 6 MHz

Solution

Note that -e ue not concemed ",.jlh the center frequency of the bandpass r~sponse just yet. We are only concerned with the relationship belween the :Ibove requirements :lOd

r-"; :J)I.: 1~) (A) N~li%ed lew-pc." filter circuit.

l.J' • J

! i \ I

\ .! ~ ~

:JOO(-dro) L. = 2."( 60 X 10&)

== 611 nH,

The remaining values are:

c. =3.3 pr . c. =4.9 pF L.=BllnH

The Bnal filter circuit Is given in Fig. 3-25C.

(B) High-po.s.s transformation.

(C) FrcquencV and impcdancc·Jcal~dfiller circuit.

Fig. :J-25. High-pass filter dC.1ign for Example J-1.

the low-p:lss response curves. Using Equation 3·14. we have:

B\V f BW. = r: =

BW . B\V ..

B MHz = 2MHz

== 3­

Therefore, tum to the Buttcn¥orth response curves ~ho~~ in Fi~. 3·9 :Ind find a prototype value that wi~l p.roV1de • dn of 3ttenulltion at an flf. ::z :l. 11,,, C'\1f"'VU JnQ'5:ate ~rJ'

"'11 ·d· L needed Jete •element Buucl'Vorth filter WI provl e UH~

ahon.

I

~."-'"---~-----_.. _--- - ----'- --------------- ­

RF Crncurr D£31C~

6 Table 3·BA. Bessel Low·Pass Prototype Element Values

2

5

(A) A representotioe ladder network.

( B) Its dUlll form.

Fig. 3·26. Duality.

(A). Low-p(Jsr prototype responte.

,I I ~

\

i I l I L

( B) Bond"o.ss response.

Fig. 3-27. Low·pass \0 bandpass transformation bandwidths.

I I

I I--_ .. -------~--

Fig 3-28. Typical bandpas1 speciucations.

n

1.000 0.576 1.111 0.508 1.250 0,443 1,429 0.380 1.667 0.319 2.000 0.260 2.500 0.203 3.333 0.1~9

5.000 0.097 10.000 0.0~7

00. 1.362 1.000 0.337 2.203 0.900 0.371 2.375 0.800 0,412 2.587 0.700 0,466 2.858 0.600 0.537 3.216 0.500 0.635 3.714 0.4.00 0.783 "lAS7 0.300 1.028 5.689 0.200 1'.518 8.HO 0.100 2.983 15.'170

00 1.463 0.293 1.000 0.233 1.082 2.240 1.111 0.209 .0.967 2.414 1.250 0.184 0.653 2.630 1,429 0.160 0.741 2.907 1.667 0.136 0.630 3.2i3 2.000 0.112 0.520 3.782 2.500 0.089 0.412 -4.5-!3 3.333 0.066 0.30G 5.805

-5.000 0.0-43 0.201 8.319 10.000 0.021 0.099 15.837

1.501 0.613 0.211

n L J

Fig. 3-29. Low·pass \0 bandpass circuit \ramformalion.

circuits, and all series elements become series-reson:lnt circuits. This process is illustrated in Fig. 3·30.

To complete the design, the transformed Blter is

•

OJ

Table j·8D. Bessel Low-Pass Prototype Element Value,

n

5

6

7

n

RsiRr. 1.000 0.900 0.800 0.700 0.600 0.500 00400 0.300 0.200 0.100

CIO

1.000 1.111 1.250. 1.429 1.667 2.000 2.500 3.J33 5.000

10.000 co

1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

00

R"IRc,

Ct

0.174 0.193 0.215 0.245 0.284 0.338 0,419 0.555 0.825 1.635 1.513

0.137 0.122 0.108 0.094 0.080 0.067 0.053 0.040 0.026 0.013 1.512 0.111 0.122 0.137 0.156 0.182 0.211 0.9.70 0.358 0.534 1.061 1.503

Lt

L'J 0.507 0.454 0.402 0.349 0.298 0.247 0.196 0.146 0.096 0.048 1.023

9-.400 0.443 0.496 0.564 0.655 0.782 0.973 1.289 1.289 3.815 1.0JJ 0.326 0.29'2 0.259 0.226 0.193 0.160 0.127 0.095 0.063 0.031 1.029

C1

l,

C.t 0.804 0.889 0.996 1.132 1.314 1.567 1.946 2.577 3.835 7.604 0.753

0.639 0.573 0.508 0.442 0.378 0.31:l 0.249 0.186 0.123 0.061 0.813

0.525 0.582 6.552­0.743 0.863 1.032 1.285 1.705 2.545 5.002 0.835

L,

lJ

1.111 0.995 0.879 0.764 0.651 0.538 0.427 0.311 0.210 0.104 0.473

0.854 0.946 1.060 1.207 1.402 1.675 2.084 2.763 4.120 8.186 0.607 0.702 0.630 0.5':>9 0.487 0.416 0.346 0.276 0.206 0.137 0.068 0.675

C~

2.258 2.433 2.650 2.927 3.295 3.808 4.573 5.843 8.375

15.949 0.162

1.113 0.995 0.881 0.767 0.653 0.541 0.4~9

0.319 0.211 0.105 0.379

0.869 0.963 1.080 ~.231

1.431 1.711 2.130 2.828 4.221 8.397 0.503

L'l

2.265 2.439 2.65.5 2.~,J3

3.300 3.812 4.577 S.847 8.378

15.951 0.129

1.105 0.990 0.875 0.762 0.649 0.537 0.427 0.318 0.210 0.104 0.Jl1

Gil

2.268 2.440 2.656 2.932 3.298 3.809 4.572 5.838 '8.362 15.917 0.105

then frequency- and impedance-scaled using the fol­I I lowing formul:u. For the .parallel-resonant branches,0r----_. (

-] dB - - - - - - - - - - - - . - - - - - - - - - - - - ­I I (Eq. 3-16) I I I 1 I (Eq. 3-17)

-~~-----------~----I I I I I I and, for the series-re.sonant branches,I I

B (Eq. J.18)C =2mozC,RFig. J..1O. Typical band-rejection filt~r curve". 1

\

----------._----- ­

(Eq. 3-19)

re, in all cases, R:: the Bnalload impedance, B:: the 3-dB bandwidth of the final designL:: the geometric .center frequency of th~ final de­

sign, . ' 1.. = the normalized inductor bandpass clement

values, .....----::--­C. = the normalized capacitor bandpass element

\-alues.

Example 3-9 furnishes one final example of the pro­ure for designing a bandpass Slter. .

SUM~fARY OF THE BANDPASS FILTER DESIGN PROCEDURE

Transform the bandpass requirements into an equivalent low-pass requirement using Eq~ation

3-14. Refer to the low-pass attenuation curves provided in order to find a response that meets the requirements iStep l. md the corresponding low-pass prototype and 'rite it down.

~

lransfonn the low-pass network into a bandpass nfiguration. Ie the bandpass configuration in both impedance

nd frequency using Equations 3-16 through 3-19.

BAND-REJECTION FILTER DESIGN

nd-rejectio'rJ filters are very similar in design ap­ch to the bandpass filter of the last section. ani}', ,is case, we want to reject a certain group of fre­cies a:; ::hcwn by the curves ,in Fig. 3-30. . e band-reject £ilter lends itself well to the low­prototype design approach using the same pToce­

as were used for the bandpass design. First, the bandstop requirements in terms of the low.

attenuation curves. This is done by using the Ie of Equation 3-14. Thus, referring to Fig. 3-30, ve.

B\\'r f.-f t

B\V = £3 - h

lets the attenuation characteristic that is needed ~Ilows you to read directly off the low-pass at­lion curves by substituting B\\'c/B\\' {or felf on ,onna1ized frequency axis. Once the number of

ts that are required in the low-pass prototype :\ is determined. the low-pass network is trans­d into a band-reject configuration as follows:

tach shunt element in the low-pas~ prototype :cuit is replaced b~' a shunt serics-rcsof1(Int IrCuit. and each series-element is replaced by leries parallel-resonant circuit.

RF Crncurr DL:Slc1\

EXAMPLE 3-9 Design a bandpass filter with the folJo\dng requirements:

f. =75 MHz B\\'... =7 MHz B\\'~. =35 MHz

Solution

Using Equation 3-1(:

B\\'..... 35 BW.... =,

=5"

Substitute this value for f/f. In the low-pass allenuation curves for the l-dB-ripple Chebyshe\' respome ~hown in Fig. 3-18. This reveals that a 3-element filter will.pr{)vtoe about5O'CIB of .Attenuation at an flI. :;:5 which is mor~ than adequat'e(Tne COfTcspondins 'cle~erit values for this Riter can be found in Table 3-1 lor an R./R~ =0.5. and an n = 3. This yields the low-pass prototype circuit oC fig. 3-32A which is transformed into the bandpass prototype circuit of Fig. 3-32B. Fin~II)', using Equation5 3-16 throu ~h 3-19, we obtain the nnal circuit that is shown in Fig. 3·32C. The calculation.. follow. Using E.Quatio05 3-16 and 3-1i:

4..43r -

Similuir . C. =504 pr L.. =8.93 nH

f- J: C. =217"(100)( 7 X lOG)

'. 1"-1= 1007 fir ~. - '_.~.,

( 10011~ x 10e)-L., = 2".( 75 x 10d ).2( ~A31 L.

=4.47 nH Using Equations 3-18 and 3-19:

Th:<; is showrl in Fig. 3-31. Note th3t both elem<.:ots in each of the resonant circuits have the same normaliz.ed value.

Once the prototype circuit has been transforml:d into its band-reject configuration, it is then scaled in im­pedance and frequency using the following formulas. For all series-resonant circuits:

Fig. 3-3}. L.ow-pass to band-reject transformation.

Passband Ripple = 1 dB L/ R. = SO ohm~ RI. =100 ohms

1000

-pass prototype C rCllit.

0."170.500 0.817

n 1""'r

~I ~."Jl :1 .000

f ( B) Band"a..ss trnnsforTT1nt!on.

150!!

1:007 pF r"" pF

~"'7 nH p'." nH 100 0l' 1 f (C)FillaL circuit rd/It frequenclj and illlTJedance scaled.

fig, 3-.12. BMdp:lsS filter desisn (or Example J-9..

I I-,"I

I,,~trllon Lon!:...-...------

I

Fi~. 3-33. The effect of Bnite-Q elements .. \>.. ~ oX

For all paral

on filter rC3pons~. (

C =-S­21TRB

nn L ::: 21Tf"zL

n

el-resonnnt circuits:

\.}J'N' f~ \

(Eq "-(0) . .J - v~ 7

. ~" (Eq. 3-21)

(Eg. 3-22)

Eq. 3-23)

where. in all C:lse~.

D=the 3-dn bandwidth . .!' =the fillalload resistance, I .. :=: the ~Crrt€"td.c~nt.:::.Jren~~<;n~y, 0

e" = th;; nnrml1lizcd cnpncitor hnnJ-rcject elcm":llt

'·nltle. L" = the normalized inductor bal1d-r~ject element

value.

6.:;

THE EFFECTS OF FINITE Q

Thus for in thi3 chC1pter. we lJa ve :lssumed the in­ductors and cJ.pacitors lIsed in the designs to be ~ossl~ss. Indeed. all of the response curves presented In this chapter :lre bnsed on that assumptioll. B\lt we know from our previous study of Chapters 1 and 1 that even though capacitors C:ln be approximated as having infinite Q. inductors cannot. :lnd the efTects of the finite-Q inductor mu~t be taken into accollnt in Clny filter design.

Thc use f)f finite element Q in " desisn intencied cor loss less elements causes the following unwanted effects (refer to Fig. 3-33) :

1. Insertion loss of the Biter is incrc:lsed whcrc:'lS the nn;-ll stopband llttenu:ltioll does not ch."\l1ge. The relativc :lttenuCltion between the two is decrc:\scd. l~

2. At frcq\lencies ill the vicinity of cutoff (f.o). the l,l~ response becomes more rOllnded and lIstl:dl\" results in nn attenuation gre:lter th:lll the .3 dB t'hat \\'<1S

originall~' intended. \1,1.1. Ripple th:lt was designed into the passb;\lld \\'ill be

I \,f

reduced: If the element Q is sufficiently low, ripple will be totallv eliminated.

4. For band-reject filters. the attel1l1a~iOIl in the stop­band becomes finite. This. couplcd with an incrc:lSc u in passlmnd insertion loss. decre:1Ses the icbth'e

"

Clttenuacioll sisnificantly.

oRegardless of the gloomy rredictions outlined , t

above. however, it is possible to design filters. usill:; the nppronch outlined in this chJpter. th~t vcry closeiy ~Iresemble the ide~l response of CJch network. The ke~' ~,'; is to lise the highest-Q inductors available for thL' ~i !

given task. Tnble 3-9 outlines the recommc~ded ~ir.i­tii

'il mum element-Q requirements for the Rlters presented ,I. in this chapter. K~ep in mind, however. that anytime !'

a low-Q component is lIsed. the nctuai attenuation )~

responsc of thc tletw~rk stray·s from the idenl re,"poll~C to n .degree dependms;..... ut?011 the elerncnt Q. 1~ IS.

therefore. highly recommended thnt yOll make ;t :\ Lhabit to use only the highest-Q components 3vailnbleo

. Table 3·9. Filter Elemcnt.Q ReQuirements

.\l;l1imllm Element () Filter Type Required .

Ol.:s~el

nu tlcr\\'ort h 0,0 lad 0 Cheb~':Ihc\' o.i -dO Ch('b\o'~h~,' O.5~d 0 Cia: b~shc:" todD Chebr~hev

J IS 2-l J9 .;; 75

The insertion loss of the filters presented in this chapter call be calculated in th~ same m::lIlncr.:15 was \Ised in Ch:lptr:r 2. Simply r(!p!;\ce e:lch rC:1ctl\'~ ~Ic­mcnt wHh re.'iistor \'~ijtlCS corre.'irLJndin~ to tne Q of the elclllcnt :Ind. then. e~crcise the v()lt:l~~ di\·isiol. 'rule (rom SO\lrce to 1011d.