Document12

RF and AF Filters 12.1

RF and AFFilters

Chapter 12

This chapter contains basic design in-formation and examples of the most com-mon filters used by radio amateurs. It wasprepared by Reed Fisher, W2CQH, andincludes a number of design approaches,tables and filters by Ed Wetherhold,

W3NQN, and others. The chapter is di-vided into two major sections. The firstsection contains a discussion of filtertheory with some design examples. It in-cludes the tools needed to predict the per-formance of a candidate filter before a

design is started or a commercial unit pur-chased. Extensive references are given forfurther reading and design information.The second section contains a number ofselected practical filter designs for imme-diate construction.

Basic ConceptsA filter is a network that passes signals

of certain frequencies and rejects or at-tenuates those of other frequencies. Theradio art owes its success to effective fil-tering. Filters allow the radio receiver toprovide the listener with only the desiredsignal and reject all others. Conversely,filters allow the radio transmitter to gen-erate only one signal and attenuate othersthat might interfere with other spectrumusers.

The simplified SSB receiver shown inFig 12.1 illustrates the use of several com-mon filters. Three of them are located be-tween the antenna and the speaker. Theyprovide the essential receiver filter func-tions. A preselector filter is placed be-tween the antenna and the first mixer.It passes all frequencies between 3.8 and4.0 MHz with low loss. Other frequencies,such as out-of-band signals, are rejectedto prevent them from overloading the firstmixer (a common problem with shortwavebroadcast stations). The preselector filteris almost always built with LC filter tech-nology.

An intermediate frequency (IF) filteris placed between the first and secondmixers. It is a band-pass filter that passesthe desired SSB signal but rejects allothers. The age of the receiver probably

Fig 12.1 — One-band SSB receiver. At least three filters are used between theantenna and speaker.

determines which of several filter tech-nologies is used. As an example, 50-kHzor 455-kHz LC filters and 455-kHzmechanical filters were used through the1960s. Later model receivers usually usequartz crystal filters with center frequen-cies between 3 and 9 MHz. In all cases, thefilter bandwidth must be less than 3 kHz toeffectively reject adjacent SSB stations.

Finally, a 300-Hz to 3-kHz audio band-pass filter is placed somewhere betweenthe detector and the speaker. It rejectsunwanted products of detection, powersupply hum and noise. Today this audio

filter is usually implemented with activefilter technology.

The complementary SSB transmitterblock diagram is shown in Fig 12.2. Thesame array of filters appear in reverseorder.

First is a 300-Hz to 3-kHz audio filter,which rejects out-of-band audio signalssuch as 60-Hz power supply hum. It isplaced between the microphone and thebalanced mixer.

The IF filter is next. Since the balancedmixer generates both lower and uppersidebands, it is placed at the mixer output

12.2 Chapter 12

to pass only the desired lower (or upper)sideband. In commercial SSB transceiv-ers this filter is usually the same as the IFfilter used in the receive mode.

Finally, a 3.8 to 4.0-MHz band-pass fil-ter is placed between transmit mixer andantenna to reject unwanted frequenciesgenerated by the mixer and prevent themfrom being amplified and transmitted.

This chapter will discuss the four mostcommon types of filters: low-pass, high-pass, band-pass and band-stop. The ideal-ized characteristics of these filters areshown in their most basic form in Fig 12.3.

A low-pass filter permits all frequen-cies below a specified cutoff frequency tobe transmitted with small loss, but will at-tenuate all frequencies above the cutofffrequency. The “cutoff frequency” is usu-ally specified to be that frequency wherethe filter loss is 3 dB.

A high-pass filter has a cutoff frequencyabove which there is small transmissionloss, but below which there is consider-able attenuation. Its behavior is oppositeto that of the low-pass filter.

A band-pass filter passes a selectedband of frequencies with low loss, but at-tenuates frequencies higher and lowerthan the desired passband. The passbandof a filter is the frequency spectrum that isconveyed with small loss. The transfercharacteristic is not necessarily perfectlyuniform in the passband, but the variationsusually are small.

A band-stop filter rejects a selectedband of frequencies, but transmits withlow loss frequencies higher and lower thanthe desired stop band. Its behavior is op-posite to that of the band-pass filter. Thestop band is the frequency spectrum inwhich attenuation is desired. The attenua-tion varies in the stop band rising to highvalues at frequencies far removed from thecutoff frequency.

FILTER FREQUENCY RESPONSEThe purpose of a filter is to pass a de-

sired frequency (or frequency band) and

Fig 12.3 — Idealized filter responses.Note the definition of fc is 3 dB downfrom the break points of the curves.

Fig 12.2 — One-band transmitter. At least three filters are needed to ensure aclean transmitted signal.

Fig 12.4 — A single-stage low-passfilter consists of a series inductor.DC is passed to the load resistorunattenuated. Attenuation increases(and current in the load decreases) asthe frequency increases.

reject all other undesired frequencies. Asimple single-stage low-pass filter isshown in Fig 12.4. The filter consists of aninductor, L. It is placed between the volt-age source eg and load resistance RL. Mostgenerators have an associated “internal”resistance, which is labeled Rg.

When the generator is switched on,power will flow from the generator to theload resistance RL. The purpose of thislow-pass filter is to allow maximum powerflow at low frequencies (below the cutofffrequency) and minimum power flow athigh frequencies. Intuitively, frequencyfiltering is accomplished because the in-ductor has reactance that vanishes at dcbut becomes large at high frequency.Thus, the current, I, flowing through theload resistance, RL, will be maximum atdc and less at higher frequencies.

The mathematical analysis of Fig 12.4 isas follows: For simplicity, let Rg = RL = R.

L

g

X2R

ei

j(1)

whereXL = 2π f Lf = generator frequency.

Power in the load, PL, is:

2L

2

L2

gL

X4R

ReP (2)

Available (maximum) power will bedelivered from the generator when:

XL = 0 and Rg = RL

g

2g

O R4

EP (3)


The filter response is:

powergeneratoravailable

loadtheinpower

P

P

O

L (4)

The filter cutoff frequency, called fc, isthe generator frequency where

L

RforXR2 cL (5)

As an example, suppose Rg = RL =50 Ω and the desired cutoff frequency is4 MHz. Equation 4 states that the cutofffrequency is where the inductive reactanceXL = 100 Ω. At 4 MHz, using the relation-ship XL = 2π f L, L = 4 μH. If this filter isconstructed, its response should follow thecurve in Fig 12.5. Note that the gentlerolloff in response indicates a poor filter.To obtain steeper rolloff a more sophisti-cated filter, containing more reactances,is necessary. Filters are designed for spe-cific value of purely resistive load imped-ance called the terminating resistance.When such a resistance is connected to theoutput terminals of a filter, the impedancelooking into the input terminals will equalthe load resistance throughout most of thepassband. The degree of mismatch acrossthe passband is shown by the SWR scale atthe left-hand side of Fig 12.5. If maximumpower is to be extracted from the genera-tor driving the filter, the generator resis-tance must equal the load resistance. Thiscondition is called a “doubly terminated”filter. Most passive filters, including theLC filters described in this chapter, aredesigned for double termination. If a filter

is not properly terminated, its passbandresponse changes.

Certain classes of filters, called “trans-former filters” or “matching networks” arespecifically designed to work betweenunequal generator and load resistances.Band-pass filters, described later, are eas-ily designed to work between unequal ter-minations.

All passive filters exhibit an undesirednonzero loss in the passband due to un-avoidable resistances associated with thereactances in the ladder network. All fil-ters exhibit undesired transmission in thestop band due to leakage around the filternetwork. This phenomenon is called the“ultimate rejection” of the filter. A typicalhigh-quality filter may exhibit an ultimaterejection of 60 dB.

Band-pass filters perform most of theimportant filtering in a radio receiver andtransmitter. There are several measures oftheir effectiveness or selectivity. Selectiv-ity is a qualitative term that arose in the1930s. It expresses the ability of a filter(or the entire receiver) to reject unwantedadjacent signals. There is no mathemati-cal measure of selectivity.

The term Q is quantitative. A band-passfilter’s quality factor or Q is expressed as Q = (filter center frequency)/(3-dB band-width). Shape factor is another way somefilter vendors specify band-pass filters. Theshape factor is a ratio of two filter band-widths. Generally, it is the ratio (60-dBbandwidth) / (6-dB bandwidth), but somemanufacturers use other bandwidths. Anideal or brick-wall filter would have a shapefactor of 1, but this would require an infi-

nite number of filter elements. The IF filterin a high-quality receiver may have a shapefactor of 2.

POLES AND ZEROSIn equation 1 there is a frequency called

the “pole” frequency that is given by fp = 0.In equation 1 there also exists a fre-

quency where the current i becomes zero.This frequency is called the zero fre-quency and is given by: f0 = infinity. Polesand zeros are intrinsic properties of allnetworks. The poles and zeros of a net-work are related to the values of induc-tances and capacitances in the network.

Poles and zero locations are of interestto the filter theorist because they allowhim to predict the frequency response of aproposed filter. For low-pass and high-pass filters the number of poles equals thenumber of reactances in the filter network.For band-pass and band-stop filters thenumber of poles specified by the filtervendors is usually taken to be half thenumber of reactances.

LC FILTERSPerhaps the most common filter found

in the Amateur Radio station is the induc-tor-capacitor (LC) filter. Historically, theLC filter was the first to be used and thefirst to be analyzed. Many filter synthesistechniques use the LC filter as the math-ematical model.

LC filters are usable from dc to approxi-mately 1 GHz. Parasitic capacitance asso-ciated with the inductors and parasiticinductance associated with the capacitorsmake applications at higher frequenciesimpractical because the filter performancewill change with the physical constructionand therefore is not totally predictable fromthe design equations. Below 50 or 60 Hz,inductance and capacitance values of LCfilters become impractically large.

Mathematically, an LC filter is a linear,lumped-element, passive, reciprocal net-work. Linear means that the ratio of outputto input is the same for a 1-V input as for a10-V input. Thus, the filter can accept aninput of many simultaneous sine waveswithout intermodulation (mixing) betweenthem.

Lumped-element means that the induc-tors and capacitors are physically muchsmaller than an operating wavelength. Inthis case, conductor lengths do not con-tribute significant inductance or capaci-tance, and the time that it takes for signalsto pass through the filter is insignificant.(Although the different times that it takesfor different frequencies to pass throughthe filter — known as group delay — isstill significant for some applications.)

The term passive means that the filterFig 12.5 — Transmission loss of a simple filter plotted against normalizedfrequency. Note the relationship between loss and SWR.

12.4 Chapter 12

does not need any internal power sources.There may be amplifiers before and/orafter the filter, but no power is necessaryfor the filter’s equations to hold. The filteralone always exhibits a finite (nonzero)insertion loss due to the unavoidable re-sistances associated with inductors and (toa lesser extent) capacitors. Active filters,as the name implies, contain internalpower sources.

Reciprocal means that the filter can passpower in either direction. Either end of thefilter can be used for input or output.

TIME DOMAIN VS FREQUENCYDOMAIN

Humans think in the time domain. Lifeexperiences are measured and recorded inthe stream of time. In contrast, AmateurRadio systems and their associated filtersare often better understood when viewedin the frequency domain, where frequencyis the relevant system parameter. Fre-quency may refer to a sine-wave voltage,current or electromagnetic field. The sine-wave voltage, shown in Fig 12.6, is awaveform plotted against time with equa-tion V = A sin(2π f t). The sine wave hasa peak amplitude A (measured in volts)and frequency, f (measured in cycles/sec-ond or Hertz). A graph showing frequencyon the horizontal axis is called a spectrum.A filter response curve is plotted on a spec-trum graph.

Historically, radio systems were bestanalyzed in the frequency domain. Theradio transmitters of Hertz (1865) andMarconi (1895) consisted of LC resonantcircuits excited by high-voltage spark gaps.The transmitters emitted packets ofdamped sine waves. The low-frequency(200-kHz) antennas used by Marconi werefound to possess very narrow bandwidths,and it seemed natural to analyze antennaperformance using sine-wave excitation. Inaddition, the growing use of 50 and 60-Hzalternating current (ac) electric power sys-tems in the 1890s demanded the use ofsine-wave mathematics to analyze thesesystems. Thus engineers trained in acpower theory were available to design andbuild the early radio systems.

In the frequency domain, the radioworld is imagined to be composed of manysine waves of different frequencies flow-ing endlessly in time. It can be shown bythe Fourier transform (Ref 7) that all peri-odic waveforms can be represented bysumming sine waves of different frequen-cies. For example, the square-wave volt-age shown in Fig 12.7 can be representedby a “fundamental” sine wave of fre-quency f = 1/t and all its odd harmonics:3f, 5f, 7f and so on. Thus, in the frequencydomain a sine wave is a narrowband sig-

Fig 12.6 — Ideal sine-wave voltage.Only one frequency is present.

Fig 12.7 — Square-wave voltage. Manyfrequencies are present, includingf ===== 1/////t and odd harmonics 3f, 5f, 7f withdecreasing amplitudes.

Fig 12.8 — Square-wave voltage filteredby a low-pass filter. By passing thesquare wave through a filter, the higherfrequencies are attenuated. The rectangu-lar shape (fast rise and fall items) arerounded because the amplitude of thehigher harmonics is decreased.

nal (zero bandwidth) and a square wave isa “wideband” signal.

If the square-wave voltage of Fig 12.7is passed through a low-pass filter, whichremoves some of its high-frequency com-ponents, the waveform of Fig 12.8 results.The filtered square wave now has a risetime, which is the time required to risefrom 10% to 90% of its peak value (A).The rise time is approximately:

cR f

0.35 (6)

where fc is the cutoff frequency of the low-pass filter.

Thus a filter distorts a time-domain sig-nal by removing some of its high-frequencycomponents. Note that a filter cannot dis-tort a sine wave. A filter can only changethe amplitude and phase of sine waves. Alinear filter will pass multiple sine waveswithout producing any intermodulation or“beats” between frequencies — this is thedefinition of linear.

The purpose of a radio system is to con-vey a time-domain signal originating at asource to some distant point with mini-mum distortion. Filters within the radiosystem transmitter and receiver may in-tentionally or unintentionally distort thesource signal. A knowledge of the sourcesignal’s frequency-domain bandwidth isrequired so that an appropriate radio sys-tem may be designed.

Table 12.1 shows the minimum neces-


modulation the transmitted RF bandwidthwill exceed the filtered source bandwidth ifinefficient (AM or FM) modulation meth-ods are employed. Thus the post-modula-tion emission bandwidth may be severaltimes the original filtered source band-width. At the receiving end of the radio link,band-pass filters are required to accept onlythe desired signal and sharply reject noiseand adjacent channel interference.

As human beings we are accustomed tooperation in the time domain. Just aboutall of our analog radio connected designoccurs in the frequency domain. This isparticularly true when it comes to filters.Although the two domains are convertible,one to the other, most filter design is per-formed in the frequency domain.

Table 12.1Typical Filter Bandwidths for Typical Signals.Source Required Bandwidth

High-fidelity speech and music 20 Hz to 15 kHzTelephone-quality speech 200 Hz to 3 kHzRadiotelegraphy (Morse code, CW) 200 HzHF RTTY 1000 Hz (varies with frequency shift)NTSC television 60 Hz to 4.5 MHzSSTV 200 Hz to 3 kHz1200 bit/s packet 200 Hz to 3 kHz

sary bandwidth of several common sourcesignals. Note that high-fidelity speechand music requires a bandwidth of 20 Hz to15 kHz, which is that transmitted by high-quality FM broadcast stations. However,telephone-quality speech requires a band-

width of only 200 Hz to 3 kHz. Thus, tominimize transmit spectrum, as required bythe FCC, filters within amateur transmit-ters are required to reduce the speechsource bandwidth to 200 Hz to 3 kHz at theexpense of some speech distortion. After

Filter SynthesisThe image-parameter method of filter

design was initiated by O. Zobel (Ref 1) ofBell Labs in 1923. Image-parameter filtersare easy to design and design techniquesare found in earlier editions of the ARRLHandbook. Unfortunately, image param-eter theory demands that the filter termi-nating impedances vary with frequency inan unusual manner. The later addition of“m-derived matching half sections” at eachend of the filter made it possible to use thesefilters in many applications. In the inter-vening decades, however, many new meth-ods of filter design have brought both betterperformance and practical component val-ues for construction.

MODERN FILTER THEORYThe start of modern filter theory is usu-

ally credited to S. Butterworth and S.Darlington (Refs 3 and 4). It is based onthis approach: Given a desired frequencyresponse, find a circuit that will yield thisresponse.

Filter theorists were aware that certainknown mathematical polynomials had“filter like” properties when plotted on afrequency graph. The challenge was tomatch the filter components (L, C and R)to the known polynomial poles and zeros.This pole/zero matching was a difficulttask before the availability of the digitalcomputer. Weinberg (Ref 5) was the firstto publish computer-generated tables ofnormalized low-pass filter componentvalues. (“Normalized” means 1-Ω resis-tor terminations and cutoff frequency ωc =2πfc = 1 radian/s.)

An ideal low-pass filter response shows

Fig 12.10 — Chebyshev approximationof an ideal low-pass filter. Notice theripple in the passband.

Fig 12.9 — Butterworth approximationof an ideal low-pass filter response.The 3-dB attenuation frequency (fc) isnormalized to 1 radian/s.

no loss from zero frequency to the cutofffrequency, but infinite loss above the cut-off frequency. Practical filters may ap-proximate this ideal response in severaldifferent ways.

Fig 12.9 shows the Butterworth or“maximally-flat” type of approximation.The Butterworth response formula is:

n2

c

O

L

1

1

P

P

(7)

whereω = frequency of interestωc = cutoff frequencyn = number of poles (reactances)PL = power in the load resistorPO = available generator power

The passband is exceedingly flat nearzero frequency and very high attenuationis experienced at high frequencies, but theapproximation for both pass and stopbands is relatively poor in the vicinity ofcutoff.

Fig 12.10 shows the Chebyshev ap-proximation. Details of the Chebyshevresponse formula can be found in (Ref 24).Use of this reference as well as similarreferences for Chebyshev filters requiresdetailed familiarity with Chebyshev poly-nomials.

IMPEDANCE AND FREQUENCYSCALING

Fig 12.11A shows normalized compo-nent values for Butterworth filters up toten poles. Fig 12.11B shows the schematicdiagrams of the Butterworth low-pass fil-ter. Note that the first reactance in Fig12.11B is a shunt capacitor C1, whereas inFig 12.11C the first reactance is a seriesinductor L1. Either configuration can beused, but a design using fewer inductors isusually chosen.

In filter design, the use of normalized

12.6 Chapter 12

values is common. Normalized generallymeans a design based on 1-Ω terminationsand a cutoff frequency (passband edge) of1 radian/second. A filter is denormalizedby applying the following two equations:

L'R

R''L (8)

C'R'

R'C (9)

whereL', C', ω' and R' are the new (desired)

valuesL and C are the values found in the

filter tables

Fig 12.13 — A 3-pole Butterworth filterscaled to 3000 Hz.

Prototype Butterworth Low-Pass FiltersC1 L2 C3 L4 C5 L6 C7 L8 C9 L10L1 C2 L3 C4 L5 C6 L7 C8 L9 C10

n1 2.00002 1.4142 1.41423 1.0000 2.0000 1.00004 0.7654 1.8478 1.8478 0.76545 0.6180 1.6180 2.0000 1.6180 0.61806 0.5176 1.4142 1.9319 1.9319 1.4142 0.51767 0.4450 1.2470 1.8019 2.0000 1.8019 1.2470 0.44508 0.3902 1.1111 1.6629 1.9616 1.9616 1.6629 1.1111 0.39029 0.3473 1.0000 1.5321 1.8794 2.0000 1.8794 1.5321 1.0000 0.3473

10 0.3129 0.9080 1.4142 1.7820 1.9754 1.9754 1.7820 1.4142 0.9080 0.3129(A)

Fig 12.11 — Component values for Butterworth low-pass filters. Greater valuesof n require more stages.

Fig 12.12 — A 3-pole Butterworth filterdesigned for a normalized frequency of1 radian/s.

Fig 12.14 — Passband loss of Butterworth low-pass filters. The horizontal axis isnormalized frequency (see text).

R = 1 Ωω = 1 radian/s.

For example, consider the design of a3-pole Butterworth low-pass filter fora transmitter speech amplifier. Let the

desired cutoff frequency be 3000 Hz andthe desired termination resistances be1000 Ω. The normalized prototype, takenfrom Fig 12.11B is shown in Fig 12.12.The new (desired) inductor value is:

H2Hz30002

cond radian/se1

1

1000'L

or L' = 0.106 H.

The new (desired) capacitor value is:

F1Hz30002

cond radian/se1

1000

1'C

or C' = 0.053 μF.The final denormalized filter is

shown in Fig 12.13. The filter response,in the passband, should obey curve n =3 in Fig 12.14. To use the normalizedfrequency response curves in Fig 12.14calculate the frequency ratio f/fc wheref is the desired frequency and fc is the cut-off frequency. For the filter just designed,the loss at 2000 Hz can be found as fol-lows: When f is 2000 Hz, the frequencyratio is: f/fc = 2000/3000 = 0.67. There-fore the predicted loss (from the n = 3


ence between 7.15 (bandwidth center) and7.147 (band-edge geometric mean) be-cause the bandwidth is small. For wide-band filters, however, there can be asignificant difference.]

Next, denormalize to a new interimlow-pass filter having R' = 50 Ω and f' =0.36 MHz.

H44.2H2100.362

1

1

50'L

6

pF8842F1100.362

1

50

1'C

6

This interim low-pass filter, shown inFig 12.15, has a cutoff frequency fc =0.36 MHz and is terminated with 50-Ω re-sistors. The desired 7.147-MHz band-pass

filter is achieved by parallel resonatingthe shunt capacitors with inductors andseries resonating the series inductor witha series capacitor. All resonators must betuned to the center frequency. Therefore,variable capacitors or inductors are re-quired for the resonant circuits. Based onthe L' and C' just calculated the parallel-resonating inductor values are:

H0.056f2C'

1L31L

2O

The series-resonating capacitor valueis:

pF11.2f2L'

12C

2O

The final band-pass filter is shown inFig 12.16. The filter should have a 3-dB

curve) is about 0.37 dB.When f is 4000 Hz, the filter is operat-

ing in the stop-band (Fig 12.17). The re-sulting frequency ratio is: f/fc = 4000/3000= 1.3. Therefore the expected loss isabout 8 dB. Note that as the number ofreactances (poles) increases the filter re-sponse approaches the low-pass responseof Fig 12.3A.

BAND-PASS FILTERS—SIMPLIFIED DESIGN

The design of band-pass filters may bedirectly obtained from the low-pass proto-type by a frequency translation. The low-pass filter has a “center frequency” (in theparlance of band-pass filters) of 0 Hz. Thefrequency translation from 0 Hz to theband-pass filter center frequency, f, isobtained by replacing in the low-pass pro-totype all shunt capacitors with paralleltuned circuits and all series inductors withseries tuned circuits.

As an example, suppose a band-passfilter is required at the front end of a home-brew 40-m QRP receiver to suppress pow-erful adjacent broadcast stations. Theproposed filter has these characteristics:

• Center frequency, fc = 7.15 MHz• 3-dB bandwidth = 360 kHz• terminating resistors = 50 Ω• 3-pole Butterworth characteristic.Start the design for the normalized

3-pole Butterworth low-pass filter (shownin Fig 12.11). First determine the centerfrequency from the band-pass limits. Thisfrequency, fO, is found by determining thegeometric mean of the band limits. In thiscase the band limits are 7.15 + 0.360/2 =7.33 MHz and 7.15 –0.360/2 = 6.97 MHz;then

MHz7.147.336.97fff hiloO

(10)where

flo = low frequency end of the band-pass(or band-stop)

fhi = high frequency end of the band-pass (or band-stop)

[Note that in this case there is little differ-

Fig 12.15 — Interim 3-pole Butterworthlow-pass filter designed for cutoff at0.36 MHz.

Fig 12.16 — Final filter design consists of the low-pass filter scaled to a centerfrequency of 7.15 MHz.

Fig 12.17 — Stop-band loss of Butterworth low-pass filters. The almost verticalangle of the lines representing filters with high values of n (10, 12, 15, 20) show theslope of the filter will be very high (sharp cutoff).

12.8 Chapter 12

bandwidth of 0.36 MHz. That is, the 3-dBloss frequencies are 6.97 MHz and 7.33MHz. The filter’s loaded Q is: Q = 7.147/0.36 or approximately 20.

The filter response, in the passband,falls on the “n = 3” curve in Fig 12.17. Touse the normalized frequency responsecurves, calculate the frequency ratio f/fc.For this band-pass case, f is the differencebetween the desired attenuation frequencyand the center frequency, while fc is theupper 3-dB frequency minus the centerfrequency. As an example the filter loss at7.5 MHz is found by using the normalizedfrequency ratio given by:

1.9287.1477.33

7.1477.5

f

f

c

Therefore, from Fig 12.17 the ex-pected loss is about 17 dB.

At 6 MHz the loss may be found by:

6.267.1477.33

67.147

f

f

c

The expected loss is approximately 47 dB.Unfortunately, awkward component val-ues occur in this type of band-pass filter.The series resonant circuit has a very largeLC ratio and the parallel resonant circuitshave very small LC ratios. The situationworsens as the filter loaded QL (QL = f0/BW) increases. Thus, this type of band-pass filter is generally used with a loadedQ less than 10.

Good examples of low-Q band-pass fil-ters of this type are demonstrated byW3NQN’s High Performance CW Filterand Passive Audio Filter for SSB in the1995 and earlier editions of this Handbook(Ref 25).

[Note: This analysis used the geomet-ric fc with the assumption that the filterresponse is symmetrical about fc, whichit is not. A more rigorous analysis yields16.9 dB at 7.5 MHz and 50.7 dB at6 MHz. — Ed.]

Q Restrictions—Band-pass FiltersMost filter component value tables as-

QU = unloaded Q of inductor:

R

L f2Q 0

U (11C)

R = inductor series resistanceL = inductanceQL = filter loaded Q

3

0L BW

fQ (11D)

BW3 = 3-dB bandwidthN = number of filter stages.

This equation assumes that all losses arein the inductors. For example, the ex-pected loss of the 7.15-MHz filter shownin Fig 12.16 is found by assuming QU =150. QL is found by equation 11D to be =7.147/0.36 = 19.8 or approximately 20.Since N = 3 then:

6

O

L

150

201

P

P

from equation (11A), which equals 0.423.Expressed as dB this is equal to 10 log(0.423) = – 3.73 dB.

Therefore this filter may not be suitablefor some applications. If the insertion lossis to be kept small there are severe restric-tions on QL/QU. With typical lumped in-ductors QU seldom exceeds 200. There-fore, LC band-pass filters are usuallydesigned with QL not exceeding 20 asshown in Fig 12.18.

This loss vs bandwidth trade-off is usu-ally why the final intermediate frequency(IF) in older radio receivers was very low.These units used the equivalent of LC fil-ters in their IF coupling. Generally, forSSB reception the desired receiver band-width is about 2.5 kHz. Then 50 kHz wasoften chosen as the final IF since this im-plies a loaded QL of 20. AM broadcast re-ceivers require a 10-kHz bandwidth anduse a 455-kHz IF, which results in QL =45. FM broadcast receivers require a200-kHz bandwidth and use a 10.7-MHzIF and QL = 22.

Fig 12.18 — Frequency range andmaximum loaded Q of band-pass filters.Crystal filters are shown with thehighest QL and LC filters the lowest.

sume lossless reactances. In practice,there are always resistance losses associ-ated with capacitors and inductors (espe-cially inductors). Lossy reactances inlow-pass filters modify the responsecurve. There is finite loss at zero fre-quency and the cutoff “knee” at fc will notbe as sharp as predicted by theoretical re-sponse curves.

The situation worsens with band-passfilters. As loaded Q is increased, themidband insertion loss may become intol-erable. Therefore, before a band-pass fil-ter design is started, estimate the expectedloss.

An approximate estimate of band-passfilter midband response is given by:

N2

U

L

O

L

Q

Q1

P

P (11A)

where:PL = power delivered to load resistor RLPO = power available from generator:

L

2g

O R4

eP (11B)

Filter Design Using Standard Capacitor Values/SoftwarePractical filters must be designed using

commercially available components.Modern computer programs are availableto aid in filter design. Originally, however,tables based upon standard value capaci-tors (SVC) were used to facilitate thisdesign process. These SVC tables are nowlocated on the CD-ROM included with thisHandbook. It is instructive to understandhow filters are designed using tables so

that you will more easily understand howto use modern computer-based designtechniques. To illustrate the process offilter design using filter design tables, theprocedure presented here uses computer-calculated tables of performance param-eters and component values for 5-elementChebyshev 50-Ω filters. The tables permitthe quick and easy selection of an equallyterminated passive LC filter for applica-

tions where the attenuation response is ofprimary interest. All of the capacitors inthe Chebyshev designs have standard, off-the-shelf values to simplify construction.Although the tables cover only the 1 to10-MHz frequency range, a simple scal-ing procedure gives standard-value ca-pacitor (SVC) designs for any impedancelevel and virtually any cutoff frequency.

Extracts from filter design tables are


off frequency exactly match the desiredcutoff frequency. A deviation of 5% or sobetween the actual and desired cutoff fre-quencies is acceptable. This permits the useof design tables based on standard capaci-tor values instead of passband ripple attenu-ation or reflection coefficient.

STANDARD VALUES IN FILTERDESIGN CALCULATIONS

Capacitors are commercially availablein special series of preferred values hav-ing designations of E12 (10% tolerance)and E24 (5% tolerance; Ref 22) The recip-rocal of the E-number is the power towhich 10 is raised to give the step multi-plier for that particular series.

First the normalized Chebyshev and el-liptic component values are calculatedbased on many ratios of standard capaci-tor values. Next, using a 50-Ω impedancelevel, the parameters of the designs arecalculated and tabulated to span the 1-10 MHz decade. Because of the large num-ber of standard-value capacitor (SVC) de-signs in this decade, the increment incutoff frequency from one design to thenext is sufficiently small so that virtuallyany cutoff frequency requirement can besatisfied. Using such a table, the selectionof an appropriate design consists ofmerely scanning the cutoff frequency col-umn to find a design having a cutoff fre-

quency that most closely matches the de-sired cutoff frequency.

CHEBYSHEV FILTERS1

Low-pass and high-pass 5-elementChebyshev designs were selected for tabu-lation because they are easy to constructand will satisfy the majority of non-stringent filtering requirements where theamplitude response is of primary interest.The precalculated 50-Ω designs are pre-sented in extracts from tables of low-passand high-pass designs with cutoff frequen-cies covering the 1-10 MHz decade. Inaddition to the component values, attenu-ation vs frequency data and SWR are alsoincluded in the table. The passband attenu-ation ripples are so low in amplitude thatthey are swamped by the filter losses andare not measurable.

LOW-PASS TABLESFig 12.19 is an extract from filter design

tables for the low-pass 5-elementChebyshev capacitor input/output con-figuration. This filter configuration is gen-erally preferred to the alternate inductorinput/output configuration because it re-quires fewer inductors. Generally, de-creasing input impedance with increasingfrequency in the stop band presents noproblems. Fig 12.20 shows the corre-sponding information for low-pass appli-

Try ELSIE for “LC”!This Handbook includes an

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reprinted in this section to illustrate thedesign procedure.

The following text by Ed Wetherhold,W3NQN, is adapted from his paper en-titled Simplified Passive LC Filter Designfor the EMC Engineer. It was presented atan IEEE International Symposium onElectromagnetic Compatibility in 1985.

The approach is based upon the fact thatfor most nonstringent filtering applica-tions, it is not necessary that the actual cut-

Fig 12.19 — A portion of a 5-element Chebyshev low-pass filter design tablefor 50-ΩΩΩΩΩ impedance, C-in/out and standard E24 capacitor values.

Fig 12.20 — A portion of a 5-element Chebyshev low-pass filter design tablefor 50-ΩΩΩΩΩ impedance, L-in/out and standard-value L and C.

—FREQUENCY (MHz)— MAX L1,5 C2,4 L3No. Fco 3 dB 20 dB 40 dB SWR (μH) (pf) (μH)1 0.744 1.15 1.69 2.60 1.027 5.60 4700 13.72 0.901 1.26 1.81 2.76 1.055 5.60 4300 12.73 1.06 1.38 1.94 2.93 1.096 5.60 3900 11.84 1.19 1.47 2.05 3.07 1.138 5.60 3600 11.25 1.32 1.58 2.17 3.23 1.192 5.60 3300 10.66 0.911 1.39 2.03 3.12 1.030 4.70 3900 11.47 1.08 1.50 2.16 3.29 1.056 4.70 3600 10.68 1.25 1.63 2.30 3.48 1.092 4.70 3300 9.929 1.42 1.77 2.46 3.68 1.142 4.70 3000 9.32

10 1.61 1.92 2.63 3.90 1.209 4.70 2700 8.7911 1.05 1.64 2.41 3.72 1.025 3.90 3300 9.63

—FREQUENCY (MHz)— MAX C1,5 L2,4 C3No. Fco 3 dB 20 dB 40 dB SWR (pF) (μH) (pF)1 1.01 1.15 1.53 2.25 1.355 3600 10.8 62002 1.02 1.21 1.65 2.45 1.212 3000 10.7 56003 1.15 1.29 1.71 2.51 1.391 3300 9.49 56004 1.10 1.32 1.81 2.69 1.196 2700 9.88 51005 1.25 1.41 1.88 2.75 1.386 3000 8.67 51006 1.04 1.37 1.94 2.94 1.085 2200 9.82 47007 1.15 1.41 1.95 2.92 1.155 2400 9.37 4700

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12.10 Chapter 12

cations, but with an inductor input/outputconfiguration. This configuration is use-ful when the filter input impedance in thestop band must rise with increasing fre-quency. For example, some RF transistoramplifiers may become unstable when ter-minated in a low-pass filter having a stop-band response with a decreasing inputimpedance. In this case, the inductor-in-put configuration may eliminate the insta-bility. (Ref 23) Because only one capacitorvalue is required in the designs of Fig12.20, it was feasible to have the inductorvalue of L1 and L5 also be a standardvalue.

HIGH-PASS TABLESA high-pass 5-element Chebyshev ca-

pacitor input/output configuration isshown in the table extract of Fig 12.21.Because the inductor input/output con-figuration is seldom used, it was not in-cluded.

SCALING TO OTHER FREQUEN-CIES AND IMPEDANCES

The tables shown are for the 1-10 MHzdecade and for a 50-Ω equally terminatedimpedance. The designs are easily scaledto other frequency decades and to otherequally terminated impedance levels,however, making the tables a universaldesign aid for these specific filter types.

Frequency ScalingTo scale the frequency and the compo-

nent values to the 10-100 or 100-1000 MHzdecades, multiply all tabulated frequenciesby 10 or 100, respectively. Then divide allC and L values by the same number. The Asand SWR data remain unchanged. To scalethe filter tables to the 0.1-1 kHz, 1-10 kHzor the 10-100 kHz decades, divide the tabu-lated frequencies by 1000, 100 or 10, re-spectively. Next multiply the componentvalues by the same number. By changingthe “MHz” frequency headings to “kHz”

and the “pF” and “μH” headings to “nF”and “mH,” the tables are easily changedfrom the 1-10 MHz decade to the 1-10 kHzdecade and the table values read directly.Because the impedance level is still at50 Ω, the component values may be awk-ward, but this can be corrected by increas-ing the impedance level by ten times usingthe impedance scaling procedure describedbelow.

Impedance ScalingAll the tabulated designs are easily

scaled to impedance levels other than50 Ω, while keeping the convenience ofstandard-value capacitors and the “scanmode” of design selection. If the desirednew impedance level differs from 50 Ω bya factor of 0.1, 10 or 100, the 50-Ω designsare scaled by shifting the decimal pointsof the component values. The other dataremain unchanged. For example, if theimpedance level is increased by ten or onehundred times (to 500 or 5000 Ω), thedecimal point of the capacitor is shifted tothe left one or two places and the decimalpoint of the inductor is shifted to the rightone or two places. With increasing imped-ance the capacitor values become smallerand the inductor values become larger.The opposite is true if the impedance de-creases.

When the desired impedance level dif-fers from the standard 50-Ω value by afactor such as 1.2, 1.5 or 1.86, the follow-ing scaling procedure is used:

1. Calculate the impedance scaling ratio:

50

ZR X (12)

where Zx is the desired new impedancelevel, in ohms.

2. Calculate the cutoff frequency (f50co)of a “trial” 50-Ω filter,

xcoco50 fRf (13)

where R is the impedance scaling ratio andfxco is the desired cutoff frequency of thefilter at the new impedance level.

3. From the appropriate SVC table se-lect a design having its cutoff frequencyclosest to the calculated f50co value. Thetabulated capacitor values of this designare taken directly, but the frequency andinductor values must be scaled to the newimpedance level.

4. Calculate the exact fxco values, where

R

'ff 50coxco (14)

and f '50co is the tabulated cutoff fre-quency of the selected design. Calculatethe other frequencies of the design in thesame way.

5. Calculate the inductor values for thenew filter by multiplying the tabulatedinductor values of the selected design bythe square of the scaling ratio, R.

—FREQUENCY (MHz)— Max C1,5 L2,4 C3No. Fco 3 dB 20 dB 40 dB SWR (pF) (H) (pF)1 1.04 0.726 0.501 0.328 1.044 5100 6.45 22002 1.04 0.788 0.554 0.366 1.081 4300 5.97 20003 1.17 0.800 0.550 0.359 1.039 4700 5.85 20004 1.07 0.857 0.615 0.410 1.135 3600 5.56 18005 1.17 0.877 0.616 0.406 1.076 3900 5.36 18006 1.33 0.890 0.609 0.397 1.034 4300 5.26 18007 1.12 0.938 0.686 0.461 1.206 3000 5.20 16008 1.25 0.974 0.693 0.461 1.109 3300 4.86 16009 1.38 0.994 0.691 0.454 1.057 3600 4.71 1600

10 1.54 1.00 0.683 0.444 1.028 3900 4.67 1600

Fig 12.21 — A portion of a 5-element Chebyshev high-pass filter design table for50-ΩΩΩΩΩ impedance, C-in/out and standard E24 capacitor values.

Notes1The Chebyshev filter is named after Pafnuty

Lvovitch Chebyshev (1821-1894), a famousRussian mathematician and academician.While touring Europe in 1852 to inspect vari-ous types of machinery, Chebyshev becameinterested in the mechanical linkage used inWatt’s steam engine. This linkage convertedthe reciprocating motion of the piston rodinto rotational motion of a flywheel needed torun factory machinery. Chebyshev notedthat Watt’s piston had zero lateral discrep-ancy at three points in its cycle. He con-cluded that a somewhat different linkagewould lead to a discrepancy of half of Watt’sand would be zero at five points in the pistoncycle. Chebyshev then wrote a paper — nowconsidered a mathematical classic — thatlaid the foundation for the topic of best ap-proximation of functions by means ofpolynomials. It is these same polynomialsthat were originally developed to improve thereciprocating-to-rotational linkage in a steamengine that now find application in the de-sign of the Chebyshev passive LC filters.From Philip J. Davis, The Thread, A Mathe-matical Yarn, 2nd edition, Harcourt BraceJovanovitch, Publishers, New York, 1989,1983; 124-page paperback.


Chebyshev Filter Design (Normalized Tables)The figures and tables in this section

provide the tools needed to designChebyshev filters including those filtersfor which the previously published stan-dard value capacitor (SVC) designs mightnot be suitable. Table 12.2 lists normal-ized low-pass designs that, in addition tolow-pass filters, can also be used to calcu-late high-pass, band-pass and band-stopfilters in either the inductor or capacitorinput/output configurations for equal im-pedance terminations. Table 12.3 pro-vides the attenuation for the resultantfilter.

This material was prepared by EdWetherhold, W3NQN, who has been theauthor of a number of articles and paperson the design of LC filters. It is a complete

Table 12.2Element values of Chebyshev low-pass filters normalized for a ripple cutoff frequency (Fap) of one radian/sec(1/2πππππ Hz) and 1-ΩΩΩΩΩ terminations.Use the top column headings for the low-pass C-in/out configuration and the bottom column headings for the low-pass L-in/outconfiguration. Fig 12.22 shows the filter schematics.

N RC Ret Loss F3/Fap C1 L2 C3 L4 C5 L6 C7 L8 C9(%) (dB) Ratio (F) (H) (F) (H) (F) (H) (F) (H) (F)

3 1.000 40.00 3.0094 0.3524 0.6447 0.35243 1.517 36.38 2.6429 0.4088 0.7265 0.40883 4.796 26.38 1.8772 0.6292 0.9703 0.62923 10.000 20.00 1.5385 0.8535 1.104 0.85353 15.087 16.43 1.3890 1.032 1.147 1.032

5 0.044 67.11 2.7859 0.2377 0.5920 0.7131 0.5920 0.23775 0.498 46.06 1.8093 0.4099 0.9315 1.093 0.9315 0.40995 1.000 40.00 1.6160 0.4869 1.050 1.226 1.050 0.48695 1.517 36.38 1.5156 0.5427 1.122 1.310 1.122 0.54275 2.768 31.16 1.3892 0.6408 1.223 1.442 1.223 0.64085 4.796 26.38 1.2912 0.7563 1.305 1.577 1.305 0.75635 6.302 24.01 1.2483 0.8266 1.337 1.653 1.337 0.82665 10.000 20.00 1.1840 0.9732 1.372 1.803 1.372 0.97325 15.087 16.43 1.1347 1.147 1.371 1.975 1.371 1.147

7 1.000 40.00 1.3004 0.5355 1.179 1.464 1.500 1.464 1.179 0.53557 1.427 36.91 1.2598 0.5808 1.232 1.522 1.540 1.522 1.232 0.58087 1.517 36.38 1.2532 0.5893 1.241 1.532 1.547 1.532 1.241 0.58937 3.122 30.11 1.1818 0.7066 1.343 1.660 1.611 1.660 1.343 0.70667 4.712 26.54 1.1467 0.7928 1.391 1.744 1.633 1.744 1.391 0.79287 4.796 26.38 1.1453 0.7970 1.392 1.748 1.633 1.748 1.392 0.79707 8.101 21.83 1.1064 0.9390 1.431 1.878 1.633 1.878 1.431 0.93907 10.000 20.00 1.0925 1.010 1.437 1.941 1.622 1.941 1.437 1.0107 10.650 19.45 1.0885 1.033 1.437 1.962 1.617 1.962 1.437 1.0337 15.087 16.43 1.0680 1.181 1.423 2.097 1.573 2.097 1.423 1.181

9 1.000 40.00 1.1783 0.5573 1.233 1.550 1.632 1.696 1.632 1.550 1.233 0.55739 1.517 36.38 1.1507 0.6100 1.291 1.610 1.665 1.745 1.665 1.610 1.291 0.61009 2.241 32.99 1.1271 0.6679 1.342 1.670 1.690 1.793 1.690 1.670 1.342 0.66799 2.512 32.00 1.1206 0.6867 1.357 1.688 1.696 1.808 1.696 1.688 1.357 0.68679 4.378 27.17 1.0915 0.7939 1.419 1.786 1.712 1.890 1.712 1.786 1.419 0.79399 4.796 26.38 1.0871 0.8145 1.427 1.804 1.713 1.906 1.713 1.804 1.427 0.81459 4.994 26.03 1.0852 0.8239 1.431 1.813 1.712 1.913 1.712 1.813 1.431 0.82399 8.445 21.47 1.0623 0.9682 1.460 1.936 1.692 2.022 1.692 1.936 1.460 0.96829 10.000 20.00 1.0556 1.025 1.462 1.985 1.677 2.066 1.677 1.985 1.462 1.0259 15.087 16.43 1.0410 1.196 1.443 2.135 1.617 2.205 1.617 2.135 1.443 1.196

N RC Ret Loss F3/Fap L1 C2 L3 C4 L5 C6 L7 C8 L9(%) (dB) Ratio (H) (F) (H) (F) (H) (F) (H) (F) (H)

revision of his previously published filterdesign material and provides both insightto the design and actual designs in just afew minutes.

For a given number of elements (N),increasing the filter reflection coefficient(RC or ρ) causes the attenuation slope toincrease with a corresponding increase inboth the passband ripple amplitude (ap)and SWR and with a decrease in the filterreturn loss. All of these parameters aremathematically related to each other. Ifone is known, the others may be calcu-lated. Filter designs having a low RC arepreferred because they are less sensitiveto component and termination impedancevariations than are designs having a higherRC. The RC percentage is used as the in-

dependent variable in Table 12.2 becauseit is used as the defining parameter in themore frequently used tables, such as thoseby Zverev and Saal (see Refs 17 and 18).

The return loss is tabulated instead ofpassband ripple amplitude (ap) because itis easy to measure using a return lossbridge. In comparison, ripple amplitudesless than 0.1 dB are difficult to measureaccurately. The resulting values of attenu-ation are contained in Table 12.3 and cor-responding values of ap and SWR may befound by referring to the Equivalent Val-ues of Reflection Coefficient, Attenua-tion, SWR and Return Loss table in theComponent Data and References chap-ter. The filter used (low pass, high-pass,band-pass and so on) will depend on the

12.12 Chapter 12

application and the stop-band attenuationneeded.

The filter schematic diagrams shown inFig 12.22 are for low-pass and high-passversions of the Chebyshev designs listedin Table 12.2. Both low-pass and high-pass equally terminated configurationsand component values of the C-in/out orL-in/out filters can be derived from thissingle table. By using a simple procedure,the low-pass and high-pass designs can betransformed into corresponding band-passand band-stop filters. The normalized ele-ment values of the low-pass C-in/out andL-in/out designs, Fig 12.22A and B, areread directly from the table using the val-ues associated with either the top or bot-tom column headings, respectively.

The first four columns of Table 12.2 listN (the number of filter elements), RC (re-flection coefficient percentage), return lossand the ratio of the 3-dB-to-Fap frequen-cies. The passband maximum ripple ampli-tude (ap) is not listed because it is difficultto measure. If necessary it can be calcu-lated from the reflection coefficient. TheF3/Fap ratio varies with N and RC; if bothof these parameters are known, the F3/Fapratio may be calculated. The remainingcolumns list the normalized Chebyshevelement values for equally terminated fil-ters for Ns from 3 to 9 in increments of 2.

Table 12.3Normalized Frequencies at Listed Attenuation Levels for Chebyshev Low-Pass Filters withN = 3, 5, 7 and 9.

Attenuation Levels (dB)N RC(%) 1.0 3.01 6.0 10 20 30 40 50 60 70 80

3 1.000 2.44 3.01 3.58 4.28 6.33 9.27 13.59 19.93 29.25 42.92 63.003 1.517 2.15 2.64 3.13 3.74 5.52 8.08 11.83 17.35 25.46 37.36 54.833 4.796 1.56 1.88 2.20 2.60 3.79 5.53 8.08 11.83 17.35 25.45 37.353 10.000 1.31 1.54 1.78 2.08 3.00 4.34 6.33 9.26 13.57 19.90 29.203 15.087 1.20 1.39 1.59 1.85 2.63 3.79 5.52 8.06 11.81 17.32 25.41

5 1.000 1.46 1.62 1.76 1.94 2.39 2.97 3.69 4.62 5.79 7.27 9.135 1.517 1.38 1.52 1.65 1.80 2.22 2.74 3.41 4.26 5.33 6.69 8.405 4.796 1.19 1.29 1.39 1.50 1.82 2.22 2.74 3.41 4.26 5.33 6.695 6.302 1.16 1.25 1.34 1.44 1.74 2.12 2.61 3.24 4.04 5.05 6.345 10.000 1.11 1.18 1.26 1.35 1.61 1.95 2.39 2.96 3.69 4.61 5.785 15.087 1.07 1.13 1.20 1.28 1.51 1.82 2.22 2.74 3.41 4.25 5.33

7 1.000 1.23 1.30 1.37 1.45 1.65 1.89 2.18 2.53 2.95 3.44 4.047 1.517 1.19 1.25 1.32 1.39 1.57 1.80 2.07 2.39 2.79 3.25 3.817 4.796 1.10 1.15 1.19 1.25 1.39 1.57 1.80 2.07 2.39 2.79 3.257 8.101 1.07 1.11 1.15 1.19 1.32 1.49 1.69 1.94 2.24 2.60 3.037 10.000 1.05 1.09 1.13 1.18 1.30 1.45 1.65 1.89 2.18 2.53 2.947 15.087 1.04 1.07 1.10 1.14 1.25 1.39 1.57 1.80 2.07 2.39 2.78

9 1.000 1.13 1.18 1.22 1.26 1.38 1.51 1.67 1.85 2.07 2.32 2.619 1.517 1.11 1.15 1.19 1.23 1.34 1.46 1.61 1.78 1.99 2.22 2.509 4.796 1.06 1.09 1.11 1.15 1.23 1.34 1.46 1.61 1.78 1.99 2.229 8.445 1.04 1.06 1.09 1.11 1.19 1.28 1.40 1.53 1.69 1.88 2.109 10.000 1.03 1.06 1.08 1.10 1.18 1.27 1.38 1.51 1.67 1.85 2.079 15.087 1.02 1.04 1.06 1.08 1.15 1.23 1.34 1.46 1.61 1.78 1.99

The Chebyshev passband ends when thepassband attenuation first exceeds themaximum ripple amplitude, ap. This fre-quency is called the “ripple cutoff fre-quency, Fap” and it has a normalized valueof unity. All Chebyshev designs in Table12.2 are based on the ripple cutoff fre-quency instead of the more familiar 3-dBfrequency of the Butterworth response.However, the 3-dB frequency of aChebyshev design may be obtained bymultiplying the ripple cutoff frequency bythe F3/Fap ratio listed in the fourth col-umn.

The element values are normalized to aripple cutoff frequency of 0.15915 Hz (oneradian/sec) and 1-Ω terminations, so thatthe low-pass values can be transformeddirectly into high-pass values. This is doneby replacing all Cs and Ls in the low-passconfiguration with Ls and Cs and by re-placing all the low-pass element valueswith their reciprocals. The normalizedvalues are then multiplied by the appro-priate C and L scaling factors to obtain thefinal values based on the desired ripplecutoff frequency and impedance level.The listed C and L element values are infarads and henries and become more rea-sonable after the values are scaled to thedesired cutoff frequency and impedancelevel.

The normalized designs presented are amixture: Some have integral values of re-flection coefficient (RC) (1% and 10%)while others have “integral” values ofpassband ripple amplitude (0.001, 0.01and 0.1 dB). These ripple amplitudes cor-respond to reflection coefficients of 1.517,4.796 and 15.087%, respectively. By hav-ing tabulated designs based on integralvalues of both reflection coefficient andpassband ripple amplitude, the correctnessof the normalized component values maybe checked against those same values pub-lished in filter handbooks whichever pa-rameter, RC or ap, is used.

In addition to the customary normalizeddesign listings based on integral values ofreflection coefficient or ripple amplitude,Table 12.2 also includes unique designshaving special element ratios that makethem more useful than previously pub-lished tables. For example, for N = 5 andRC = 6.302, the ratio of C3/C1 is 2.000.This ratio allows 5-element low-pass fil-ters to be realized with only one capacitorvalue because C3 may be obtained by us-ing parallel-connected capacitors eachhaving the same value as C1 and C5.

In a similar way, for N = 7 and RC =8.101, C3/C1 and C5/C1 are also 2.000.Another useful N = 7 design is that forRC = 1.427%. Here the L4/L2 ratio is


Fig 12.22 — The schematic diagrams shown are low-pass and high-passChebyshev filters with the C-in/out and L-in/out configurations. For all normalizedvalues see Table 12.2.A: C-in/out low-pass configuration. Use the C and L values associated with the topcolumn headings of the Table.B: L-in/out low-pass configuration. For normalized values, use the L and C valuesassociated with the bottom column headings of the Table.C: L-in/out high-pass configuration is derived by transforming the C-in/out low-pass filter in A into an L-in/out high-pass by replacing all Cs with Ls and all Lswith Cs. The reciprocals of the lowpass component values become the highpasscomponent values. For example, when n = 3, RC = 1.00% and C1 = 0.3524 F, L1and L3 in C become 2.838 H.D: The C-in/out high-pass configuration is derived by transforming the L-in/outlow-pass in B into a C-in/out high-pass by replacing all Ls with Cs and all Cs withLs. The reciprocals of the low-pass component values become the high-passcomponent values. For example, when n = 3, RC = 1.00% and L1 = 0.3524 H, C1and C3 in D become 2.838 F.

1.25, which is identical to 110/88. Thismeans a seventh-order C-in/out low-passaudio filter can be realized with four sur-plus 88-mH inductors. Both L2 and L6can be 88 mH while L4 is made up of a

series connection of 22 mH and 88 mH.The 22-mH value is obtained by connect-ing the two windings of one of the foursurplus inductors in parallel. Other use-ful ratios also appear in the N = 9 listing

for both C3/C1 and L4/L2.Except for the first two N = 5 designs,

all designs were calculated for a reflectioncoefficient range from 1% to about 15%.The first two N = 5 designs were includedbecause of their useful L3/L1 ratios. De-signs with an RC of less than 1% are notnormally used because of their poor selec-tivity. Designs with RC greater than 15%yield increasingly high SWR values withcorrespondingly increased objectionablereflective losses and sensitivity to termi-nation impedance and component valuevariations.

Low-pass and high-pass filters may berealized in either a C-input/output or an L-input/output configuration. The C-input/output configuration is usually preferredbecause fewer inductors are required,compared to the L-input/output configu-ration. Inductors are usually more lossy,bulky and expensive than capacitors. Theselection of the filter order or number offilter elements, N, is determined by thedesired stop-band attenuation rate of in-crease and the tolerable reflection coeffi-cient or SWR. A steeper attenuation sloperequires either a design having a higherreflection coefficient or more circuit ele-ments. Consequently, to select an opti-mum design, the builder must determinethe amount of attenuation required in thestop band and the permissible maximumamount of reflection coefficient or SWR.

Table 12.3 shows the theoretical nor-malized frequencies (relative to the ripplecutoff frequency) for the listed attenua-tion levels and reflection coefficient per-centages for Chebyshev low-pass filtersof 3, 5, 7 and 9 elements. For example, forN = 5 and RC = 15.087%, an attenuation of40 dB is reached at 2.22 times the ripplecutoff frequency (slightly more than oneoctave). The tabulated data are also appli-cable to high-pass filters by simply takingthe reciprocal of the listed frequency. Forexample, for the same previous N and RCvalues, a high-pass filter attenuation willreach 40 dB at 1/2.22 = 0.450 times theripple cutoff frequency.

The attenuation levels are theoreticaland assume perfect components, no cou-pling between filter sections and no signalleakage around the filter. A workingmodel should follow these values to the 60or 70-dB level. Beyond this point, the ac-tual response will likely degrade some-what from the theoretical.

Fig 12.23 shows four plotted attenua-tion vs normalized frequency curves for N= 5 corresponding to the normalized fre-quencies in Table 12.3 At two octavesabove the ripple cutoff frequency, fc, theattenuation slope gradually becomes 6 dBper octave per filter element.

12.14 Chapter 12

LOW-PASS AND HIGH-PASSFILTERSLow-Pass Filter

Let’s look at the procedure used to cal-culate the capacitor and inductor valuesof low-pass and high-pass filters by us-ing two examples. Assume a 50-Ω low-pass filter is needed to give more than40 dB of attenuation at 2fc or one octaveabove the ripple-cutoff frequency of4.0 MHz. Referring to Table 12.3, we seefrom the 40-dB column that a filter with7 elements (N = 7) and a RC of 4.796%will reach 40 dB at 1.80 times the cutofffrequency or 1.8 × 4 = 7.2 MHz. Sincethis design has a reasonably low reflec-tion coefficient and will satisfy the at-tenuation requirement, it is a goodchoice. Note that no 5-element filters aresuitable for this application because40 dB of attenuation is not achieved oneoctave above the cutoff frequency.

From Table 12.2, the normalized com-ponent values corresponding to N = 7 andRC = 4.796% for the C-in/out configura-tion are: C1, C7 = 0.7970 F, L2, L6 =1.392 H, C3, C5 = 1.748 F and L4 =1.633 H. See Fig 12.23A for the corre-sponding configuration. The C and L nor-malized values will be scaled from a ripple

cutoff frequency of one radian/sec and animpedance level of 1 Ω to a cutoff fre-quency of 4.0 MHz and an impedancelevel of 50 Ω. The Cs and Ls scaling fac-tors are calculated:

fR2

1Cs (15)

f2

RLs (16)

where:R = impedance levelf = cutoff frequency.

In this example:

126s 105.879

104502

1

fR2

1C

66s 101.989

1042

50

f2

RL

Using these scaling factors, thecapacitor and inductor normalized val-ues are scaled to the desired cutoff fre-quency and impedance level:

C1, C7 = 0.797 × 795.8 pF = 634 pFC3, C5 = 1.748 × 795.8 pF = 1391 pFL2, L6 = 1.392 × 1.989 μH = 2.77 μHL4 = 1.633 × 1.989 μH = 3.25 μH

High-Pass FilterThe procedure for calculating a high-

pass filter is similar to that for a low-passfilter, except a low-pass-to-high-passtransformation must first be performed.Assume a 50-Ω high-pass filter is neededto give more than 40 dB of attenuation oneoctave below (fc/2) a ripple cutoff fre-quency of 4.0 MHz. Referring to Table12.3, we see from the 40-dB column that a7-element low-pass filter with RC of4.796% will give 40 dB of attenuation at1.8fc. If this filter is transformed into ahigh-pass filter, the 40-dB level is reachedat fc/1.80 or at 0.556fc = 2.22 MHz. Sincethe 40-dB level is reached before oneoctave from the 4-MHz cutoff frequency,this design will be satisfactory.

From Fig 12.22, we choose the low-passL-in/out configuration in B and transformit into a high-pass filter by replacing allinductors with capacitors and all capaci-tors with inductors. Fig 12.22D is thefilter configuration after the transforma-tion. The reciprocals of the low-pass valuesbecome the high-pass values to completethe transformation. The high-pass valuesof the filter shown in Fig 12.22D are:

F1.2550.7970

1C7,1C

H0.71841.392

1L6,2L

F0.57211.748

15C,3C

and

H0.61241.633

14L

Using the previously calculated C and Lscaling factors, the high-pass componentvalues are calculated the same way asbefore:

C1, C7 = 1.255 × 795.8 pF = 999 pFC3, C5 = 0.5721× 795.8 pF = 455 pFL2, L6 = 0.7184 × 1.989 μH = 1.43 μHL4 = 0.6124 × 1.989 μH = 1.22 μH

BAND-PASS FILTERSBand-pass filters may be classified as

either narrowband or broadband. If theratio of the upper ripple cutoff frequencyto the lower cutoff frequency is greaterthan two, we have a wideband filter. Forwideband filters, the band-pass filter(BPF) requirement may be realized bysimply cascading separate high-pass andlow-pass filters having the same designimpedance. (The assumption is that thefilters maintain their individual responseseven though they are cascaded.) For thisto be true, it is important that both filters

Fig 12.23 — The graph shows attenuation vs frequency for four 5-element low-pass filters designed with the information obtained from Table 12.2. This graphdemonstrates how reflection coefficient percentage (RC), maximum passbandripple amplitude (ap), SWR, return loss and attenuation rolloff are all related. Theexact frequency at a specified attenuation level can be obtained from Table 12.3.


have a relatively low reflection coefficientpercentage (less than 5%) so the SWRvariations in the passband will be small.

For narrowband BPFs, where the sepa-ration between the upper and lower cutofffrequencies is less than two, it is neces-sary to transform an appropriate low-passfilter into a BPF. That is, we use thelow-pass normalized tables to designnarrowband BPFs.

We do this by first calculating a low-pass filter (LPF) with a cutoff frequencyequal to the desired bandwidth of the BPF.The LPF is then transformed into the de-sired BPF by resonating the low-pass com-ponents at the geometric center frequencyof the BPF.

For example, assume we want a 50-ΩBPF to pass the 75/80-m band and attenu-ate all signals outside the band. Based onthe passband ripple cutoff frequencies of3.5 and 4.0 MHz, the geometric center fre-quency = (3.5 × 4.0)0.5 = (14)0.5 = 3.741657or 3.7417 MHz. Let’s slightly extend thelower and upper ripple cutoff frequenciesto 3.45 and 4.058 MHz to account for pos-sible component tolerance variations andto maintain the same center frequency.We’ll evaluate a low-pass 3-element pro-totype with a cutoff frequency equal to theBPF passband of (4.058–3.45)MHz =0.608 MHz as a possible choice for trans-formation.

Further, assume it is desired to attenu-ate the second harmonic of 3.5 MHz by atleast 40 dB. The following calculationsshow how to design an N = 3 filter toprovide the desired 40-dB attenuation at7 MHz and above.

The bandwidth (BW) between 7 MHzon the upper attenuation slope (call it “f+”)of the BPF and the corresponding fre-quency at the same attenuation level onthe lower slope (call it “f–”) can be calcu-lated based on (f+)(f–) = (fc)

2 or

MHz27

14f

Therefore, the bandwidth at this un-known attenuation level for 2 and 7 MHzis 5 MHz. This 5-MHz BW is normalizedto the ripple cutoff BW by dividing5.0 MHz by 0.608 MHz:

8.22608.0

0.5

We now can go to Table 12.3 and searchfor the corresponding normalized fre-quency that is closest to the desired nor-malized BW of 8.22. The low-pass designof N = 3 and RC = 4.796% gives 40 dB fora normalized BW of 8.08 and 50 dB for11.83. Therefore, a design of N = 3 and RC= 4.796% with a normalized BW of 8.22 is

C1, C3 = 0.6292 × 5235 pF = 3294 pF andL2 = 0.9703 × 13.09 μH = 12.70 μH.

The LPF (in a pi configuration) is trans-formed into a BPF with 3.7417-MHz cen-ter frequency by resonating the low-passelements at the center frequency. Theresonating components will take the sameidentification numbers as the componentsthey are resonating.

H0.5493329414

25330

C1F

253303L,1L

2c

pF142.512.714

25330

L2F

253302C

2c

where L, C and f are in μH, pF and MHzrespectively.

The BPF circuit after transformation(for N = 3, RC = 4.796%) is shown inFig 12.24.

The component-value spread is

23549.0

7.12

and the reactance of L1 is about 13 Ω at thecenter frequency. For better BPF perfor-mance, the component spread should bereduced and the reactance of L1 and L3should be raised to make it easier toachieve the maximum possible Q for thesetwo inductors. This can be easily done bydesigning the BPF for an impedance levelof 200 Ω and then using the center taps onL1 and L3 to obtain the desired 50-Ω ter-minations. The result of this approach isshown in Fig 12.25

The component spread is now a morereasonable

5.7720.2

7.12

and the L1, L3 reactance is 51.6 Ω. Thishigher reactance gives a better chance toachieve a satisfactory Q for L1 and L3 witha corresponding improvement in the BPFperformance.

As a general rule, keep reactance valuesbetween 5 Ω and 500 Ω in a 50-Ω circuit.When the value falls below 5 Ω, either theequivalent series resistance of the induc-tor or the series inductance of the capaci-tor degrades the circuit Q. When theinductive reactance is greater than 500 Ω,the inductor is approaching self-resonanceand circuit Q is again degraded. Inpractice, both L1 and L3 should be bifilar wound on a powdered-iron toroidalcore to assure that optimum coupling is

Fig 12.25 — A filter designed for 200-ΩΩΩΩΩsource and load provides better values.By tapping the inductors, we can use a200-ΩΩΩΩΩ filter design in a 50-ΩΩΩΩΩ system.

Fig 12.24 — After transformation of theband-pass filter, all parallel elementsbecome parallel LCs and all serieselements become series LCs.

at an attenuation level somewhere between40 and 50 dB. Consequently, a low-passdesign based on 3 elements and a 4.796%RC will give slightly more than the desired40 dB attenuation above 7 MHz. The nextstep is to calculate the C and L values of thelow-pass filter using the normalized com-ponent values in Table 12.2.

From this table and for N = 3 and RC =4.796%, C1, C3 = 0.6292 F and L2 =0.9703 H. We calculate the scaling factorsas before and use 0.608 MHz as the ripplecutoff frequency:

126

s

105235100.608502

1

fR2

1C

66s 1013.09

100.6082

50

f2

RL

12.16 Chapter 12

obtained between turns over the entirewinding. The junction of the bifilar wind-ing serves as a center tap.

Side-Slope Attenuation CalculationsThe following equations allow the cal-

culation of the frequencies on the upper andlower sides of a BPF response curve at anygiven attenuation level if the bandwidth atthat attenuation level and the geometriccenter frequency of the BPF are known:

22clo XfXf (17)

BWff lohi(18)

whereBW = bandwidth at the given attenua-

tion level,fc = geometric center frequency

2

BWX

For example, if f = 3.74166 MHz andBW = 5 MHz, then

2.5X2

BW

and:

MHz2.002.53.741662.5f 22lo

MHz752BWff lohi

BAND-STOP FILTERSBand-stop filters may be classified as

either narrowband or broadband. If theratio of the upper ripple cutoff frequencyto the lower cutoff frequency is greaterthan two, the filter is considered wide-band. A wideband band-stop filter (BSF)requirement may be realized by simplyparalleling the inputs and outputs of sepa-rate low-pass and high-pass filters hav-ing the same design impedance and withthe low-pass filter having its cutoff fre-quency one octave or more below thehigh-pass cutoff frequency.

In order to parallel the low-pass andhigh-pass filter inputs and outputs with-out one affecting the other, it is essentialthat each filter have a high impedance inthat portion of its stop band that lies inthe passband of the other. This means thateach of the two filters must begin and endin series branches. In the low-pass filter,the input/output series branches mustconsist of inductors and in the high-passfilter, the input/output series branchesmust consist of capacitors.

When the ratio of the upper to lowercutoff frequencies is less than two, theBSF is considered to be narrowband, anda calculation procedure similar to that ofthe narrowband BPF design procedure isused. However, in the case of the BSF,the design process starts with the designof a high-pass filter having the desiredimpedance level of the BSF and a ripplecutoff frequency the same as that of thedesired ripple bandwidth of the BSF.After the HPF design is completed, everyhigh-pass element is resonated to the cen-ter frequency of the BSF in the samemanner as if it were a BPF, except that allshunt branches of the BSF will consist ofseries-tuned circuits, and all seriesbranches will consist of parallel-tunedcircuits — just the opposite of the reso-nant circuits in the BPF. The reason forthis becomes obvious when the imped-ance characteristics of the series and par-allel circuits at resonance are consideredrelative to the intended purpose of thefilter, that is, whether it is for a band-passor a band-stop application.

The design, construction and test ofband-stop filters for attenuating high-level broadcast-band signals is describedin reference 30 at the end of this chapter.

Quartz Crystal FiltersPractical inductor Q values effectively

set the minimum achievable bandwidthlimits for LC band-pass filters. Higher-Qcircuit elements must be employed toextend these limits. These high-Q resona-tors include PZT ceramic, mechanical andcoaxial devices. However, the quartz crys-tal provides the highest Q and best stabil-ity with temperature and time of allavailable resonators. Quartz crystals suit-able for filter use are fabricated over a fre-quency range from audio to VHF.

The quartz resonator has the equivalentcircuit shown in Fig 12.26. Ls, Cs and Rsrepresent the motional reactances and lossresistance. Cp is the parallel plate capaci-tance formed by the two metal electrodesseparated by the quartz dielectric. Quartzhas a dielectric constant of 3.78.Table 12.4 shows parameter values fortypical moderate-cost quartz resonators.QU is the resonator unloaded Q.

ssU rf2Q (19)

QU is very high, usually exceeding25,000. Thus the quartz resonator is anideal component for the synthesis of ahigh-Q band-pass filter.

Fig 12.26 — Equivalent circuit of a quartz crystal. The curve plots the crystalreactance against frequency. At fp, the resonance frequency, the reactance curvegoes to infinity.


Table 12.4Typical Parameters for AT-Cut Quartz Resonators

Freq Mode rs Cp Cs L QU(MHz) n (Ω) (pF) (pF) (mH)

1.0 1 260 3.4 0.0085 2900 72,000 5.0 1 40 3.8 0.011 100 72,000 10.0 1 8 3.5 0.018 14 109,000 20 1 15 4.5 0.020 3.1 26,000 30 3 30 4.0 0.002 14 87,000 75 3 25 4.0 0.002 2.3 43,000110 5 60 2.7 0.0004 5.0 57,000150 5 65 3.5 0.0006 1.9 27,000200 7 100 3.5 0.0004 2.1 26,000

Courtesy of Piezo Crystal Co, Carlisle, Pennsylvania

Fig 12.27 — A: Series test circuit for acrystal. In the test circuit the output ofa variable frequency generator, eg, isused as the test signal. The frequencyresponse in B shows the highestattenuation at resonance (fp). See text.

Fig 12.28 — The practical one-stagecrystal filter in A has the responseshown in B. The phasing capacitor isadjusted for best response (see text).

A quartz resonator connected betweengenerator and load, as shown in Fig 12.27A,produces the frequency response ofFig 12.27B. There is a relatively low loss atthe series resonant frequency fs and highloss at the parallel resonant frequency fp.The test circuit of Fig 12.27A is useful fordetermining the parameters of a quartz

series resonant circuits, if properly offsetin frequency, will produce an approximate2-pole Butterworth or Chebyshev response.Crystals A and B are usually chosen so thatthe parallel resonant frequency (fp) of oneis the same as the series resonant frequency(fs) of the other.

Half-lattice filter sections can be cascadedto produce a composite filter with many poles.Until recently, most vendor- supplied com-mercial filters were lattice types. Ref 11 dis-cusses the computer design of half-latticefilters.

Fig 12.29 — A half-lattice crystal filter. No phasing capacitor is needed in thiscircuit.

resonator, but yields a poor filter.A crystal filter developed in the 1930s is

shown in Fig 12.28A. The disturbing effectof Cp (which produces fp) is canceled bythe phasing capacitor, C1. The voltage-reversing transformer T1 usually consistsof a bifilar winding on a ferrite core. Volt-ages Va and Vb have equal magnitude but180° phase difference. When C1 = Cp, theeffect of Cp will disappear and a well-behaved single resonance will occur asshown in Fig 12.28B. The band-pass filterwill exhibit a loaded Q given by:

L

SSL R

Lf2Q (20)

This single-stage “crystal filter,” oper-ating at 455 kHz, was present in almost allhigh-quality amateur communications re-ceivers up through the 1960s. When thefilter was switched into the receiver IFamplifier the bandwidth was reduced to afew hundred Hz for Morse code reception.

The half-lattice filter shown in Fig 12.29is an improvement in crystal filter design.The quartz resonator parallel-plate capa-citors, Cp, cancel each other. Remaining

12.18 Chapter 12

Monolithic Crystal Filters

SAW Filters

Many quartz crystal filters producedtoday use the ladder network design shownin Fig 12.30. In this configuration, all reso-nators have the same series resonant fre-quency fs. Inter-resonator coupling isprovided by shunt capacitors such as C12and C23. Refs 12 and 13 provide good lad-der filter design information. A test set forevaluating crystal filters is presented else-where in this chapter.

Fig 12.30 — A four-stage crystal ladderfilter. The crystals must be chosenproperly for best response.

A monolithic (Greek: one-stone) crys-tal filter has two sets of electrodes depos-ited on the same quartz plate, as shown inFig 12.31. This forms two resonators withacoustic (mechanical) coupling betweenthem. If the acoustic coupling is correct, a2-pole Butterworth or Chebyshev re-sponse will be achieved. More than tworesonators can be fabricated on the sameplate yielding a multipole response.Monolithic crystal filter technology ispopular because it produces a low parts

Fig 12.31 — Typical two-pole monolithiccrystal filter. This single small (1/2 to3/4-inch) unit can replace 6 to 12, ormore, discrete components.

count, single-unit filter at lower cost thana lumped-element equivalent. Monolithiccrystal filters are typically manufacturedin the range from 5 to 30 MHz for the fun-damental mode and up to 90 MHz for thethird-overtone mode. QL ranges from 200to 10,000.

Fig 12.32 — The interdigitated transducer, on the left, launches SAW energy to a similar transducer on the right (see text).

The resonators in a monolithic crystalfilter are coupled together by bulk acous-tic waves. These acoustic waves are gen-erated and propagated in the interior of aquartz plate. It is also possible to launch,by an appropriate transducer, acoustic

waves that propagate only along the sur-face of the quartz plate. These are called“surface-acoustic-waves” because theydo not appreciably penetrate the interiorof the plate.

A surface-acoustic-wave (SAW) filter

consists of thin aluminum electrodes,or fingers, deposited on the surface ofa piezoelectric substrate as shown in Fig12.32. Lithium Niobate (LiNbO3) is usu-ally favored over quartz because it yieldsless insertion loss. The electrodes make


up the filter’s transducers. RF voltageis applied to the input transducer and gen-erates electric fields between the fingers.The piezoelectric material vibrateslaunching an acoustic wave along the sur-face. When the wave reaches the outputtransducer it produces an electric fieldbetween the fingers. This field generates avoltage across the load resistor.

Since both input and output transducersare not entirely unidirectional, someacoustic power is lost in the acoustic ab-

sorbers located behind each transducer.This lost acoustic power produces amidband electrical insertion loss typicallygreater than 10 dB. The SAW filter fre-quency response is determined by thechoice of substrate material and fingerpattern. The finger spacing, (usually one-quarter wavelength) determines the filtercenter frequency. Center frequencies areavailable from 20 to 1000 MHz. The num-ber and length of fingers determines thefilter loaded Q and shape factor.

Loaded Qs are available from 2 to 100,with a shape factor of 1.5 (equivalent to adozen poles). Thus the SAW filter can bemade broadband much like the LC filtersthat it replaces. The advantage is substan-tially reduced volume and possibly lowercost. SAW filter research was driven bymilitary needs for exotic amplitude-response and time-delay requirements.Low-cost SAW filters are presently foundin television IF amplifiers where highmidband loss can be tolerated.

Transmission-Line FiltersLC filter calculations are based on the

assumption that the reactances arelumped—the physical dimensions of thecomponents are considerably less than theoperating wavelength. Therefore the un-avoidable interturn capacitance associ-ated with inductors and the unavoidable

Fig 12.33 — Transmission lines. A:Coaxial line. B: Coupled stripline,which has two ground planes. C:Microstripline, which has only oneground plane.

TEM, or Transverse ElectromagneticMode, the electric and magnetic fields as-sociated with a transmission line are at rightangles (transverse) to the direction of wavepropagation. Coaxial cable, stripline andmicrostrip are examples of TEM compo-nents. Waveguides and waveguide resona-tors are not TEM components.

TRANSMISSION LINES FOR FILTERSFig 12.33 shows three popular trans-

mission lines used in transmission-linefilters. The circular coaxial transmissionline (coax) shown in Fig 12.33A consistsof two concentric metal cylinders sepa-rated by dielectric (insulating) material.

Fig 12.34 — Coaxial-line impedance varies with the ratio of the inner- and outer-conductor diameters. The dielectric constant, εεεεε, is 1.0 for air and 2.32 forpolyethylene.

series inductance associated with capaci-tors are neglected as secondary effects. Ifcareful attention is paid to circuit layoutand miniature components are used,lumped LC filter technology can be usedup to perhaps 1 GHz.

Transmission-line filters predominatefrom 500 MHz to 10 GHz. In addition theyare often used down to 50 MHz whennarrowband (QL > 10) band-pass filteringis required. In this application they exhibitconsiderably lower loss than their LCcounterparts.

Replacing lumped reactances with se-lected short sections of TEM transmissionlines results in transmission-line filters. In

12.20 Chapter 12

Fig 12.36 — Stub reactance for variouslengths of transmission line. Values arefor Z0 = 50 ΩΩΩΩΩ. For Z0 = 100 ΩΩΩΩΩ, double thetabulated values.

Coaxial transmission line possesses acharacteristic impedance given by:

d

Dlog

138Z0 (21)

A plot of Z0 vs D/d is shown in Fig 12.34.At RF, Z0 is an almost pure resistance. Ifthe distant end of a section of coax is ter-minated in Z0, then the impedance seenlooking into the input end is also Z0 at allfrequencies. A terminated section of coaxis shown in Fig 12.35A. If the distant endis not terminated in Z0, the input imped-ance will be some other value. In Fig12.35B the distant end is short-circuitedand the length is less than 1/4 λ. The inputimpedance is an inductive reactance asseen by the notation +j in the equation inpart B of the figure.

The input impedance for the case of theopen-circuit distant end, is shown in Fig12.35C. This case results in a capacitivereactance (–j). Thus, short sections of co-axial line (stubs) can replace the inductorsand capacitors in an LC filter. Coax lineinductive stubs usually have lower lossthan their lumped counterparts.

XL vs for shorted and open stubs isshown in Fig 12.36. There is an optimumvalue of Z0 that yields lowest loss, givenby

Fig 12.35 — Transmission line stubs.A: A line terminated in its characteristicimpedance. B: A shorted line less than1/4-λλλλλ long is an inductive stub. C: Anopen line less than 1/4-λλλλλ long is acapacitive stub.

Fig 12.37 — The Z0 of stripline varies with w, b and t (conductor thickness). SeeFig 12.33B. The conductor thickness is t and the plots are normalized in terms of t/b.

75Z0 (22)

If the dielectric is air, Z0 = 75 Ω. If thedielectric is polyethylene (ε = 2.32) Z0 =50 Ω. This is the reason why polyethylenedielectric flexible coaxial cable is usuallymanufactured with a 50-Ω characteristicimpedance.

The first transmission-line filters werebuilt from sections of coaxial line. Theirmechanical fabrication is expensive and itis difficult to provide electrical couplingbetween line sections. Fabrication diffi-culties are reduced by the use of shieldedstrip transmission line (stripline) shownin Fig 12.33B. The outer conductor ofstripline consists of two flat parallel metalplates (ground planes) and the inner con-ductor is a thin metal strip. Sometimes the

inner conductor is a round metal rod. Thedielectric between ground planes and stripcan be air or a low-loss plastic such as poly-ethylene. The outer conductors (groundplanes or shields) are separated from eachother by distance b.

Striplines can be easily coupled to-gether by locating the strips near eachother as shown in Fig 12.33B. StriplineZ0 vs width (w) is plotted in Fig 12.37.Air-dielectric stripline technology is bestfor low bandwidth (QL > 20) band-passfilters.

The most popular transmission line ismicrostrip (unshielded stipline), shown inFig 12.33C. It can be fabricated with stan-dard printed-circuit processes and is theleast expensive configuration. Unfortu-nately, microstrip is the lossiest of the threelines; therefore it is not suitable for narrowband-pass filters. In microstrip the outerconductor is a single flat metal ground-plane. The inner conductor is a thin metalstrip separated from the ground-plane by asolid dielectric substrate. Typical sub-strates are 0.062-inch G-10 fiberglass (ε =4.5) for the 50- MHz to 1-GHz frequencyrange and 0.031-inch Teflon (ε = 2.3) forfrequencies above 1 GHz.

Conductor separation must be mini-mized or free-space radiation and un-wanted coupling to adjacent circuits maybecome problems. Microstrip character-istic impedance and the effective dielec-tric constant (ε) are shown in Fig 12.38.Unlike coax and stripline, the effectivedielectric constant is less than that of thesubstrate since a portion of the electro-magnetic wave propagating along themicrostrip “sees” the air above the sub-strate.

The least-loss characteristic impedance


Fig 12.38 — Microstrip parameters(after H. Wheeler, IEEE Transactions onMTT, March 1965, p 132). εεεεεe is theeffective εεεεε.

Fig 12.40 — This Butterworth filter is constructed in combline. It was originallydiscussed by R. Fisher in December 1968 QST.

Fig 12.39 — This 146-MHz striplineband-pass filter has been measured tohave a QL of 63 and a loss ofapproximately 1 dB.

Dimension 52 MHz 146 MHz 222 MHzA 9" 7" 7"B 7" 9" 9"L 73/8" 6" 6"S 1" 11/16" 13/8"W 1" 15/8" 15/8"

Capacitance 52 MHz 146 MHz 222 MHz(pF)C1 110 22 12C2 135 30 15C3 110 22 12Cc 35 6.5 2.8QL 10 29 36PerformanceBW3 (MHz) 5.0 5.0 6.0Loss (dB) 0.6 0.7 —

for stripline and microstrip-lines is not75 Ω as it is for coax. Loss decreases as linewidth increases, which leads to clumsy,large structures. Therefore, to conservespace, filter sections are often constructedfrom 50-Ω stripline or microstrip stubs.

Transmission-Line Band-PassFilters

Band-pass filters can also be constructedfrom transmission-line stubs. At VHF thestubs can be considerably shorter than aquarter wavelength yielding a compactfilter structure with less midband lossthan its LC counterpart. The single-stage146-MHz stripline band-pass filter shownin Fig 12.39 is an example. This filter con-sists of a single inductive 50-Ω strip-linestub mounted into a 2 × 5 × 7-inch alumi-num box. The stub is resonated at 146 MHzwith the “APC” variable capacitor, C1.Coupling to the 50-Ω generator and load isprovided by the coupling capacitors Cc. Themeasured performance of this filter is: fo =146 MHz, BW = 2.3 MHz (QL = 63) andmidband loss = 1 dB.

Single-stage stripline filters can becoupled together to yield multistage fil-ters. One method uses the capacitorcoupled band-pass filter synthesis tech-nique to design a 3-pole filter. Anothermethod allows closely spaced inductive

stubs to magnetically couple to each other.When the coupled stubs are grounded onthe same side of the filter housing, thestructure is called a “combline filter.”Three examples of combline band-passfilters are shown in Fig 12.40. These fil-ters are constructed in 2 × 7 × 9-inch chas-sis boxes.

Quarter-Wave Transmission-LineFilters

Fig 12.41 shows that when = 0.25 λg,the shorted-stub reactance becomes infi-nite. Thus, a 1/4-λ shorted stub behaves likea parallel-resonant LC circuit. Properinput and output coupling to a 1/4-λ reso-nator yields a practical band-pass filter.Closely spaced 1/4-λ resonators will coupletogether to form a multistage band-passfilter. When the resonators are groundedon opposite walls of the filter housing, thestructure is called an “interdigital filter”because the resonators look like interlacedfingers. Two examples of 3-pole UHFinterdigital filters are shown in Fig 12.41.Design graphs for round-rod interdigitalfilters are given in Ref 16. The 1/4-λ reso-nators may be tuned by physically chang-ing their lengths or by tuning the screwopposite each rod.

If the short-circuited ends of two 1/4-λresonators are connected to each other, the

12.22 Chapter 12

Fig 12.41 — These 3-pole Butterworthfilters (upper: 432 MHz, 8.6 MHzbandwidth, 1.4 dB pass-band loss;lower: 1296 MHz, 110 MHz bandwidth,0.4 dB pass-band loss) are constructedas interdigitated filters. The material isfrom R. E. Fisher, March 1968 QST.

resulting 1/2-λ stub will remain in reso-nance, even when the connection toground-plane is removed. Such a floating1/2-λ microstrip line, when bent into aU-shape, is called a “hairpin” resonator.Closely coupled hairpin resonators can bearranged to form multistage band-passfilters. Microstrip hairpin band-pass fil-ters are popular above 1 GHz because theycan be easily fabricated using photo-etch-ing techniques. No connection to theground-plane is required.

Transmission-Line FiltersEmulating LC Filters

Low-pass and high-pass transmission-linefilters are usually built from short sections oftransmission lines (stubs) that emulatelumped LC reactances. Sometimes low-losslumped capacitors are mixed with transmis-sion-line inductors to form a hybrid compo-nent filter. For example, consider the720-MHz, 3-pole microstrip low-pass filtershown in Fig 12.42A that emulates the LCfilter shown in Fig 12.42B. C1 and C3 arereplaced with 50-Ω open-circuit shunt stubs

C long. L2 is replaced with a short section of100-Ω line Llong. The LC filter, Fig 12.42B,was designed for fc = 720 MHz. Such a filtercould be connected between a 432-MHztransmitter and antenna to reduce harmonicand spurious emissions. A reactance chartshows that XC is 50 Ω, and the inductor reac-tance is 100 Ω at fc. The microstrip version isconstructed on G-10 fiberglass 0.062-inchthick, with ε = 4.5. Then, from Fig 12.38, w is0.11 inch and C = 0.125 λg for the 50-Ωcapacitive stubs. Also, from Fig 12.38, w is0.024 inch and L is 0.125 g for the 100-Ωinductive line. The inductive line length isapproximate because the far end is not a shortcircuit. g is 300/(720)(1.75) = 0.238 m, or9.37 inches. Thus C is 1.1 inch and L is1.1 inch.

This microstrip filter exhibits about20 dB of attenuation at 1296 MHz. Itsresponse rises again, however, around3 GHz. This is because the fixed-lengthtransmission-line stubs change in terms ofwavelength as the frequency rises. Thisparticular filter was designed to eliminatethird-harmonic energy near 1296 MHzfrom a 432-MHz transmitter and does abetter job in this application than theButterworth filter in Fig 12.41, which hasspurious responses in the 1296-MHz band.


Fig 12.42 — A microstrip 3-poleemulated-Butterworth low-pass filterwith a cutoff frequency of 720 MHz. A:Microstrip version built with G-10fiberglass board (εεεεε = 4.5, h = 0.062inches). B: Lumped LC version of thesame filter. To construct this filter withlumped elements very small values of Land C must be used and straycapacitance and inductance must bereduced to a tiny fraction of thecomponent values.

Helical ResonatorsEver-increasing occupancy of the radio

spectrum brings with it a parade of re-ceiver overload and spurious responses.Overload problems can be minimized byusing high-dynamic-range receiving tech-niques, but spurious responses (such as theimage frequency) must be filtered outbefore mixing occurs. Conventional tunedcircuits cannot provide the selectivity nec-essary to eliminate the plethora of signalsfound in most urban and many suburbanneighborhoods. Other filtering techniquesmust be used.

Helical resonators are usually a betterchoice than 1/4-λ cavities on 50, 144 and222 MHz to eliminate these unwantedinputs. They are smaller and easier to build.In the frequency range from 30 to 100 MHzit is difficult to build high-Q inductors, andcoaxial cavities are very large. In this fre-quency range the helical resonator is anexcellent choice. At 50 MHz for example,a capacitively tuned, 1/4-λ coaxial cavitywith an unloaded Q of 3000 would be about4 inches in diameter and nearly 5 ft long.

On the other hand, a helical resonator withthe same unloaded Q is about 8.5 inches indiameter and 11.3 inches long. Even at432 MHz, where coaxial cavities are com-mon, the use of helical resonators results insubstantial size reductions.

The helical resonator was described byW1HR in a QST article as a coil sur-rounded by a shield, but it is actually ashielded, resonant section of helicallywound transmission line with relativelyhigh characteristic impedance and lowaxial propagation velocity. The electricallength is about 94% of an axial 1/4-λ or84.6°. One lead of the helical winding isconnected directly to the shield and theother end is open circuited as shown inFig 12.43. Although the shield may be anyshape, only round and square shields willbe considered here.

DESIGNThe unloaded Q of a helical resonator is

determined primarily by the size of theshield. For a round resonator with a copper

Fig 12.43 — Dimensions of round andsquare helical resonators. Thediameter, D (or side, S) is determinedby the desired unloaded Q. Otherdimensions are expressed in terms ofD or S (see text).

coil on a low-loss form, mounted in a cop-per shield, the unloaded Q is given by

0U f50DQ (23)

whereD = inside diameter of the shield, in

inchesfo = frequency, in MHz.

D is assumed to be 1.2 times the widthof one side for square shield cans. Thisformula includes the effects of losses andimperfections in practical materials. Ityields values of unloaded Q that are easilyattained in practice. Silver plating theshield and coil increases the unloaded Q

12.24 Chapter 12

Fig 12.44 — The design nomograph for round helical resonators starts by selecting QU and the required shield diameter. Aline is drawn connecting these two values and extended to the frequency scale (example here is for a shield of about 3.8inches and QU of 500 at 7 MHz). Finally the number of turns, N, winding pitch, P, and characteristic impedance, Z0, aredetermined by drawing a line from the frequency scale through selected shield diameter (but this time to the scale on theright-hand side. For the example shown, the dashed line shows P ≈≈≈≈≈ 0.047 inch, N = 70 turns, and Zn = 3600 ΩΩΩΩΩ).

by about 3% over that predicted by theequation. At VHF and UHF, however, it ismore practical to increase the shield sizeslightly (that is, increase the selected QUby about 3% before making the calcula-tion). The fringing capacitance at theopen-circuit end of the helix is about0.15 D pF (that is, approximately 0.3 pFfor a shield 2 inches in diameter). Oncethe required shield size has been deter-mined, the total number of turns, N, wind-ing pitch, P and characteristic impedance,Z0, for round and square helical resona-tors with air dielectric between the helixand shield, are given by:

Df

1908N

0(24A)

2312

DfP

20 (24B)

Df

99,000Z

00 (24C)

Sf

1590N

0(24D)

1606

SfP

20 (24E)

Sf

82,500Z

00 (24F)

In these equations, dimensions D andS are in inches and f0 is in megahertz.The design nomograph for round helicalresonators in Fig 12.44 is based on theseformulas.

Although there are many variables toconsider when designing helical resona-tors, certain ratios of shield size to lengthand coil diameter to length, provide opti-mum results. For helix diameter, d =0.55 D or d = 0.66 S. For helix length,b = 0.825D or b = 0.99S. For shieldlength, B = 1.325 D and H = 1.60 S.

Fig 12.45 simplifies calculation of thesedimensions. Note that these ratios resultin a helix with a length 1.5 times its diam-eter, the condition for maximum Q. Theshield is about 60% longer than the helix— although it can be made longer —to completely contain the electric field atthe top of the helix and the magnetic fieldat the bottom.

The winding pitch, P, is used primarily

to determine the required conductor size.Adjust the length of the coil to that givenby the equations during construction. Con-ductor size ranges from 0.4 P to 0.6 P forboth round and square resonators and areplotted graphically in Fig 12.46.

Obviously, an area exists (in terms offrequency and unloaded Q) where the de-signer must make a choice between a con-ventional cavity (or lumped LC circuit)and a helical resonator. The choice is af-fected by physical shape at higher frequen-cies. Cavities are long and relatively smallin diameter, while the length of a helicalresonator is not much greater than its di-ameter. A second consideration is thatpoint where the winding pitch, P, is lessthan the radius of the helix (otherwise thestructure tends to be nonhelical). Thiscondition occurs when the helix has fewerthan three turns (the “upper limit” on thedesign nomograph of Fig 12.44).

CONSTRUCTIONThe shield should not have any seams

parallel to the helix axis to obtain as highan unloaded Q as possible. This is usuallynot a problem with round resonators be-cause large-diameter copper tubing is used


Fig 12.46 — This chart provides the design information ofhelix conductor size vs winding pitch, P. For example, awinding pitch of 0.047 inch results in a conductor diameterbetween 0.019 and 0.028 inch (#22 or #24 AWG).

Fig 12.45 — The helical resonator is scaled from this designnomograph. Starting with the shield diameter, the helix diameter,d, helix length, b, and shield length, B, can be determined withthis graph. The example shown has a shield diameter of3.8 inches. This requires a helix mean diameter of 2.1 inches,helix length of 3.1 inches, and shield length of 5 inches.

for the shield, but square resonators re-quire at least one seam and usually more.The effect on unloaded Q is minimum ifthe seam is silver soldered carefully fromone end to the other.

Results are best when little or no dielec-tric is used inside the shield. This is usuallyno problem at VHF and UHF because theconductors are large enough that a support-ing coil form is not required. The lower endof the helix should be soldered to the near-est point on the inside of the shield.

Although the external field is mini-mized by the use of top and bottom shieldcovers, the top and bottom of the shieldmay be left open with negligible effect onfrequency or unloaded Q. Covers, if pro-vided, should make electrical contact withthe shield. In those resonators where thehelix is connected to the bottom cover, thatcover must be soldered solidly to theshield to minimize losses.

TUNINGA carefully built helical resonator de-

signed from the nomograph of Fig 12.44will resonate very close to the design fre-quency. Slightly compress or expand thehelix to adjust resonance over a smallrange. If the helix is made slightly longerthan that called for in Fig 12.45, the reso-

nator can be tuned by pruning the openend of the coil. However, neither of thesemethods is recommended for wide fre-quency excursions because any major de-viation in helix length will degrade theunloaded Q of the resonator.

Most helical resonators are tuned bymeans of a brass tuning screw or high-quality air-variable capacitor across theopen end of the helix. Piston capacitorsalso work well, but the Q of the tuningcapacitor should ideally be several timesthe unloaded Q of the resonator. Varactordiodes have sometimes been used whereremote tuning is required, but varactorscan generate unwanted harmonics andother spurious signals if they are excitedby strong, nearby signals.

When a helical resonator is to be tunedby a variable capacitor, the shield size isbased on the chosen unloaded Q at theoperating frequency. Then the number ofturns, N and the winding pitch, P, arebased on resonance at 1.5 f0. Tune theresonator to the desired operating fre-quency, f0.

INSERTION LOSSThe insertion loss (dissipation loss), IL,

in decibels, of all single-resonator circuitsis given by

Fig 12.47 — The ratio of loaded (QL) tounloaded (QU) Q determines the inser-tion loss of a tuned resonant circuit.

U

L10L

Q

Q1

1log20I (25)

whereQL = loaded QQU = unloaded Q

This is plotted in Fig 12.47. For the mostpractical cases (QL > 5), this can be closelyapproximated by IL ≈ 9.0 (QL/QU) dB. Theselection of QL for a tuned circuit is dic-tated primarily by the required selectivityof the circuit. However, to keep dissipa-tion loss to 0.5 dB or less (as is the case forlow-noise VHF receivers), the unloadedQ must be at least 18 times the QL.

12.26 Chapter 12

Band-Pass FilterPick K, Q, ωo = 2πfcwhere fc = center freq.Choose CThen

CKQ

1R00

C00K22Q

QR2

C2Q

3R0

Example:K = 2, fo = 800 Hz, Q = 5 and C = 0.022 μFR1 = 22.6 kΩ (use 22 kΩ)R2 = 942 Ω (use 910 Ω)R3 = 90.4 kΩ (use 91 kΩ)

High-Pass Filter

C1K8aa

4R

C2

1

122

C2

RC

1R

1K1K

KRR 1

3

14 KRR

whereK = gainfc = –3 dB cutoff frequencyωc = 2πfcC = a standard value near 10/fc

(in μF)

Note: For unity gain, short R4 and omitR3.

Example:a = 0.765 (see table, first of two stages)K = 4f = 250 Hzωc = 1570.8C = 0.04 μF (use 0.039 μF)R1 = 11,123.2 Ω (use 11 kΩ)R2 = 22,722 Ω (use 22 kΩ)R3 = 14,830.9 Ω (use 15 kΩ)R4 = 44,492.8 Ω (use 47 kΩ)

Fig 12.48 — This response curve fora single-resonator 432-MHz filtershows the effects of capacitive andinductive input/output coupling. Theresponse curve can be madesymmetrical on each side ofresonance by combining the twomethods (inductive input andcapacitive output, or vice versa).

Unless otherwise specified, values of R are in ohms, C is in farads, F in hertz and ωωωωω in radians per second. Calculationsshown here were performed on a scientific calculator.

Low-Pass Filter

4C1K4a

C 22

1

C2C14C2

2C1K42a2aC

21R

2C121

2RCC

1R

1K1K

RRKR 21

3

214 RRKR

whereK = gainfc = –3 dB cutoff frequencyωc = 2πfcC2 = a standard value near 10/fc (in μF)

Note: For unity gain, short R4 and omit R3.

Example:a = 1.414 (see table, one stage)K = 2f = 2700 Hzωc = 16,964.6 rads/secC2 = 0.0033 μFC1 ≤ 0.00495 μF (use 0.0050 μF)R1 ≤ 25,265.2 Ω (use 24 kΩ)R2 = 8,420.1 Ω (use 8.2 kΩ)R3 = 67,370.6 Ω (use 68 kΩ)R4 = 67,370.6 Ω (use 68 kΩ)

Fig 12.49 — Equations for designing a low-pass RC active audio filter are given at A. B, C and D show design information forhigh-pass, band-pass and band-reject filters, respectively. All of these filters will exhibit a Butterworth response. Values of Kand Q should be less than 10.

COUPLINGSignals are coupled into and out of he-

lical resonators with inductive loops at thebottom of the helix, direct taps on the coilor a combination of both. Although thecorrect tap point can be calculated easily,coupling by loops and probes must be de-termined experimentally.

The input and output coupling is oftenprovided by probes when only one resona-tor is used. The probes are positioned onopposite sides of the resonator for maxi-mum isolation. When coupling loops are


Factor “a” for Low- and High-Pass Filters

No. ofStages Stage 1 Stage 2 Stage 3 Stage 4

1 1.414 — — —2 0.765 1.848 — —3 0.518 1.414 1.932 —4 0.390 1.111 1.663 1.962

These values are truncated from those of Appendix C of Ref 21, for even-orderButterworth filters.

Band-Reject Filter

R1C121

F0

4Q

11K

R1K1R

where

0fF10

2C3

C21C

R1 = R2 = 2R3R4 = (1 – K)RR5 = K × R

Example:f0 = 500 Hz, Q=10K = 0.975C1 = C2 = 0.02 μF (or use 0.022 μF)C3 = 0.04 μF (or use 0.044 μF)R1 = R2 = 15.92 kΩ (use 15 kΩ)R3 = 7.96 kΩ (use 8.2 kΩ)R >> 398 Ω (390 Ω)R4 = 9.95 Ω (use 10 Ω)R5 = 388.1 Ω (use 390 Ω)

used, the plane of the loop should beperpendicular to the axis of the helixand separated a small distance from thebottom of the coil. For resonators withonly a few turns, the plane of the loop canbe tilted slightly so it is parallel with theslope of the adjacent conductor.

Helical resonators with inductive cou-pling (loops) exhibit more attenuation tosignals above the resonant frequency(as compared to attenuation below reso-nance), whereas resonators with capaci-tive coupling (probes) exhibit more

attenuation below the passband, asshown for a typical 432-MHz resonatorin Fig 12.48. Consider this characteristicwhen choosing a coupling method. Thepassband can be made more symmetricalby using a combination of coupling meth-ods (inductive input and capacitive out-put, for example).

If more than one helical resonator isrequired to obtain a desired band-passcharacteristic, adjacent resonators may becoupled through apertures in the shieldwall between the two resonators. Unfortu-nately, the size and location of the aper-ture must be found empirically, so thismethod of coupling is not very practicalunless you’re building a large number ofidentical units.

Since the loaded Q of a resonator isdetermined by the external loading, thismust be considered when selecting a tap(or position of a loop or probe). The ratioof this external loading, Rb, to the charac-teristic impedance, Z0, for a 1/4-λ resona-tor is calculated from:

UL0

b

Q

1

Q

10.785

Z

RK (26)

Even when filters are designed and builtproperly, they may be rendered totallyineffective if not installed properly. Leak-age around a filter can be quite high atVHF and UHF, where wavelengths areshort. Proper attention to shielding andgood grounding is mandatory for mini-mum leakage. Poor coaxial cable shieldconnection into and out of the filter is oneof the greatest offenders with regard tofilter leakage. Proper dc-lead bypassingthroughout the receiving system is goodpractice, especially at VHF and above.Ferrite beads placed over the dc leads mayhelp to reduce leakage. Proper filter ter-mination is required to minimize loss.

Most VHF RF amplifiers optimized fornoise figure do not have a 50-Ω input im-pedance. As a result, any filter attached tothe input of an RF amplifier optimized for

noise figure will not be properly termi-nated and filter loss may rise substantially.As this loss is directly added to the RFamplifier noise figure, carefully chooseand place filters in the receiver.

ACTIVE FILTERSPassive HF filters are made from com-

binations of inductors and capacitors.These may be used at low frequencies, butthe inductors often become a limiting fac-tor because of their size, weight, cost andlosses. The active filter is a compact, low-cost alternative made with op amps, resis-tors and capacitors. They often occupy afraction of the space required by an LCfilter. While active filters have been tradi-tionally used at low and audio frequen-cies, modern op amps with small-signalbandwidths that exceed 1 GHz have ex-tended their range into MF and HF.

Active filters can perform any commonfilter function: low pass, high pass,bandpass, band reject and all pass (usedfor phase or time delay). Responses suchas Butterworth, Chebyshev, Bessel andelliptic can be realized. Active filters canbe designed for gain, and they offer excel-lent stage-to-stage isolation.

Despite the advantages, there are alsosome limitations. They require power, andperformance may be limited by the opamp’s finite input and output levels, gainand bandwidth. While LC filters can bedesigned for high-power applications,active filters usually are not.

The design equations for various fil-ters are shown in Fig 12.49. Fig 12.50shows a typical application of a two-stage, bandpass filter. A two-stage filteris considered the minimum acceptable forCW, while three or four stages will provemore effective under some conditions ofnoise and interference.

CRYSTAL-FILTER EVALUATIONCrystal filters, such as those described

earlier in this chapter, are often con-structed of surplus crystals or crystals

12.28 Chapter 12

Fig 12.50 — Typical application of a two-stage active filter in the audio chain of a QRP CW tranceiver. The filter can bebypassed, or another filter can be switched in by S1.

whose characteristics are not exactlyknown. Randy Henderson, WI5W, devel-oped a swept frequency generator for test-ing these filters. It was first described inMarch 1994 QEX. This test instrumentadds to the ease and success in quicklybuilding filters from inexpensive micro-processor crystals.

A template, containing additional infor-mation, is available on the CD includedwith this book.

An OverviewThe basic setup is shown in Fig 12.51A.

The VCO is primarily a conventionalLC-tuned Hartley oscillator with its fre-quency tuned over a small range by avaractor diode (MV2104 in part B of thefigure). Other varactors may be used aslong as the capacitance specificationsaren’t too different. Change the 5-pF cou-pling capacitor to expand the sweep widthif desired.

The VCO signal goes through a bufferamplifier to the filter under test. The filter isfollowed by a wide-bandwidth amplifier andthen a detector. The output of the detector isa rectified and filtered signal. This varying

dc voltage drives the vertical input of anoscilloscope. At any particular time, thedeflection and sweep circuitry commandsthe VCO to “run at this frequency.” Thesame deflection voltage causes the oscillo-scope beam to deflect left or right to a posi-tion corresponding to the frequency.

Any or all of these circuits may be elimi-nated by the use of appropriate commer-cial test equipment. For example, acommercial sweep generator would elimi-nate the need for everything but the wide-band amplifier and detector. Motorola,Mini-Circuits Labs and many others selldevices suitable for the wide-band ampli-fiers and detector.

The generator/detector system coversapproximately 6 to 74 MHz in threeranges. Each tuning range uses a separateRF oscillator module selected by switchS1. The VCO output and power-supplyinput are multiplexed on the “A” lead toeach oscillator. The tuning capacitance foreach VCO is switched into the appropriatecircuit by a second set of contacts on S1.CT is the coarse tuning adjustment for eachoscillator module.

Two oscillator coils are wound on PVC

plastic pipe. The third, for the highest fre-quency range, is self-supporting #14 cop-per wire. Although PVC forms with SuperGlue dope may not be “state of the art”technology, frequency stability is com-pletely adequate for this instrument.

The oscillator and buffer stage operateat low power levels to minimize frequencydrift caused by component heating. Crys-tal filters cause large load changes as thefrequency is swept in and out of the pass-band. These large changes in impedancetend to “pull” the oscillator frequency andcause inaccuracies in the passband shapedepicted by the oscilloscope. Therefore abuffer amplifier is a necessity. The wide-band amplifier in Fig 12.52 is derivedfrom one in ARRL’s Solid State Designfor the Radio Amateur.

S2 selects a 50-Ω, 10-dB attenuator inthe input line. When the attenuator is inthe line, it provides a better output matchfor the filter under test. The detector usessome forward bias for D2. A simple unbi-ased diode detector would offer about50 dB of dynamic range. Some dc biasincreases the dynamic range to almost70 dB. D3, across the detector output (the


Fig 12.51 — The test set block diagram, lower left, starts with a swept frequency oscillator, shown in the schematic. If acommercial swept-frequency oscillator is available, it can be substituted for the circuit shown.

scope input), increases the vertical-am-plifier sensitivity while compressing orlimiting the response to high-level sig-nals. With this arrangement, high levelsof attenuation (low-level signals) areeasier to observe and low attenuation lev-els are still visible on the CRT. The diodeonly kicks in to provide limiting at highersignal levels.

The horizontal-deflection sweep cir-cuit uses a dual op-amp IC (see Fig12.53). One section is an oscillator; theother is an integrator. The integrator out-put changes linearly with time, giving auniform brightness level as the trace ismoved from side to side. Increasing C1decreases the sweep rate. Increasing C2decreases the slope of the output wave-form ramp.

OperationThe CRT is swept in both directions,

left to right and right to left. The displayed

Table 12.5VCO Coils

Coil Inside Diameter Length Turns, Wire Inductance(inches) (inches) (μH)

large 0.85 1.1 18 t, #28 5.32medium 0.85 0.55 7 t, #22 1.35small 0.5 0.75 5 t, #14 0.27

The two larger coils are wound on 3/4-inch PVC pipe and the smaller one on a 1/2-inchdrill bit. Tuning coverage for each oscillator is obtained by squeezing or spreading theturns before gluing them in place. The output windings connected to A1, A2 and A3 areeach single turns of #14 wire spaced off the end of the tapped coils. The taps areapproximate and 25 to 30% of the full winding turns — up from the cold (ground) end ofthe coils.

curve is a result of changes in frequency,not time. Therefore it is unnecessary toincorporate the usual right-to-left, snap-back and retrace blanking used in oscillo-scopes.

S3 in Fig 12.53 disables the automaticsweep function when opened. This per-mits manual operation. Use a frequencycounter to measure the VCO output, fromwhich bandwidth can be calculated. Turn

12.30 Chapter 12

Fig 12.53 — The sweep generator provides both an up and down sweep voltage(see text) for the swept frequency generator and the scope horizontal channel.

Fig 12.52 — The filter under test is connected to Q3 onthe right side of the schematic. The detector output, onthe left side, connects to the oscilloscope vertical input.A separate voltage regulator, an LM 317, is used to powerthis circuit. Q3, Q4, Q5 and Q6 are 25C1424 or 2N2857.

the fine-tune control to position the CRTbeam at selected points of the pass-bandcurve. The difference in frequency read-

ings is the bandwidth at that particularpoint or level of attenuation.

Substitution of a calibrated attenuator

for the filter under test can providereference readings. These reference read-ings may be used to calibrate an other-wise uncalibrated scope vertical displayin dB.

The buffer amplifier shown here is setup to drive a 50-Ω load, and the wide-bandwidth amplifier input impedance isabout 50 Ω. If the filter is not a 50-Ω unit,however, various methods can be used toaccommodate the difference. For ex-ample, a transformer may be used forwidely differing impedance levels,whereas a minimum-loss resistive padmay be preferable where impedancelevels differ by a factor of approximately1.5 or less — presuming some loss is ac-ceptable.

ReferencesA. Ward, “Monolithic Microwave In-

tegrated Circuits,” Feb 1987 QST, pp23-29.

Z. Lau, “A Logarithmic RF Detector forFilter Tuning,” Oct 1988 QEX, pp 10-11.

BAND-PASS FILTERS FOR 144 OR 222 MHZ

Spectral purity is necessary duringtransmitting. Tight filtering in a receivingsystem ensures the rejection of out-of-band signals. Unwanted signals that leadto receiver overload and increasedintermodulation-distortion (IMD) prod-

ucts result in annoying in-band “birdies.”One solution is the double-tuned band-pass filters shown in Fig 12.54. They weredesigned by Paul Drexler, WB3JYO. Eachincludes a resonant trap coupled betweenthe resonators to provide increased rejec-

tion of undesired frequencies.Many popular VHF conversion

schemes use a 28-MHz intermediate fre-quency (IF), yet proper filtering of theimage frequency is often overlooked inamateur designs. The low-side injection


Fig 12.56 — Filter response plot of the222-MHz band-pass filter, with animage-reject notch for a 28 MHz IF.

Fig 12.54 — Schematic of the band-pass filter. Components must bechosen to work with the power levelof the transmitter.

Fig 12.55 — Filter response plot of the144-MHz band-pass filter, with animage-reject notch for a 28 MHz IF.

Component Values

144 MHz 220 MHzC2 1 pF 1 pFC1, C3 1-7 pf piston 1-7 pF pistonL2 27t no. 26 enam 15t no. 24

on T37-10 enam on T44-10L1, L3 7t no. 18, 1/4-in 4t no. 18, 1/4-in

ID, tap 11/2t ID, tap 11/2t

Switched Capacitor FiltersThe switched capacitor filter, or SCF,

uses an IC to synthesize a high-pass,low-pass, band-pass or notch filter. Theperformance of multiple-pole filters isavailable, with Q and bandwidth set byexternal resistors. An external clock fre-

quency sets the filter center frequency,so this frequency may be easily changedor digitally controlled. Dynamic rangeof 80 dB, Q of 50, 5-pole equivalentdesign and maximum usable frequencyof 250 kHz are available for such uses as

audio CW and RTTY filters. In addition,all kinds of digital tone signaling suchas DTMF and modem encoding and de-coding are being designed with thesecircuits.

frequency used in 144-MHz mixingschemes is 116 MHz and the image fre-quency, 88 MHz, falls in TV channel 6.Inadequate rejection of a broadcast carrierat this frequency results in a strong,wideband signal at the low end of the 2-mband. A similar problem on the transmitside can cause TVI. These band-pass fil-ters have effectively suppressed undesired

mixing products. See Fig 12.55 and 12.56.The circuit is constructed on a double-

sided copper-clad circuit board. Mini-mize component lead lengths to eliminateresistive losses and unwanted stray cou-pling. Mount the piston trimmers throughthe board with the coils soldered to theopposite end, parallel to the board. Theshield between L1 and L3 decreasesmutual coupling and improves the fre-quency response. Peak C1 and C3 foroptimum response.

L1, C1, L3 and C3 form the tank cir-cuits that resonate at the desired fre-quency. C2 and L2 reject the undesired

energy while allowing the desired signalto pass. The tap points on L1 and L3 pro-vide 50-Ω matching; they may be ad-justed for optimum energy transfer.Several filters have been constructedusing a miniature variable capacitor inplace of C2 so that the notch frequencycould be varied.

AN EASY-TO-BUILD, HIGH-PERFORMANCE PASSIVE CW FILTERModern commercial receivers for ama-

teur radio applications have featured CWfilters with digital signal processing(DSP) circuits. These DSP filters provideexceptional audio selectivity with theadded advantages of letting the userchange the filter’s center frequency andbandwidth. Yet in spite of these improve-ments, many hams are dissatisfied withDSP filters due to increased distortion of

the CW signal and the presence of a con-stant low-level, wide-band noise at theaudio output. One way to avoid this dis-tortion and noise is to switch to a selectivepassive filter that generates no noise! Al-though the center frequency and band-width of the passive filter is fixed andcannot be changed, this is not a seriousproblem once a center frequency preferredby the user is chosen. The bandwidth can

be made narrow enough for good selectiv-ity with no ringing that frequently occurswhen the bandwidth is too narrow. Thispassive CW filter project was designed,built and refined over many years byARRL Technical Advisor Edward E.Wetherhold, W3NQN.

The effectiveness of an easy-to-build,high-performance passive CW filter inproviding distortion-free and noise-free

12.32 Chapter 12

Fig 12.57 — Schematic diagram of the five-resonator CW filter. See Table 12.6 for capacitor and inductor values to build afilter with a center frequency of 546, 600, 700, 750 or 800 Hz.

P1 — Phone plug to match yourreceiver audio output jack.

J1 — Phone jack to match yourheadphone.

R1 — 6.8 to 50 ohms, 1/4-W, 10%resistor (see text).

S1 — DPDT switch.T1, T2 — 200 to 8-ΩΩΩΩΩ impedance-

matching transformers, 0.4-W,Miniature Core Type EI-24, MouserNo. 42TU200.

Note: The circled numbers identify thecircuit nodes corresponding to thesame nodes labeled in the pictorialdiagram in Fig 12.58.

CW reception — when compared withseveral commercial amateur receivers us-ing DSP filtering — was experienced bySteve Root, KØSR. He reported that whenhe replaced his DSP filter with the passiveCW filter that he assembled, he had theimpression that the signals in the filterpassband were amplified. In reality, thenoise floor appeared to drop one or twodB. When attempting to hear low-levelDX CW signals, Steve now prefers thepassive CW filter over DSP filters.1 TheCW filter assembled and used by KØSR isthe passive five-resonator CW filter thathas been widely published in many Hand-books and magazines since 1980, and mostrecently in Rich Arland’s, K7SZ, QRPcolumn in the May 2002 issue of QST (seereferences 2-11 at the end of this text).

If you want to build the high-perfor-mance passive five-resonator CW filter andexperience no-distortion and no-noise CWreception, this article will show you how.

This inductor-capacitor CW filter usesone stack of 85-mH inductors and twomodified separate inductors in a five-reso-nator circuit that is easy to assemble, giveshigh performance and is low cost. Al-though these inductors have been referredto as “88 mH” over the past 25 years, theiractual value is closer to 85 mH, and forthat reason the designs presented in thisarticle are based on an inductor value of85 mH.

Five bandpass filter designs for centerfrequencies between 546 Hz and 800 Hzare listed in Table 12.6. Select the centerfrequency that matches your transceiversidetone frequency. If you are using a di-

rect conversion receiver or an old receiverwith a BFO, you may select any of thedesigns having a center frequency that youfind easy on your ears. The author canprovide a kit of parts with detailed instruc-tions for assembling this filter at a nomi-nal cost. For contact information, see theend of this text.

The actual 3-dB bandwidth of the filtersis between 250 and 270 Hz depending onthe center frequency. This bandwidth isnarrow enough to give good selectivity,and yet broad enough for easy tuning withno ringing. Five high-Q resonators pro-vide good skirt selectivity that is adequatefor interference-free CW reception.Simple construction, low cost and goodperformance make this filter an ideal first

project for anyone interested in puttingtogether a useful station accessory, pro-vided you operate CW mode of course!

DESIGNS AND INTERFACINGFig 12.57 shows the filter schematic

diagram. Component values are given inTable 12.6 for five center-frequency de-signs. All designs are to be terminated inan impedance between 200 and 230 Ωand standard commercial 8-Ω to 200-Ωaudio transformers are used to match thefilter input and output to the 8-Ω audiooutput jack on your receiver — and to an8-Ω headset. Details are discussed a bitlater in this text to interface using head-phones with other than 8-Ω impedancesthat are now quite common.

Table 12.6CW Filter Using One 85-mH Inductor Stack and Two Modified 85-mHInductors

Center Freq. (Hz) 546 600 700 750 800

C1, C5 (nF) 1000 828 608 530 466C2, C4 (μF) 1.0 1.0 1.0 1.0 1.0C3 (nF) 333 276 202.7 176.5 155L2, L4 (mH) 85 70.36 51.69 45.0 39.6Remove Turns* NONE 66 160 200 232

*The total number of turns removed, split equally from each of the two windings of L2. Do thesame also for L4. (E.g., for a 700-Hz center frequency, remove 80 turns from each of the twowindings of L2, for a total of 160 turns removed from L2. Repeat exactly for L4.)

For all designs: L1, L5 = 85 mH; L3 = 3 (85 mH) = 255 mH; T1,T2 = 8/200 Ω CT; R1 = 6.8 to50 Ω. Although the surplus inductors are commonly considered to be 88 mH, the actual valueis closer to 85 mH. For this reason, all designs are based on the 85-mH value. L2 and L4have white cores, Magnetic Part No. 55347, OD Max = 24.3 mm, ID Min = 13.77 mm, HT =9.70 mm; μ = 200, AL = 169 mH/1000T ±8%. The calculated 3-dB BW is 285 Hz and is the samefor all designs; however, the actual bandwidth is 5 to 10-percent narrower dependingon the inductor Q at the edges of the filter passband.


Fig 12.58 — Part A shows a pictorial diagram of the lead-connection details for L2 and L4. Part B shows the filterwiring diagram, including the inductor stack wiring of L1,L3 and L5. Part C is a photo of the assembled filterinstalled in a Jameco H2581 plastic box. The bypassswitch (S1) and input/output transformers (T1, T2) are onthe right side of the box.

(C)

CONSTRUCTIONThe encircled numbers in Fig 12.57 indi-

cate the filter circuit nodes for reference.Fig 12.58A shows the L2 and L4 inductorlead connections for the 546-Hz designwhere no turns need to be removed; the twoinductors are used in their original condi-tion. For all other designs, turns need to beremoved from each of the windings. Thenumber of turns requiring removal from theL2 and L4 windings is listed in Table 12.6.

Fig 12.58B shows a pictorial of the filterassembly and the connections between thecapacitors and the 85-mH stack terminals.Inductors L1, L3 and L5 are containedwithin the inductor stack and are intercon-nected using the terminal lugs on the stackas shown in the pictorial diagram. The en-circled numbers show the circuit nodescorresponding to those in Fig 12.57.

After the correct number of turns are re-moved from L2 and L4, the leads are gentlyscraped until you see copper and then thestart lead (with sleeving) of one winding isconnected to the finish lead of the otherwinding to make the center tap. The centertap lead and the other start and finish leadsof L2 and L4 are connected as indicated inFig 12.58B. L2 and L4 are fastened to op-posite ends of the stack with clear siliconesealant that is available in a small tube atlow cost from your local hardware store.Use the silicone sealant to fasten C2 and C4to the side of the stack. The capacitor leads

Table 12.7Node-to-Node Resistances for the 546-Hz CW Filter

Nodes Component ResistanceFrom To Designation (ohms ±20%)1 GND T1 hi-Z winding 122 GND L1 + 1/2(L2) 123 GND L2 84 GND 1/2(L2) 45 GND L3 + 1/2(L4) 286 GND 1/2(L4) 47 GND L4 88 GND L5 + 1/2(L4) 129 GND T2 hi-Z winding 122 4 L1 85 6 L3 246 8 L5 82 3 L1 + 1/2(L2) 128 7 L5 + 1/2(L4) 12

Notes1. See Figs 12.57 and 12.58 for the filter node locations.2. Check your wiring using the resistance values in this table. If there is a significant difference

between your measured values and the table values, you have a wiring error that must becorrected!

3. The resistances of L2 and L4 in the four other filters will be somewhat less than the 546-Hzvalues.

For accurate measurements, use a high-quality digital ohmmeter.

of C1, C3 and C5 are adequate to supportthe capacitors when their leads are solderedto the stack terminals. Fig 1C is a photo ofthe assembled filter installed in a Jamecoplastic box. Transformers T1 and T2 aresecured to the bottom of the plastic box withmore silicone sealant and are placed onopposite sides of the DPDT switch. See the

photograph for the placement of the phonejack and plug.

After the stack and capacitor wiring iscompleted, the correctness of the wiring ischecked before installing the stack in thebox. To do this, check the measured node-to-node resistances of the filter with thevalues listed in Table 12.7.

12.34 Chapter 12

Fig 12.59 — Measured attenuation responses of the 546- and 750-Hz filters. Theresponses are plotted relative to the zero dB attenuation levels at the centerfrequencies of the filters. The other filter response curves are similar, but centeredat their design frequency.

INTERFACING TO SOURCE ANDLOAD

The T1 and T2 transformers match thefilter to the receiver low-impedance audiooutput and to an 8-ohm headset or speaker.If your headset impedance is greater than200 Ω, omit T2 and connect a 1/2-watt re-sistor from node 9 (C5 output lead) toground. Choose the resistor so the parallelcombination of the headset impedance andthe resistor gives the correct filter termi-nation impedance (within about 10% of230 Ω).

PERFORMANCEThe measured 30-dB and 3-dB band-

widths of the 750-Hz filter are about 567and 271 Hz, respectively. The 30/3-dBshape factor is 2.09. Use this factor tocompare the selectivity performance ofthis filter with others. Fig 12.59 showsthe measured relative attenuation re-sponses of the 546-Hz and 750-Hz filters.

These responses were measured in a200-Ω system without the transformers.All attenuation levels were measuredrelative to a zero-dB attenuation level atthe filter center frequency.

The measured insertion loss of thesepassive filters with transformers isslightly less than 3 dB and this is typical offilters of this type. This small loss is com-pensated by slightly increasing the re-ceiver audio gain.

R1 is selected to maintain a relativelyconstant audio level when the filter isswitched in or out of the circuit. The correctvalue of R1 for your audio system shouldbe determined by experiment and probablywill be between 6.8 and 50 Ω. Start with ashort circuit across the S1A and B termi-nals and gradually increase the resistanceuntil the audio level appears to be the samewith the filter in or out of the circuit.

Thousands of hams have constructedthis five-resonator filter, and many have

commented on its ease of assembly, ex-cellent performance and lack of hiss andringing!

ORDERING PARTS/CONTACTINGTHE AUTHOR

The author can provide a kit of partswith detailed instructions for assemblingthis filter at a nominal cost. The kit in-cludes an inductor stack and two induc-tors, a pre-punched plastic box with aplastic mounting clip for the inductorstack, five matched capacitors, two trans-formers, a phone plug and jack and aminiature DPDT switch. Write to EdWetherhold, W3NQN, 1426 Catlyn Place,Annapolis, MD 21401-4208 for detailsabout parts and prices. Be sure to includea self-addressed, stamped 91/2 × 4-inch en-velope with your request.

Notes1Private correspondence from Steve Root,

KØSR, in his letter to the author dated 5September 2002. (Permission was re-ceived to publish his comments.)

21994 ARRL Handbook, 71st edition, RobertSchetgen, KU7G, Editor, pp 28-1,-2,Simple High-Performance CW Filter.

3Radio Handbook, 23rd edition, W. Orr,W6SAI, Editor, Howard W. Sams & Co.,1987 (1-Stack CW Filter), p 13-4,-5,-6.

4Wetherhold, “Modern Design of a CW Filterusing 88- and 44-mH Surplus Inductors,”QST, Dec. 1980, pp 14-19 and Feedback,QST, Jan 1981, p 43.

5Wetherhold, “High-Performance CW Filter,”Ham Radio, Apr 1981, pp 18-25.

6Wetherhold, “CW and SSB Audio FiltersUsing 88-mH Inductors,” QEX, Dec 1988,pp 3-10.

7Wetherhold, “A CW Filter for the RadioAmateur Newcomer,” RADIO COMMUNI-CATION (Radio Society of Great Britain),Jan 1985, pp 26-31.

8Wetherhold, “Easy-to-Build One-Stack CWFilter Has High Performance and LowCost,” SPRAT (Journal of the G-QRPClub), Issue No. 54, Spring 1988, p 20.

9Piero DeGregoris, I3DGF, “Un Facile FiltroCW ad alte prestazioni e basso costo,”Radio Rivista 12-93, pp 44, 45.

10QRP Power, Rich Arland, K7SZ, contribut-ing editor, May 2002 QST, p 96, PassiveCW Filters.

11Ken Kaplan, WB2ART, “Building theW3NQN Passive Audio Filter,” The Key-note, Issue 7, 2002, pp 16-17, Newsletterof FISTS CW Club.

12MPP Cores for Filter and Inductor Applica-tions, MAGNETICS 1991 Catalog, Butler,PA, p 64.


A BC-BAND ENERGY-REJECTION FILTERInadequate front-end selectivity or

poorly performing RF amplifier and mixerstages often result in unwanted cross-talkand overloading from adjacent commer-cial or amateur stations. The filter shownis inserted between the antenna and re-ceiver. It attenuates the out-of-band sig-nals from broadcast stations but passessignals of interest (1.8 to 30 MHz) withlittle or no attenuation.

The high signal strength of local broad-cast stations requires that the stop-bandattenuation of the high-pass filter also behigh. This filter provides about 60 dB ofstop-band attenuation with less than 1 dBof attenuation above 1.8 MHz. The filterinput and output ports match 50 Ω with amaximum SWR of 1.353:1 (reflection co-efficient = 0.15). A 10-element filter yields

Fig 12.61 — The filter fits easily in a 2 ×2 × 5-inch enclosure. The version in thephoto was built on a piece of perfboard.

Fig 12.60 — Schematic, layout and response curve of the broadcast bandrejection filter.

adequate stop-band attenuation and a rea-sonable rate of attenuation rise. The designuses only standard-value capacitors.

BUILDING THE FILTERThe filter parts layout, schematic dia-

gram, response curve and component val-ues are shown in Fig 12.60. The standardcapacitor values listed are within 2.8% ofthe design values. If the attenuation peaks(f2, f4 and f6) do not fall at 0.677, 1.293and 1.111 MHz, tune the series-resonantcircuits by slightly squeezing or separat-ing the inductor windings.

Construction of the filter is shownin Fig 12.61. Use Panasonic NP0 ceramicdisk capacitors (ECC series, class 1) orequivalent for values between 10 and270 pF. For values between 330 pF and

0.033 μF, use Panasonic P-series poly-propylene (type ECQ-P) capacitors. Thesecapacitors are available through Digi-Keyand other suppliers. The powdered-ironT-50-2 toroidal cores are availablethrough Amidon, Palomar Engineers andothers.

For a 3.4-MHz cutoff frequency, dividethe L and C values by 2. (This effectivelydoubles the frequency-label values in Fig12.60.) For the 80-m version, L2 throughL6 should be 20 to 25 turns each, woundon T-50-6 cores. The actual turns requiredmay vary one or two from the calculatedvalues. Parallel-connect capacitors asneeded to achieve the nonstandard capaci-tor values required for this filter.

FILTER PERFORMANCEThe measured filter performance is

shown in Fig 12.60. The stop-band attenu-ation is more than 58 dB. The measuredcutoff frequency (less than 1 dB attenua-tion) is under 1.8 MHz. The measuredpassband loss is less than 0.8 dB from 1.8to 10 MHz. Between 10 and 100 MHz, theinsertion loss of the filter gradually in-creases to 2 dB. Input impedance wasmeasured between 1.7 and 4.2 MHz. Overthe range tested, the input impedance ofthe filter remained within the 37 to 67.7-Ωinput-impedance window (equivalent to amaximum SWR of 1.353:1).

12.36 Chapter 12

A WAVE TRAP FOR BROADCAST STATIONS

Nearby medium-wave broadcast stationscan sometimes cause interference to HFreceivers over a broad range of frequencies.This being the case, set a trap to catch theunwanted frequencies.

OPERATIONThe way the circuit works is quite

simple. Referring to Fig 12.62, you can seethat it consists essentially of only two com-ponents, a coil L1 and a variable capacitorC1. This series-tuned circuit is connectedin parallel with the antenna circuit of thereceiver. The characteristic of a series-tuned circuit is that the coil and capacitorhave a very low impedance (resistance) tofrequencies very close to the frequency towhich the circuit is tuned. All other fre-quencies are almost unaffected. If the cir-cuit is tuned to 1530 kHz, for example, thesignals from a broadcast station on that fre-quency will flow through the filter toground, rather than go on into the receiver.All other frequencies will pass straight intothe receiver. In this way, any interferencecaused in the receiver by the station on1530 kHz is significantly reduced.

CONSTRUCTIONThis is a series-tuned circuit that is adjust-

able from about 540 kHz to 1600 kHz. Itis built into a metal box, Fig 12.63, to shieldit from other unwanted signals and is con-nected as shown in Fig 12.62. To make theinductor, first make a former by winding twolayers of paper on the ferrite rod. Fix thisin place with black electrical tape. Next, layone end of the wire for the coil on top ofthe former, leaving about an inch of wireprotruding beyond the end of the ferrite rod.Use several turns of electrical tape to securethe wire to the former. Now, wind the coilalong the former, making sure the turns arein a single layer and close together. Leavean inch or so of wire free at the end of thecoil. Once again, use a couple of turns ofelectrical tape to secure the wire to theformer. Finally, remove half an inch ofenamel from each end of the wire.

Alternatively, if you have an old AMtransistor radio, a suitable coil can usuallybe recovered already wound on a ferriterod. Ignore any small coupling coils. Drillthe box to take the components, then fitthem in and solder together as shown inFig 12.64. Make sure the lid of the box isfixed securely in place, or the wave trap’sperformance will be adversely affected bypick-up on the components.

CONNECTION AND ADJUSTMENTConnect the wave trap between the

antenna and the receiver, then tune C1

Components List

InductorL1 80 turns of 30 SWGenamelled wire, woundon a ferrite rod

Fig 12.62 — The wave trap consistsof a series tuned circuit, which‘shunts’ signals on an unwantedfrequency to ground.

Fig 12.63 — The wave trap can beroughly calibrated to indicate thefrequency to which it is tuned.

Fig 12.64 — Wiring of the wave trap. The ferrite rod is held in place withcable clips.

CapacitorC1 300 pF polyvariconvariable

Associated itemsCase (die-cast box)Knob to suitSockets to suitNuts and boltsPlastic cable clips

until the interference from the offendingbroadcast station is a minimum. You maynot be able to eliminate interference com-pletely, but this handy little device shouldreduce it enough to listen to the amateurbands. Lets say you live near an AM trans-mitter on 1530 kHz, and the signals breakthrough on your 1.8-MHz receiver. Bytuning the trap to 1530 kHz, the problem isgreatly reduced. If you have problemsfrom more than one broadcast station, theproblem needs a more complex solution.


SECOND-HARMONIC-OPTIMIZED (CWAZ) LOW-PASS FILTERS1

The FCC requires transmitter spuriousoutputs below 30 MHz to be attenuated by40 dB or more for power levels between5 and 500 W. For power levels greaterthan 5 W, the typical second-harmonicattenuation (40-dB) of a seven-elementChebyshev low-pass filter (LPF) is mar-ginal. An additional 10 dB of attenuationis needed to ensure compliance with theFCC requirement.

Jim Tonne, WB6BLD, solved theproblem of significantly increasing the sec-ond-harmonic attenuation of the seven-element Chebyshev LPF while maintain-ing an acceptable return loss (> 20 dB) overthe amateur passband. Jim’s idea was pre-sented in February 1999 QST by EdWetherhold, W3NQN. These filters aremost useful with single-band, single-devicetransmitters. Common medium-powermultiband transceivers use push-pull poweramplifiers because such amplifiers inher-ently suppress the second harmonic.

Tonne modified a seven-elementChebyshev standard-value capacitor (SVC)LPF to obtain an additional 10 dB of stop-band loss at the second-harmonic fre-quency. He did this by adding a capacitoracross the center inductor to form a reso-nant circuit. Unfortunately, return loss(RL) decreased to an unacceptable level,less than 12.5 dB. He needed a way to addthe resonant circuit, while maintainingan acceptable RL level over the passband.

The typical LPF, and the ChebyshevSVC designs listed in this chapter all haveacceptable RL levels that extend from thefilter ripple-cutoff frequency down to dc.For many Amateur Radio applications, weneed an acceptable RL only over the ama-teur band for which the LPF is designed.We can trade RL levels below the amateurband for improved RL in the passband, andsimultaneously increase the stop-bandloss at the second-harmonic frequency.

THE CWAZ LOW-PASS FILTERThis new eight-element LPF has a to-

pology similar to that of the seven-elementChebyshev LPF, with two exceptions: Thecenter inductor is resonated at the secondharmonic in the filter stop band, and thecomponent values are adjusted to main-tain a more than acceptable RL across theamateur passband. To distinguish this newLPF from the SVC Chebyshev LPF,Wetherhold named it the “Chebyshev withAdded Zero” or “CWAZ” LPF design.

You should understand that CWAZLPFs are output filters for single-bandtransmitters. They provide optimum sec-ond and higher harmonic attenuationwhile maintaining a suitable level of

Table 12.8CWAZ 50-ΩΩΩΩΩ Low-Pass FiltersDesigned for second-harmonic attenuation in amateur bands below 30 MHz.

StartBand Frequency C1,7 C3,5 C4 L2,6 L4 F4(m) (MHz) (pF) (pF) (pF) (μH) (μH) (MHz)— 1.00 2986 4556 680.1 9.377 8.516 2.091

1659 2531 378 3.76

160 1.80 1450 + 220 2100 + 470 5.21 4.731500 + 150 2200 + 330 330 + 47 3.78853 1302 194 7.32

80 3.50 1150 + 150 2.68 2.43470 + 390 1200 + 100 150 + 47 7.27427 651 97.2 14.6

40 7.00 1.34 1.22330 + 100 330 + 330 100 14.4296 451 67.3 21.1

30 10.1 0.928 0.843150 + 150 470 68 21.0213 325 48.6 29.3

20 14.0 0.670 0.608220 330 47 29.8165 252 37.6 37.8

17 18.068 0.519 0.47182 + 82 100 + 150 39 37.1142 217 32.4 43.9

15 21.0 0.447 0.406150 220 33 43.5120 183 27.3 52.0

12 24.89 0.377 0.342120 180 27 52.4107 163 24.3 58.5

10 28.0 0.335 0.304100 82 + 82 27 55.6

NOTE: The CWAZ low-pass filters are designed for a single amateur band to provide more than50-dB attenuation to the second harmonic of the fundamental frequency and to the higherharmonics. All component values for any particular band are calculated by dividing the 1-MHzvalues in the first row (included for reference only) by the start frequency of the selected band.The upper capacitor values in each row show the calculated design values obtained by dividingthe 1-MHz capacitor values by the amateur-band start frequency in megahertz. The lowerstandard-capacitor values are suggested as a convenient way to realize the design values. Themiddle capacitor values in the 160- and 80-meter-band designs are suggested values when thehigh-value capacitors (greater than 1000 pF) are on the low side of their tolerance range. Thedesign F4 frequency (see upper value in the F4 column) is calculated by multiplying the 1-MHzF4 value by the start frequency of the band. The lower number in the F4 column is the F4frequency based on the suggested lower capacitor value and the listed L4 value.

Fig 12.65 — Schematic diagram of a CWAZ low-pass filter designed for maximumsecond-harmonic attenuation. See Table 12.8 for component values of CWAZ 50-ΩΩΩΩΩdesigns. L4 and C4 are tuned to resonate at the F4 frequency given in Table 12.8.For an output power of 10 W into a 50-ΩΩΩΩΩ load, the RMS output voltage is

V22.45010 Consequently, a 100 V dc capacitor derated to 60 V (for RFfiltering) is adequate for use in these LPFs if the load SWR is less than 2.5:1.

.

12.38 Chapter 12

Fig 12.66 — Schematic diagram of a 20-meter SVC Chebyshev LPF.

Fig 12.67 — The plots show the ELSIE computer-calculatedreturn- and insertion-loss responses of the seventh-orderChebyshev SVC low-pass filter shown in Fig 12.66. The20-meter passband RL is about 21 dB, and the insertion lossover the second-harmonic frequency band ranges from 35 to39 dB. A listing of the component values is included.

Fig 12.68 — The plots show the ELSIE computer-calculatedreturn- and insertion-loss responses of the eight-elementlow-pass filter using the CWAZ capacitor and inductorvalues listed in Table 12.8 for the 20-meter low-pass filter.Notice that the calculated attenuation to second-harmonicsignals is greater than 60 dB, while RL over the 20-meterpassband is greater than 25 dB.

return loss over the amateur band forwhich they’re designed.

Fig 12.65 shows a schematic diagramof a CWAZ LPF design. Table 12.8 listssuggested capacitor and inductor valuesfor all amateur bands from 160 through10 meters. If you want to calculateCWAZ values for different bands, sim-ply divide the first-row C and L values(for 1 MHz) by the start frequency of thedesired band. For example, C1, 7 for the160-meter design is equal to 2986/1.80 =1659 pF. The other component values forthe 160-meter LPF are calculated in asimilar manner.

CWAZ VERSUS SEVENTH-ORDERSVC1

The easiest way to demonstrate the supe-riority of a CWAZ LPF over the ChebyshevLPF is to compare the RL and insertion-loss responses of these two designs.Fig 12.66 shows a 20-meter SVC ChebyshevLPF design based on the SVC tables on theHandbook CD-ROM. Fig 12.67 shows thecomputer-calculated return- and insertion-loss responses of the LPF shown inFig 12.66. The plotted responses weremade using Jim Tonne’s ELSIE filterdesign and analysis software. The Win-dows-based program is available from this

web site — http://tonnesoftware.com.1

Fig 12.68 shows the computer-calculatedreturn- and insertion-loss responses of aCWAZ LPF intended to replace the seven-element 20-meter Chebyshev SVC LPF. Thestop-band attenuation of the CWAZ LPFin the second-harmonic band is more than60 dB and is substantially greater than thatof the Chebyshev LPF. Also, the pass-bandRL of the CWAZ LPF is quite satisfactory,at more than 25 dB. The disadvantages ofthe CWAZ design are that an extra capacitoris needed across L4, and several of the de-signs listed in Table 12.8 require paralleledcapacitors to realize the design values. Nev-ertheless, these disadvantages are minor incomparison to the increased second-har-monic stop-band attenuation that is possiblewith a CWAZ design.

Notes1Those seriously interested in passive LC fil-

ter design can experience the capabilitiesof ELSIE software. The student version ofthis software permits filter design configu-rations up to the 7th Order, 7th Stage level.The software can be downloaded from theweb site of Jim Tonne (WB6BLD) at thisURL: http://tonnesoftware.com/.

http://tonnesoftware.com.1

http://tonnesoftware.com/


THE DIPLEXER FILTERThis section, covering diplexer filters,

was written by William E. Sabin, WØIYH.The diplexer is helpful in certain applica-tions, and Chapter 11 shows them used asfrequency mixer terminations.

Diplexers have a constant filter-inputresistance that extends to the stop band aswell as the passband. Ordinary filters thatbecome highly reactive or have an open orshort-circuit input impedance outside thepassband may degrade performance.

Fig 12.69 shows a normalized prototype5-element, 0.1-dB Chebyshev low-pass/high-pass (LP/HP) filter. This idealized fil-ter is driven by a voltage generator with zerointernal resistance, has load resistors of1.0 Ω and a cutoff frequency of 1.0 radianper second (0.1592 Hz). The LP prototypevalues are taken from standard filter tables.1

The first element is a series inductor. TheHP prototype is found by:

a) replacing the series L (LP) with aseries C (HP) whose value is 1/L, and

b) replacing the shunt C (LP) with ashunt L (HP) whose value is 1/C.

For the Chebyshev filter, the return lossis improved several dB by multiplying theprototype LP values by an experimentallyderived number, K, and dividing the HPvalues by the same K. You can calculatethe LP values in henrys and farads for a50-Ω RF application with the followingformulas:

Rf2

KCC;

f2

RKLL

CO

P(LP)LP

CO

P(LP)LP

whereLP(LP) and CP(LP) are LP prototype

valuesK = 1.005 (in this specific example)R = 50 ΩfCO = the cutoff (–3-dB response) fre-

quency in Hz.For the HP segment:

KRf2

CC;

f2

RLL

CO

P(HP)HP

CO

P(HP)HP K

where LP(HP) and CP(HP) are HP prototypevalues.

Fig 12.70 shows the LP and HP re-sponses of a diplexer filter for the 80-meterband. The following items are to be noted:• The 3 dB responses of the LP and HP

meet at 5.45 MHz.• The input impedance is close to 50 Ω at

all frequencies, as indicated by the highvalue of return loss (SWR <1.07:1).

• At and near 5.45 MHz, the LP input re-actance and the HP input reactance areconjugates; therefore, they cancel andproduce an almost perfect 50-Ω inputresistance in that region.

Fig 12.69 — Low-pass and high-pass prototype diplexer filter design. The low-passportion is at the top, and the high-pass at the bottom of the drawing. See text.

Fig 12.70 — Response for the low-pass and high-pass portions of the 80-meterdiplexer filter. Also shown is the return loss of the filter.

• Because of the way the diplexer filter isderived from synthesis procedures, thetransfer characteristic of the filter ismostly independent of the actual valueof the amplifier dynamic output imped-ance.2 This is a useful feature, since theRF power amplifier output impedanceis usually not known or specified.

• The 80-meter band is well within the LPresponse.

• The HP response is down more than20 dB at 4 MHz.

• The second harmonic of 3.5 MHz isdown only 18 dB at 7.0 MHz. Becausethe second harmonic attenuation of theLP is not great, it is necessary that theamplifier itself be a well-balanced push-pull design that greatly rejects thesecond harmonic. In practice this is nota difficult task.

• The third harmonic of 3.5 MHz is downalmost 40 dB at 10.5 MHz.Fig 12.71A shows the unfiltered of a

solid-state push-pull power amplifier forthe 80-meter band. In the figure you cansee that:• The second harmonic has been sup-

pressed by a proper push-pull design.• The third harmonic is typically only

15 dB or less below the fundamental.

The amplifier output goes through ourdiplexer filter. The desired output comesfrom the LP side, and is shown in Fig 12.71B.In it we see that:

• The fundamental is attenuated onlyabout 0.2 dB.

• The LP has some harmonic content;however, the attenuation exceeds FCCrequirements for a 100-W amplifier.

12.40 Chapter 12

Fig 12.71 — At A, the output spectrum of a push-pull 80-meter amplifier. At B, the spectrum after passing through thelow-pass filter. At C, the spectrum after passing through the high-pass filter.

(A) (B) (C)

Fig 12.71C shows the HP output of thediplexer that terminates in the HP load ordump resistor. A small amount of the funda-mental frequency (about 1%) is also lost inthis resistor. Within the 3.5 to 4.0 MHzband, the filter input resistance is almostexactly the correct 50-Ω load resistancevalue. This is because power that wouldotherwise be reflected back to the amplifieris absorbed in the dump resistor.

Solid state power amplifiers tend tohave stability problems that can be diffi-cult to debug.3 These problems may beevidenced by level changes in: load im-pedance, drive, gate or base bias, B+, etc.Problems may arise from:

• The reactance of the low-pass filter out-side the desired passband. This is espe-cially true for transistors that aredesigned for high-frequency operation.

• Self resonance of a series inductor atsome high frequency.

• A stopband impedance that causes volt-age, current and impedance reflectionsback to the amplifier, creating instabili-ties within the output transistors.

Intermodulation performance can alsobe degraded by these reflections. Thestrong third harmonic is especially both-ersome for these problems.

The diplexer filter is an approach thatcan greatly simplify the design process,especially for the amateur with limitedPA-design experience and with limitedhome-lab facilities. For these reasons, theamateur homebrew enthusiast may wantto consider this solution, despite itsslightly greater parts count and expense.

The diplexer is a good technique fornarrowband applications such as the HFamateur bands.4 From Fig 12.70, we seethat if the signal frequency is movedbeyond 4.0 MHz the amount of desiredsignal lost in the dump resistor becomeslarge. For signal frequencies below3.5 MHz the harmonic reduction may be

inadequate. A single filter will not sufficefor all the HF amateur bands.

This treatment provides you with theinformation to calculate your own filters.A QEX article has detailed instructions forbuilding and testing a set of six filters fora 120-W amplifier. These filters cover allnine of the MF/HF amateur bands.5 CheckARRLWeb at: www.arrl.org/qex/.

You can use this technique for other fil-ters such as Bessel, Butterworth, linearphase, Chebyshev 0.5, 1.0, etc.6 However,the diplexer idea does not apply to the el-liptic function types.

The diplexer approach is a resource thatcan be used in any application where aconstant value of filter input resistanceover a wide range of passband andstopband frequencies is desirable for somereason. The ARRL Radio Designer pro-gram is an ideal way to finalize the designbefore the actual construction.7 The coildimensions and the dump resistor wattageneed to be determined from a consider-ation of the power levels involved, as illus-trated in Fig 12.71.

Another significant application of thediplexer is for elimination of EMI, RFI andTVI energy. Instead of being reflected andvery possibly escaping by some otherroute, the unwanted energy is dissipatedin the dump resistor.7

Notes1Williams, A. and Taylor, F., Electronic Filter

Design Handbook, any edition, McGraw-Hill.

2Storer, J.E., Passive Network Synthesis,McGraw-Hill 1957, pp 168-170. This bookshows that the input resistance is ideallyconstant in the passband and the stopbandand that the filter transfer characteristic isideally independent of the generator im-pedance.

3Sabin, W. and Schoenike, E., HF RadioSystems and Circuits, Chapter 12, NoblePublishing, 1998. This publication is avail-able from ARRL as Order no. 7253. It canbe ordered at: www.arrl.org/catalog/.

Also the previous edition of this book,Single-Sideband Systems and Circuits,McGraw-Hill, 1987 or 1995.

4Dye, N. and Granberg, H., Radio FrequencyTransistors, Principles and Applications,Butterworth-Heinemann, 1993, p 151.

5Sabin, W.E. WØIYH, “Diplexer Filters for theHF MOSFET Power Amplifier,” QEX, Jul/Aug, 1999. Also check ARRLWeb at:www.arrl.org/qex/.

6See note 1. Electronic Filter Design Hand-book has LP prototype values for variousfilter types, and for complexities from 2 to10 components.

7Weinrich, R. and Carroll, R.W., “AbsorptiveFilters for TV Harmonics,” QST, Nov 1968,pp 10-25.

OTHER FILTER PROJECTSFilters for specific applications may be

found in other chapters of this Handbook.Receiver input filters, transmitter filters,interstage filters and others can be sepa-rated from the various projects and builtfor other applications. Since filters are afirst line of defense against electromag-netic interference (EMI) problems, thefollowing filter projects appear in theEMI/Direction Finding chapter:• Differential-mode high-pass filter for

75-Ω coax (for TV reception)• Brute-force ac-line filter• Loudspeaker common-mode choke• LC filter for speaker leads• Audio equipment input filter

REFERENCES1. O. Zobel, “Theory and Design of Elec-

tric Wave Filters,” Bell System Techni-cal Journal, Jan 1923.

2. ARRL Handbook, 1968, p 50.3. S. Butterworth, “On the Theory of Filter

Amplifiers,” Experimental Wirelessand Wireless Engineer, Oct 1930, pp536-541.

4. S. Darlington, “Synthesis of Reactance4-Poles Which Produce Prescribed In-sertion Loss Characteristics,” Journalof Mathematics and Physics, Sep 1939,pp 257-353.


5. L. Weinberg, “Network Design by use ofModern Synthesis Techniques andTables,” Proceedings of the NationalElectronics Conference, vol 12, 1956.

6. Laplace Transforms: P. Chirlian, BasicNetwork Theory, McGraw Hill, 1969.

7. Fourier Transforms: Reference Data forEngineers, Chapter 7, 7th edition,Howard Sams, 1985.

8. Cauer Elliptic Filters: The Design of Fil-ters Using the Catalog of NormalizedLow-Pass Filters, Telefunken, 1966.Also Ref 7, pp 9-5 to 9-11.

9. M. Dishal, “Top Coupled Band-pass Fil-ters,” IT&T Handbook, 4th edition,American Book, Inc, 1956, p 216.

10. W. E. Sabin, WØIYH, “Designing Nar-row Band-Pass Filters with a BASICProgram,” May 1983, QST, pp 23-29.

11. U. R. Rohde, DJ2LR, “Crystal FilterDesign with Small Computers” May1981, QST, p 18.

12. J. A. Hardcastle, G3JIR, “Ladder Crys-tal Filter Design,” Nov 1980, QST, p 20.

13. W. Hayward, W7ZOI, “A Unified Ap-proach to the Design of Ladder CrystalFilters,” May 1982, QST, p 21.

13a. J. Makhinson, N6NWP, “Designingand Building High-Performance CrystalLadder Filters,” Jan 1995, QEX, pp 3-17.

13b. W. Hayward, W7ZOI, “Refinementsin Crystal Ladder Filter Design,” June1995, QEX, pp 16-21.

14. R. Fisher, W2CQH, “Combline VHFBand-pass Filters,” Dec 1968, QST, p 44.

15. R. Fisher, W2CQH, “Interdigital Band-pass Filters for Amateur VHF/UHFApplications,” Mar 1968, QST, p 32.

16. W. S. Metcalf, “Graphs Speed Inter-digitated Filter Design,” Microwaves,Feb 1967.

17. A. Zverev, Handbook of Filter Synthe-sis, John Wiley and Sons.

18. R. Saal, The Design of Filters Using theCatalog of Normalized Low-Pass Fil-

ters, Telefunken.19. P. Geffe, Simplified Modern Filter

Design (New York: John F. Rider, a di-vision of Hayden Publishing Co, 1963).

20. A Handbook on Electrical Filters(Rockville, Maryland: White Electro-magnetics, 1963).

21. A. B. Williams, Electronic Filter De-sign Handbook (New York: McGraw-Hill, 1981).

22. Reference Data for Radio Engineers,6th edition, Table 2, p 5-3 (Indianapolis,IN: Howard W. Sams & Co, 1981).

23. R. Frost, “Large-Scale S ParametersHelp Analyze Stability,” Electronic De-sign, May 24, 1980.

24. Edward E. Wetherhold, W3NQN,“Modern Design of a CW Filter Using 88and 44-mH Surplus Inductors,” Dec1980, QST, pp14-19. See also Feedbackin Jan 1981, QST, p 43.

25. E. Wetherhold, W3NQN, “ CW andSSB Audio Filters Using 88-mH Induc-tors,” QEX-82, pp 3-10, Dec 1988; RadioHandbook, 23rd edition, W. Orr, editor,p 13-4, Howard W. Sams and Co., 1987;and, “A CW Filter for the Radio AmateurNewcomer,” Radio Communication, pp26-31, Jan 1985, RSGB.

26. M. Dishal, “Modern Network TheoryDesign of Single Sideband Crystal Lad-der Filters,” Proc IEEE, Vol 53, No 9,Sep 1965.

27. J.A. Hardcastle, G3JIR, “Computer-Aided Ladder Crystal Filter Design,”Radio Communication, May 1983.

28. John Pivnichny, N2DCH, “LadderCrystal Filters,” MFJ Publishing Co, Inc.

29. D. Jansson, WD4FAB and E.Wetherhold, W3NQN, “High-Pass Fil-ters to Combat TVI,” Feb 1987, QEX,pp 7-8 and 13.

30. E. Wetherhold, W3NQN, “Band-StopFilters for Attenuating High-LevelBroadcast-Band Signals,” Nov 1995,

QEX, pp 3-12.31. T. Moliere, DL7AV, “Band-Reject Fil-

ters for Multi-Multi Contest Opera-tions,” Feb 1996, CQ Contest, pp 14-22.

32. T. Cefalo Jr, WA1SPI, “Diplexers,Some Practical Applications,” Fall 1997,Communications Quarterly, pp 19-24.

33. R. Lumachi, WB2CQM. “How toSilverplate RF Tank Circuits,” Dec 1997,73 Amateur Radio Today, pp 18-23.

34. W. Hayward, W7ZOI, “Extending theDouble-Tuned Circuit to Three Resona-tors,” Mar/Apr 1998, QEX, pp 41-46.

35. P. R. Cope, W2GOM/7, “The Twin-TFilter,” July/Aug 1998, QEX, pp 45-49.

36. J. Tonne, WB6BLD, “Harmonic Filters,Improved,” Sept/Oct 1998, QEX, pp 50-53.

37. E. Wetherhold, W3NQN, “Second-Har-monic-Optimized Low-Pass Filters,”Feb 1999, QST, pp 44-46.

38. F. Heemstra, KT3J, “A Double-TunedActive Filter with Interactive Coupling,”Mar/Apr 1999, QEX, pp 25-29.

39. E. Wetherhold, W3NQN, “Clean UpYour Signals with Band-Pass Filters,”Parts 1 and 2, May and June 1998, QST,pp 44-48 and 39-42.

40. B. Bartlett, VK4UW and J. Loftus,VK4EMM, “Band-Pass Filters for Con-testing,” Jan 2000, National ContestJournal, pp 11-15.

41. W. Sabin, WØIYH, “Diplexer Filtersfor the HF MOSFET Power Amplifier,”July/Aug 1999, QEX, pp 20-26.

42. W. Sabin, WØIYH, “Narrow Band-PassFilters for HF,” Sept/Oct 2000, QEX, pp13-17.

43. Z. Lau, W1VT, “A Narrow 80-MeterBand-Pass Filter,” Sept/Oct 1998, QEX,p 57.

44. E. Wetherhold, W3NQN, “ReceiverBand-Pass Filters Having MaximumAttenuation in Adjacent Bands,” July/Aug 1999, QEX, pp 27-33.

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