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Chapter 12

RF and AF

Filters

This chapter contains basic design in-

formation and examples of the most com-

mon filters used by radio amateurs. It was

prepared by Reed Fisher, W2CQH, and

includes a number of design approaches,

tables and filters by Ed Wetherhold,

W3NQN, and others. The chapter is di-

vided into two major sections. The first

section contains a discussion of filter

theory 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 for

further reading and design information.

The second section contains a number of

selected practical filter designs for imme-

diate construction.

Basic Concepts

A filter is a network that passes signals

of certain frequencies and rejects or at-

tenuates those of other frequencies. The

radio art owes its success to effective fil-

tering. Filters allow the radio receiver to

provide the listener with only the desired

signal and reject all others. Conversely,

filters allow the radio transmitter to gen-

erate only one signal and attenuate others

that might interfere with other spectrum

users.

The simplified SSB receiver shown in

Fig 12.1

illustrates the use of several com-

mon filters. Three of them are located be-

tween the antenna and the speaker. They

provide 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 and

4.0 MHz with low loss. Other frequencies,

such as out-of-band signals, are rejected

to prevent them from overloading the first

mixer (a common problem with shortwave

broadcast stations). The preselector filter

is almost always built with LC filter tech-

nology.

An intermediate frequency (IF) filter

is placed between the first and second

mixers. It is a band-pass filter that passes

the desired SSB signal but rejects all

others. The age of the receiver probably

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

antenna and speaker.

determines which of several filter tech-

nologies is used. As an example, 50-kHz

or 455-kHz LC filters and 455-kHz

mechanical filters were used through the

1960s. Later model receivers usually use

quartz crystal filters with center frequen-

cies between 3 and 9 MHz. In all cases, the

filter bandwidth must be less than 3 kHz to

effectively reject adjacent SSB stations.

Finally, a 300-Hz to 3-kHz audio band-

pass filter is placed somewhere between

the detector and the speaker. It rejects

unwanted products of detection, power

supply hum and noise. Today this audio

filter is usually implemented with active

filter technology.

The complementary SSB transmitter

block diagram is shown in

Fig 12.2.

The

same array of filters appear in reverse

order.

First is a 300-Hz to 3-kHz audio filter,

which rejects out-of-band audio signals

such as 60-Hz power supply hum. It is

placed between the microphone and the

balanced mixer.

The IF filter is next. Since the balanced

mixer generates both lower and upper

sidebands, it is placed at the mixer output

RF and AF Filters

12.1

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

clean transmitted signal.

to pass only the desired lower (or upper)

sideband. In commercial SSB transceiv-

ers this filter is usually the same as the IF

filter used in the receive mode.

Finally, a 3.8 to 4.0-MHz band-pass fil-

ter is placed between transmit mixer and

antenna to reject unwanted frequencies

generated by the mixer and prevent them

from being amplified and transmitted.

This chapter will discuss the four most

common types of filters: low-pass, high-

pass, band-pass and band-stop. The ideal-

ized characteristics of these filters are

shown in their most basic form in

Fig 12.3.

A low-pass filter permits all frequen-

cies below a specified cutoff frequency to

be transmitted with small loss, but will at-

tenuate all frequencies above the cutoff

frequency. The “cutoff frequency” is usu-

ally specified to be that frequency where

the filter loss is 3 dB.

A high-pass filter has a cutoff frequency

above which there is small transmission

loss, but below which there is consider-

able attenuation. Its behavior is opposite

to that of the low-pass filter.

A band-pass filter passes a selected

band of frequencies with low loss, but at-

tenuates frequencies higher and lower

than the desired passband. The passband

of a filter is the frequency spectrum that is

conveyed with small loss. The transfer

characteristic is not necessarily perfectly

uniform in the passband, but the variations

usually are small.

A band-stop filter rejects a selected

band of frequencies, but transmits with

low loss frequencies higher and lower than

the desired stop band. Its behavior is op-

posite to that of the band-pass filter. The

stop band is the frequency spectrum in

which attenuation is desired. The attenua-

tion varies in the stop band rising to high

values at frequencies far removed from the

cutoff frequency.

Fig 12.3 — Idealized filter responses.

Note the definition of f

is 3 dB down

from the break points of the curves.

reject all other undesired frequencies. A

simple single-stage low-pass filter is

shown in

Fig 12.4.

The filter consists of an

inductor, L. It is placed between the volt-

age source e

and load resistance R

. Most

generators have an associated “internal”

resistance, which is labeled R

When the generator is switched on,

power will flow from the generator to the

load resistance R

. The purpose of this

low-pass filter is to allow maximum power

flow at low frequencies (below the cutoff

frequency) and minimum power flow at

high frequencies. Intuitively, frequency

filtering is accomplished because the in-

ductor has reactance that vanishes at dc

but becomes large at high frequency.

Thus, the current, I, flowing through the

load resistance, R

, will be maximum at

dc and less at higher frequencies.

The mathematical analysis of Fig 12.4 is

as follows: For simplicity, let R

= R

= R.

(1)

where

= 2π f L

f = generator frequency.

Power in the load, P

, is:

(2)

Available (maximum) power will be

delivered from the generator when:

= 0 and R

= R

(3)

FILTER FREQUENCY RESPONSE

The purpose of a filter is to pass a de-

sired frequency (or frequency band) and

Fig 12.4 — A single-stage low-pass

filter consists of a series inductor.

DC is passed to the load resistor

unattenuated. Attenuation increases

(and current in the load decreases) as

the frequency increases.

12.2

Chapter 12

The filter response is:

power in the load

available generator power

(4)

The filter cutoff frequency, called f

, is

the generator frequency where

or f

(5)

and the desired cutoff frequency is

4 MHz. Equation 4 states that the cutoff

frequency is where the inductive reactance

= 100

Ω.

At 4 MHz, using the relation-

ship X

= 2π f L, L = 4 μH. If this filter is

constructed, its response should follow the

curve in

Fig 12.5.

Note that the gentle

rolloff 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 the

output terminals of a filter, the impedance

looking into the input terminals will equal

the load resistance throughout most of the

passband. The degree of mismatch across

the passband is shown by the SWR scale at

the left-hand side of Fig 12.5. If maximum

power is to be extracted from the genera-

tor driving the filter, the generator resis-

tance must equal the load resistance. This

condition is called a “doubly terminated”

filter. Most passive filters, including the

LC filters described in this chapter, are

designed for double termination. If a filter

As an example, suppose R

= R

is not properly terminated, its passband

response changes.

Certain classes of filters, called “trans-

former filters” or “matching networks” are

specifically designed to work between

unequal generator and load resistances.

Band-pass filters, described later, are eas-

ily designed to work between unequal ter-

minations.

All passive filters exhibit an undesired

nonzero loss in the passband due to un-

avoidable resistances associated with the

reactances in the ladder network. All fil-

ters exhibit undesired transmission in the

stop band due to leakage around the filter

network. This phenomenon is called the

“ultimate rejection” of the filter. A typical

high-quality filter may exhibit an ultimate

rejection of 60 dB.

Band-pass filters perform most of the

important filtering in a radio receiver and

transmitter. There are several measures of

their effectiveness or

selectivity.

Selectiv-

ity is a qualitative term that arose in the

1930s. It expresses the ability of a filter

(or the entire receiver) to reject unwanted

adjacent signals. There is no mathemati-

cal measure of selectivity.

The term

is quantitative. A band-pass

filter’s

quality factor

or Q is expressed as

Q = (filter center frequency)/(3-dB band-

width).

Shape factor

is another way some

filter vendors specify band-pass filters. The

shape factor is a ratio of two filter band-

widths. Generally, it is the ratio (60-dB

bandwidth) / (6-dB bandwidth), but some

manufacturers use other bandwidths. An

ideal or

brick-wall

filter would have a shape

factor of 1, but this would require an infi-

nite number of filter elements. The IF filter

in a high-quality receiver may have a shape

factor of 2.

POLES AND ZEROS

In equation 1 there is a frequency called

the “pole” frequency that is given by f

= 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: f

= infinity. Poles

and zeros are intrinsic properties of all

networks. 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 interest

to the filter theorist because they allow

him to predict the frequency response of a

proposed filter. For low-pass and high-

pass filters the number of poles equals the

number of reactances in the filter network.

For band-pass and band-stop filters the

number of poles specified by the filter

vendors is usually taken to be half the

number of reactances.

LC FILTERS

Perhaps the most common filter found

in the Amateur Radio station is the induc-

tor-capacitor (LC) filter. Historically, the

LC filter was the first to be used and the

first to be analyzed. Many filter synthesis

techniques 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 parasitic

inductance associated with the capacitors

make applications at higher frequencies

impractical because the filter performance

will change with the physical construction

and therefore is not totally predictable from

the design equations. Below 50 or 60 Hz,

inductance and capacitance values of LC

filters become impractically large.

Mathematically, an LC filter is a linear,

lumped-element, passive, reciprocal net-

work. Linear means that the ratio of output

to input is the same for a 1-V input as for a

10-V input. Thus, the filter can accept an

input of many simultaneous sine waves

without intermodulation (mixing) between

them.

Lumped-element

means that the induc-

tors and capacitors are physically much

smaller than an operating wavelength. In

this case, conductor lengths do not con-

tribute significant inductance or capaci-

tance, and the time that it takes for signals

to pass through the filter is insignificant.

(Although the different times that it takes

for different frequencies to pass through

the filter — known as group delay — is

still significant for some applications.)

The term

passive

means that the filter

RF and AF Filters

12.3

Fig 12.5 — Transmission loss of a simple filter plotted against normalized

frequency. Note the relationship between loss and SWR.

does not need any internal power sources.

There may be amplifiers before and/or

after the filter, but no power is necessary

for the filter’s equations to hold. The filter

alone always exhibits a finite (nonzero)

insertion loss due to the unavoidable re-

sistances associated with inductors and (to

a lesser extent) capacitors. Active filters,

as the name implies, contain internal

power sources.

Reciprocal

means that the filter can pass

power in either direction. Either end of the

filter can be used for input or output.

TIME DOMAIN VS FREQUENCY

DOMAIN

Humans think in the time domain. Life

experiences are measured and recorded in

the stream of time. In contrast, Amateur

Radio systems and their associated filters

are often better understood when viewed

in the frequency domain, where frequency

is 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 a

waveform plotted against time with equa-

tion V = A sin(2π f t). The sine wave has

a peak amplitude A (measured in volts)

and frequency, f (measured in cycles/sec-

ond or Hertz). A graph showing frequency

on the horizontal axis is called a spectrum.

A filter response curve is plotted on a spec-

trum graph.

Historically, radio systems were best

analyzed in the frequency domain. The

radio transmitters of Hertz (1865)

and

Marconi (1895) consisted of LC resonant

circuits excited by high-voltage spark gaps.

The transmitters emitted packets of

damped sine waves. The low-frequency

(200-kHz) antennas used by Marconi were

found to possess very narrow bandwidths,

and it seemed natural to analyze antenna

performance using sine-wave excitation. In

addition, the growing use of 50 and 60-Hz

alternating current (ac) electric power sys-

tems in the 1890s demanded the use of

sine-wave mathematics to analyze these

systems. Thus engineers trained in ac

power theory were available to design and

build the early radio systems.

In the frequency domain, the radio

world is imagined to be composed of many

sine waves of different frequencies flow-

ing endlessly in time. It can be shown by

the Fourier transform (Ref 7) that all peri-

odic waveforms can be represented by

summing sine waves of different frequen-

cies. For example, the square-wave volt-

age shown in

Fig 12.7

can be represented

by a “fundamental” sine wave of fre-

quency f = 1/t and all its odd harmonics:

3f, 5f, 7f and so on. Thus, in the frequency

domain a sine wave is a

narrowband

sig-

12.4

Chapter 12

nal (zero bandwidth) and a square wave is

a “wideband” signal.

If the square-wave voltage of Fig 12.7

is passed through a low-pass filter, which

removes some of its high-frequency com-

ponents, the waveform of

Fig 12.8

results.

The filtered square wave now has a rise

time, which is the time required to rise

from 10% to 90% of its peak value (A).

The rise time is approximately:

0.35

(6)

where f

is the cutoff frequency of the low-

pass filter.

Thus a filter distorts a time-domain sig-

nal by removing some of its high-frequency

components. Note that a filter cannot dis-

tort a sine wave. A filter can only change

the amplitude and phase of sine waves. A

linear filter will pass multiple sine waves

without producing any intermodulation or

“beats” between frequencies — this is the

definition of

linear.

The purpose of a radio system is to con-

vey a time-domain signal originating at a

source to some distant point with mini-

mum distortion. Filters within the radio

system transmitter and receiver may in-

tentionally or unintentionally distort the

source signal. A knowledge of the source

signal’s frequency-domain bandwidth is

required so that an appropriate radio sys-

tem may be designed.

Table 12.1

shows the minimum neces-

Fig 12.7 — Square-wave voltage. Many

frequencies are present, including

1/t and odd harmonics 3f, 5f, 7f with

decreasing amplitudes.

Fig 12.6 — Ideal sine-wave voltage.

Only one frequency is present.

Fig 12.8 — Square-wave voltage filtered

by a low-pass filter. By passing the

square wave through a filter, the higher

frequencies are attenuated. The rectangu-

lar shape (fast rise and fall items) are

rounded

because the amplitude of the

higher harmonics is decreased.

Table 12.1

Typical Filter Bandwidths for Typical Signals.

Source

High-fidelity speech and music

Telephone-quality speech

Radiotelegraphy (Morse code, CW)

HF RTTY

NTSC television

SSTV

1200 bit/s packet

Required Bandwidth

20 Hz to 15 kHz

200 Hz to 3 kHz

200 Hz

1000 Hz (varies with frequency shift)

60 Hz to 4.5 MHz

200 Hz to 3 kHz

sary bandwidth of several common source

signals. Note that high-fidelity speech

and music requires a bandwidth of 20 Hz to

15 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, to

minimize transmit spectrum, as required by

the FCC, filters within amateur transmit-

ters are required to reduce the speech

source bandwidth to 200 Hz to 3 kHz at the

expense of some speech distortion. After

modulation the transmitted RF bandwidth

will exceed the filtered source bandwidth if

inefficient (AM or FM) modulation meth-

ods are employed. Thus the post-modula-

tion

emission bandwidth

may be several

times the original filtered source band-

width. At the receiving end of the radio link,

band-pass filters are required to accept only

the desired signal and sharply reject noise

and adjacent channel interference.

As human beings we are accustomed to

operation in the time domain. Just about

all of our analog radio connected design

occurs in the frequency domain. This is

particularly 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.

Filter Synthesis

The image-parameter method of filter

design was initiated by O. Zobel (Ref 1) of

Bell Labs in 1923. Image-parameter filters

are easy to design and design techniques

are found in earlier editions of the ARRL

Handbook.

Unfortunately, image param-

eter theory demands that the filter termi-

nating impedances vary with frequency in

an unusual manner. The later addition of

“m-derived matching half sections” at each

end of the filter made it possible to use these

filters in many applications. In the inter-

vening decades, however, many new meth-

ods of filter design have brought both better

performance and practical component val-

ues for construction.

MODERN FILTER THEORY

The start of modern filter theory is usu-

ally credited to S. Butterworth and S.

Darlington (Refs 3 and 4). It is based on

this approach: Given a desired frequency

response, find a circuit that will yield this

response.

Filter theorists were aware that certain

known mathematical polynomials had

“filter like” properties when plotted on a

frequency graph. The challenge was to

match the filter components (L, C and R)

to the known polynomial poles and zeros.

This pole/zero matching was a difficult

task before the availability of the digital

computer. Weinberg (Ref 5) was the first

to publish computer-generated tables of

normalized low-pass filter component

values. (“Normalized” means 1-Ω resis-

tor terminations and cutoff frequency

2πf

= 1 radian/s.)

An ideal low-pass filter response shows

no loss from zero frequency to the cutoff

frequency, but infinite loss above the cut-

off frequency. Practical filters may ap-

proximate this ideal response in several

different ways.

Fig 12.9

shows the Butterworth or

“maximally-flat” type of approximation.

The Butterworth response formula is:

(7)

where

= number of poles (reactances)

= power in the load resistor

= available generator power

The passband is exceedingly flat near

zero frequency

and

very high attenuation

is experienced at high frequencies, but the

approximation for both pass and stop

bands is relatively poor in the vicinity of

cutoff.

Fig 12.10

shows the Chebyshev ap-

proximation. Details of the Chebyshev

response formula can be found in (Ref 24).

Use of this reference as well as similar

references for Chebyshev filters requires

detailed familiarity with Chebyshev poly-

nomials.

IMPEDANCE AND FREQUENCY

SCALING

Fig 12.11A

shows normalized compo-

nent values for Butterworth filters up to

ten poles. Fig 12.11B shows the schematic

diagrams of the Butterworth low-pass fil-

ter. Note that the first reactance in Fig

12.11B is a shunt capacitor C1, whereas in

Fig 12.11C the first reactance is a series

inductor L1. Either configuration can be

used, but a design using fewer inductors is

usually chosen.

In filter design, the use of

normalized

RF and AF Filters

12.5

= frequency of interest

= cutoff frequency

Fig 12.9 — Butterworth approximation

of an ideal low-pass filter response.

The 3-dB attenuation frequency (f

) is

normalized to 1 radian/s.

Fig 12.10 — Chebyshev approximation

of an ideal low-pass filter. Notice the

ripple in the passband.

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