Chapter 13: Filters

A circuit that passes signals of some frequencies and blocks others

A filter treats frequencies differently, letting some through while blocking others; a circuit that weakens every frequency equally is just an attenuator.

The inductor in series with the load, the capacitor across the load

The inductor in the signal path passes low frequencies easily while the capacitor across the load shunts high frequencies away; swapping the two gives the highpass filter.

A voltage divider

The inductor and the capacitor-and-load share the input voltage like the two arms of a voltage divider whose ratio changes with frequency; it is not used as a tuned circuit.

To let signals below its cutoff frequency through and attenuate those above it

A lowpass filter passes everything below its cutoff with little loss, DC included, since its inductor is then a short and its capacitor open, and it attenuates higher frequencies.

The inductor is almost an open circuit and the capacitor almost a short

XL = 2πfL keeps rising and XC = 1 ÷ (2πfC) falls towards zero, so the inductor nearly blocks and the capacitor nearly shorts; the reverse is true at DC.

How its gain varies with frequency

A frequency response plots the filter’s gain, usually in decibels, against frequency, so one graph shows the passband, transition band and stopband, not just one point.

The range of frequencies it lets through with little attenuation

The passband is the whole range a filter passes with relatively little attenuation, for a lowpass filter everything below FC, not one lossless frequency.

The range of frequencies it attenuates heavily

The stopband is where signals are greatly attenuated, from FS onwards, by 60 dB or more in the manual’s graphs; where attenuation only starts rising is the transition band.

The transition band

Between FC and FS a signal is only partly attenuated, which is the transition band; the stopband, with 60 dB or more of attenuation, begins only at FS, 55 MHz.

3 dB

The cutoff frequency is the half-power point, 3 dB down; the −6 dB points, a quarter of the power, are used instead when a bandpass filter’s bandwidth is specified.

To reduce harmonics of the transmitted frequency

The amplifier inevitably produces harmonics, at whole-number multiples of the wanted frequency, and the filter after it cuts them to a low level; it leaves the signal’s bandwidth alone.

Distortion in the power amplifier

A power amplifier that distorts the waveform adds signals at whole-number multiples of the wanted frequency; driving it too little gives less distortion, not more.

It is very hard to do, and the amplifier would probably be inefficient

A harmonic-free power amplifier is very hard to design and would probably waste much of its power; the obstacle is difficulty and efficiency, not price.

21,12 MHz

The nth harmonic is n times the fundamental, so the third is 3 × 7,04 = 21,12 MHz; 28,16 MHz, four times the fundamental, is the fourth harmonic.

5th

18,25 ÷ 3,65 = 5, so 18,25 MHz is the 5th harmonic; calling 7,3 MHz the first harmonic would wrongly make it the 4th.

430 MHz

The nth harmonic is n times the fundamental: 3 × 145,3 = 435,9 MHz, in the 70 cm band and nearest 430 MHz; 580 MHz would be the fourth harmonic.

17,75 MHz

Harmonics are whole-number multiples: 14,2, 21,3 and 35,5 MHz are 2, 3 and 5 times 7,1 MHz, but 17,75 MHz is 2,5 times, so it is no harmonic.

80 Ω

XL = 2πfL is proportional to frequency, so doubling f doubles XL to 80 Ω; halving it to 20 Ω is what a capacitor’s reactance would do.

2 µW

60 dB is a power ratio of 10⁶, a 1 with six zeros, so 2 W ÷ 1 000 000 = 2 µW; dividing by 1 000 wrongly uses the voltage ratio.

0,18 V

For voltage, 35 dB is a ratio of 56,2, since 20 log₁₀ 56,2 = 35, so 10 ÷ 56,2 = 0,178 V, nearest 0,18 V; the power ratio, 3 162, gives 3,2 mV.

It lets signals above its cutoff frequency through and attenuates those below

A highpass filter passes signals above its cutoff with little loss and attenuates lower ones, DC included, because its series capacitor will not pass DC.

The capacitor in series with the load, the inductor across the load

The series capacitor passes high frequencies but blocks low ones and DC, while the inductor across the load shunts low frequencies away; the swapped layout is the lowpass filter.

The capacitor’s reactance is high and the inductor’s is low

At low frequencies XC = 1 ÷ (2πfC) is large, holding the signal back, and XL = 2πfL is small, diverting what passes from the load; far above cutoff the reverse holds.

100 Ω

XC = 1 ÷ (2πfC) is inversely proportional to frequency, so four times the frequency gives a quarter of the reactance, 100 Ω; 1,6 kΩ treats the capacitor like an inductor.

25 Ω

XL = 2πfL is proportional to frequency, so halving f from 1,8 MHz to 900 kHz halves XL to 25 Ω; doubling it to 100 Ω is how a capacitor would behave.

48,4 Ω

XL = 2πfL = 2π × 3,5 × 10⁶ × 2,2 × 10⁻⁶ = 48,4 Ω; leaving out the 2π gives 7,7 Ω.

30 W

The cutoff frequency is the half-power point, where the attenuation is 3 dB, so 60 ÷ 2 = 30 W reaches the load; 42 W wrongly applies 0,707, the voltage factor.

Very strong medium-wave broadcast signals

Medium-wave broadcast stations, roughly 530 kHz to 1,6 MHz, are strong enough to overload a receiver, so the filter blocks them while passing the amateur bands from 1,8 MHz; static is broadband.

About 500 kHz to 1,6 MHz

Medium-wave broadcasting runs from about 500 kHz to 1,6 MHz, just below the 160 m amateur band at 1,8 MHz; 88 to 108 MHz is the FM broadcast band.

It passes a band of frequencies and attenuates those above and below it

A bandpass filter passes the band between its low and high cutoff frequencies, FL and FH, and attenuates signals above and below that band, not merely one frequency.

A series tuned circuit in line with the source, a parallel one across the load

L1C1 is a series tuned circuit in the signal path and L2C2 a parallel one across the load; swapped round, the series circuit would short the load at resonance.

The series tuned circuit’s impedance is low and the parallel one’s very high

At resonance a series tuned circuit has almost no impedance and a parallel one a very high impedance, so the signal passes unloaded; far from resonance the reverse holds.

The series tuned circuit’s impedance is high and the parallel one’s low

Away from resonance the series circuit’s impedance is high and the parallel circuit’s low, the reverse of resonance, so the divider leaves little voltage across the load.

160 kHz

Bandwidth = FH − FL = 3 780 − 3 620 = 160 kHz; 3 700 kHz is the centre frequency, a point inside the passband, not its width.

14 175 kHz

The centre frequency lies halfway between the cutoffs: (14 000 + 14 350) ÷ 2 = 14 175 kHz; 175 kHz is half the bandwidth, with the lower cutoff left out.

120 kHz

FL is 60 kHz below the centre, so FH is 60 kHz above it at 7 160 kHz, and the bandwidth is FH − FL = 120 kHz, twice the 60 kHz offset.

800 Hz

The cutoffs lie half the bandwidth either side of the centre, so FH = 700 + 200 ÷ 2 = 800 Hz; adding the whole bandwidth gives 900 Hz.

7 MHz

Both tuned circuits resonate at the same frequency, f = 1 ÷ (2π√(LC)) = 1 ÷ (2π√(3,3 × 10⁻⁶ × 150 × 10⁻¹²)) = 7,15 MHz, nearest 7 MHz; omitting 2π gives 45 MHz.

Preselectors

Preselectors are bandpass filters ahead of the receiver that admit the chosen amateur band and reject the others; preamplifiers also sit near the input but amplify rather than select.

4

Slope factor = −60 dB width ÷ −6 dB width = 1 900 Hz ÷ 450 Hz = 4,22, nearest 4; dividing the other way gives 0,24, and leaving the kilohertz unconverted 0,004.

1,4

Slope factor = −60 dB width ÷ −6 dB width = 3,5 ÷ 2,5 = 1,4; the wider figure goes on top, so it is never below 1, unlike 0,71.

18 kHz

Since slope factor = −60 dB width ÷ −6 dB width, the −60 dB width = 1,5 × 12 = 18 kHz; dividing gives 8 kHz, narrower than the passband itself, which cannot be.

The attenuation it gives to signals far outside its passband

Ultimate rejection is the attenuation deep in the stopband, perhaps 94 dB, and good filters have a very high figure; at the cutoff frequencies the loss is only 3 dB.

A small slope factor and a very high ultimate rejection

Steep skirts give a small slope factor, close to 1, and strong stopband attenuation a very high ultimate rejection; a large slope factor means gently sloping skirts.

Its attenuation rises steeply just outside the passband

The slope factor compares the −60 dB width with the −6 dB width, so a value near 1 means skirts that fall almost vertically; a large value means gentle slopes.

70 dB

Both levels are referred to 1 mW, so the attenuation is their difference, +10 − (−60) = 70 dB; adding them, 10 + (−60) = −50, gives the wrong 50 dB.

100 dB

The power ratio is 4 mW ÷ 0,4 pW = 10¹⁰, a 1 with ten zeros, so the attenuation is 10 log₁₀ 10¹⁰ = 100 dB; 200 dB uses 20 log, the voltage formula.

A voltage across it distorts it slightly, and distorting it produces a voltage

The piezoelectric effect works both ways: a voltage slightly distorts the crystal, and distorting or vibrating the crystal produces a voltage across it.

They behave electrically very much like tuned circuits

A quartz crystal behaves like a tuned circuit with an extremely sharp response, so crystals give bandpass filters of great selectivity; a crystal does not pass all AC equally.

Its ability to separate one signal from others nearby

Selectivity is the ability to tell one signal apart from its neighbours, which crystal filters provide; picking up very weak signals is sensitivity, a different property.

It is a very small fraction of it

Crystal filters are extremely narrow: a 9 MHz filter may be only 2,4 kHz wide, under 0,03 % of its centre frequency, far less than a tenth of it.

0,1 %

12 kHz ÷ 10,7 MHz = 0,0011 = 0,11 %, nearest 0,1 %, a tiny fraction; treating the kilohertz as megahertz gives 112 %, nearest 100 %.

It works on only one fixed frequency and cannot be tuned

A crystal filter is fixed on one frequency and cannot be tuned at all, so retuning it is impossible; the superhet receiver, covered later, gets round this limitation.

2,4 kHz

A 2,4 kHz filter suits normal SSB phone, 1,8 kHz helps in difficult conditions and 250 to 500 Hz suits CW; a 3 kHz filter would admit more interference.

250 Hz to 500 Hz

A CW signal occupies very little spectrum, so filters of 250 Hz to 500 Hz are used; 25 Hz to 50 Hz would be too narrow to tune easily or to follow the keying.

To cope with difficult conditions, such as a crowded contest band

A narrower 1,8 kHz filter trims the SSB passband to shut out strong signals close by, useful in difficult conditions such as contests, though the audio sounds less natural.

It attenuates a band of frequencies and passes those above and below it

A bandstop filter attenuates the band between its two cutoff frequencies and passes signals above and below it; attenuating those as well would leave nothing to pass.

The notch filter

A notch filter is a narrow, adjustable bandstop filter tuned onto the carrier to cut it out; a narrower crystal filter would also clip the wanted speech.

A narrow bandstop filter that the operator can adjust

A notch filter is a very narrow, manually adjustable bandstop filter that cuts a deep notch at one frequency; a narrow bandpass filter would keep that frequency instead.

A filter made of several stages connected one after another

Several filter stages connected in a row, each strengthening the filtering, make a multi-pole filter; it still has a single passband, not several separate bands.

A better slope factor and better ultimate rejection

Adding stages strengthens the filtering, so a multi-pole filter can achieve better bandwidth, slope factor and ultimate rejection together, though it may bring complications.

Variation of a filter’s attenuation within its passband

Passband ripple is the rise and fall of attenuation across the wanted passband that complex filters can show, not a single loss figure at its centre.

Unwanted phase shifts

Besides ripple in the passband, complex filters can shift the phase of signals in unwanted ways; a passive filter does not create harmonics.

Standard filter configurations

Butterworth, Chebyshev and Bessel are standard filter configurations, each with its own balance of ripple, slope factor, rejection, ringing and loss; none is a crystal cut.

80 dB

Losses in decibels add when stages follow one another, 40 + 40 = 80 dB, a power ratio of 10⁸; adding 3 dB would only double the loss of one filter.

Like a resistor in series with the inductor

The winding’s resistance carries the same current as the inductance, so it acts like a resistor in series with an ideal inductor, not one in parallel.

Stray capacitance

Unwanted capacitance, such as between the turns of a coil or between neighbouring tracks, is stray capacitance; decoupling capacitance is added on purpose.

Like a capacitor

Far enough up in frequency, the stray capacitance between the turns outweighs the inductance, so the coil acts as a capacitor rather than an ever-larger inductor.

Their winding gives them a lot of inductance

Wire wound into a coil gives a wirewound resistor much inductance, so at high enough frequencies it acts like an inductor; its stray capacitance is the smaller problem.

They have small amounts of unwanted inductance and capacitance

Carbon-film resistors are far better than wirewound ones at RF, but they still have some stray inductance and capacitance, which matters at high enough frequencies.

They cannot follow rapid changes of voltage

Electrolytic capacitors cannot follow quick changes in voltage, so however large their value they work poorly at RF; their capacitance does not rise with frequency.

They give large values of capacitance in a compact size

Power-supply filters need large capacitance, and electrolytics give large values in a compact size; they are polarised, so they must be connected the right way round.

Inductance

Every lead is a short length of wire with some inductance, which matters at high frequencies; leakage current comes from the dielectric, not the leads.

Like transmission lines

At high frequencies PCB tracks have both stray inductance and stray capacitance and start acting as transmission lines, not as pure inductors; fast boards are laid out by software.

With automated design software

Fast computer boards are laid out by automated software that allows for the stray inductance, capacitance and transmission-line effects of their tracks, too many to handle by hand.

It depends: some degrade from about 1 MHz, others work beyond 1 GHz

There is no single answer: parts made for UHF behave well even above 1 GHz, while others start to degrade at around 1 MHz.

When its resistance is about equal to its reactance

A reactive part should be almost pure reactance; once its resistance grows to match its reactance it no longer does its job, whereas a large reactance is its purpose.

When it starts to show noticeable reactance

A resistor that develops reactance, and with it a phase shift between voltage and current, is no longer a pure resistor; a slight change with temperature is a different matter.

Make them well soldered or crimped

Joints in RF circuits should be well soldered or crimped; clipped or loose connections are unreliable and add resistance and other effects that are hard to trace.

Its parts or wiring do not suit RF, adding unwanted reactance

The checked design rules out a calculation error; parts not made for RF, long interconnections and a crowded layout add stray inductance and capacitance that spoil the filter.