# Filters in Electronics

# Filters in Electronics

A filter is a frequency-selective circuit filters are designed to pass some frequencies and reject others.

There are numerous ways to implement filter circuits.

Simple filters in electronics create by using resistors and capacitors or inductors and capacitors are call passive filters because they use passive components that do not amplify.

In communication work, many filters in electronics are of the passive LC variety, although many other types are use.

Some special types of filters are active filters that use RC networks with feedback in op amp circuits,

switched capacitor filters, crystal and ceramic filters, surface acoustic wave (SAW) filters,

and digital filters implemented with digital signal processing (DSP) techniques.

#### The five basic kinds of filter circuits are as follows:

**Low-pass filter:**

Passes frequencies below a critical frequency called the cutoff frequency and greatly attenuates those above the cutoff frequency.

**High-pass filter:**

Passes frequencies above the cutoff but rejects those below it.

**Bandpass filter:**

Passes frequencies over a narrow range between lower and upper cutoff frequencies.

**Band-reject filter:**

Rejects or stops frequencies over a narrow range but allows frequencies above and below to pass.

**All-pass filter:**

Passes all frequencies equally well over its design range but has a fixed or predictable phase shift characteristic.

**RC Filters**

A low-pass filter allows the lower-frequency components of the applied voltage to develop output voltage across the load resistance, whereas the higher-frequency components are attenuate, or reduce, in the output.

A high-pass filter does the opposite, allowing the higher-frequency components of the applied voltage to develop voltage across the output load resistance Filters.

#### Read More-Filter Circuits

The case of an RC coupling circuit is an example of a high-pass filter because the ac component of the input voltage is developed across R and the dc voltage is block by the series capacitor. Furthermore, with higher frequencies in the ac component, more ac voltage is couple.

##### Any low-pass or high-pass filter can be think of as a frequency-dependent voltage divider because the amount of output voltage is a function of frequency.

RC filters in electronics use combinations of resistors and capacitors to achieve the desired response. Most RC filters are of the low-pass or high-pass type. Some band-reject or notch filters are also make with RC circuits. Bandpass filters in electronics can be make by combining low-pass and high-pass RC sections, but this is rarely do.

**Low-Pass Filter:**

A low-pass filter is a circuit that introduces no attenuation at frequencies below the cutoff frequency.

But completely eliminates all signals with frequencies above the cutoff.

Low-pass filters are sometimes refer to as high cut filters.

**Figure 1 **Ideal response curve of a low-pass filter.

The ideal response curve for a low-pass filter is show in Fig-1.

This response curve cannot be realize in practice.

In practical circuits, instead of a sharp transition at the cutoff frequency,

there is a more gradual transition between little or no attenuation and maximum attenuation.

**Figure 2 **RC low-pass filter. (a) Circuit. (b) Low-pass filter.

The simplest form of low-pass filter is the RC circuit show in Fig. 2a).

The circuit forms a simple voltage divider with one frequency-sensitive component, in this case the capacitor.

At very low frequencies, the capacitor has very high reactance compared to the resistance and therefore the attenuation is minimum.

As the frequency increases, the capacitive reactance decreases.

When the reactance becomes smaller than the resistance, the attenuation increases rapidly.

The frequency response of the basic circuit is illustrate in Fig. 2(b).

The cutoff frequency of this filter is that point where R and X_{C} are equal.

#### The cutoff frequency, also known as the critical frequency, is determine by the expression

For example, if R 5 4.7 kΩ and C 5 560 pF, the cutoff frequency is

**Figure 3 **Two stages of RC filter improve the response but increase signal loss. (a) Circuit. (b) Response curve.

At the cutoff frequency, the output amplitude is 70.7 percent of the input amplitude at lower frequencies.

This is the so-called 3-dB down point.

In other words, this filter has a voltage gain of 23 dB at the cutoff frequency.

At frequencies above the cutoff frequency, the amplitude decreases at a linear rate of 6 dB per octave or 20 dB per decade.

An octave is define as a doubling or halving of frequency, and a decade represents a one-tenth or times-10 relationship.

If the frequency increased by a factor of 10 from 600 Hz to 6 kHz, the attenuation would increase by a factor of 20 dB from 3 dB at cutoff to 23 dB at 6 kHz.

Assume that a filter has a cutoff of 600 Hz. If the frequency doubles to 1200 Hz, the attenuation will increase by 6 dB, or from 3 dB at cutoff to 9 dB at 1200 Hz.

If a faster rate of attenuation is require,

two RC sections set to the same cutoff frequency can be use. Such a circuit is show in Fig-3(a).

#### With this circuit, the rate of attenuation is 12 dB per octave or 40 dB per decade.

Two identical RC circuits are used, but an isolation or buffer amplifier such as an emitter-follower (gain < 1) is used between them to prevent the second section from loading the first.

Cascading two RC sections without the isolation will give an attenuation rate less than the theoretically ideal 12-dB octave because of the loading effects.

If the cutoff frequency of each RC section is the same, the overall cutoff frequency for the complete filter is somewhat less.

##### This is cause by added attenuation of the second section.

With a steeper attenuation curve, the circuit is say to be more selective.

The disadvantage of cascading such sections is that higher attenuation makes the output signal considerably smaller.

This signal attenuation in the passband of the filter is call insertion loss.

**Figure 4: **A low-pass filter implemented with an inductor.

A low-pass filter can also be implement with an inductor and a resistor, as shown in Fig-4. The response curve for this RL filter is the same as that shown in Fig. 2(b).

##### The cutoff frequency is determine by using the formula

The RL low-pass filters are not as widely used as RC filters because inductors are usually larger, heavier, and more expensive than capacitors. Inductors also have greater loss than capacitors because of their inherent winding resistance.

**Figure 5 **Frequency response curve of a high-pass filter. (a) Ideal. (b) Practical.

**High-Pass Filter:**

A high-pass filter passes frequencies above the cutoff frequency with little or no attenuation but greatly attenuates those signals below the cutoff.

The ideal high-pass response curve is show in Fig-5 (a).

Approximations to the ideal response curve shown in Fig-5(b) can be obtain with a variety of RC and LC filters in electronics.

The basic RC high-pass filter is show in Fig-6(a).

Again, it is nothing more than a voltage divider with the capacitor serving as the frequency-sensitive component in a voltage divider.

At low frequencies, X_{C} is very high. When X_{C} is much higher than R, the voltage divider effect provides high attenuation of the low-frequency signals.

As the frequency increases, the capacitive reactance decreases.

When the capacitive reactance is equal to or less than the resistance, the voltage divider gives very little attenuation.

##### Therefore, high frequencies pass relatively unattenuated.

The cutoff frequency for this filter is the same as that for the low-pass circuit and is derive from setting X_{C} equal to R and solving for frequency:

#### The roll-off rate is 6 dB per octave or 20 dB per decade.

A high-pass filter can also be implement with a coil and a resistor, as shown in Fig. 6(b). The cutoff frequency is

The response curve for this filter is the same as that shown in Fig. 5(b). The rate of attenuation is 6 dB per octave or 20 dB per decade, as was the case with the low-pass filter. Again, improved attenuation can be obtain by cascading filter sections.

**Figure 6 **(a) RC high-pass filter. (b) RL high-pass filter.

##### RC Notch Filter:

Notch filters are also refer to as band stop or band-reject filters in electronics. Band-reject filters are use to greatly attenuate a narrow range of frequencies around a center point. Notch filters accomplish the same purpose, but for a single frequency.

**Figure 7: **RC notch filter.

A simple notch filter that is implement with resistors and capacitors as show in Fig- 7(a) is call a parallel-T or twin-T notch filter.

This filter is a variation of a bridge circuit.

Recall that in a bridge circuit the output is zero if the bridge is balance.

If the component values are precisely match,

the circuit will be in balance and produce an attenuation of an input signal at the design frequency as high as 30 to 40 dB.

A typical response curve is show in Fig-7(b).

The center notch frequency is compute with the formula

For example, if the values of resistance and capacitance are 100 kV and 0.02 μF, the notch frequency is

Twin-T notch filters are use primarily at low frequencies, audio and below.

A common use is to eliminate 60-Hz power line hum from audio circuits and low-frequency medical equipment amplifiers.

The key to high attenuation at the notch frequency is precise component values.

The resistor and capacitor values must be match to achieve high attenuation.

#### Note: Twin-T notch filters in electronics are use at low frequencies to eliminate power line hum from audio circuits and medical equipment amplifiers.

**LC Filters** in Electronics

RC filters in electronics are use primarily at the lower frequencies. They are very common at audio frequencies but are rarely use above about 100 kHz. At radio frequencies, their passband attenuation is just too great, and the cutoff slope is too gradual. It is more common to see LC filters in electronics made with inductors and capacitors. Inductors for lower frequencies are large, bulky, and expensive, but those used at higher frequencies are very small, light, and inexpensive. Over the years, a multitude of filter types have been develop. Filter design methods have also changed over the years, thanks to computer design.

**Filter Terminology:**

When working with filters, you will hear a variety of terms to describe the operation and characteristics of filters. The following definitions will help you understand filter specifications and operation.

#### 1.**Passband:**

This is the frequency range over which the filter passes signals. It is the frequency range between the cutoff frequencies or between the cutoff frequency and zero (for low-pass) or between the cutoff frequency and infinity (for high-pass).

#### 2.**Stop band:**

This is the range of frequencies outside the passband, i.e., the range of frequencies that is greatly attenuate by the filter. Frequencies in this range are reject.

#### 3.**Attenuation:**

This is the amount by which undesired frequencies in the stop band are reduce. It can be express as a power ratio or voltage ratio of the output to the input. Attenuation is usually give in decibels.

#### 4.**Insertion loss:**

Insertion loss is the loss the filter introduces to the signals in the passband. Passive filters in electronics introduce attenuation because of the resistive losses in the components. Insertion loss is typically give in decibels.

#### 5.**Impedance:**

Impedance is the resistive value of the load and source terminations of the filter. Filters are usually design for specific driving source and load impedances that must be present for proper operation.

#### 6.**Ripple:**

Amplitude variation with frequency in the passband, or the repetitive rise and fall of the signal level in the passband of some types of filters in electronics, is know as ripple. It is usually state in decibels. There may also be ripple in the stop bandwidth in some types of filters.

#### 7.**Shape factor. **

Shape factor, also known as bandwidth ratio, is the ratio of the stop bandwidth to the pass bandwidth of a bandpass filter. It compares the bandwidth at minimum attenuation, usually at the 23-dB points or cutoff frequencies, to that of maximum attenuation and thus gives a relative indication of attenuation rate or selectivity. The smaller the ratio, the greater the selectivity. The ideal is a ratio of 1, which in general cannot be obtain with practical filters.

The filter in Fig-8 has a bandwidth of 6 kHz at the 23-dB attenuation point and a bandwidth of 14 kHz at the 240-dB attenuation point. The shape factor then is 14 kHz/6 kHz 5 2.333. The points of comparison vary with different filters and manufacturers. The points of comparison may be at the 6-dB down and 60-dB down points or at any other designated two levels.

**Figure 8: Shape factor.**

#### 8.**Pole: **

A pole is a frequency at which there is a high impedance in the circuit. It is also use to describe one RC section of a filter. A simple low-pass RC filter such as that in Fig. 2-24(a) has one pole. The two-section filter in Fig. 2-25 has two poles. For LC low- and high-pass filters in electronics, the number of poles is equal to the number of reactive components in the filter. For bandpass and band-reject filters in electronics, the number of poles is generally assume to be one-half the number of reactive components use.

#### 9.**Zero. **

This term refers to a frequency at which there is zero impedance in the circuit.

**Envelope delay. **

Also known as time delay, envelope delay is the time it takes for a specific point on an input waveform to pass through the filter.

Roll-off-Also called the attenuation rate, roll-off is the rate of change of amplitude with frequency in a filter. The faster the roll-off, or the higher the attenuation rate, the more selective the filter is, i.e., the better able it is to differentiate between two closely spaced signals, one desired and the other not.

##### Any of the four basic filter types can be easily implement with inductors and capacitors.

Such filters can be make for frequencies up to about several hundred megahertz before the component values get too small to be practical. At frequencies above this frequency, special filters made with microstrip techniques on printed-circuit boards, surface acoustic wave filters, and cavity resonators are common. Because two types of reactances are use, inductive combined with capacitive, the roll-off rate of attenuation is greater with LC filters than with RC filters. The inductors make such filters larger and more expensive, but the need for better selectivity makes them necessary.

**Figure 2-31 **Low-pass filter configurations and response. (a) L section. (b) T section. (c) π section. (d) Response curve.

**Figure 2-32 **Low-pass Butterworth filter attenuation curves beyond the cutoff frequency f_{c}.

**Low- and High-Pass LC Filters**

Fig. 2-31 shows the basic low-pass filter configurations. The basic two-pole circuit in Fig. 2-31(a) provides an attenuation rate of 12 dB per octave or 20 dB per decade. These sections may be cascade to provide an even greater roll-off rate. The chart in Fig. 2-32 shows the attenuation rates for low-pass filters in electronics with two through seven poles. The horizontal axis f /f_{c} is the ratio of any give frequency in ratio to the filter cutoff frequency f_{c}. The value n is the number of poles in the filter.

Assume a cutoff frequency of 20 MHz. The ratio for a frequency of 40 MHz would be 40/20 5 2. This represents a doubling of the frequency, or one octave. The attenuation on the curve with two poles is 12 dB. The π and T filters in electronics in Fig. 2-31(b) and (c) with three poles give an attenuation rate of 18 dB for a 2:1 frequency ratio. Fig. 2-33 shows the basic high-pass filter configurations. A curve similar to that in Fig. 2-32 is also use to determine attenuation for filters in electronics with multiple poles. Cascading these sections provides a greater attenuation rate. Those filter configurations using the least number of inductors are prefer for lower cost and less space.

**Figure 2-33 **High-pass filters. (a) L section. (b) T section (b) π section.

**Types of Filters**

The major types of LC filters in use are name after the person who discover and develop the analysis and design method for each filter. The most widely used filters in electronics are Butterworth, Chebyshev, Cauer (elliptical), and Bessel. Each can be implement by using the basic low- and high-pass configurations shown previously. The different response curves are achieve by selecting the component values during the design.

**Butterworth:**

The Butterworth filter effect has maximum flatness in response in the pass band and a uniform attenuation with frequency. The attenuation rate just outside the passband is not as great as can be achieve with other types of filters in electronics. See Fig. 2-34 for an example of a low-pass Butterworth filter.

**Chebyshev:**

Chebyshev (or Tchebyschev) filters have extremely good selectivity; i.e., their attenuation rate or roll-off is high, much higher than that of the Butterworth filter (see Fig. 2-34). The attenuation just outside the passband is also very high—again, better than that of the Butterworth. The main problem with the Chebyshev filter is that it has ripple in the passband, as is evident from the figure. The response is not flat or constant, as it is with the Butterworth filter. This may be a disadvantage in some applications.

**Cauer (Elliptical):**

Cauer filters produce an even greater attenuation or roll-off rate than do Chebyshev filters and greater attenuation out of the passband. However, they do this with an even higher ripple in the passband as well as outside of the passband.

**Bessel:**

Also called Thomson filters in electronics, Bessel circuits provide the desired frequency response (i.e., low-pass, bandpass, etc.) but have a constant time delay in the passband.

**Figure 2-34 **Butterworth, elliptical, Bessel, and Chebyshev response curves.

Bessel filters have what is know as a flat group delay: as the signal frequency varies in the passband, the phase shift or time delay it introduces is constant. In some applications, constant group delay is necessary to prevent distortion of the signals in the passband due to varying phase shifts with frequency. Filters that must pass pulses or wideband modulation are examples. To achieve this desired response, the Bessel filter has lower attenuation just outside the passband.

**Mechanical Filters:**

An older but still useful filter is the mechanical filter. This type of filter uses resonant vibrations of mechanical disks to provide the selectivity. The signal to be filter is apply to a coil that interacts with a permanent magnet to produce vibrations in the rod connected to a sequence of seven or eight disks whose dimensions determine the center frequency of the filter. The disks vibrate only near their resonant frequency, producing movement in another rod connected to an output coil. This coil works with another permanent magnet to generate an electrical output. Mechanical filters are design to work in the 200- to 500-kHz range and have very high Qs. Their performance is comparable to that of crystal filters in electronics.

Whatever the type, passive filters are usually design and built with discrete components although they may also be putt into integrated-circuit form. A number of filter design software packages are available to simplify and speed up the design process. The design of LC filters in electronics is specialize and complex and beyond the scope of this text. However, filters in electronics can be purchase as components. These filters are predesign and package in small sealed housings with only input, output, and ground terminals and can be use just as integrated circuits are. A wide range of frequencies, response characteristics, and attenuation rates can be obtain.

**Bandpass Filters in Electronics:**

A bandpass filter is one that allows a narrow range of frequencies around a center frequency fc to pass with minimum attenuation but rejects frequencies above and below this range. The ideal response curve of a bandpass filter is show in Fig. 2-35(a). It has both upper and lower cutoff frequencies f2 and f1, as indicated. The bandwidth of this filter is the difference between the upper and lower cutoff frequencies, or BW 5 f2 2 f1. Frequencies above and below the cutoff frequencies are eliminate.

The ideal response curve is not obtainable with practical circuits, but close approximations can be obtain. A practical bandpass filter response curve is show in Fig. 2-35(b). The simple series and parallel resonant circuits described in the previous section have a response curve like that in the figure and make good bandpass filters. The cutoff frequencies are those at which the output voltage is down 0.707 percent from the peak output value. These are the 3-dB attenuation points.

**Figure 2-35 **Response curves of a bandpass filter. (a) Ideal. (b) Practical.

Two types of bandpass filters are show in Fig. 2-36. In Fig. 2-36(a), a series resonant circuit is connect in series with an output resistor, forming a voltage divider. At frequencies above and below the resonant frequency, either the inductive or the capacitive reactance will be high compare to the output resistance. Therefore, the output amplitude will be very low. However, at the resonant frequency, the inductive and capacitive reactances cancel, leaving only the small resistance of the inductor. Thus most of the input voltage appears across the larger output resistance. The response curve for this circuit is show in Fig. 2-35(b). Remember that the bandwidth of such a circuit is a function of the resonant frequency and Q: BW 5 f_{c} /Q.

A parallel resonant bandpass filter is show in Fig. 2-36(b). Again, a voltage divider is form with resistor R and the tuned circuit. This time the output is take from across the parallel resonant circuit.

##### At frequencies above and below the center resonant frequency, the impedance of the parallel tuned circuit is low compare to that of the resistance.

Therefore, the output voltage is very low. Frequencies above and below the center frequency are greatly attenuate. At the resonant frequency, the reactances are equal and the impedance of the parallel tuned circuit is very high compare to that of the resistance. Therefore, most of the input voltage appears across the tuned circuit. The response curve is similar to that shown in

Improved selectivity with steeper “skirts” on the curve can be obtain by cascading several bandpass sections. Several ways to do this are show in Fig. 2-37. As sections are cascade, the bandwidth becomes narrower and the response curve becomes steeper. An example is show in Fig. 2-38. As indicated earlier, using multiple filter sections greatly improves the selectivity but increases the passband attenuation (insertion loss), which must be offsett by added gain.

**Figure 2-36 **Simple bandpass filters.

**Figure 2-37 **Some common bandpass filter circuits.

**Fig- 2-38 **How cascading filter sections narrow the bandwidth and improve selectivity.

**Figure 2-39 **LC tuned band stop filters. (a) Shunt. (b) Series. (c) Response curve.

**Fig-**** 2-40 **Bridge-T notch filter.

**Band-Reject Filters:**

Band-reject filters, also known as band stop filters, reject a narrow band of frequencies around a center or notch frequency. Two typical LC band stop filters are show in Fig. 2-39. In Fig. 2-39(a), the series LC resonant circuit forms a voltage divider with input resistor R. At frequencies above and below the center rejection or notch frequency, the LC circuit impedance is high compare to that of the resistance. Therefore, signals at frequencies above and below center frequency are pass with minimum attenuation. At the center frequency, the tuned circuit resonates, leaving only the small resistance of the inductor. This forms a voltage divider with the input resistor. Since the impedance at resonance is very low compare to the resistor, the output signal is very low in amplitude. A typical response curve is show in Fig. 2-39(c).

#### A parallel version of this circuit is show in Fig. 2-39(b), where the parallel resonant circuit is connect in series with a resistor from which the output is take.

At frequencies above and below the resonant frequency, the impedance of the parallel circuit is very low; there is, therefore, little signal attenuation, and most of the input voltage appears across the output resistor. At the resonant frequency, the parallel LC circuit has an extremely high resistive impedance compared to the output resistance, and so minimum voltage appears at the center frequency. LC filters used in this way are often refer to as traps.

Another bridge-type notch filter is the bridge-T filter show in Fig. 2-40. This filter, which is widely use in RF circuits, uses inductors and capacitors and thus has a steeper response curve than the RC twin-T notch filter. Since L is variable, the notch is tunable. Fig. 2-41 shows common symbols used to represent RC and LC filters or any other type of filter in system block diagrams or schematics.

**Figure 2-41 **Block diagram or schematic symbols for filters.

#### Active Filters in Electronics

Active filters are frequency-selective circuits that incorporate RC networks and amplifiers with feedback to produce low-pass, high-pass, bandpass, and band stop performance. These filters can replace standard passive LC filters in many applications. They offer the following advantages over standard passive LC filters.

**Gain:**Because active filters in electronics use amplifiers, they can be design to amplify as well as filter, thus offsetting any insertion loss.**No inductors:**Inductors are usually larger, heavier, and more expensive than capacitors and have greater losses. Active filters use only resistors and capacitors.**Easy to tune:**Because selected resistors can be make variable, the filter cutoff frequency, center frequency, gain, Q, and bandwidth are adjustable.**Isolation:**The amplifiers provide very high isolation between cascaded circuits because of the amplifier circuitry, thereby decreasing interaction between filter sections.**Easier impedance matching:**Impedance matching is not as critical as with LC filters.

Fig. 2-42 shows two types of low-pass active filters and two types of high-pass active filters. Note that these active filters use op amps to provide the gain. The voltage divider, made up of R1 and R2, sets the circuit gain in the circuits of Fig. 2-42(a) and (c) as in any noninverting op amp. The gain is sett by R3 and/or R1 in Fig. 2-42(b) and by C3 and/or C1 in Fig. 2-42(d). All circuits have what is call a second-order response, which means that they provide the same filtering action as a two-pole LC filter. The roll-off rate is 12 dB per octave, or 40 dB per decade. Multiple filters can be cascade to provide faster roll-off rates.

**Figure 2-42 **Types of active filters. (a) Low-pass. (b) Low-pass. (c) High-pass. (d) High-pass.

**Figure 2-43 **Active bandpass and notch filters. (a) Bandpass. (b) Bandpass. (c) High-Q notch.

Two active bandpass filters and a notch filter are show in Fig. 2-43. In Fig. 2-43(a), both RC low-pass and high-pass sections are combine with feedback to give a bandpass result. In Fig. 2-43(b), a twin-T RC notch filter is use with negative feedback to provide a bandpass result. A notch filter using a twin-T is illustrate in Fig. 2-43(c). The feedback makes the response sharper than that with a standard passive twin-T.

Active filters are make with integrated-circuit (IC) op amps and discrete RC networks. They can be design to have any of the responses discussed earlier, such as Butterworth and Chebyshev, and they are easily cascade to provide even greater selectivity.

Active filters are also available as complete packaged components. The primary disadvantage of active filters is that their upper frequency of operation is limit by the frequency response of the op amps and the practical sizes of resistors and capacitors. Most active filters are use at frequencies below 1 MHz, and most active circuits operate in the audio range and slightly above. However, today op amps with frequency ranges up to one microwave (>1 GHz) mated with chip resistors and capacitors have made RC active filters practical for applications up to the RF range.

**Crystal and Ceramic Filters**

The selectivity of a filter is limit primarily by the Q of the circuits, which is generally the Q of the inductors used. With LC circuits, it is difficult to achieve Q values over 200. In fact, most LC circuit Qs are in the range of 10 to 100, and as a result, the roll-off rate is limit. In some applications, however, it is necessary to select one desired signal, distinguishing it from a nearby undesired signal (see Fig. 2-44).

A conventional filter has a slow roll-off rate, and the undesired signal is not, therefore, fully attenuated. The way to gain greater selectivity and higher Q, so that the undesirable signal will be almost completely reject, is to use filters that are made of thin slivers of quartz crystal or certain types of ceramic materials.

These materials exhibit what is call piezoelectricity. When they are physically bent or otherwise distorted, they develop a voltage across the faces of the crystal. Alternatively, if an ac voltage is apply across the crystal or ceramic, the material vibrates at a very precise frequency, a frequency that is determined by the thickness, shape, and size of the crystal as well as the angle of cut of the crystal faces. In general, the thinner the crystal or ceramic element, the higher the frequency of oscillation.

**Figure 2-44: **How selectivity affects the ability to discriminate between signals.

Crystals and ceramic elements are widely use in oscillators to set the frequency of operation to some precise value, which is held despite temperature and voltage variations that may occur in the circuit.

Crystals and ceramic elements can also be use as circuit elements to form filters, specifically bandpass filters. The equivalent circuit of a crystal or ceramic device is a tuned circuit with a Q of 10,000 to 1,000,000, permitting highly selective filters to be build.

**Crystal Filters:**

Crystal filters are make from the same type of quartz crystals normally used in crystal oscillators. When a voltage is apply across a crystal, it vibrates at a specific resonant frequency, which is a function of the size, thickness, and direction of cut of the crystal. Crystals can be cut and ground for almost any frequency in the 100-kHz to 100-MHz range. The frequency of vibration of crystal is extremely stable, and crystals are therefore widely use to supply signals on exact frequencies with good stability.

The equivalent circuit and schematic symbol of a quartz crystal are show in Fig. 2-45. The crystal acts as a resonant LC circuit. The series LCR part of the equivalent circuit represents the crystal itself, whereas the parallel capacitance C_{P }is the capacitance of the metal mounting plates with the crystal as the dielectric.

**Figure 2-45 **Quartz crystal. (a) Equivalent circuit. (b) Schematic symbol.

**Figure 2-46 **Impedance variation with frequency of a quartz crystal.

Fig. 2-46 shows the impedance variations of the crystal as a function of frequency. At frequencies below the crystal’s resonant frequency, the circuit appears capacitive and has a high impedance. However, at some frequency, the reactances of the equivalent inductance L and the series capacitance C_{S} are equal, and the circuit resonates. The series circuit is resonant when X_{L}= ^{X}C_{S}. At this series resonant frequency f_{S}, the circuit is resistive. The resistance of the crystal is extremely low, giving the circuit an extremely high Q. Values of Q in the 10,000 to 1,000,000 range are common. This makes the crystal a highly selective series resonant circuit.

If the frequency of the signal applied to the crystal is above f_{S}, the crystal appears inductive. At some higher frequency, the reactance of the parallel capacitance C_{P} equals the reactance of the net inductance. When this occurs, a parallel resonant circuit is form. At this parallel resonant frequency f_{P}, the impedance of the circuit is resistive but extremely high.

Because the crystal has both series and parallel resonant frequencies that are close together, it makes an ideal component for use in filters. By combining crystals with selected series and parallel resonant points, highly selective filters with any desired bandpass can be construct.

The most commonly used crystal filter is the full crystal lattice show in Fig. 2-47. It is a bandpass filter. Note that transformers are use to provide the input to the filter and to extract the output. Crystals Y_{1} and Y_{2} resonate at one frequency, and crystals Y_{3} and Y_{4} resonate at another frequency. The difference between the two crystal frequencies determines the bandwidth of the filter. The 3-dB down bandwidth is approximately 1.5 times the crystal frequency spacing. For example, if the Y_{1} to Y_{2} frequency is 9 MHz and the Y_{3} to Y_{4} frequency is 9.002 MHz, the difference is 9.002μ9.000= 0.002 MHz =2 kHz. The 3-dB bandwidth is, then, 1.5×2 kHz= 3 kHz.

##### The crystals are also chose so that the parallel resonant frequency of Y_{3} to Y_{4} equals the series resonant frequency of Y_{1} to Y_{2}.

The series resonant frequency of Y_{3} to Y_{4} is equal to the parallel resonant frequency of Y_{1} to Y_{2}. The result is a passband with extremely steep attenuation. Signals outside the passband are reject as much as 50 to 60 dB below those inside the passband. Such a filter can easily discriminate between very closely spaced desired and undesired signals.

**Figure 2-47 **Crystal lattice filter.

Another type of crystal filter is the ladder filter show in Fig. 2-48, which is also a bandpass filter. All the crystals in this filter are cutt for exactly the same frequency. The number of crystals used and the values of the shunt capacitors set the bandwidth. At least six crystals must usually be cascade to achieve the kind of selectivity needed in communication applications.

**Figure 2-48 **Crystal ladder filter.

**Ceramic Filters:**

Ceramic is a manufactured crystallike compound that has the same piezoelectric qualities as quartz. Although the Q of ceramic does not have as high an upper limit as that of quartz, it is typically several thousand, which is very high compared to the Q obtainable with LC filters. Ceramic disks can be make so that they vibrate at a fixed frequency, thereby providing filtering actions. Typical ceramic filters are of the bandpass type with center frequencies of 455 kHz and 10.7 MHz.Ceramic filters are very small and inexpensive and are, therefore, widely used in transmitters and receivers. These are available in different bandwidths depending upon the application. Such ceramic filters are widely use in communication receivers.

A schematic diagram of a ceramic filter is show in Fig. 2-49. For proper operation, the filter must be driven from a generator with an output impedance of Rg and be terminated with a load of RL. The values of Rg and RL are usually 1.5 or 2 kV.

**Surface Acoustic Wave Filters****:**

A special form of a crystal filter is the surface acoustic wave (SAW) filter. This fixed tuned bandpass filter is designed to provide the exact selectivity required by a given application. Fig. 2-50 shows the schematic design of a SAW filter. SAW filters are made on a piezoelectric ceramic substrate such as lithium niobate. A pattern of interdigital fingers on the surface converts the signals into acoustic waves that travel across the filter surface. By controlling the shapes, sizes, and spacings of the interdigital fingers, the response can be tailored to any application. Interdigital fingers at the output convert the acoustic waves back to electrical signals.

**Figure 2-49 **Schematic symbol for a ceramic filter.

**Figure 2-50 **A surface acoustic wave filter.

SAW filters are normally bandpass filters used at very high radio frequencies where selectivity is difficult to obtain. Their common useful range is from 10 MHz to 3 GHz. They have a low shape factor, giving them exceedingly good selectivity at such high frequencies. They do have a significant insertion loss, usually in the 10- to 35-dB range, which must be overcome with an accompanying amplifier. SAW filters are widely used in modern TV receivers, radar receivers, wireless LANs, and cell phones.

**Switched Capacitor Filters**

Switched capacitor filters (SCFs) are active IC filters made of op amps, capacitors, and transistor switches. Also known as analog sampled data filters or commutating filters, these devices are usually implemented with MOS or CMOS circuits. They can be designed to operate as high-pass, low-pass, bandpass, or band stop filters. The primary advantage of SCFs is that they provide a way to make tuned or selective circuits in an IC without the use of discrete inductors, capacitors, or resistors.

Switched capacitor filters are made of op amps, MOSFET switches, and capacitors.

##### All components are fully integrated on a single chip, making external discrete components unnecessary.

The secret to the SCF is that all resistors are replaced by capacitors that are switched by MOSFET switches. Resistors are more difficult to make in IC form and take up far more space on the chip than transistors and capacitors. With switched capacitors, it is possible to make complex active filters on a single chip. Other advantages are selectibility of filter type, full adjustability of the cutoff or center frequency, and full adjustability of bandwidth. One filter circuit can be used for many different applications and can be set to a wide range of frequencies and bandwidths.

**Switched Integrators:**

The basic building block of SCFs is the classic op amp integrator, as shown in Fig. 2-51(a). The input is applied through a resistor, and the feedback is provided by a capacitor. With this arrangement, the output is a function of the integral of the input:

With ac signals, the circuit essentially functions as a low-pass filter with a gain of 1/RC.

**Figure 2-51 **IC integrators. (a) Conventional integrator. (b) Switched capacitor integrator.

To work over a wide range of frequencies, the integrator RC values must be changed. Making low and high resistor and capacitor values in IC form is difficult. However, this problem can be solved by replacing the input resistor with a switched capacitor, as shown in Fig. 2-51(b). The MOSFET switches are driven by a clock generator whose frequency is typically 50 to 100 times the maximum frequency of the ac signal to be filtered. The resistance of a MOSFET switch when on is usually less than 1000Ω. When the switch is off, its resistance is many megohms.

The clock puts out two phases, designated ϕ_{1} and ϕ_{2}, that drive the MOSFET switches. When S_{1} is on, S_{2} is off and vice versa. The switches are of the break- before make type, which means that one switch opens before the other is closed. When S_{1} is closed, the charge on the capacitor follows the input signal. Since the clock period and time duration that the switch is on are very short compared to the input signal variation, a brief “sample” of the input voltage remains stored on C_{1} and S_{1} turns off.

#### Now S_{2} turns on.

The charge on capacitor C1 is applied to the summing junction of the op amp. It discharges, causing a current to flow in the feedback capacitor C2. The resulting output voltage is proportional to the integral of the input. But this time, the gain of the integrator is

where f is the clock frequency.

#### Capacitor C_{1}, which is switched at a clock frequency off with period T, is equivalent to a resistor value of R=T/C_{1}.

The beauty of this arrangement is that it is not necessary to make resistors on the IC chip. Instead, capacitors and MOSFET switches, which are smaller than resistors, are used. Further, since the gain is a function of the ratio of C_{1} to C_{2}, the exact capacitor values are less important than their ratio. It is much easier to control the ratio of matched pairs of capacitors than it is to make precise values of capacitance.

By combining several such switching integrators, it is possible to create low-pass, high-pass, bandpass, and band-reject filters of the Butterworth, Chebyshev, elliptical, and Bessel type with almost any desired selectivity. The center frequency or cutoff frequency of the filter is set by the value of the clock frequency. This means that the filter can be tuned on the fly by varying the clock frequency.

A unique but sometimes undesirable characteristic of an SCF is that the output signal is really a stepped approximation of the input signal. Because of the switching action of the MOSFETs and the charging and discharging of the capacitors, the signal takes on a stepped digital form. The higher the clock frequency compared to the frequency of the input signal, the smaller this effect. The signal can be smoothed back into its original state by passing it through a simple RC low-pass filter whose cutoff frequency is set to just above the maximum signal frequency.

#### Various SCFs are available in IC form, both dedicated single-purpose or universal versions.

Some models can be configured as Butterworth, Bessel, Eliptical, or other formats with as many as eight poles. They can be used for filtering signals up to about 100 kHz. Manufacturers include Linear Technology, Maxim Integrated Products, and Texas Instruments. One of the most popular is the MF10 made by Texas Instruments. It is a universal SCF that can be set for low-pass, high-pass, bandpass, or band-reject operation. It can be used for center or cutoff frequencies up to about 20 kHz. The clock frequency is about 50 to 100 times the operating frequency.

**Commutating Filters:**

An interesting variation of a switched capacitor filter is the commutating filter shown in Fig. 2-52. It is made of discrete resistors and capacitors with MOSFET switches driven by a counter and decoder. The circuit appears to be a low-pass RC filter, but the switching action makes the circuit function as a bandpass filter. The operating frequency f_{out} is related to the clock frequency fc and the number N of switches and capacitors used.

**Figure 2-52 **A commutating SCF.

The bandwidth of the circuit is related to the RC values and number of capacitors and switches used as follows:

#### For the filter in Fig. 2-52, the bandwidth is BW 5 1/(8πRC).

Very high Q and narrow bandwidth can be obtained, and varying the resistor value makes the bandwidth adjustable.

The operating waveforms in Fig. 2-52 show that each capacitor is switched on and off sequentially so that only one capacitor is connected to the circuit at a time. A sample of the input voltage is stored as a charge on each capacitor as it is connected to the input. The capacitor voltage is the average of the voltage variation during the time the switch connects the capacitor to the circuit.

Fig. 2-53(a) shows typical input and output waveforms, assuming a sine wave input. The output is a stepped approximation of the input because of the sampling action of the switched capacitors. The steps are large, but their size can be reduced by simply using a greater number of switches and capacitors. Increasing the number of capacitors from four to eight, as in Fig. 2-53(b), makes the steps smaller, and thus the output more closely approximates the input. The steps can be eliminated or greatly minimized by passing the output through a simple RC low-pass filter whose cutoff is set to the center frequency value or slightly higher.

**Figure 2-53 **Input and output for commutating filter. (a) Four-capacitor filter. (b) Eight-capacitor filter.

**Figure 2-54 **Comb filter response of a commutating filter.

One characteristic of the commutating filter is that it is sensitive to the harmonics of the center frequency for which it is designed. Signals whose frequency is some integer multiple of the center frequency of the filter are also passed by the filter, although at a somewhat lower amplitude. The response of the filter, called a comb response, is shown in Fig. 2-54. If such performance is undesirable, the higher frequencies can be eliminated with a conventional RC or LC low-pass filter connected to the output.