What Are Active Filters?
Active filters are frequency selective circuits that incorporate RC networks and amplifiers with feedback to produce a low-pass, high-pass, bandpass, and bandstop 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 use amplifiers, they can be designed 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 made 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 filter 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 set 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 called 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 cascaded to provide faster roll-off rates.
Two active bandpass filters and a notch filter are shown in Fig. 2-43. In Fig. 2-43(a), both RC low-pass and high-pass sections are combined with feedback to give a bandpass result. In Fig. 2-43(b), a twin-T RC notch filter is used with negative feedback to provide a bandpass result. A notch filter using a twin-T is illustrated in Fig. 2-43(c).
The feedback makes the response sharper than that with a standard passive twin-T. Active filters are made with integrated-circuit (IC) op-amps and discrete RC networks. They can be designed to have any of the responses discussed earlier, such as Butterworth and Chebyshev, and they are easily cascaded 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 limited by the frequency response of the op-amps and the practical sizes of resistors and capacitors. Most active filters are used at frequencies below 1 MHz, and most active circuits operate in the audio range and slightly above. However, today op-amps with a frequency range 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.
What is Crystal Filters?
The selectivity of a filter is limited 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 limited. 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 rejected, is to use filters that are made of thin slivers of quartz crystal or certain types of ceramic materials. These materials exhibit what is called 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 applied 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. Crystals and ceramic elements are widely used 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 used 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 built.
Crystal filters are made from the same type of quartz crystals normally used in crystal oscillators. When a voltage is applied across a crystal, it vibrates at a specific resonant frequency, which is a function of the size, thickness, and direction of the 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 a crystal is extremely stable, and crystals are therefore widely used to supply signals on exact frequencies with good stability. The equivalent circuit and schematic symbol of a quartz crystal are shown 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 CP is the capacitance of the metal mounting plates with the crystal as the dielectric.
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 CS are equal, and the circuit resonates. The series circuit is resonant when XL = XCS. At this series resonant frequency fS, 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 fS, the crystal appears inductive. At some higher frequency, the reactance of the parallel capacitance CP equals the reactance of the net inductance. When this occurs, a parallel resonant circuit is formed. At this parallel resonant frequency fP, 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 constructed. The most commonly used crystal filter is the full crystal lattice shown in Fig. 2-47. It is a bandpass filter. Note that transformers are used to provide the input to the filter and to extract the output. Crystals Y1 and Y2 resonate at one frequency, and crystals Y3 and Y4 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 Y1 to Y2 frequency is 9 MHz and the Y3 to Y4 frequency is 9.002 MHz, the difference is 9.002 – 9.000 = 0.002 MHz = 2 kHz. The 3-dB bandwidth is, then, 1.5×2 kHz = 3 kHz. The crystals are also chosen so that the parallel resonant frequency of Y3 to Y4 equals the series resonant frequency of Y1 to Y2. The series resonant frequency of Y3 to Y4 is equal to the parallel resonant frequency of Y1 to Y2. The result is a passband with extremely steep attenuation.
Signals outside the passband are rejected 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. Another type of crystal filter is the ladder filter shown in Fig. 2-48, which is also a bandpass filter. All the crystals in this filter are cut 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 cascaded to achieve the kind of selectivity needed in communication applications.
What Are Ceramic Filters?
Ceramic is a manufactured crystalline compound that has the same piezoelectric qualities as quartz. Ceramic disks can be made so that they vibrate at a fixed frequency, thereby providing filtering actions. Ceramic filters are very small and inexpensive and are, therefore, widely used in transmitters and receivers. 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. Typical ceramic filters are of the bandpass type with center frequencies of 455 kHz and 10.7 MHz. These are available in different bandwidths depending upon the application. Such ceramic filters are widely used in communication receivers. A schematic diagram of a ceramic filter is shown 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. 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 the 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.
What is 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 bandstop 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 selectability 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.
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 fi lter with a gain of 1/RC.
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 1000 V. When the switch is off, its resistance is many megohms. The clock puts out two phases, designated ϕ1 and ϕ2, that drive the MOSFET switches. When S1 is on, S2 is off and vice versa. The switches are of the break- before making type, which means that one switch opens before the other is closed. When S1 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 C1 and S1 turns off. Now S2 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 C1, which is switched at a clock frequency of f with period T, is equivalent to a resistor value of R = T/C1. 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 C1 to C2, 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 turned 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, Elliptical, 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
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 fout is related to the clock frequency fc and the number N of switches and capacitors used.
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 = 1y(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 shows 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.
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.
Switched Capacitor Filters | Commutating Filters ( Active Filters | Crystal Filters | Ceramic Filters | Surface Acoustic Wave )
Crystal Filters and Ceramic Filters ( Active Filters | Crystal Filters | Ceramic Filters | Surface Acoustic Wave )
Active Filters | Schematic symbols | Characteristics ( Active Filters | Crystal Filters | Ceramic Filters | Surface Acoustic Wave )
Types of Filters | Mechanical | Band Reject | Bandpass ( Active Filters | Crystal Filters | Ceramic Filters | Surface Acoustic Wave )
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