Table of Contents

**What is Filter?**

A filter is a frequency-selective circuit. Filters are designed to pass some frequencies and reject others. The series and parallel resonant circuits reviewed in Section 2-2 are examples of filters. There are numerous ways to implement filter circuits.

Simple filters created by using resistors and capacitors or inductors and capacitors are called passive filters because they use passive components that do not amplify. In communication work, many filters are of the passive LC variety, although many other types are used.

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 Filter**

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 attenuated, or reduced, in the output. A high-pass filter does the opposite, allowing the higher-frequency components of the applied voltage to develop a voltage across the output load resistance.

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 blocked by the series capacitor. Furthermore, with higher frequencies in the ac component, more ac voltage is coupled. Any low-pass or high-pass filter can be thought of as a frequency-dependent voltage divider because the amount of output voltage is a function of frequency.

RC filters 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 made with RC circuits. Bandpass filters can be made by combining low-pass and high-pass RC sections, but this is rarely done.

**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 referred to as high cut filters. The ideal response curve for a low-pass filter is shown in Fig. 2-23. This response curve cannot be realized 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.

The simplest form of a low-pass filter is the RC circuit shown in Fig. 2-24(a). 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 illustrated in Fig. 2-24(b). The cutoff frequency of this filter is that point where R and XC are equal. The cutoff frequency, also known as the critical frequency, is determined by the expression.

**Example 2-23** What is the cutoff frequency of a single-section RC low-pass filter with R = 8.2 kV

and C = 0.0033 μF?

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 defined as a doubling or halving of frequency, and a decade represents a one-tenth or times-10 relationship. 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 the cutoff to 9 dB at 1200 Hz. 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 the cutoff to 23 dB at 6 kHz.

If a faster rate of attenuation is required, two RC sections set to the same cutoff frequency can be used. Such a circuit is shown in Fig. 2-25(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 caused by added attenuation of the second section. With a steeper attenuation curve, the circuit is said 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 called insertion loss. A low pass filter can also be implemented with an inductor and a resistor, as shown in Fig. 2-26. The response curve for this RL filter is the same as that shown in Fig. 2-24(b).

The cutoff frequency is determined 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.

**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 shown in Fig. 2-27(a). Approximations to the ideal response curve shown in Fig. 2-27(b) can be obtained with a variety of RC and LC filters. The basic RC high pass filter is shown in Fig. 2-28(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, XC is very high.

When XC 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 derived from setting XC 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 implemented with a coil and a resistor, as shown in Fig. 2-28(b). The cutoff frequency is

The response curve for this filter is the same as that shown in Fig. 2-27(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 obtained by cascading filter sections.

**Example 2-24** What is the closest standard EIA resistor value that will produce a cutoff frequency of

3.4 kHz with a 0.047-μF capacitor in a high-pass RC filter?

**RC Notch Filter**

Notch filters are also referred to as bandstop or band-reject filters. Band-reject filters are used to greatly attenuate a narrow range of frequencies around a center point. Notch filters accomplish the same purpose, but for a single frequency. A simple notch filter that is implemented with resistors and capacitors as shown in Fig. 2-29(a) is called 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 balanced. If the component values are precisely matched, 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 shown in Fig. 2-29(b). The center notch frequency is computed 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 used 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 matched to achieve high attenuation.

**Example 2-25** What values of capacitors would you use in an RC twin-T notch filter to remove 120 Hz if R 5 220 kV?

**LC Filter**

RC filters are used primarily at the lower frequencies. They are very common at audio frequencies but are rarely used 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 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 has been developed. 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.

**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).

**Stopband**

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

**Attenuation**

This is the amount by which undesired frequencies in the stopband are reduced. It can be expressed as a power ratio or voltage ratio of the output to the input. Attenuation is usually given in decibels.

**Insertion loss**

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

**Impedance**

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

**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, is known as ripple. It is usually stated in decibels. There may also be a ripple in the stop bandwidth in some types of filters.

**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 obtained with practical filters. The filter in Fig. 2-30 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 = 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.

**Pole**

A pole is a frequency at which there is high impedance in the circuit. It is also used 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, the number of poles is equal to the number of reactive components in the filter. For bandpass and band-reject filters, the number of poles is generally assumed to be one-half of the number of reactive components used.

**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 implemented with inductors and capacitors. Such filters can be made 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 used, 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.

**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 cascaded to provide an even greater roll-off rate. The chart in Fig. 2-32 shows the attenuation rates for low-pass filters with two through seven poles. The horizontal axis f/FC is the ratio of any given frequency in ratio to the filter cutoff frequency Fc.

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 = 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 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 is similar to that in Fig. 2-32 is also used to determine attenuation for filters with multiple poles. Cascading these sections provides a greater attenuation rate. Those filter configurations using the least number of inductors are preferred for lower cost and less space.

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