14 Filter I

Vinay Gupta

    Introduction

Filters are the electrical circuits that allow a specific band of frequencies to pass through it without any attenuation (i.e. output is approximately equal to the input) and attenuate others lying outside this band (i.e. output is nearly zero). Thus, all the filters have an atleast one pass-band and one attenuation-band (stop-band). The frequencies that separate the pass and attenuation bands are called cut-off frequencies. Filters are used widely in the circuits which require separation of signals according to their frequencies. Some of the common applications include audio-signal shaping and sound enhancement, channel separation and signal enhancement in communication circuits, smoothing of digitally generated analog signals etc.

Classification of filters

Electrical filters may be classified in various categories as:

1. Analog or digital

2. Active or passive

3. Audio or Radio-frequency (RF)

Analog filters are the filters used to process analog signals using analog techniques while the digital filters are used to process analog signals using digital techniques. Filters are classified as active and passive depending on the type of elements used in the construction. A passive filter is designed using passive components such as resistors, capacitors, inductors while an active filter is built using active components such as transistors or op-amps in addition to the passive components. The operating frequency of the filter depends largely on the type of element used in the filter. For example, RC filters are used for audio or low frequency signals whereas, LC or crystal filters are used for radio frequencies. Depending on the range of operation of the filter, it may be classified as audio-frequency (AF) or radio-frequency (RF).

Active versus passive filters

The simplest approach to build a filter is using passive components such as resistors, capacitors, inductors etc. Passive filters operate well for radio frequencies but for audio frequencies, inductors are not preferred as they are large, costly and dissipate more power. Also, at low frequencies, in order to increase the inductance, more turns of wire are required which add up in the series resistance, thereby degrading the performance of inductor. Despite the simplicity in design of filters using passive components, active filters are more preferred for practical applications due to the following advantages:

1.   Gain and frequency adjustment flexibility

Since, transistors or op-amps are capable of providing the gain in the circuit, therefore the input signal is transferred to the output without any attenuation. This is in contrast to a passive filter where some drop in voltage takes place across the passive components. Also, it is easier to adjust the frequency range in active filters.

2.   It prevents loading

A passive filter has low input impedance which loads the source. Also, it has high output impedance which provides poor load driving capability. Load impedance is not isolated from the passive filter components. Therefore, any variation in load alters the filter response characteristics. On the other hand, op-amp offers high input impedance and low output impedance, which prevents the loading of the source or load. Also, the load is isolated from the frequency determining network and thus, will not affect the characteristics of the filter.

3.   Low cost

Active filters are more economical than passive filters because of the easy availability of cost-effective op-amps in comparison to inductors which are costly.

Active filters are widely used in the field of communication and signal processing. Infact, they are an essential component in almost all electronic systems such as radio, television, telephone, radar, space satellites, biomedical equipment etc. and are thus, discussed in this module.

Active filters

As discussed before, active filters use op-amp as active element and resistor and capacitor as passive element. Depending on the frequency range allowed to pass through the filter, active filters are classified as:

1.  Low-pass filter

2. High-pass filter

3. Band-pass filter

4. Band-reject filter

5. All-pass filter

The frequency response characteristics of various types of active filters are shown in figure 1. The ideal response of filter is shown by dashed line whereas, practical response is shown by a solid line.

Figure 1: Frequency response of (a) Low-pass (b) High-pass (c) Band-pass (d) Band reject filter (e) Phase shift between input and output voltages of an all-pass filter

A low-pass filter allows all frequencies below the high cut-off frequency (fH) to pass through it and attenuates all others. Thus, it has a constant gain from 0 to fH Hz as shown in figure 1(a). Bandwidth is given by fH. At fH, the gain reduces by 3 dB and at frequencies f > fH, the gain decreases with increase in input frequency. The frequencies lying between 0 and fH are called pass-band frequencies and those lying beyond fH are attenuated and hence, called stop-band frequencies. The filter response for an ideal low-pass filter is also shown by a dashed line in figure 1(a) for comparison. It may be seen that an ideal filter has zero loss in its pass-band (0 < f < fH) whereas, it has infinite loss in its stop-band (f>fH). It is difficult to realize the ideal filter response in practical applications because linear networks cannot produce discontinuities. However, it is possible to obtain a filter response that approximates the ideal one by using high-speed op-amps, special design techniques and precision components etc. Some of the most widely used practical filters that approximate the ideal response includes Butterworth, Chebyshev and Cauer filters. Butterworth filter has a distinct characteristic of having a flat pass-band and a flat stop-band due to which it also called as maximally flat or flat-flat filter. Chebyshev filter has a ripple pass-band and a flat-stop-band. It has a sharper cut-off frequency than Butterworth filter but because of the presence of ripples in the pass-band, it has the drawback of appearance of various gain maxima and minima below the cut-off frequency. Cauer filter also known as elliptic filter, has ripple pass-band and a ripple stop-band. Generally, Cauer filter gives the best stop-band response among all the three filters. Also, it has the sharpest roll-off in the transition region. The frequency response of various types of practical active filters is shown in figure 2. Out of the various types of filters, Butterworth filters function as excellent general purpose filters and are most widely used. In addition, Butterworth filters offer the advantage of simplicity in design and are thus, discussed in this module.

Figure 2: Frequency response of various practical active filters

A high-pass filter as shown in figure 1(b) allows higher frequencies to pass through it without attenuation. The gain increases with increase in frequency till the lower cut-off frequency (fL) and has a constant gain thereafter. Thus, a high-pass filter has a stop-band from 0 to fL Hz and it has a pass-band for frequencies f > fL.

A band-pass filter allows a particular band of frequencies lying between a lower cut-off frequency (fL) and a higher cut-off frequency (fH) (fH > fL) to pass through it. As shown in figure 1(c), the gain increases with increase in input frequency till fL, becomes constant for frequencies lying between fL and fH and decreases thereafter with further increase in frequency. Thus, it has a pass-band for fL < f < fH and two stop-bands for f > fH and 0 < f < fL. Bandwidth of a band-pass filter is (fH – fL).

A band-reject filter works exactly opposite to that of a band-pass filter. It rejects (does not allow) a specific range of frequencies lying between the cut-off frequencies, fL and fH (where fH > fL) and allows other frequencies as shown in figure 1(d). Thus, it has two pass bands for 0 < f < fL and f > fH and a stop-band for fL < f < fH. Centre of the stop-band (pass-band) in band-reject (band-pass) filter is called as the centre frequency (fC). A band reject filter is also called as band elimination or band-stop filter.

An all-pass filter, as the name suggests allows all frequencies to pass through it without attenuation as evident from the equal amplitudes of input and output voltages in figure 1(e). There is a phase shift between the input and output signals and is a function of frequency. The highest frequency upto which the input and output voltages remain equal in magnitude is dependent on the unity gain-bandwidth of the op-amp. However, at this frequency, the phase shift between input and output is maximum.

Order of a filter

The frequency response of all types of filters in its stop-band either steadily decreases or increases or both depending on the type of filter as shown in figure 1(a – e).The order of a filter is determined by the rate of change in gain in going from pass-band to stop-band or vice-versa. For example, in a low-pass filter for f >  fH, the filter is called as a first order low-pass filter if the gain rolls-off at a rate of 20 dB/decade. For a second-order low-pass filter, the gain decreases at a rate of 40 dB/decade and so on. In a high-pass filter, the opposite occurs. The gain in a first order high-pass filter increases at a rate of 20 dB/decade for frequencies f < fL in its stop-band and in a second-order high-pass filter, the rate of increase in gain is 40 dB/decade. Frequency response of low-pass and high-pass filters for three orders is shown in figure 3(a) and 3(b) respectively. It may be seen from figure 3, that as the order of filter increases, the frequency response approaches to that of an ideal filter.

Figure 3: Frequency response of various orders of (a) Low-pass filter and (b) High-pass filter

First-order low-pass Butterworth filter

The circuit diagram for a first-order low-pass Butterworth filter is shown in figure 4(a). The circuit uses a RC network for filtering. Op-amp is used in the non-inverting configuration so that it does not load the R-C network. Gain of the filter is determined by the resistors R1 and Rf.

Figure 4: (a) Circuit diagram of a first-order low-pass Butterworth filter (b) Its frequency response

According to the voltage-divider rule, the voltage at the non-inverting terminal (v1) can be written as

The frequency response of first order low-pass Butterworth filter is shown in figure 4(b). The designed filter has a constant gain equal to that of a non-inverting amplifier in the frequency range varied from 0 to fH Hz. At high cut-off frequency, i.e f=fH, the gain reduces to 0.707 of its constant value and after fH, it decreases with increase in frequency at a constant rate. When the frequency in increased ten times (i.e. by a decade), voltain gain is reduced to one-tenth value (i.e it reduces by 20 dB). Hence, the voltage gain rolls-off at a rate of 20 dB/decade for frequencies f > fH. The frequency f=fH is called the cut-off frequency as at this frequency, the gain of the filter is reduced by 3 dB and it is the minimum change in power that can be resolved by human ear. Cut-off frequency is also called as corner frequency, break frequency or -3 dB frequency.

Second-order low-pass filter

Second-order filters are important in electronics because they can be used to design higher-order filters having very high roll-off rates. By cascading first and second order filters, filters of nth order for both even and odd values of n can be easily constructed. The easiest procedure of making a second order filter is to cascade two first order active filters but this is not preferred as the circuit uses two op-amps. Alternately, a  first-order low-pass filter can be converted into a second-order low-pass filter (where the gain rolls off at the rate of 40 dB/decade) by including an additional RC network as shown in figure 5(a).

Figure 5: (a) Circuit diagram of a second order low-pass filter (b) Input circuit of the low-pass filter transformed into the s-domain (c) frequency response

As for the first order low-pass filter, the gain of the second order low-pass filter is decided by the resistors R1 and RF, however, the cut-off frequency fH is a function of resistances, R and R’ and capacitors C and C’. In order to derive an expression for the cut-off frequency of the filter, the circuit is transformed into s-domain where s=jω as shown in figure 5(b). Applying Kirchoff’s current law at node VA, we have,

For frequencies f > fH, the gain of second order low-pass filter rolls off at the rate of 40 dB/decade. Therefore, the denominator (which is quadratic in s) must have real and equal roots.

The frequency response of the second order filter is shown in figure 5(c). It is similar to the first order high-pass filter, however, here the gain rolls off at a rate of 40 dB/decade.

First-order high-pass Butterworth filter

The design of high-pass filter is complementary to that of a low-pass filter and can be formed using a non-inverting op-amp by simply interchanging the position of frequency determining resistor and capacitor in low-pass Butterworth filter. The circuit diagram of a high-pass filter is shown in figure 6.

Figure 6: (a) Circuit diagram of a first-order high-pass Butterworth filter (b) Its frequency response

    According to the voltage-divider rule, the voltage at the non-inverting terminal (v1) can be written as

The frequency response of the high-pass Butterworth filter is shown in figure 6(b). High-pass Butterworth filter allows all frequencies greater than fL to pass through it with a gain equal to that of a non-inverting amplifier (AF). The frequency, fL is called the low cut-off frequency and corresponds to the frequency at which gain reduces to (1/√2) of its constant value. At frequencies lower than fL, gain of the high-pass filter decreases with a constant rate.

Second order high-pass filter

Similar to the first order high-pass filter, a second order high pass filter can be formed from second order low-pass filter by interchanging the frequency determining resistors and capacitors or it can be viewed as formed from a first order high-pass filter by including an additional RC arm. The circuit diagram of a second order high-pass filter is shown in figure 7(a) and the corresponding frequency response is given in figure 7(b).

Figure 7: (a) Circuit diagram of a second order high-pass filter (b) Frequency response

The cut-off frequency, fL can be obtained in a similar way as done for a second-order low-pass filter by transforming the circuit into s-domain. The voltage gain-magnitude equation of the second-order high-pass filter is given as

References:

• Op-Amps and Linear Integrated Circuit, R. A. Gayakwad, 4th edition, 2000, Prentice Hall. Operational Amplifiers, 5th Edition by George Clayton, Steve Winder, Elsevier India, 2012,
• Operational Amplifiers & Linear ICs, David A. Bell, Oxford University press, 3rd Edition, (2011).
• Operational Amplifiers and Linear Integrated Circuits, Robert F. Coughlin, Frederick F. Driscoll, 6th Edition, Pearson.