LM833,LMF100,MF10 Application Note 779 A Basic Introduction to Filters - Active, Passive,and Switched Capacitor Literature Number: SNOA224A A Basic Introduction to Filters—Active, Passive, and Switched-Capacitor National Semiconductor A Basic Introduction to Application Note 779 Kerry Lacanette Filters—Active, Passive, April 21, 2010 and Switched-Capacitor 1.0 Introduction and the most widely-used mathematical tools are based in the frequency domain. Filters of some sort are essential to the operation of most The frequency-domain behavior of a filter is described math- electronic circuits. It is therefore in the interest of anyone in- ematically in terms of its transfer function or network volved in electronic circuit design to have the ability to develop function. This is the ratio of the Laplace transforms of its out- filter circuits capable of meeting a given set of specifications. put and input signals. The voltage transfer function H(s) of a Unfortunately, many in the electronics field are uncomfortable filter can therefore be written as: with the subject, whether due to a lack of familiarity with it, or a reluctance to grapple with the mathematics involved in a complex filter design. (1) This Application Note is intended to serve as a very basic in- troduction to some of the fundamental concepts and terms where VIN(s) and VOUT(s) are the input and output signal volt- associated with filters. It will not turn a novice into a filter de- ages and s is the complex frequency variable. signer, but it can serve as a starting point for those wishing to The transfer function defines the filter's response to any ar- learn more about filter design. bitrary input signal, but we are most often concerned with its effect on continuous sine waves. Especially important is the 1.1 Filters and Signals: What Does a magnitude of the transfer function as a function of frequency, which indicates the effect of the filter on the amplitudes of Filter Do? sinusoidal signals at various frequencies. Knowing the trans- In circuit theory, a filter is an electrical network that alters the fer function magnitude (or gain) at each frequency allows us amplitude and/or phase characteristics of a signal with re- to determine how well the filter can distinguish between sig- spect to frequency. Ideally, a filter will not add new frequen- nals at different frequencies. The transfer function magnitude cies to the input signal, nor will it change the component versus frequency is called the amplitude response or some- frequencies of that signal, but it will change the relative am- times, especially in audio applications, the frequency re- plitudes of the various frequency components and/or their sponse. phase relationships. Filters are often used in electronic sys- Similarly, the phase response of the filter gives the amount tems to emphasize signals in certain frequency ranges and of phase shift introduced in sinusoidal signals as a function reject signals in other frequency ranges. Such a filter has a of frequency. Since a change in phase of a signal also rep- gain which is dependent on signal frequency. As an example, resents a change in time, the phase characteristics of a filter consider a situation where a useful signal at frequency f1 has become especially important when dealing with complex sig- been contaminated with an unwanted signal at f2. If the con- nals where the time relationships between signal components taminated signal is passed through a circuit (Figure 1) that at different frequencies are critical. has very low gain at f2 compared to f1, the undesired signal By replacing the variable s in (1) with jω, where j is equal to can be removed, and the useful signal will remain. Note that and ω is the radian frequency (2πf), we can find the in the case of this simple example, we are not concerned with filter's effect on the magnitude and phase of the input signal. the gain of the filter at any frequency other than f1 and f2. As The magnitude is found by taking the absolute value of (1): long as f2 is sufficiently attenuated relative to f1, the perfor- mance of this filter will be satisfactory. In general, however, a filter's gain may be specified at several different frequencies, or over a band of frequencies. (2) Since filters are defined by their frequency-domain effects on and the phase is: signals, it makes sense that the most useful analytical and graphical descriptions of filters also fall into the frequency do- main. Thus, curves of gain vs frequency and phase vs fre- (3) quency are commonly used to illustrate filter characteristics, AN-779 1122101 FIGURE 1. Using a Filter to Reduce the Effect of an Undesired Signal at Frequency f2, while Retaining Desired Signal at Frequency f1 © 2010 National Semiconductor Corporation 11221 www.national.com As an example, the network of Figure 2 has the transfer func- The phase is: tion: AN-779 (4) (6) The above relations are expressed in terms of the radian fre- quency ω, in units of radians/second. A sinusoid will complete one full cycle in 2π radians. Plots of magnitude and phase versus radian frequency are shown in Figure 3. When we are more interested in knowing the amplitude and phase re- sponse of a filter in units of Hz (cycles per second), we convert from radian frequency using ω = 2πf, where f is the frequency 1122102 in Hz. The variables f and ω are used more or less inter- changeably, depending upon which is more appropriate or FIGURE 2. Filter Network of Example convenient for a given situation. Figure 3(a) shows that, as we predicted, the magnitude of the This is a 2nd order system. The order of a filter is the highest transfer function has a maximum value at a specific frequency power of the variable s in its transfer function. The order of a (ω0) between 0 and infinity, and falls off on either side of that filter is usually equal to the total number of capacitors and frequency. A filter with this general shape is known as a band- inductors in the circuit. (A capacitor built by combining two or pass filter because it passes signals falling within a relatively more individual capacitors is still one capacitor.) Higher-order narrow band of frequencies and attenuates signals outside of filters will obviously be more expensive to build, since they that band. The range of frequencies passed by a filter is use more components, and they will also be more complicat- known as the filter's passband. Since the amplitude response ed to design. However, higher-order filters can more effec- curve of this filter is fairly smooth, there are no obvious bound- tively discriminate between signals at different frequencies. aries for the passband. Often, the passband limits will be Before actually calculating the amplitude response of the net- defined by system requirements. A system may require, for work, we can see that at very low frequencies (small values example, that the gain variation between 400 Hz and 1.5 kHz of s), the numerator becomes very small, as do the first two be less than 1 dB. This specification would effectively define terms of the denominator. Thus, as s approaches zero, the the passband as 400 Hz to 1.5 kHz. In other cases though, numerator approaches zero, the denominator approaches we may be presented with a transfer function with no pass- one, and H(s) approaches zero. Similarly, as the input fre- band limits specified. In this case, and in any other case with quency approaches infinity, H(s) also becomes progressively no explicit passband limits, the passband limits are usually smaller, because the denominator increases with the square assumed to be the frequencies where the gain has dropped of frequency while the numerator increases linearly with fre- by 3 decibels (to or 0.707 of its maximum voltage gain). quency. Therefore, H(s) will have its maximum value at some These frequencies are therefore called the −3 dB frequen- frequency between zero and infinity, and will decrease at fre- cies or the cutoff frequencies. However, if a passband gain quencies above and below the peak. variation (i.e., 1 dB) is specified, the cutoff frequencies will be To find the magnitude of the transfer function, replace s with the frequencies at which the maximum gain variation specifi- jω to yield: cation is exceeded. (5) 1122105 (b) 1122103 (a) FIGURE 3. Amplitude (a) and phase (b) response curves for example filter. Linear frequency and gain scales. www.national.com 2 AN-779 The precise shape of a band-pass filter's amplitude response frequency axis. A logarithmic frequency scale is very useful curve will depend on the particular network, but any 2nd order in such cases, as it gives equal weight to equal ratios of fre- band-pass response will have a peak value at the filter's cen- quencies. ter frequency. The center frequency is equal to the geometric Since the range of amplitudes may also be large, the ampli- mean of the −3 dB frequencies: tude scale is usually expressed in decibels (20log|H(jω)|). Figure 4 shows the curves of Figure 3 with logarithmic fre- (7) quency scales and a decibel amplitude scale. Note the im- where f is the center frequency proved symmetry in the curves of Figure 4 relative to those of c Figure 3. fI is the lower −3 dB frequency f is the higher −3 dB frequency h 1.2 The Basic Filter Types Another quantity used to describe the performance of a filter is the filter's “Q”.