A Basic Introduction to Filters - Active, Passive and ...

1 TL/H/11221A Basic Introduction to Filters active , Passive , and Switched-CapacitorAN-779 National SemiconductorApplication Note 779 Kerry LacanetteApril 1991A Basic Introduction toFilters active , Passive ,and INTRODUCTIONF ilters of some sort are essential to the operation of mostelectronic circuits. It is therefore in the interest of anyoneinvolved in electronic circuit design to have the ability todevelop filter circuits capable of meeting a given set ofspecifications. Unfortunately, many in the electronics fieldare uncomfortable with the subject, whether due to a lack offamiliarity with it, or a reluctance to grapple with the mathe-matics involved in a complex filter Application Note is intended to serve as a very basicintroduction to some of the fundamental concepts andterms associated with Filters .

2 It will not turn a novice into afilter designer, but it can serve as a starting point for thosewishing to learn more about filter Filters and Signals: What Does a filter Do?In circuit theory, a filter is an electrical network that altersthe amplitude and/or phase characteristics of a signal withrespect to frequency. Ideally, a filter will not add new fre-quencies to the input signal, nor will it change the compo-nent frequencies of that signal, but it will change the relativeamplitudes of the various frequency components and/ortheir phase relationships. Filters are often used in electronicsystems to emphasize signals in certain frequency rangesand reject signals in other frequency ranges. Such a filterhas againwhich is dependent on signal frequency. As anexample, consider a situation where a useful signal at fre-quency f1has been contaminated with an unwanted signalat f2.

3 If the contaminated signal is passed through a circuit(Figure 1)that has very low gain at f2compared to f1, theundesired signal can be removed, and the useful signal willremain. Note that in the case of this simple example, we arenot concerned with the gain of the filter at any frequencyother than f1and f2. As long as f2is sufficiently attenuatedrelative to f1, the performance of this filter will be satisfacto-ry. In general, however, a filter s gain may be specified atseveral different frequencies, or over a band of Filters are defined by their frequency-domain effectson signals, it makes sense that the most useful analyticaland graphical descriptions of Filters also fall into the fre-quency domain. Thus, curves of gain vs frequency andphase vs frequency are commonly used to illustrate filtercharacteristics,and the most widely-used mathematicaltools are based in the frequency frequency-domain behavior of a filter is described math-ematically in terms of itstransfer functionornetworkfunction.

4 This is the ratio of the Laplace transforms of itsoutput and input signals. The voltage transfer function H(s)of a filter can therefore be written as:(1)H(s)eVOUT(s)VIN(s)where VIN(s) and VOUT(s) are the input and output signalvoltages and s is the complex frequency transfer function defines the filter s response to anyarbitrary input signal, but we are most often concerned withits effect on continuous sine waves. Especially important isthe magnitude of the transfer function as a function of fre-quency, which indicates the effect of the filter on the ampli-tudes of sinusoidal signals at various frequencies. Knowingthe transfer function magnitude (or gain) at each frequencyallows us to determine how well the filter can distinguishbetween signals at different frequencies. The transfer func-tion magnitude versus frequency is called theamplituderesponseor sometimes, especially in audio applications,thefrequency , thephase responseof the filter gives the amountofphase shiftintroduced in sinusoidal signals as a functionof frequency.

5 Since a change in phase of a signal also rep-resents a change in time, the phase characteristics of a filterbecome especially important when dealing with complexsignals where the time relationships between signal compo-nents at different frequencies are replacing the variable s in (1) with j0, where j is equal to0b1 , and0is the radian frequency (2qf), we can find thefilter s effect on the magnitude and phase of the input sig-nal. The magnitude is found by taking the absolute value of(1):(2)lH(j0)le VOUT(j0)VIN(j0) and the phase is:(3)arg H(j0)eargVOUT(j0)VIN(j0)TL/H/11221 1 FIGURE 1. Using a filter to Reduce the Effect of an Undesired Signal atFrequency f2, while Retaining Desired Signal at Frequency f1C1995 National Semiconductor CorporationRRD-B30M75/Printed in U. S. an example, the network ofFigure 2has the transferfunction:(4)H(s)ess2asa1TL/H/112 21 2 FIGURE 2.

6 filter Network of ExampleThis is a 2nd order system. Theorderof a filter is the high-est power of the variable s in its transfer function. The orderof a filter is usually equal to the total number of capacitorsand inductors in the circuit. (A capacitor built by combiningtwo or more individual capacitors is still one capacitor.)Higher-order Filters will obviously be more expensive tobuild, since they use more components, and they will alsobe more complicated to design. However, higher-order fil-ters can more effectively discriminate between signals atdifferent actually calculating the amplitude response of thenetwork, we can see that at very low frequencies (smallvalues of s), the numerator becomes very small, as do thefirst two terms of the denominator. Thus, as s approacheszero, the numerator approaches zero, the denominator ap-proaches one, and H(s) approaches zero.

7 Similarly, as theinput frequency approaches infinity, H(s) also becomes pro-gressively smaller, because the denominator increases withthe square of frequency while the numerator increases lin-early with frequency. Therefore, H(s) will have its maximumvalue at some frequency between zero and infinity, and willdecrease at frequencies above and below the find the magnitude of the transfer function, replace s withj0to yield:(5)A(0)elH(s)le j0b02aj0a1 e0002a(1b02)2 The phase is:(6)i(0)earg H(s)e90 btanb102(1b02)The above relations are expressed in terms of the radianfrequency0, in units of radians/second. A sinusoid willcomplete one full cycle in 2qradians. Plots of magnitudeand phase versus radian frequency are shown inFigure we are more interested in knowing the amplitude andphase response of a filter in units of Hz (cycles per second),we convert from radian frequency using0e2qf, where f isthe frequency in Hz.

8 The variables f and0are used more orless interchangeably, depending upon which is more appro-priate or convenient for a given 3(a)shows that, as we predicted, the magnitude ofthe transfer function has a maximum value at a specific fre-quency (00) between 0 and infinity, and falls off on eitherside of that frequency. A filter with this general shape isknown as aband-passfilter because it passes signals fall-ing within a relatively narrow band of frequencies and atten-uates signals outside of that band. The range of frequenciespassed by a filter is known as the filter spassband. Sincethe amplitude response curve of this filter is fairly smooth,there are no obvious boundaries for the passband. Often,the passband limits will be defined by system system may require, for example, that the gain variationbetween 400 Hz and kHz be less than 1 dB.

9 This specifi-cation would effectively define the passband as 400 Hz kHz. In other cases though, we may be presented with atransfer function with no passband limits specified. In thiscase, and in any other case with no explicit passband limits,the passband limits are usually assumed to be the frequen-cies where the gain has dropped by 3 decibels (to02/2 of its maximum voltage gain). These frequencies aretherefore called theb3 dB frequenciesor thecutoff fre-quencies. However, if a passband gain variation ( , 1 dB)is specified, the cutoff frequencies will be the frequencies atwhich the maximum gain variation specification is 3(a)TL/H/11221 5(b)FIGURE 3. Amplitude (a) and phase (b) response curvesfor example filter . Linear frequency and gain precise shape of a band-pass filter s amplitude re-sponse curve will depend on the particular network, but any2nd order band-pass response will have a peak value at thefilter scenter frequency.

10 The center frequency is equal tothe geometric mean of theb3 dB frequencies:fce0fIfh(8)where fcis the center frequencyfIis the lowerb3 dB frequencyfhis the higherb3 dB frequencyAnother quantity used to describe the performance of a filteris the filter s Q . This is a measure of the sharpness ofthe amplitude response. The Q of a band-pass filter is theratio of the center frequency to the difference between the2b3 dB frequencies (also known as theb3 dB bandwidth).Therefore:(9)QefcfhbfIWhen evaluating the performance of a filter , we are usuallyinterested in its performance overratiosof we might want to know how much attenuation occursat twice the center frequency and at half the center frequen-cy. (In the case of the 2nd-order bandpass above, the atten-uation would be the same at both points).