Filter Design Guide

Contents

Introduction

This guide presents our active and passive filter circuits, as well as those of OSI. We assume you have Microsoft Exel, so you can use our Filter Tool. The filter tool calculates and plots the frequency response of a variety of filters. At times in our discussion, we assume you understand transfer functions, and the way their poles and zeros determine their frequency response.

Active Filters

An active filter contains an amplifier whose output is connected to its input through passive components, usually capacitors and resistors. This feedback of the output to the input allows us to build filters with imaginary poles using capacitors and resistors alone. Without feedback, a filter with imaginary poles must have both inductors and capacitors. The main purpose of active filters is to eliminate inductors and decrease the value of the filter's capacitors.

In a classic, passive filter made of inductors, capacitors, and resistores, the filter's frequency response is the result of the impedance of inductors and capacitors changing with respect to one another, and with respect to the resistors in the filter. At the cut-off frequency of a filter, the impedances of all its elements will be of the same order of magnitude. Suppose the resistors are of order 1 kΩ. If the filter's cut-off frequency is 1 kHz, the inductors must be of order 100 mH. The 181LY-104J from Toko is out of stock right now at Digi-Key, but they list a price of $3.75 US, and the part itself is 14 mm hight and 10 mm in diameter. Suppose we drop the resistors to 10 Ω. The inductors are of order 1 mH, and we can get these in surface mount packages for around $1 US. By reducing the values of the resistors, we reduce the inductors also, but we increase the capacitors. A capacitor with impedance 10 Ω at 1 kHz would be of order 10 μF. We can get a 10-μF surface-mount capacitor for around $1 US also. Now we drive the circuit with a 1-kHz, 1-V sine wave. Because the resistors are of order 10 &Omega, we will draw of order 100 mA from our voltage source. So we could decrease the size of the signal to around 10 mV, and now we draw only 1 mA. We have, however, overlooked one thing: the precision of the components. The precision needs to be at least 5% in a 4-pole filter, or else the poles will be wrong, and the filter response will not be sharp. A 5% 1-mH inductor costs more than $1, as does a 5% 10-μF capacitor. And if we have to drop the frequency of the filter to 100 Hz, our cost increases tenfold, not to mention the size.

With the help of feedback around a low-cost, low-power operational amplifiers (op-amps), we can eliminate inductors and use the same capacitance throughout our filter. If we use capacitors from the same reel of parts, their values are similar. It is the relative values of the capacitors that is most important to the filter response shape. We use resistors of different values, but precise resistors are inexpensive compared to precise capacitors. It is hard to find a 1% accurate capacitor, but a 1% accurate resistor costs only a few cents. If we want to change the filter function, we just buy new resistors. With the help of an Excel spreadsheet, we can try out different resistor values and see what the response will look like. The result is a compact, versatile, and precise filter response. Here is an example 150-Hz low-pass filter that occupies 10-mm square area on one side of a printed circuit board, consumes less than 5 μA, all for unter $5.

We discuss operating frequency below, but let us begin with an overview of how operating frequency affects your choice of filter implementation.

At frequencies between 1 kHz and 10 MHz, we might use an active filter, or we might use an inductor. It depends upon the amount of power we can provide to the active filter's amplifier. The more power we can provide, the faster the selection of amplifiers we can choose from, and the higher the frequency at which the active filter will be effective. These days, for a budget of 10 mA, you can build an active low-pass filter with cut-off frequency 10 MHz. In this schematic you will find a 1.6-MHz four-pole low-pass filter. After reading the sections below, you will know how to enter the gain and time constants of the filter stages into our filter tool, and see the filter's response for yourself.

Below 100 kHz, you have a choice of active filter implementation: you can build them with op-amps, capacitors, and resistors, or you can use programmable analog circuits like Lattice Semiconductor's PAC series of chips.

At frequencies higher than 10 MHz, inductors are small and inexpensive, while op-amps are no longer fast enough to implement filter functions efficiently. Aside from classic inductor-capacitor (LC) filters, like those we buy from Minicircuits, you can get ceramic filters, SAW filters, and crystal filters. A SAW bandpass filter at 9000 MHz costs only a few dollars, and has a response as sharp as a ten-pole LC band-pass filter.

Active Low-Pass Filters

There are many low-pass active filter circuits. We use the one shown in Figure 1. As you can see, the output of an op-amp is fed back to its positive input by a capactitor. This capactor acts as an inductor in the circuit equations. The output is also fed back to the negative input by a resistor divider, and this division gives the active filter its gain, A.

The single stage shown in Figure 1 provides a transfer function with two imaginary poles. Here is an abbreviated analysis of the circuit in the Laplace Domain. We indicate the Laplace transform of the voltage at each node in Figure 1 with the node letter. So X represents the Laplace transform of the voltage at node X.

As we vary A, while leaving RC constant, the poles of its transfer function move off the negative real axis of the s-plane, and along the circumference of a circle to the imaginary axis. The radius of this circle is 1/RC, and its center is the origin. Figure 3 shows the circle's upper-left quadrant.

A Butterworth filter is the filter with maximally flat amplitude response in its pass-band. By cut-off frequencey we mean the frequency at which the Butterworth filter output drops to 71% (1/√2) of its maximum amplitude at lower frequencies. The Butterworth filter's maximum amplitude occurs at 0 rad/s, but the Chebyshev filter's maximum amplitude occurs at several other frequencies below the cut-off frequency.

It so happens that the poles of a Butterworth low-pass filter with cut-off frequency ω_c are evenly-spaced around the circumference of a half-circle of radius ω_c centered upon the origin of the s-plane. The poles of a two-pole filter are at ±45°. Those of a four-pole filter are at ±22.5° and ±67.5°. The following table gives the poles of the low-pass Butterworth filters with one to eight poles and cut-off frequency 1 rad/s. These are called the poles of the normalized Butterworth polynomials.

Turn to the first three sheets in our filter tool, you will see places where you can enter the poles of two, three, and four-pole filters. You specify a conjuate pair of poles by giving its real part and the absolute value of its imaginary part. You specify a solitary pole in the three-pole filter by giving its real part only. We provide a table of example pole sets for various filter functions. As you enter new pole values, the spreadsheet plots the amplitude response of your poles.

We implement each conjugate pair of poles with a single op-amp stage, as shown above. We can implement solitary poles with an RC network, or in another op-amp stage with a capacitor across its feedback resistor. This single-pole op-amp stage allows us to amplify and filter at the same time. With two op-amps we can implement a three-pole low-pass filter with amplification. In this circuit, you see a three-pole filter made out of two op-amps. It has a 10-MΩ input resistance with a 0.15-Hz high-pass filter, followed by a three-pole filter, and provides a total gain of 25. We implement the single pole with a capacitor across the first op-amp's feedback resistor. At high frequencies, the capacitor reduces the ×11 gain of the first op-amp stage to ×1. Ideally, the capacitor should reduce the gain to ×0, which it would if we arranged the op-amp as an inverting amplifier. But a reduction in gain from ×11 to ×1 is a good enough approximation for our filter to work well.

The frequency plot extends from 0 rad/s to 3 rad/s. The plot is intended for use with the poles of normalized filter polynomials, which are polynomials whose cut-off frequency is 1 rad/s. If we want to make a 4-pole Butterworth low-pass filter with cut-off frequency ω_c using active filter stages like the ones in Figure 1, we build two stages, each with RC=1/ω_c, and pick the value of A of each stage to put its two conjugate poles at the correct place on the circult of radius ω_c in the s-plane. These values of A are independent of ω_c, so we can look at Figure 3 and Table 1, and figure out the gain we need to produce the poles of the normalized Butterworth filter. We use these values of A in our filter with cut-off frequency ω_c. You can determine A with sufficient accuracy by interpolating between points in the locus of Figure 3, knowing that these points represent steps of 0.1 in A, starting with 1.0 with the two poles together on the negative real axis.

Instead of using Figure 3 to determine the correct values of A for your Butterworth filter stages, select the Pole Locus sheet in our filter tool. There you will find the data for Figure 3, and a place where you can enter values of A and RC, and get the pole real and imaginary parts of the resulting conjugate pole pair. We have done this for you, and present the correct values of A in the table below. Note that filters with an odd number of poles require only an R-C network to implement the solitary pole on the negative real axis. In case you are wondering: the ordering of the stages is not important. The filter will work just as well with any sequence of correct gain values.

On page five of our filter tool you will see how you can vary A and RC of one-stage, one-and-a-half stage, and two-stage active filter to create a two-, three-, and four-pole low-pass filters. If you set RC to 1, and enter the Butterworth values of A from Table 2, you will see the Butterworth maximally flat amplitude response, with cut-off frequency 1 rad/s. All stages of a Butterworth filter have RC=1/ω_c. Vary RC for each stage by a factor of two or so, as well as varying A, and you will obtain a variety of other responses, especially with the two-stage filter. We provide RC and A values for ω_c = 1 rad/s. Try the Chebyshev 0.5-dB ripple response, which we show in Figure 4.

If you want to build a filter with a response of the same shape as the one you see above, but with cut-off frequency ω_c, divide each stage's RC by ω_c, but leave A the same. You will now have the same response shape, but with the frequency axis scaled by ω_c.

You can experiment with values of A and RC in the spreadsheet to see what the resulting filter response looks like. You will find that trying to obtain the Chebyshev 3-dB Ripple response by trial and error is difficult. That's why we tend to use tables of pre-calculated poles to build our filters. In The Art of Electronics, you will find a table of A and RC values for Chebyshev filters of several ripple magnitudes, all normalized to a cut-off frequency of 1 rad/s. For the poles of a variety of Chebyshev filters, look here.

Table 3 gives some example low-pass filters. You can plot the frequency response of each filter using the Active LPF page of our filter tool. Calculate the gain of each op-amp stage, enter it into the corresponding gain cells. Calculate RC for each stage. Multiply all the RC values by the same scaling factor so that they are all of order 1 s. Enter the scaled RC values into the corresponding time constant cells. You will see the filter response plotted as you make your changes.

Active High-Pass Filters

You can change the low-pass filter into a high-pass filter by exchanging the resistors and capacitors, as show in Figure 5. The poles of the transfer function remain fixed, but we introduce two zeros at the origin.

As we can see from its transfer function, the high-pass filter is equivalent to a double-differentiator in series with the same low-pass filter we would have obtained with the same resistor and capacitor values. At high frequencies, the amplification of the double-differentiator is cancelled by the attenuition of the low-pass filter, giving us flat response. At low frequencies, the attenuition of the double-differentiator is dominant, and causes a reduction in the output amplitude with decreasing frequency. When the angular frequency of the input, ω, is equal to 1/RC, the response of the high-pass filter and the low-pass filter are equal. Another way of considering the transfer function is to say that the high-pass filter response is the low-pass filter response subjected to the transformation RCs = 1/RCs', so that the response of the high-pass filter at angular frequency 1/RC equals that of the low-pass filter at the same frequency, and at angular frequency 2/RC it equals that of the low-pass filter at frequency 1/2RC.

You will find a poor drawing of a 19-kHz two-pole Butterworth high-pass filter we designed in 1988 here, along with accompanying calculations. The capacitors are 4.7 nF, R is 1.8 kΩ, and A = 1.59.

Operating Frequency

Filters made with passive components get larger and heavier as their cut-off frequency decreases. A 10-kHz high-pass filter made with inductors and capacitors, feeding a 50-Ω load, must contain inductors whose impedance is of order 50 / 2π.10kHz ≈ 1 mH. A 1-mH inductor with series resistance less than 5 Ω (10% of 50 Ω) is hard to find for less than $5, and it will be at least 15 mm on each side. A capacitor with impedance 50 Ω at 10 kHz is roughly 330 μF. Capacitors that size tend to be electrolytic, and therefore polarized, so that you can't connect them to an alternating voltage. By the time we get down to 100 Hz, the inductors and capacitors are huge, and we tend to see them used these days only for filtering power supplies.

Passive filters are cumbersome at low frequencies, but active filters stop working at high frequencies. The amplifier in most active filters is an operational amplifier, or op-amp. Op-amps are high-gain differential amplifiers with a negative and positive input. They have their own internal gain, which is large at low frequencies, but drops as frequency increases. Most op-amps have internal compensation, which means they have a capacitor somewhere inside that acts as a single-pole low-pass filter in series with their large gain. Without this compensation, op-amps are tend to be unstable in feedback loops, and designers don't like to worry about instability in op-amp circuits. Even without the capacitor, the op-amp gain drops with frequency, but it does so in an irregular and unpredictable way.

Because of the compensating capacitor, the gain of an op-amp feedback amplifier is inversely proportional to its half-power bandwidth, and so we have the gain-bandwidth product of an op-amp, which is the product of the gain of any feedback amplifier you make out of the op-amp and the frequency at which loss of op-amp internal gain causes a 29% (3 dB) drop in output amplitude.

Suppose the gain-bandwidth product of our op-amp is 1 MHz, which is the case for the OPA2277 we use in this filter. The second stage of the filter has gain 2.27, so its bandwidth is 1 MHz / 2.27 ≈ 400 kHz. The cut-off frequency of the filter, f_c is 10 kHz. The amplifier bandwidth is fourty times greater than the cut-off frequency. Our rule of thumb is that the amplifier bandwidth should be at least ten times greater than the filter cut-off frequency. With stage gains of around two (2), the OPA2277 allows us to build filters with cut-off frequencies up to 40 kHz.

Another of our favorite op-amps, the EL2244, has gain-bandwidth product 120 MHz, and would allow us to build filters with cut-off frequency up to 10 MHz. At higher frequencies, there's not much point in using an active filter, because inductors are simpler, provided that you know how to design passive filters. Consider a passive 10-MHz low-pass filter driven by a 50-Ω source, and terminated by a 50-Ω load. The inductors in the filter should have impedance comparable to 50 Ω at 10 MHz, which means their inductance, L, should be of order 50 / 2π.10MHz ≈ 1 μH. We can buy a 1-μH inductor in a P0805 surface-mount package with series resistance less than 0.2 Ω and self-resonant frequency 100 MHz for less than ten cents. The filter won't work so well at 1000 MHz, becuase the parasitic capacitance of the cheap inductor will be passing the 1000 MHz through the filter, but you can block frequencies ten times higher than the cut-off with an auxilliary single-pole low-pass filter with cut-off frequency 100 MHz. You can make that filter out of a capacitor and a resistor. You can find tables of inductor and capacitor value for normalized filters on the web. A table for low-pass Chebyshev filters is here

Programmable analog circuits, such as those provided by Maxim Semiconductor and Lattice Semiconductor, use analog switches and capacitors to make switched capacitor filters. By connecting a capacitor to two terminals only 10% of the time, you create a capacitor 10% as large. The speed of these circuits is limited by the capacitor switching frequency. The latest circuits appear to support switching frequencies up to 1 MHz, which means we can use them to make filters that operate up to about 100 kHz.

Speaking for ourselves, we use active filters below 1 MHz, passive filters above 10 MHz, and debate with ourselves for the range in between. We have not yet seen a need for programmable filters, but when we do, we look forward to using them at frequencies below 100 kHz.

Component Accuracy

Because the value of RC for a Butterworth filter is the same for every stage, we can use resistor and capacitor arrays to implement the RC values, and we will be sure that the values of RC will be within 1% of one another. Even if the absolute accuracy of the array resistance of capacitance is only 5%, we find that the variation within the array is less than one tenth of the absolute accuracy.

We find the same relative-accuracy effect within a reel of surface-mount resistors or capactitors, so that the relative accuracy of a reel of 5% resistors will by 0.5%, and of a reel of 20% capacitors will be 2%. (We measured twenty resistors and capacitors from each of several reels before we made this claim.) The relative accuracy of surface-mount parts within the same reel makes the VCVS circuit far more practical than it was twenty years ago, when we had to buy 1% accurate capacitors to be confident in a 4-pole Butterworth filter response.

It will rarely matter if our active filter's cut-off frequency is moved by ±5% by component variation. It is more important that the relative position of the filter poles remains accurate, so that the poles work together to give the correct filter shape. A 5% variation in component values can re-arrange the poles of a Butterworth filter so as to introduce a 5-dB ripple in the pass-band. But we can build a Butterworth filter with flat response by using an array of resistors and capacitors for the common value of R and C in each stage, and by using 1% resistors in the voltage divider that determines the gain of each stage.

The same is not quite true for a Chebyshev filter, where each stage has a different value of RC. But we can still use an array of capacitors, so that all the capacitors are the same to within 1%, and then use 1% resistors, which are freely avaliable and inexpensive.

Filter Polynomials

By varying the poles of the four-pole filter on the second page of our filter tool, you will see how sensitive the response is to the pole values. If you try to obtain the Butterworth response by guesswork and successive changes, you quickly get lost. How did anyone figure out the correct pole values?

The Butterworth Polynomials provide us with maximally flat amplitude response in the pass-band. The ratio of the output amplitude to the input amplitude is (1 + ω²ⁿ/ω_c²ⁿ)^-½, where n is the number of poles in the filter, ω is the frequency of the input in rad/s, and ω_c is the cut-off frequency in rad/s. This amplitude relation defines the Butterworth Polynomials. Figure 6 shows how the normalized second-order Butterworth Polynomial, s² + s√2 + 1, provides the specified frequency response. The normalized polynomal is the polynomial for ω_c = 1 rad/s. The poles of this normalised second-order polynomial are the poles we give in row two of Table 1.

Figure 6: Normalized Second-Order Butterworth Polynomial in Normalized Second-Order Low-Pass Transfer Function. By normalized we mean ω_c = 1 rad/s.

The Chebyshev Polynomials provide us with the Chebyshev filter poles. The Chebyshev polynomials allow you to accept variation in the pass-band amplitude response in exchange for sharper cut-off just outside the pass band. The normalized second-order Butterworth amplitude response reduces to (1 + ω⁴)^-½. There is no term in ω². Because there are no term in ω², the amplitude response is always decreasing as frequency increases, and contains no ripples. But if we can accept some ripples in the pass-band amplitude response, then we can add an ω² term that cooperates with the ω⁴ term to drop the response more sharply above ω_c.

You can see the basis of the Chebyshev polynomial derivation here, and we give you the the normalized 3-dB passband ripple Chebyshev polynomials in Table 4, where you can compare them to the normalized Butterworth polynomials.

All the polynomials in Table 4 have value 1 for s = 0. If our transfer function has the polynomial in its denominator, its gain for ω = 0 rad/s will be 1. At ω = 1 rad/s, all the polynomials produce a gain 3-dB less than the maximum gain for ω ≤ 1 rad/s. The maximum gain of the Butterworth filter is 1, and occurs at ω = 0 rad/s, but the maximum gain of the 3-dB Chebyshev filter is √2, or 3 dB greater than unity, and occurs at one or more values of ω between 0 rad/s and 1 rad/s. The gain of the Chebyshev filter at ω = 1 rad/s is 1.

By comparing the Chebyshev and Butterworth polynomials, you can see why the Chebyshev provides a sharper cut-off outside its pass-band. The s⁵ term in the fifth-order Chebyshev polynomial has coefficient 15.9, while the s⁵ term in the Butterworth polynomial has coefficient 1. For frequencies less than ω_c, the fifth-order Chebyshev polynomial balances its large s⁵ term with large s⁴, s³, s², and s terms. This balancing is done in such a way that we produce the minimum amount of ripple in the pass-band response for the maximum coefficient in s⁵. Once ω rises above ω_c, the s⁵ term overtakes all other terms rapidly, and gives a sudden drop in response. Not only does the Chebyshev filter always give us a sharper cut-off than the Butterworth filter, but the advantage grows with the order of the filter, as you can see in Figure 7.

Passive Filters

A passive filter is one made up of inductors, capacitors, and resistors. In most cases, the only resistors in the filter are the source and load impedances. These resistors rarely exist in your circuit as separate resistor components, but are instead an inherent property of the amplifier that provides the signal, and the amplifier that receives the output of the filter. In the section above on filter polynomials we show how you arrive at a polynomial function of frequency that best matches your requirements. Passive filters implement these polynomial frequency responses with capacitors and inductors that interact with your source and load impedances.

One way to start off learning about passive filters is to use a passive filter calculator like this one. You enter your source and load resistances, pick a classic polynomal frequency response (Butterworth, Chebyshev, Bessel, as described above), pick the order of the polynomial (the highest power of frequency in the transfer function), and a characteristic frequency for the response, such as the −3 dB point for a Butterworth low-pass or high-pass filter. The calculator gives you a circuit diagram and gives you the inductor (L) and capacitor (C) values in Henries (H) and Farhads (F).

A classice passive filter, such as the ones designed by the calculator linked to above, takes the form of a sideways latter, in which the bottom rail is an un-interrupted signal ground, and the top rail is a series of inductors or a series of capacitors. It will be inductors in a low-pass filter and capacitors in a high-pass filter. The steps of the ladder (if the ladder were vertical they would be the steps) are capacitors in a low-pass filter and inductors in a high-pass filter. The ladder is the favored structure for passive filters, because it is easy to convert a polynomial frequency function into a ladder circuit. The total number of capacitors and inductors is equal to the highest power of frequency in the frequency polynomial, and gives us the order of the filter.

Given that the filter is a ladder, another thing you need to specify for a passive filter is whether it is a "shunt" or "series" filter. A shunt filter is one in which the first element connects to signal ground (0 V). A series filter is one in which the first element connects to the second element, and the second element connects to ground. If your source impedance is zero, a shunt filter does not make sense, because no component to ground can affect a signal with zero source impedance. Nor does it makes sense to have a filter with a series element at the end connected to an infinite load impedance, because no series impedance can affect the signal seen by the infinite load.

When you are thinking about these passive filter circuits, and how they interact with your amplifiers, keep in mind that a voltage source, V, in series with a resistor, R, is equivalent to a current source I = V/R in parallel with the same resistance, R. If your signal source is the collector of an NPN transistor driving a 50 Ω load, it is equivalent to a voltage source in series with this resistor. You can shunt the output to ground through a capacitor to start a low-pass filter, or you can put an inductor in series with it. On the other hand, you cannot shunt it to ground with an inductor to start a high-pass filter because you will ruin the DC bias of the transistor. But you could start with a large series capacitance, whose impedance is negligable compared to the source impedance, to isolate the transistor from the filter at DC, and then begin your high-pass filter with an inductive shunt to ground.

In all cases, and throughout the filter, you will find that the impedance of each individual capacitor and inductor is of the same order as the source and load impedances at your desired high-pass or low-pass cut-off frequency. If your source impedance is 50 Ω and your load impedance is 500 Ω, the impedance of the components will increase from through the circuit so that those closest to the source are of order 50 Ω, and those closest to the load are of order 500 Ω.

These days, you are likely to be using active filters in the cases where you have high source and load impedances, or mis-matched impedances. Your most likely application for a passive filter will be when you have a radio-frequency (RF) circuit with 50 Ω source and load impedances.

SAW Filter

A SAW (surface acoustic wave) filter is a small, passive device made out of peizoelectric crystal. Most SAW filters are band-pass filters. The Demodulating Receiver A3005, by OSI uses a 950-MHz SAW band-pass filter to reject RF power outside the frequency range 930 MHz to 970 MHz. You will find a description of the SAW filter and its behavior in the circuit here.

Imagine a thin, rectangular piece of piezo-electric crystal. Its top surface is flat. The crystal expands and contracts with applied electric field. If we tap the crystal, it will shiver for a short time, like a crystal wine glass, but at a frequency too high for us to hear. As the crystal shivers, it developes a voltage across its surfaces that matches the frequency and amplitude of the shiver. If we put electrodes on the top and bottom surface, the crystal will resonate when we apply a sinusoidal voltage of the correct frequency. Crystal oscillators use this resonance to create a frequency that is as accurate as the dimensions and flatness of the crystal.

A SAW filter does not use bulk resonance of a crystal. It uses surface acoustic waves. Instead of electrodes on opposite faces of a crystal, the SAW filter has two interlocking combs of electrodes on one end of the crystal surface, and another two such combs at the opposite end, but on the same crystal surface. We connect our input across these combs, as show in Figure 8.

The combs set up electic fields along the surface of the crystal, instead within the bulk of the crystal. The electric fields cause the surface of the crystal to expand and contract. If we apply a sinusoidal voltage across the input electrodes of just the right frequency, we will create a surface wave that propagates across the crystal, where it creates a voltage across the output electrodes, which are also interlocking combs. You can see that only frequencies with a whole number of wavelengths between the teeth of each comb will be reinforced as they propagate across the comb fingers. The result is a filter: only frequencies that match the combs at both ends will be transformed from electrical signals into surface wave signals and back to electrical signals again. If you would like a more detailed mathematical analysis of the surface wave propagation, try this.

Waves travel along the surface of the crystal at a velocity dictated by the crystal's Young's modulus, and is of order 4000 m/s for piezo-electric crystals. The wavelength of a 950-MHz wave propagating at this speed is only 4.2 μm. At such high frequencies, SAW filters tend to use harmonics of the fundamental resonant frequency of its combs. They do this by starting with a comb designed for an integer fraction of the desired band-pass frequency, and then inserting cuts in the comb fingers that make the combs inefficient at the fundamental frequency, but efficient at the desired band-pass frequency.

Let's compare the frequency response of the DSF947.5 SAW filter to that of polynomial filters.

If we imagine that the SAW filter is a combination of a high-pass filter at the low end of its pass-band, and a low-pass filter at the high end, then we can compare its response at the high-end to that of a polynomial low-pass filter. We see that within 10 MHz of the cut-off frequency, the SAW filter's virtual low-pass filter response drops by 30 dB. Suppose we were to try to build a polynomial low-pass filter with the same response. A ten-pole 3-dB ripple Chebyshev filter response drops by 30 dB after a 10% change in frequency. In other words, the SAW filter cut-off at both ends of its pass-band is ten times sharper than that of a ten-pole 3-dB ripple Chebyshev filter. Perhaps we could match the SAW filter response with a 100-pole filter made with 0.1% tolerant inductors and capacitors. The result would cost hundreds of dollars, and take up a hundred square centimeters of board space. The DSF=947.5, on the other hand, is less than $3, and comes in a 5-mm square package.

Conclusion

We presented the work we have done when designing our own filters, and described our design method. We have provided links to our filter circuits. While we may not give you the circuits you need directly, we hope that you can now embark upon your own filter designs with confidence and enthusiasm.