This is the first in a series of articles describing applications of the LTC1562 connected as a lowpass, highpass or bandpass filter with added stopband notches to increase selectivity. Part 1 covers lowpass filters.
Lowpass filters with stopband notches are useful in applications seeking steep attenuation in the vicinity of the cutoff frequency. When compared to classical all-pole realizations (such as Butterworth or Chebyshev) they are more “efficient”; that is, they meet a given attenuation requirement with the least number of poles.
Lowpass filters with stopband notches (broadly referred to as “elliptics” or “Cauers”) can be designed with the aid of some literature or with commercially available software. The new FilterCAD™ for Windows® program, supplied free of charge by Linear Technology Corporation, is an excellent example.
For instance, a 100kHz lowpass filter with a 0.1dB passband ripple and 40dB attenuation at 200kHz can be realized with a 6th order Chebyshev or a 4th order textbook elliptic. Curves A and B of Figure 1 illustrate the respective amplitude responses.
When considering the practical implementation of the filters of Figure 1 (curves A and B), in the author’s experience, it is easier to implement the higher order all-pole filter (curve A), rather than the 4th order version with the stopband notches. The realization of deep stopband notches may result in hardware complexity. This is especially true if a discrete R-C active implementation is chosen and if a single 5V supply and a wide input dynamic range are required.
Nevertheless, curve C of Figure 1 is of particular interest because of its rather simple hardware implementation. Curve C is derived from the classical elliptic response, curve B, where the high frequency notch is “pushed” to infinity and the highest Q pole pair is readjusted to maintain passband flattness. The penalty is the slight gain roll-off at the cutoff frequency, which, for many applications, is acceptable. For sake of simplicity the amplitude response of the filter of Figure 1, curve C, is called a “p-e” (pseudo-elliptic) response.
Figure 2 illustrates the group delay responses of the three filters of Figure 1, with the same curve letter designations. The group delay of curve C is the flattest.
High order filter realizations were a subject of passionate interest in the 1960s and ’70s. One very popular method, which stems from the simplicity of its hardware implementation, consists of decomposing a high order filter polynomial into cascaded second and first order polynomials. Each polynomial is then implemented with commercially available active and passive components. The major drawback of the “cascaded” method is the relatively high Q of at least one of the 2nd order sections and the resulting requirement for precision components for its realization.
Using the “cascading” principle outlined above, the “p-e” response of curve c, Figure 1, can be treated as a self-contained 4th order block and two ( or more) of these blocks can be cascaded to form an 8th order (or higher) lowpass filter with two (or more) stop-band notches. This interesting novelty is driven by the simplicity of its hardware realization; it requires, however, the transformation of an 8th order classical elliptic lowpass response into two cascaded 4th order “p-e” responses.
Figure 3 shows a compact hardware implementation of the 4th order “p-e” filter using one half of the LTC1562 quad Operational Filter IC, which was introduced in the February 1998 issue of Linear Technology magazine.1 Two 2nd order sections form the 4th order filter function. A phase-shifting external capacitor, CIN1, and a feedforward path through resistor RFF2, create the desired notch.
To make the circuit technique of Figure 3 intuitively obvious, consider the following:
A signal of a given frequency can be notched if it is phase shifted by 180 degrees and then summed with itself. If the summation is governed by equal gains, a complete signal cancellation occurs and the notch depth, at least in theory, is infinite.
A phase shift of 180 degrees at a single frequency, fO, is easily provided by a second order inverting bandpass filter; hence, in Figure 3, if CIN1 equals zero, a notch is formed as the bandpass output (pin 2) is summed with the input via (RIN2, RFF2). Moreover, if the summation has equal gains (1), the notch should, in theory, have infinite depth.
In Figure 3, an external capacitor, CIN1, is added to provide additional phase lead, so that the frequency of the notch is higher than the center frequency, fO1, of the second order section used to create it.
The notch frequency, fn1, is directly proportional to the center frequency, fO1, and indirectly proportional to the time constant (RIN1 • CIN1) divided by the (RQ1 • C) (C is an internal capacitor of 159pF); therefore:
A step-by-step algorithm for building compact “p-e” lowpass filters with the new LTC1562 quad Operational Filter building block is outlined below:
Start with a set of two (lowpass) pole pairs and one finite stopband notch. Arrange the pole pairs in ascending order of Q values.
1. Calculate the frequency-setting resistor, R21:
2. Calculate the Q-setting resistor, RQ1:
Note: The calculations for R21 and RQ1 are from the LTC1562 Data Sheet; they are applicable to any 2nd order section using the LTC1562 proprietary architecture.
3. Calculate the input resistor, RIN1, from the following expression:
Make sure that RIN1 > R21; If not, make RIN1 = R21 and proceed to Step 4a.
Note: R21/RIN1 is the DC gain from the input to the lowpass output of the first building block, pin 3. The expression for RIN1 ensures optimum dynamic behavior of all nodes of the LTC1562.
4a. Use the value of RIN1, calculated above, and calculate the value of the input capacitor CIN1 from the notch equation (2).
Use a commercially available NPO-type 0402 surface mount capacitor with the value nearest the ideal value of CIN1 calculated above. For instance, if CIN1 (ideal) is 60.14pF, choose an off-the-shelf 56pF standard value.
4b. Recalculate the value of RIN1 after a CIN1 of 56pF is chosen.
5. Calculate the frequency- and Q-setting resistors R22, RQ2, as done in steps 1 and 2, above. Choose the closest 1% standard resistor values.
6. Calculate the feedforward resistor, RFF2:
7. Calculate the input resistor RIN2, to satisfy the gain condition for the notch (1).
Make the practical value of RIN2 as close as possible to the value calculated above; otherwise, the stopband notch depth will be affected.
An Example Using FilterCAD
The following is a comprehensive example of how to synthesize and realize a complex lowpass filter using two “p-e” 4th order sections in cascade. FilterCAD for Windows will be used to synthesize the filter.
A classical 8th order, 100kHz lowpass elliptic filter with theoretical passband ripple, AMAX, of 0.005dB, and a minimum stopband attenuation, AMIN, of 85dB at twice cutoff, can be synthesized by cascading four biquadratic 2nd order sections, as shown in Table 1. Each biquadratic section comprises a complex pole pair of center frequency fO, and an imaginary zero pair of notch frequency fn. The amplitude response is shown in Figure 4, curve A. The filter above is easily transformed into two cascadable 4th order “p-e” sections by performing the following steps.
- Set the two highest notch frequencies to infinity and expect a decrease in stopband attenuation as well as gain peaking in the vicinity of the cutoff frequency (Figure 4, curve B).
- Use the interactive capability of FilterCAD to increase the frequency of the right hand notch (Figure 5 curve C), until the stopband ripple has equal peaks.
- Use the interactive capability of FilterCAD to flatten the passband by lowering the Qs. Start with the highest Q, then proceed with the second highest, then the third.
Table 2 illustrates the parameters of the transformed filter. Compared to Table 1, two notch frequencies are set to infinity, one notch frequency has been increased and the three highest Qs have been reduced. Figure 5, curve C, illustrates the amplitude response of the transformed filter. The original filter shown in Figure 4 is also shown in Figure 5, curve A, for comparison. The main difference between curves A and C is the theoretical stopband attenuation. Curve C, with its lower Q, will also exhibit improved transient behavior.
A Practical Case
The high Qs of the previous synthesized filters ensure, at least in theory, passband flatness all the way up to the cutoff frequency. In practice, errors occur in the vicinity of the filter cutoff. They are most often manifested as gain peaking and they are caused by the tolerances of the passive components and the finite bandwidth of the active circuitry. The gain peaking at the filter cutoff can be addressed by predistorting the high Q section, that is, by intentionally lowering the Q so that the theoretical response will show some gain rolloff at the cutoff frequency.
The synthesized filter of Table 2 can be efficiently realized by two cascaded “p-e” 4th order sections, as illustrated in the block diagram, Figure 6. Note the arrangement of the pole-zero pairs of Figure 6 and compare it with Table 2. In Table 2, the sections appear in order of increasing fO and Q. In Figure 6, within each 4th order “p-e” filter, the 2nd order section with the highest Q is placed first; the 4th order “p-e” filter containing the highest Q is cascaded last. The notches (fn1 and fn2) are so arranged that the highest frequency notch is formed from the pole pair whose center frequency (fO) is closest to the filter cutoff frequency. For example, the 250kHz notch is placed with the 99.99kHz pole pair. This nonobvious arrangement allows for a stopband attenuation approaching the theoretical values. The highest Q of 5.87 is reduced to 3.97 for reasons mentioned above and for improving the transient response of the circuit. See Figure 7 for the amplitude response; note the slight rolloff at the cutoff frequency. Figure 8 shows the complete hardware realization using all four sections of an LTC1562 continuous-time quad Operational Filter IC. The algorithm outlined above was followed to calculate the values of the external passive components. The circuit occupies as much real estate as a U.S. dime. This is quite significant considering the cumbersome alternative of a fully discrete realization with op amps, Rs and Cs.
Figure 7, curve A, shows the amplitude response of the filter hardware illustrated in Figure 8. No attempt was made to adjust any component. Both notches are fully resolved, but due to the tolerances of the components and the finite bandwidth of the active circuitry, the stopband attenuation, although impressive, is 2dB above the theoretical value. Subsequently, the value of RQ1 was lowered to 16.2k (curve B) to better define the first notch. The filter reaches attenuation levels beyond 85dB all the way up to 0.5MHz input frequencies. The measured attenuation at 1MHz was still better than 78dB. The dynamic range of the circuit is quite impressive: the measured wideband noise was 40µVRMS and the THD for 1VRMS and 50kHz input signal was better than –80dB.
1 Hauser, Max. “Universal Continuous-Time Filter Challenges Discrete Designs.” Linear Technology VIII:1 (February 1998), p.1.