Amplifiers Texas Instruments Incorporated Designing active analog filters in minutes By Bonnie Baker Senior WEBENCH® Applications Engineer Introduction National Semiconductor’s WEBENCH Active Filter Active analog filters can be found in almost every electronic Designer software. It uses the approach of these programs circuit. Audio systems use filters for frequency-band limit- and formulas verbatim when generating active filters. But ing and equalization. Designers of communication systems it goes beyond these two programs by allowing in-depth use filters for tuning specific frequencies and eliminating adjustments to filter variables, optimizing the filter, finding others. To attenuate high-frequency signals, every data- appropriate TI operational amplifiers (op amps) for the acquisition system has either an anti-aliasing (low-pass) filter circuits, and providing SPICE simulation capability. filter before the analog-to-digital converter (ADC) or an Key design parameters for a low-pass analog filter anti-imaging (low-pass) filter after the digital-to-analog The frequency-domain specifications of a low-pass analog converter (DAC). This analog filtering can also remove filter include four fundamental parameters: higher-frequency noise superimposed on the signal before it reaches the ADC or after it leaves the DAC. If an input • fc, the filter’s –3-dB cutoff frequency signal to an ADC is beyond half of the converter’s sampling • Ao, the gain of the filter frequency, the magnitude of that signal is converted reli- • Asb, the stop-band attenuation ably; but the frequency is modified as it aliases back into • fs, the frequency of the intercept to the stop-band the digital output. attenuation A low-pass, high-pass, band-pass, or band-stop filter can These parameters are shown in WEBENCH Filter be efficiently designed with the WEBENCH® Filter Designer’s Filter Type window in Figure 1. The frequency Designer software from Texas Instruments (TI). This span from DC to the cutoff frequency (fc) is the pass-band application replaces TI’s FilterPro™ and the former Figure 1. WEBENCH® Filter Designer’s key analog-filter parameters 28 High-Performance Analog Products www.ti.com/aaj 4Q, 2013 Analog Applications Journal Texas Instruments Incorporated Amplifiers region. The magnitude of the response in the pass band is Ao in Figure 1. The response in the pass band can be flat Figure 2. Increased number of poles in Butterworth filter creates sharper rolloff with no ripple, as it is with a Butterworth or Bessel filter. Conversely, a Chebyshev filter has a ripple up to the cutoff frequency. The magnitude of the ripple error of a 1.0 Chebyshev filter is 2DA . MAX IN As the response of the filter goes beyond fc, it falls /V M = 1 through the transition band to the stop-band region. The OUT filter approximation (Butterworth, Chebyshev, Bessel, etc.) V 0.1 determines the bandwidth of the transition band and the M = 2 order (M) of the filter. The number of poles in the transfer function determines the filter order. For instance, if a M = 4 filter has three poles in its transfer function, it is a third- 0.01 M = 16 order filter. Generally, the transition band becomes smaller when Amplitude Response, more poles are used to implement the filter design, as M = 32 M = 8 shown in Figure 2 for a Butterworth low-pass filter. Ideally, 0.001 a low-pass, anti-aliasing filter should perform with a 0.1 1.0 10 “brick-wall” style of response, with an extremely small Normalized Frequency transition band. Practically speaking, this is not the best approach for an anti-aliasing solution. With active-filter design, every two poles require an op amp. For instance, a 32nd-order filter requires 16 op amps, 32 capacitors, and identified by examining amplitude versus frequency up to 48 resistors. domain and amplitude versus time domain. Analog filter-approximation types Butterworth filter Figure 3 shows the low-pass-filter types available in the The transfer function of a Butterworth filter consists of all Solutions window from the WEBENCH Filter Designer’s poles and no zeroes and is represented by Visualizer screen. This screen appears after the user clicks V Ao on the Start Filter Design button shown in Figure 1. OUT = . V n n− 1 n − 2 2 The more popular filter-approximation types are the IN as0+ as 1 + as 2 . an− 1 s+ as1 n + Butterworth, Chebyshev, and Bessel. Filters can be Figure 3. Low-pass-filter types in WEBENCH® Filter Designer 29 Analog Applications Journal 4Q, 2013 www.ti.com/aaj High-Performance Analog Products Amplifiers Texas Instruments Incorporated Figure 4 shows that the response of a fourth-order, low- Figure 5 shows that the step response of the same pass Butterworth filter is flat in the pass-band portion. The fourth-order Butterworth filter has some overshoot and technical term for this characteristic is maximally flat. ringing in the time domain. If the filter order were higher, Later it will be shown that the rate of attenuation in the this overshoot would also be higher. If this filter is used transition band is not as good as with the Chebyshev filter. after a multiplexer, its settling time should be considered. Figure 4. Frequency response of fourth-order, low-pass Butterworth filter 20 400 Gain 0 200 Phase ) –20 0 ) dB ( degrees –40 –200 ( –60 –400 Gain Simulation –80 –600 Phase Simulation –100 –800 –120 –1000 1 10 100 1 k 10 k 100 k 1 M Frequency (Hz) Figure 5. Step response of fourth-order, low- pass Butterworth filter 0.8 0.6 0.4 ) V ( 0.2 0 Simulation –0.2 OUT V –0.4 –0.6 –0.8 0 10 20 30 40 50 60 70 80 90 100 110 Time, t (ms) 30 High-Performance Analog Products www.ti.com/aaj 4Q, 2013 Analog Applications Journal Texas Instruments Incorporated Amplifiers Chebyshev filter Figure 6. Frequency response of fourth-order, low-pass The transfer function of the Chebyshev filter is Chebyshev filter similar to the Butterworth filter only in that it has all poles and no zeroes: 20 400 VOUT Ao = Gain V 2 n− 1 0 200 IN a0+ as 1 + as 2 + . an− 1 s+ s n Phase Figure 6 shows that the frequency response of a –20 0 ) fourth-order, low-pass Chebyshev filter has a ) 0.2-dB ripple in the pass-band region. The pole dB ( –40 –200 degrees placement in the circuit design determines this ( ripple. In general, an increase in ripple magni- –60 –400 tude lessens the width of the transition band. The ripple magnitude of 2DAMAX (Figure 1) theoretically can be as large or as small as –80 –600 desired. A high-ripple magnitude generally Gain Simulation results in more error in the pass-band region but –100 –800 Phase Simulation a faster attenuation in the transition band. The rate of attenuation in the transition band –120 –1000 is steeper than for a Butterworth filter. For instance, to meet the transition bandwidth of a –140 –1200 third-order Chebyshev with a 0.5-dB ripple, a 1 10 100 1 k 10 k 100 k 1 M fourth-order Butterworth filter is required. Frequency (Hz) Although with the Chebyshev filter there is ring- ing in the pass-band region, the stop band is devoid of ringing. The step response of a fourth-order, low-pass Figure 7. Step response of fourth-order, low- Chebyshev filter with a 0.2-dB ripple has a fair degree of pass Chebyshev filter overshoot and ringing (Figure 7). The overshoot and ringing phenomena are a conse- 0.8 quence of the phase response in the frequency domain. Recall that the Fourier analysis of a step response (or square wave) shows that a square wave can be constructed 0.6 by adding odd harmonic sinusoidal signals. Consequently, the higher frequencies from the step input arrive at the 0.4 output of the filter before the lower frequencies. This is ) V called a distortion group delay. This group delay in sec- ( 0.2 onds is calculated as Change in phase/Change in frequency 0 . 360 Simulation –0.2 OUT V –0.4 –0.6 –0.8 0 10 20 30 40 50 60 70 80 90 100 110 Time, t (ms) 31 Analog Applications Journal 4Q, 2013 www.ti.com/aaj High-Performance Analog Products Amplifiers Texas Instruments Incorporated Comparison of filter-approximation types from, such as Chebyshev, Butterworth, Bessel, linear phase, For low-pass filters, the type of filter approximation affects and transitional Gaussian. The filter response best suited the frequency response before and beyond the filter’s cut- for the design is determined by optimizing for pulse off frequency. Since the inverse of frequency (in hertz) is response, settling time, lowest cost, pass-band ripple, seconds, the filter type inversely impacts the time domain. and stop-band attenuation. Table 1 compares low-pass Butterworth, Chebyshev, and Sallen-Key or multiple feedback topologies are design Bessel filters in the frequency domain (pass-band and options for each filter stage, and the best op amps for the transition regions) and the time domain (step response). design are chosen by evaluating gain bandwidth versus current versus cost and other parameters. The resistor/ capacitor tolerances can be specified as ideal, 0.5, 1, 2, 5, Table 1. Comparison of filter-approximation types 10, or 20%. Experimenting with user-defined capacitor TRANSITION seed values adjusts the range of resistor values in the filter FILTER TYPE PASS BAND REGION STEP RESPONSE design. Filter topologies can also be optimized for sensitiv- Maximally flat Steeper than Some overshoot ity, lowest cost, and smallest footprint.