How Much Does The Phase Shift For A 1st Order Band Pass Filter
Phase Response in Agile Filters
Function three—The Band-Pass Response
Introduction
In the first commodity of this serial,ane I examined the relationship of the filter phase to the topology of the implementation of the filter. In the second article,2 I examined the phase shift of the filter transfer function for the low-pass and loftier-pass responses. This article volition concentrate on the ring-pass response. While filters are designed primarily for their amplitude response, the phase response can be important in some applications.
For purposes of review, the transfer part of an active filter is actually the cascade of the filter transfer function and the amplifier transfer part (meet Effigy ane).
The Ring-Pass Transfer Function
Changing the numerator of the low-laissez passer epitome to
volition convert the filter to a ring-pass function. This will put a zip in the transfer function. An s term in the numerator gives us a zero and an s term in the numerator gives us a pole. A nix will give a rise response with frequency while a pole will requite a falling response with frequency.
The transfer function of a second-gild band-pass filter is then:
ω0 here is the frequency (F0 = two π ω0) at which the proceeds of the filter peaks.
H0 is the circuit gain (Q peaking) and is defined as:
where H is the gain of the filter implementation.
Q has a particular meaning for the band-pass response. It is the selectivity of the filter. It is defined as:
where FL and FH are the frequencies where the response is –3 dB from the maximum.
The bandwidth (BW) of the filter is described as:
It tin can be shown that the resonant frequency (F0) is the geometric hateful of FL and FH, which ways that F0 will announced one-half fashion between FL and FH on a logarithmic scale.
Also, note that the skirts of the band-pass response will always be symmetrical around F0 on a logarithmic scale.
The amplitude response of a band-pass filter to various values of Q is shown in Figure 2. In this effigy, the gain at the heart frequency is normalized to one (0 dB).
Once more, this article is primarily concerned with the phase response, but it is useful to have an thought of the aamplitude response of the filter.
A word of caution is advisable here. Band-pass filters tin can be defined two different ways. The narrow-band example is the classic definition that nosotros have shown above. In some cases, nonetheless, if the high and low cutoff frequencies are widely separated, the band-pass filter is constructed out of separate high-pass and low-pass sections. Widely separated, in this context, means separated past at to the lowest degree two octaves (×iv in frequency). This is the wideband case. We are primarily concerned with the narrow-band case for this commodity. For the wideband case, evaluate the filter as separate high-pass and depression-pass sections.
While a band-pass filter can be defined in terms of standard responses, such as Butterworth, Bessel, or Chebyshev, they are also ordinarily divers past their Q and F0.
The phase response of a band-pass filter is:
Note that in that location is no such thing equally a unmarried-pole ring-pass filter.
Figure 3 evaluates Equation vi from two decades beneath the center frequency to two decades above the center frequency. The middle frequency has a phase shift of 0°. The centre frequency is i and the Q is 0.707. This is the same Q used in the previous article, although in that commodity nosotros used α. Retrieve α = 1/Q.
Inspection shows the shape of this curve is basically the same as that of the low-pass (and the high-pass for that matter). In this example, however, the phase shift is from xc°, below the eye frequency going to 0° at the center frequency to –90° to a higher place the centre frequency.
In Effigy iv nosotros examine the stage response of the band-pass filter with varying Q. If we take a wait at the transfer office, nosotros tin see that the phase change can have place over a relatively large frequency range, and that the range of the alter is inversely proportional to the Q of the circuit. Again, inspection shows that the curves have the same shape as those for the low-pass (and high-pass) responses, just with a different range.
The Amplifier Transfer Function
It has been shown in previous installments that the transfer function is basically that of a single-pole filter. While the stage shift of the amplifier is generally ignored, it tin touch on the overall transfer of the composite filter. The AD822 was arbitrarily chosen to use in the simulations of the filters in this article. It was chosen partially to minimize the consequence on the filter transfer function. This is because the phase shift of the amplifier is considerably higher in frequency than the corner frequency of the filter itself. The transfer function of the AD822 is shown in Figure v, which is data taken directly from the data canvass.
Example 1: A i kHz, 2-Pole Band-Pass Filter with a Q = xx
The first case volition be a filter designed every bit a ring-pass from the beginning. Nosotros arbitrarily choose a center frequency of ane kHz and a Q of 20. Since the Q is on the higher side, we will use the dual amplifier band-pass (DABP) configuration. Over again, this is an arbitrary selection.
We use the design equations from Reference 1. The resultant circuit is shown in Figure 6:
We are primarily concerned with phase in this article, just I recollect information technology useful to examine the amplitude response.
We see the phase response in Figure 8:
Note that the DABP configuration is noninverting. Figure viii matches Figure three.
Instance 2: A 1 kHz, 3-Pole 0.5 dB Chebyshev Low-Laissez passer to Ring-Laissez passer Filter Transformation
Filter theory is based on a low-pass prototype that tin can then be manipulated into the other forms. In this example, the prototype that will exist used is a i kHz, 3-pole, 0.5 dB Chebyshev filter. A Chebyshev filter was called considering information technology would evidence more conspicuously if the responses were not correct. The ripples in the laissez passer ring, for instance, would non line up. A Butterworth filter would probably be also forgiving in this instance. A 3-pole filter was called so that a pole pair and a single pole would be transformed.
The pole locations for the LP paradigm (from Reference 1) are:
| Phase | α | β | F0 | α |
| one | 0.2683 | 0.8753 | ane.0688 | 0.5861 |
| 2 | 0.5366 | 0.6265 |
The first phase is the pole pair and the second phase is the unmarried pole. Notation the unfortunate convention of using α for ii entirely divide parameters. The α and β on the left are the pole locations in the s plane. These are the values that are used in the transformation algorithms. The α on the right is 1/Q, which is what the blueprint equations for the physical filters want to see.
The low-laissez passer prototype is now converted to a ring-pass filter. The equation string outlined in Reference 1 is used for the transformation. Each pole of the prototype filter will transform into a pole pair. Therefore, the three-pole prototype, when transformed, will accept six poles (iii-pole pairs). In improver, there will be half-dozen zeros at the origin. There is no such thing equally a single-pole band-pass.
Part of the transformation process is to specify the three dB bandwidth of the resultant filter. In this case this bandwidth will exist set up to 500 Hz. The results of the transformation yield:
| Stage | F0 | Q | A0 |
| 1 | 804.five | seven.63 | 3.49 |
| 2 | 1243 | seven.63 | 3.49 |
| 3 | thousand | 3.73 | 1 |
In practise, it might exist useful to put the lower proceeds, lower Q department first in the string, to maximize betoken level handling. The reason for the gain requirement for the offset ii stages is that their center frequencies will exist adulterate relative to the eye frequency of the full filter (that is, they will be on the skirt of other sections).
Since the resultant Qs are moderate (less than 20), the multiple feedback topology will exist chosen. The pattern equations for the multiple feedback ring-pass filter from Reference i are used to design the filter. Figure ix shows the schematic of the filter itself.
In Figure 10 we expect at the phase shift of the complete filter. The graph shows the phase shift of the showtime section alone (Department 1), of the showtime two sections together (Section 2), and of the complete filter (Section 3). These show the phase shift of the "existent" filter sections, including the phase shift of the amplifier and the inversion of the filter topology.
In that location are a couple of details to note on Effigy ten. First, the phase response is cumulative. The first section shows a modify in stage of 180° (the phase shift of the filter office, disregarding the phase shift of the filter topology). The second section shows a phase change of 360° due to having two sections, 180° from each of the two sections. Remember that 360° = 0°. And the tertiary section shows 540° of phase shift, 180° from each of the sections. Also note that at the frequencies above 10 kHz nosotros are starting to see the phase scroll-off slightly due to the amplifier response. We can see that the curl-off is once more cumulative, increasing for each section.
In Effigy 11 we see the aamplitude response of the complete filter.
Conclusion
This article considers the stage shift of band-laissez passer filters. In previous articles in this serial, nosotros examined the stage shift in relation to filter topology and for high-pass and low-pass topologies. In future articles, we will look at notch and all-pass filters. In the final installment, we will necktie information technology all together and examine how the phase shift affects the transient response of the filter, looking at the group delay, impulse response, and step response, and what that means to the bespeak.
Endnotes:
iHank Zumbahlen. "Phase Relations in Agile Filters." Analog Dialogue, Volume 41, Number 4, 2007.
twoHank Zumbahlen. "Stage Response in Agile Filters Part two, the Low-Pass and High-Pass Responses." Analog Dialogue, Volume 43, Number 3, 2009.
References
Daryanani, One thousand. Principles of Agile Network Synthesis and Designorthward. John Wiley & Sons, 1976.
Graeme, J., G. Tobey, and 50. Huelsman. Operational Amplifiers Blueprint and Applications. McGraw-Loma, 1971.
Van Valkenburg, Mac. Analog Filter Pattern. Holt, Rinehart and Winston, 1982.
Williams, Arthur B. Electronic Filter Design Handbook. McGraw-Hill, 1981.
Zumbahlen, Hank. Basic Linear Design. Ch. 8. Analog Devices, Inc, 2006.
Zumbahlen, Hank. "Chapter v: Analog Filters." Op Amp Applications Handbook. Newnes-Elsevier, 2006.
Zumbahlen, Hank. Linear Circuit Design Handbook. Newnes-Elsevier, 2008.
Zumbahlen, Hank. "Phase Relations in Active Filters." Analog Dialogue, Volume 41, 2007
Zverev, Anatol I. Handbook of Filter Synthesis. John Wiley & Sons, 1967.
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