Volume 41 – October 2007 Download this article in PDF format. (628K) Phase Relations in Active Filters
In applications that use filters, the amplitude response is generally of greater interest than the phase response. But in some applications, the phase response of the filter is important. An example of this might be where a filter is an element of a process control loop. Here the total phase shift is of concern, since it may affect loop stability. Whether the topology used to build the filter produces a sign inversion at some frequencies can be important. It might be useful to visualize the active filter as two cascaded filters. One is the ideal filter, embodying the transfer equation; the other is the amplifier used to build the filter. This is illustrated in Figure 1. An amplifier used in a closed negative-feedback loop can be considered as a simple low-pass filter with a first-order response. The gain rolls off with frequency above a certain breakpoint. In addition, there will be, in effect, an additional 180° phase shift at all frequencies if the amplifier is used in the inverting configuration.
First, the filter response is chosen; then, a circuit topology is selected to implement it. The filter response refers to the shape of the attenuation curve. Often, this is one of the classical responses such as Butterworth, Bessel, or some form of Chebyshev. Although these response curves are usually chosen to affect the amplitude response, they will also affect the shape of the phase response. For the purpose of the comparisons in this discussion, the amplitude response will be ignored and considered essentially constant. Filter complexity is typically defined by
the filter “order,” which is related to the number of energy storage elements
(inductors and capacitors). The order of the filter transfer function’s
denominator defines the attenuation rate as frequency increases. The asymptotic
filter rolloff rate is –6
Frequency in radians per second is equal
to 2π times frequency in Hz ( The center frequency
can also be referred to as the
Similarly, the phase response of a single-pole, high-pass filter is given by:
Figure 3 evaluates Equation 2 from two decades below to two decades above the center frequency. The normalized center frequency (=1) has a phase shift of +45°. It is evident that the high-pass and the low-pass phase responses are similar, only shifted by 90° (π/2 radians).
For the second-order, low-pass case, the transfer function has a phase shift that can be approximated by
where α is the damping ratio of the
filter. It will determine the peaking in the amplitude response and the
sharpness of the phase transition. It is the inverse of the
Figure 4 evaluates this equation (using α = 1.414) from two decades below to two decades above the center frequency. Here the center frequency (=1) shows a phase shift of –90°.The phase response of a 2-pole, high-pass filter can be approximated by
In Figure 5 this equation is evaluated (again using α = 1.414), from two decades below to two decades above the center frequency (=1), which shows a phase shift of –90°.
Again, it is evident that the high-pass and low-pass phase responses are similar, just shifted by 180° (π radians). In higher-order filters, the phase response of each additional section is cumulative, adding to the total. This will be discussed in greater detail later. In keeping with common practice, the displayed phase shift is limited to the range of ±180°. For example, –181° is really the same as +179°, 360° is the same as 0°, and so on.
in-phase,
first-order, low-pass response. The buffer will add no phase shift,
as long as its bandwidth is significantly greater than that of the filter.
Remember that the frequency in these plots
is An alternative structure
is shown in Figure 7. This circuit, which adds resistance in parallel
to continuously discharge an integrating capacitor, is basically a
The circuits shown above, which attenuate the high frequencies and pass the low frequencies, are low-pass filters. Similar circuits also exist to pass high frequencies. The passive configuration for a first-order, high-pass filter is shown in Figure 8; and its phase variation with normalized frequency is shown in Figure 3 (in-phase response).
The plot in Figure 3
(left axis) will be referred to as the
Sallen-Key, the multiple-feedback,
the state-variable, and its close cousin, the biquad. They
are the most common and are relevant here. More complete information on
the various topologies is given in the References.
The amplifier gain in Sallen-Key filters can be increased by connecting a resistive attenuator in the feedback path to the inverting input of the op amp. However, changing the gain will affect the equations for the frequency-determining network, and the component values will have to be recalculated. Also, the amplifier’s dynamics are more likely to need scrutiny, since they introduce gain into the loop.
Q, high-frequency sections because of the limited open-loop
gain of the op amp at high frequencies. A guideline is that the open-loop
gain of the op amp should be at least 20 dB (i.e., ×10) above the amplitude
response at the resonant (or cutoff) frequency, including the peaking
caused by the Q of the filter. The peaking due to Q will have an
amplitude of magnitude A_{0}:
where
The multiple-feedback
filter inverts the phase of the signal. This is equivalent to adding 180°
to the phase shift of the filter itself. The variation of phase vs. frequency
is shown in Figure 4 (right axis).
This will be referred to as the
inverted,
second-order, high-pass response.
This type of filter may be more difficult to implement stably at high frequencies because it is based on a differentiator, which, like all differentiator circuits, maintains greater closed-loop gain at higher frequencies and tends to amplify noise.
_{0}) can be adjusted independently; and low-pass,
high-pass, and band-pass outputs are available simultaneously. The gain
of the filter is also independently variable.Since all parameters of the state variable filter can be adjusted independently, component spread can be minimized. Also, mismatches due to temperature and component tolerances are minimized. The op amps used in the integrator sections will have the same limitations on op amp gain-bandwidth as described in the multiple-feedback section.
The phase shift vs. frequency of the low-pass section will be an inverted second-order response (see Figure 4, right axis) and the high-pass section will have the inverted high-pass response (see Figure 5, right axis).
With the addition of a fourth amplifier section, high-pass, notch (low-pass, standard, and high-pass), and all-pass filters can be realized. A schematic for a biquad with a high-pass section is shown in Figure 16.
The phase shift vs. frequency of the LOWPASS1 section will be the in-phase, second-order, low-pass response (see Figure 4, left axis). The LOWPASS2 section will have the inverted second-order response (see Figure 4, right axis). The HIGHPASS section has a phase shift that inverts (see Figure 5, right axis).
Similarly, the various high-pass topologies are compared in Table 2.
Q of 0.707.
Figure 17 shows the effect on phase response of a low-pass filter (the
results for high-pass are similar) as Q is varied. The phase responses
for values of Q = 0.1, 0.5, 0.707, 1, 2, 5, 10, and 20 are plotted.
It’s worth noting that the phase can start to change well below the cutoff
frequency at low values of Q.
Although not the subject of this article,
the variation of amplitude response with The peaking that occurs in high-
Q available with a single second-order
section.A fourth-order filter cascade of transfer functions is shown in Figure 19. Here we see that the filter is built of two second-order sections.
Figure 20 shows the
effect on phase response of building a fourth-order filter in three different ways. The first is built with
two Sallen-Key (SK) Butterworth sections. The second consists of two multiple-feedback
(MFB) Butterworth sections. The third is built with one SK section and
one MFB section. But just as two cascaded first-order sections don’t make
a second-order section, two cascaded second-order Butterworth sections
do not equal a fourth-order Butterworth section. The first section of
a Butterworth filter has an As noted earlier, the SK section is noninverting, while the MFB section inverts. Figure 20 compares the phase shifts of these three fourth-order sections. The SK and the MFB filters have the same response because two inverting sections yield an in-phase response (–1 × –1 = +1). The filter built with mixed topologies (SK and MFB) yields a response shifted by 180° (+1 × –1 = –1).
Note that the total phase shift is twice that of a second-order section (360° vs. 180°), as expected. High-pass filters would have similar phase responses, shifted by 180°. This cascading idea can be carried out for higher-order filters, but anything over eighth-order is difficult to assemble in practice. Future articles will examine phase relationships in band-, notch- (band-reject), and all-pass filters.
- Daryanani,
G.
*Principles of Active Network Synthesis and Design*. J. Wiley & Sons. 1976. ISBN: 0-471-19545-6. - Graeme, J., G. Tobey, and L. Huelsman.
*Operational Amplifiers Design and Applications*. McGraw-Hill. 1971. ISBN 07-064917-0. - Sallen,
R. P., and E. L. Key. “A Practical Method of Designing RC Active Filters.”
*IRE Trans. Circuit Theory. 1955*. Vol. CT-2, pp. 74-85. - Thomas, L. C. “The Biquad: Part II—A Multipurpose
Active Filtering System.”
*IEEE Trans. Circuits and Systems*. 1971. Vol. CAS-18. pp. 358-361. - Thomas, L. C. “The Biquad: Part I—Some Practical
Design Considerations.”
*IEEE Trans*.*Circuits and Systems*. 1971. Vol. CAS-18. pp. 350-357. - Tow, J. “Active RC Filters—A State-Space
Realization.”
*Proc. IEEE. 1968. Vol. 56. pp. 1137-1139*. - Van Valkenburg, M. E.
*Analog Filter Design*. Holt, Rinehart & Winston. 1982. - Williams, A. B.
*Electronic Filter Design Handbook*. McGraw-Hill. 1981. - Zumbahlen, H. “Analog Filters.” Chapter 5, in Jung,
W.,
*Op Amp Applications Handbook*. Newnes-Elsevier (2006). (Original chapter from ADI Seminar Notes is available online.) - Zumbahlen,
H.
*Basic Linear Design*. Ch. 8. Analog Devices Inc. 2006. (Available soon).
Copyright 1995- Analog Devices, Inc. All rights reserved. |