Download this article in PDF format. (609KB) Phase Response in Active Filters A previous
article This article will concentrate on the low-pass and high-pass responses. Future articles in this series will examine the band-pass and notch (band-reject) responses, the all-pass response, and the impulse and step responses of the filter. To review, the transfer function of an active filter can be viewed as the cascaded response of the filter transfer function and an amplifier transfer function (Figure 1).
For the single-pole low-pass case, the transfer function has a phase shift given by:
where ω
represents a radian frequency (ω
= 2π _{0}.Figure 2 (left axis)
evaluates Equation 1 from two decades below the center frequency to two
decades above the center frequency. Since a single-pole low-pass has a
90°
range of phase shift—from 0°
to 90°—the
center frequency has a phase shift of –45°. At ω
= ω
Similarly, the phase response of a single-pole high-pass filter is given by:
Figure 2 (right axis) evaluates Equation 2 from two decades below to two decades above the center frequency. The center frequency (=1) has a phase shift of +45°. If the low-pass
In the low-pass case, the output of the filter lags the input (negative phase shift); in the high-pass case the output leads the input (positive phase shift). Figure 3 shows waveforms: an input sine-wave signal (center trace), the output of a 1-kHz-cutoff single-pole high-pass filter (top trace), and the output of a 1-kHz-cutoff single-pole low-pass filter (bottom trace). The signal frequency is also 1 kHz—the cutoff frequency of both filters. The 45° lead and lag of the waveforms are clearly evident.
For the second order low-pass case, the transfer function’s phase shift can be approximated by:
Figure 4 (left axis) evaluates this equation (using α = √2 = 1.414) from two decades below the center frequency to two decades above the center frequency. Here the center frequency is 1, with a phase shift of –90°.
In Equation 3, α,
the The phase response of a 2-pole high-pass filter can be approximated by:
In Figure 4 (right axis), this equation is evaluated with α = 1.414 from two decades below the center frequency to two decades above the center frequency. At the center frequency (=1), the phase shift is 90°. Figure 2 and Figure 4 use single curves because the high-pass and the low-pass phase responses are similar, just shifted by 90° and 180° (π/2 and π radians). This is equivalent to a change of the sign of the phase, causing the outputs of the low-pass filter to lag and the high-pass filter to lead. In practice, a high-pass filter is really a wideband band-pass filter because the amplifier’s response introduces at least a single low-pass pole. Figure 5 shows the
phase- and gain response of a 2-pole low-pass filter, plotted as a function
of
Note that each 2-pole
section provides a maximum 180°
of phase shift; and at the extremities, a phase shift of –180°,
though lagging by 360°,
is an angle with the same properties as a phase shift of 180°.
For this reason, a multistage filter will often be graphed in a restricted
range, say 180°
to –180°,
to improve the accuracy of reading the graph (see Figures 9 and 11). In
such cases, it must be realized that the angle graphed is actually the
true angle plus or minus Figure 6 shows the
gain- and phase response of a 2-pole high-pass filter with varying
1) Unlike
the Butterworth case, the center frequencies of the individual sections
are all different. This allows a graph 2) The
3) An odd number of poles emphasizes the difference between single- and two-pole sections. The filter sections were designed using the Filter Design Wizard, available on the Analog Devices website. The
Figure 8 shows the
schematic of the complete filter. The filter topology chosen—
Figure 9 shows phase shifts at each stage of the complete filter. The graph shows the phase shift of the first section alone (Section 1—blue), the first two sections (Sections 1 and 2—red) and the complete filter (Sections 1, 2, and 3—green). These include the basic phase shifts of the filter sections, the 180° contributed by each inverting amplifier, and the effects of amplifier frequency response on overall phase shift.
A few details of
interest: First the phase response, being a net lag, accumulates negatively.
The first 2-pole section starts with
voltage-controlled
voltage source (VCVS) sections rather than multiple-feedback (MFB).
Though an arbitrary choice, VCVS requires only two capacitors per 2-pole
section, rather than MFB’s three capacitors per section, and the first
two sections are noninverting.
Figure 11 shows
the phase response at each section of the filter. The first section’s
phase shift starts at 180°
at low frequencies, dropping to 0°
at high frequencies. The second section, adding 180°
at low frequencies, starts at 360°
The transfer function of a single-pole low-pass filter:
where The transfer function of a two-pole active low-pass filter:
where The transfer function of a single-pole high-pass filter:
The transfer function of a two-pole active high-pass filter:
The values of
For a more detailed discussion, see References 6, 7, and 8.
- Zumbahlen, H.
“Phase Relations in Active Filters.”
*Analog Dialogue*. Volume 41, No. 4. October 2007. - 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. - Van Valkenburg,
M.E.
*Analog Filter Design*. Holt, Rinehart & Winston (1982). ISBN: 0-03-059246-1 - 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 (2006). - Zumbahlen, H.
*Linear Circuit Design Handbook.*Newnes-Elsevier (2008). ISBN: 978-0-7506-8703-4
Copyright 1995- Analog Devices, Inc. All rights reserved. |