Simple state variable active filter
The state variable active filter (SVAF) is an active filter you don’t see mentioned much today; however, it’s been a valuable asset for us old analog types in the past. This became especially true when cheap dual and quad op-amps became common place, as one can “roll their own” SVAF with just one IC package and still have an op-amp left over for other tasks!
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The unique features of this filter are having low-pass (LP), high-pass (HP), and band-pass (BP) filter results simultaneously available, with low component sensitivity, and an independent filter “Q” while creating a quadratic 2nd order filter function with 40-dB/decade slope factors. The main drawback is requiring three op-amps and a few more resistors than other active filter types.
The SVAF employs dual series-connected and scaled op-amp integrators with dual independent feedback paths, which creates a highly flexible filter architecture with the mentioned “extra” components as the downside.
With the three available LP, HP, and BP outputs, this filter seemed like a nice candidate for investigating with the Bode function available in modern DSOs. This is especially so for the newer Siglent DSO implementations that can plot three independent channels, which allows a single Bode plot with three independent plot variables: LP, HP, and BP.
Creating a SVAF with a couple of LM358 duals (didn’t have any DIP-type quad op-amps like the LM324 directly available, which reminds me, I need to order some soon!!), a couple of 0.01-µF mylar Caps, and a few 10 kΩ and 1 kΩ resistors seemed like a fun project.
The SVAF natural frequency corner is simply 1/RC, as shown in the notebook image in Figure 1 as ~1.59 kHz with the mentioned component values. The filter’s “Q” was set by changing R4 and R5.
Figure 1 The author’s hand-drawn schematic with R1=R2, R3=R6, and C1=C2, resistor values are 1 kΩ and 10 kΩ, and capacitors are 0.01 µF.
This produced plots of a Q of 1, 2, and 4 shown in Figure 2, Figure 3, and Figure 4, respectively, along with supporting LTspice simulations.
The DSO Bode function was set up with DSO CH1 as the input, CH2 (red) as the HP, CH3 (cyan) as the LP, and CH4 (green) as the BP. The phase responses can also be seen as the dashed color lines that correspond to the colors of the HP, LP, and BP amplitude responses.
While it is possible to include all the DSO channel phase responses, this clutters up the display too much, so on the right-hand side of each image, the only phase response I show is the BP phase (magenta) in the DSO plots.
Figure 2 The left side shows the Q =1 LTspice plot of the SVAF with the amplitude and phase of the HP (magenta + dashed magenta), the amplitude and phase of the LP (cyan + dashed cyan), and the amplitude and phase of the BP (green + dashed green). The right side shows the Q =1 DSO plot of the SVAF with HP (red), LP (cyan), BP (green), and phase of the BP (magenta).
Figure 3 The left side shows the Q =2 LTspice plot of the SVAF with the amplitude and phase of the HP (magenta + dashed magenta), the amplitude and phase of the LP (cyan + dashed cyan), and the amplitude and phase of the BP (green + dashed green). The right side shows the Q =2 DSO plot of the SVAF with HP (red), LP (cyan), BP (green), and phase of the BP (magenta).
Figure 4 The left side shows the Q =4 LTspice plot of the SVAF with the amplitude and phase of the HP (magenta + dashed magenta), the amplitude and phase of the LP (cyan + dashed cyan), and the amplitude and phase of the BP (green + dashed green). The right side shows the Q =4 DSO plot of the SVAF with HP (red), LP (cyan), BP (green), and phase of the BP (magenta).
The Bode frequency was swept with 33 pts/dec from 10 Hz to 100 kHz using a 1-Vpp input stimulus from a LAN-enabled arbitrary waveform generator (AWG). Note how the three responses all cross at ~1.59 kHz, and the BP phase, or the magenta line for the images on the right side, crosses zero degrees here.
If we extend the frequency of the Bode sweep out to 1 MHz, as shown in Figure 5, well beyond where you would consider utilizing an LM358. The simulation and DSO Bode measurements agree well, even at this range. Note how the simulation depicts the LP LM358 op-amp output resonance ~100 kHz (cyan) and the BP Phase (magenta) response.
Figure 5 The left side shows the Q =7 LTspice plot of the SVAF with the amplitude and phase of the HP (magenta + dashed magenta), the amplitude and phase of the LP (cyan + dashed cyan), and the amplitude and phase of the BP (green + dashed green). The right side shows the Q =7 DSO plot of the SVAF with HP (red), LP (cyan), BP (green), and phase of the BP (magenta).
I’m honestly surprised the simulation agrees this well, considering the filter was crudely assembled on a plug-in protoboard and using the LM358 op-amps. This is likely due to the inverting configuration of the SVAF structure, as our experience has shown that inverting structures tend to behave better with regard to components, breadboard, and prototyping, with all the unknown parasitics at play!
Anyway, the SVAF is an interesting active filter capable of producing simultaneous LP, HP, and BP results. It is even capable of producing an active notch filter with an additional op-amp and a couple of resistors (requires 4 total, but with the LM324, a single package), which the interested reader can discover.
Michael A Wyatt is a life member with IEEE and has continued to enjoy electronics ever since his childhood. Mike has a long career spanning Honeywell, Northrop Grumman, Insyte/ITT/Exelis/Harris, ViaSat and retiring (semi) with Wyatt Labs. During his career he accumulated 32 US Patents and in the past published a few EDN Articles including Best Idea of the Year in 1989.
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