Non-ideal circuit operation: Up to now, we've assumed that the circuit was ideal, but there comes a time (or actually a frequency) when this is no longer valid. Simple logic tells us that the amplifier must be an active component at the frequencies of interest or else we have problems. But what are these problems?
As mentioned previously, there are three basic modes of operation: below cutoff, above cutoff, and in the area of cutoff. Assuming that the amplifier has adequate frequency response beyond cutoff, the filter works as expected. At frequencies well above cutoff, the high-frequency model depicted in Figure 3 is used to show the expected circuit operation. The assumption made here is that C1 and C2 are effective shorts when compared to the impedance of R1 and R2, so the amplifier's input is at ac ground. In response, the amplifier generates an ac ground at its output limited only by its output impedance, ZO. The formula shows the transfer function of this particular model.
ZO is the closed-loop output impedance. It depends on the loop transmission and the open-loop output impedance, zo:
where a(f) is the open-loop gain of the amplifier and b is the feedback factor. This feedback factor is constant—set by resistors R3 and R4. But the open-loop gain, a(f), depends on frequency.
With dominant-pole compensation, the amplifier's open-loop gain decreases by 20 dB/decade over the usable frequencies of operation. Assuming zo is mainly resistive (usually a valid assumption up to a few hundred megahertz), ZO increases at a rate of 20 dB/decade. The transfer function appears to be a first-order high pass.
At frequencies above 100 MHz (or so), the parasitic inductance in the output starts playing a role and the transfer function transitions to a second-order high pass. Plus, at higher frequency, the high-pass transfer function will roll off due to stray capacitance.
Simulation and lab data: To show the effects described above, a Sallen-Key low-pass filter was simulated in Spice and lab tested using a THS3001 operational amplifier from Texas Instruments. The THS3001 is a high-speed, current-feedback amplifier with an advertised bandwidth of 420 MHz. Choosing R1 = R2 = 1 kΩ, C1 = C2 = 1 nF, R3 = open, and R4 = 1 kΩ results in a low-pass filter with fC = 159 kHz and Q = 1/2.
Figure 4a depicts the simulation circuit with the Spice model modified so that the output impedance of the amplifier is 0 Ω. In Figure 5, curve (a) shows the frequency response as simulated in Spice. It also reveals that with zero output impedance, the attenuation of the signal continues to increase as frequency rises.
Figure 4b depicts the high-frequency model as exemplified in Figure 3, where the input is at ground and the output impedance controls the transfer function. The Spice model used for the THS3001 includes an LRC network for the output impedance. Again, Figure 5 shows the frequency response as simulated in Spice, but this time it's symbolized by curve (b). The magnitude of the signal at the output is seen to cross curve (a) at about 7 MHz. Above this frequency, the output impedance causes the switch in the transfer function, which is described above.
Look to Figure 4c for the simulation circuit using the Spice model with the LCR output impedance. Figure 5's curve (c) shows the frequency response for this model. With the output impedance, the attenuation caused by the circuit follows curve (a) until it crosses curve (b), at which point it follows curve (b). Figure 4d reveals the circuit as tested in the lab, with curve (d) in Figure 5 showing that the measured data agrees with the simulated data.
Comments about component selection: Until now, the choosing of resistor and capacitor values has been left without mention. Theoretically, any values of R and C that satisfy the equations may be used. But practical considerations call for certain guidelines to be followed. Given a specific corner frequency, the values of C and R are inversely proportional to one another. By making C larger, R becomes smaller and vice versa.
In the case of the low-pass Sallen-Key filter, the ratio between the output impedance of the amplifier and the value of filter component R sets the transfer functions seen at frequencies well above cutoff. The larger the resistor's value, the lower the transmission of signals at high frequency. Making R too large may result in C becoming so small that the parasitic capacitors, including the input capacitance of the amplifier, cause errors. The best choice of component values depends on the particulars of your circuit and the tradeoffs you're willing to make.
Here are some general recommendations for capacitors and resistors: Engineers should avoid capacitors with values less than 100 pF. If at all possible, use an NPO type. X7R is okay in a pinch, but avoid Z5U and other low-quality dielectrics. In critical applications, even higher-quality dielectrics, like polyester, polycarbonate, Mylar, etc., may be required. As for resistors, values in the range of a few hundred to a few thousand ohms are the best bet. You also should choose metal-film resistors that possess low temperature coefficients. Finally, use 1%-tolerance capacitors and resistors, preferably those of the surface-mount variety.