Since professors R.P. Sallen and E.L. Key described it in 1955, the Sallen-Key low-pass filter has become one of the most widely used filters in electronic systems. Perhaps because the mathematics can be somewhat daunting, however, little has been written to help working engineers specify the correct components to achieve their objectives. For example, few realize the limitations of Sallen-Key filters at high frequencies.
The following describes the basic operations of a Sallen-Key low-pass filter and offers a simplified way of working with such circuits. Based on laboratory research, it also demonstrates some of this filter's limitations at high frequencies.
Sallen-Key basics: The two-stage RC network shown in Figure 1 forms a second-order low-pass filter. This circuit has the limitation that its Q is always less than one-half. With R1 = R2 and C1 = C2, then Q = 1/3. Q approaches the maximum value of one-half when the impedance of the second RC is much larger than the first. But most filters usually require larger Qs than one-half.
Q can be enhanced with an amplifier in positive feedback. With that feedback localized to the filter's cutoff frequency, almost any Q can be realized. Mostly, it's only limited by the physical constraints of the power supply and component tolerances. The Sallen-Key low-pass filter shown in Figure 2 is an example of how an amplifier is used in this manner. C2 is no longer connected to ground, but rather provides a positive feedback path around the amplifier.
The operation can be described qualitatively. At low frequencies, where C1 and C2 appear as open circuits, the signal is simply amplified to the output. R3 and R4 are chosen to give the desired gain. At high frequencies, C1 and C2 appear as short circuits, and the signal is shunted to ground at the amplifier's input. The amplifier amplifies this input to its output, and the signal doesn't appear at VO. Near the cutoff frequency, where the impedance of C1 and C2 are on the same order as R1 and R2, positive feedback via C2 provides Q enhancement of the signal.
Ideal operation: The standard frequency-domain equation for a second-order low-pass filter is:
where fC is the corner frequency and Q is the quality factor.
When f<<fC, Equation 1 reduces to HLP = K, and the circuit passes signals multiplied by gain factor K. When f = fC, Equation 1 reduces to: HLP = −jKQ, and the signals are enhanced by the factor Q. When f>>fC, Equation 1 reduces to:
and the signals are attenuated by the square of the frequency ratio. With attenuation at higher frequencies increasing by a power of two, the formula describes a second-order low-pass filter.
Deriving the transfer function of the circuit in Figure 2, the Sallen-Key ideal low-pass transfer function is defined by Equation 2.
By letting
Equation 2 follows the same form as Equation 1. With simplifications, you can deal with the equation more easily.
Simplification 1: Set filter components as ratios. Letting R1 = mR, R2 = R, C1 = C, and C2 = nC, results in:
This simplifies things somewhat, but there's interaction between fC and Q.
The design should start by setting the gain and Q based on m, n, and K, and then selecting C and calculating R to set fC. It may be observed that K = 1 +[(m+1)/(mn)] results in Q = ∞. With larger values, Q becomes negative. In other words, the poles move into the right half of the s-plane and the circuit oscillates. The most frequently designed filters require low Q values, so this should rarely be a design issue.
Simplification 2: Set filter components as ratios and gain = 1. Letting R1 = mR, R2 = R, C1 = C, C2 = nC, and K = 1 results in:
This keeps the gain equal to 1 in the pass band. But again, there's interaction between fC and Q. Design should start by choosing the ratios m and n to set Q, and then selecting C and calculating R to set fC.
Simplification 3: Set resistors as ratios and capacitors equal. Letting R1 = mR, R2 = R, and C1 = C2 = C, results in:
The main motivation behind setting the capacitors equal is the limited selection of values in comparison with resistors. Interaction exists between setting fC and Q. Design should start with choosing m and K to set the gain and Q of the circuit before choosing C and calculating R to set fC.
Simplification 4: Set filter components equal. Letting R1 = R2 = R and C1 = C2 = C results in:
Now fC and Q are independent of one another. Design is greatly simplified, although it's simultaneously limited. Q is now determined by the gain of the circuit. The choice of RC sets fC. The capacitor should be chosen, and the resistor calculated. One minor drawback is that because the gain controls the Q of the circuit, further gain or attenuation may be necessary to achieve the desired signal gain in the passband.
Values of K that are very close to 3 result in high Qs that are sensitive to variations in the component values of R3 and R4. Setting K = 2.9 results in a nominal Q of 10. A worst-case analysis with 1% resistors results in Q = 16. In contrast, if setting K = 2 for a Q of 1, worst-case analysis with the same 1% resistors results in Q = 1.02. Resistor values where K = 3 leads to Q = ∞. And with larger values, Q becomes negative. The poles move into the right half of the s-plane and the circuit will oscillate. The most frequently designed filters require low Q values, so this should rarely become a design issue.