Sallen-Key Analog Filters

Posted on September 5, 2010 by webmaster

Of all circuits, the one that has most influenced me is the 2nd order “biquad” using one opamp. This is commonly called the “Sallen-Key” biquad. This was the circuit I tried to understand and explain before even taking a class on circuit analysis.

Sallen Key Schematic

Sallen Key Schematic

The single opamp biquad presented here isn’t a true biquad. As shown, it can’t solve an arbitrary biquad equation, as there are restrictions on the numerator. The circuit I show here is suitable for lowpass filtering, as it has a fairly arbitrary 2nd order denominator.

The circuit can be analyzed using nodal analysis. Its also helpful to realize that the inner opamp circuit, using R2 and R1, is a non-inverting amplifier with gain k = 1+ \frac{R2}{R1}. A few of the other resistors/capacitors will eventually be renamed in the analysis, so please avoid skipping sections.

One Elegant Case

The two main equations used to describe the system are:

  • \frac{V_a - V_i}{R_a}+\frac{V_a - \frac{V_o}{k}}{R_b} + \frac{V_a - V_o}{C_b} = 0
  • \frac{\frac{V_o}{k}}{C_a} + \frac{\frac{V_o}{k}-V_a}{R_b} = 0

These can be solved (by hand or by computer) to give:

  • \frac{V_o}{V_i} = \frac{k}{C_a C_b R_a R_b s^2+\left( C_a R_b+\left( -C_b k+C_b+C_a\right) R_a\right) s+1}

For filter design, it is better to express the denominator in a different form:

  • D(s) = s^2 + \frac{\omega_0}{Q} s + \omega_0^2

This will allow the developer to design the filter in terms of cutoff frequency and quality factor.  This allows for a cascade of 2nd order stages to be used to generated higher order filters.  Higher order filters typically use pairs of “poles” with different frequency-Q coordinates.  For this implementation, the starting equations are:

  • \omega_0 = \frac{1}{\sqrt{R_a R_b C_a C_b}}
  • \frac{\omega_0}{Q} = \frac{C_a R_b+\left( -C_b k+C_b+C_a\right) Ra}{C_a C_b R_a R_b}

The latter I’ve left more abstract, as it will be beneficial at a later stage.

Now comes the “art” side of the design.  From a practical perspective, it’s very desirable to make the capacitors equal valued.  This is because capacitors are only released in a few values compared to resistors.  Further, depending on capacitor technology, different capacitors are different in size.  Resistors are typically available in any size.

The second thing to do is to convert Ra,Rb into a different representation.  In the cutoff frequency equation, Ra is multiplied by Rb.  A more generic R can be the geometric mean of Ra and Rb.  A second parameter, \alpha, can be used to determine the spread.  This only works for positive resistances, but that’s not an issue here.

  • C_a = C_b = C
  • R_a = \alpha R
  • R_b = \frac{R}{\alpha}
  • R_a \cdot R_b = R^2

Using the above relations, the frequency,Q coordinates become a bit easier to define:

  • \omega_0 = \frac{1}{R C}
  • Q = -\frac{\alpha}{{\alpha}^{2} k-2 {\alpha}^{2}-1}

The final step is to realize that its easy to get a gain of 2.  In that case, the denominator becomes simplified.  In fact it gives a stunningly simple expression:

  • k = 2
  • Q = \alpha

Again, the capacitors are usually the thing that will be chosen.  For this reason, it can be useful for the engineer to specific \omega_o,Q for the mathematical description, and C for the the implementation.  This can be done using a few manipulations:

  • \omega_0 = \frac{1}{R C}
  • R = \frac{1}{\omega_0 C}
  • R_a = \frac{Q}{\omega_0 C}
  • R_b = \frac{1}{Q \omega_0 C}

Other Notes

The above implementation isn’t the only choice.  This method has the disadvantage that a precise gain of 2 is required to get the Q factor’s denominator to become simple.  As the targeted Q increases, the spread of the resistors increases as well.  When the gain doesn’t exactly cancel the the -2\alpha^2 term in the denominator, it might become a significant term compared to the remaining additive term in the denominator.  Thus, if Q needs to be high, or very exact, this implementation should not be used.

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