# Gain in Sallen-Key filter

How can we increase the passband gain in a Sallen-Key type low pass / high pass filter, preferably without using additional op-amps.

The Sallen-Key filter can be realized with gain but component value adjustments are necessary to reduce Q. Q rises as gain increases from unity and the feedback capacitor is usually the component to lower. Another way of looking at this is by thinking about the feedback capacitor - it's the only component that is affected by gain - if the gain were (say) 10 and the feedback signal to the cap were reduced by 10 the circuit would perform the same theoretically. Reducing the cap or potting down the input to the cap has the same effect.

Here is a diagram from TI source: -

As you can see, the frequency cut-off is exactly the same as the unity gain circuit but Q is now dependent of gain 1+R4/R3.

The source (TI) linked above should allow you to derive the high pass version

• +1 Nicely done. Was about to say something to this effect, then read your answer ;-) – DrFriedParts Jul 4 '13 at 9:10

A sallen-Key lowpass for a fixed gain of "2" even has advantages if compared with the classical unity gain approach: Now it is possible to use two equal-valued capacitors.

There are three basic design strategies: Gain $A=1$ (unity gain), $A=2$ or equal components with $A<3$.

Based on the element allocation as shown in the circuit diagram provided by Andy aka (with $R1,R2,R3,R4,C2,C2$) we have simple design formulas:

• $C1=C2=C ; R3=R4=R$
• $R2/R1=k$
• Gain: $Ao=2$
• Pole frequency: $wp=\dfrac{1}{R1 \times C \times \sqrt{k}}$
• Pole Q: $Qp=\sqrt{1/k}$ or $k=1/Qp^2$

Corresponding values for wp and Qp depend on the particular lowpass specification (desired approximation and passband edge).

Example: Butterworth response with $Qp=0.7071$ and $wp=w(3 dB)$ results in $k=R2/R1=2$

• I'm glad you liked it. I enjoy seeing the results of turning regular equations into LaTeX format. But I had to resist the urge of turning the numbers into subscripts (from $C1$ to $C_1$, for example), but I thought it would be better to stick to the source representation. – Ricardo Jun 5 '14 at 19:56