The capacitor ratio increases as the square of Q.

While there are many types of capacitors providing values from a picoFarad to many Farads, most types of capacitors have voltage coefficients that cause excessive distortion in most signal processing applications. Many types also vary too much over temperature, time and other conditions. For decent precision over environmental conditions along with good linearity, the most cost-effective and volumetrically efficient choice is COG (or NPO) ceramic. Many film capacitors also can be used if you have the space and dollar budget for them. While larger values are available today, there is a substantial price premium for them. A good limit for keeping costs down is 0.01uF (10nF).

At the lower end, the limitation is circuit stray capacitances. A good rule of thumb is to not stray too much below 100pF, maybe 50pF. Using equation 32b, with 10nF for C₂ and 100pF for C₄ we come up with a practical limit of five for Q. This is not a hard limit; just a rule of thumb. Of course, if you can get good capacitors larger than 10nF within budget, then this limit on Q increases.

Can we reduce the capacitor ratio in this topology? If we remove the restriction of R₁=R₃, can we improve the capacitor ratios? Recall that we will not take a large sensitivity hit by removing this restriction on the resistor value, and we will suffer no increase in sensitivity if the resistors track each other.

Recall that the equation for Q is:

(33)

Let's call R₃/R₁ = n and C₄/C₂ = m. Then the equation for Q becomes:

(34)

If R₃=R₁ gives us the highest Q, then the derivative of this equation should be zero when R₃=R₁ or n=1. Since there are no terms in the numerator, the denominator would be at a minimum in this case and, thus, its derivative would also be zero. So let's take the derivative of the denominator with respect to n, which is:

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(35)

We have either a maximum or minimum Q at R₁=R₃. Simply plug in ratios of n=2 and then n=1/2 to show that this is a maximum, indicating that keeping the resistors of equal value is the optimal choice.

Back to multiple feedback filters
Recall that in Part 1, we looked at both the Sallen-Key and MFB lowpass filters and found that the MFB filter always had natural frequency and Q sensitivities to the components with magnitude less than one. Now that we know how to make Sallen-Key filters with similar or perhaps even lower sensitivities, we compare an MFB implementation of the same nominal filter characteristics as the Sallen-Key filters. Figure 6 is a unity gain MFB lowpass filter with the same natural frequency, and Q as the Sallen-Key filters we designed earlier.

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