I want to make a Sallen & Key 2nd-Order Butterworth highpass filter with a defined gain. Let's say gain is +6 db.

[![enter image description here](https://i.stack.imgur.com/EuwOs.png)](https://i.stack.imgur.com/EuwOs.png)

Then,

R3 = R4 = 1 kOhm

F = 1000 Hz

C = 100 nF

R2 = 2 \* Q / ( 2 \* 3.14 \* F \* C )

Now, I would assume that Q = 1 / ( 3 - A), where A = 1 + R3 / R4.
But this table here doesn't agree.

[![enter image description here](https://i.stack.imgur.com/nrxGm.png)](https://i.stack.imgur.com/nrxGm.png)

What am i missing?
['https://electronics.stackexchange.com/questions/562193', 'https://electronics.stackexchange.com', 'https://electronics.stackexchange.com/users/262542/']
**Transfer function** for two **equal capacitors C** and gain **A=2**:

H(s)=N(s)/D(s) with

**N(s)=(s² \* 2 \* R1 \* R2 C²)** and **D(s)=[1+s(R1 \* 2C - R2 \* C) + s² \* R1 \* R2 \* C²]**

From this (comparison with the general transfer function) with **k=R2/R1**

Pole frequency **wp=1/[R1 \* C \* SQRT(k)]** and

Pole-Q **Qp=SQRT(k)/(2-k)**=0.7071 with

k1,2=2 + [1 (-+)SQRT(1+4)]

**k1**=2 + 1 - 2.236 =**0.764** (for k2 Qp is negative).

**R2=0.764 \* R1**

---

**EDIT** (completion):

Here is the general transfer function for a Sallen-Key filter (lowpass, highpass) with gain **A=1+(R3/R4)** and the horizontal elements Y0 and Y5:

H(s)=N(s)/D(s) with

**N(s)=A \* Y0 \* Y5** and

**D(s)=Y2(Y0+Y1+Y5) + Y5[Y0 \* Y1 \* (1-A)]**

Fot a **highpass** we select: **Y0=sC0, Y5=sC5, Y1=1/R1 and Y2=1/R2**

For **C0=C5** and **A=2** the highpass function results as given above (after some mathematical manipulatons.)
From "A handbook of Active Filters", by D.E. Johnson, et al., reference VCVS high-pass filter. This uses tables of coefficients for the various types of filters (butterworth, cheby, elliptic, ...). Warning, C is a coefficient, C1 are the capacitor values.

For 2nd order butterworth coefficients:

```
B = sqrt(2)
C = 1.0

```

Equations:

```
R2 = 4 * C / {[B + sqrt(B^2 + 8 * C * (K + 1)] * w * C1}
R1 = C / {w^2 * C1^2 * R2}
R4 = K * R2 / {K - 1}
R3 = K * R2

```

where:

```
K = gain
w = 2*π*f
C1 = capacitors (arbitrary value, same value for both)

```

R4 & R3 equations are given to optimize DC offset. You can make them any reasonable value if you don't care about DC offset.

Choosing C1 = 100nF:

```
R2 = 1391.77 -> nearest 1% value = 1.40k
R1 = 1821.85 -> nearest 1% value = 1.82k
R3 = R4 = 2783.54 -> nearest 1% value = 2.87k

```

Spice simulation shows this to be very close to what you want. Watch out for your choice of dielectric for C1. Either use plastic or COG dielectrics. If using COG (NPO) dielectric, you may want to make C1 = 10nF. X7R, and other class 2 ceramic dielectrics, will introduce distortion, have a strong temperature dependence, and change value over time.