4th order Butterworth LPF with two Sallen-Key biquads (10 kHz)

Butterworth n = 4 (pole table): $p = -0.3827 \pm j0.9239$ and $-0.9239 \pm j0.3827$, so

$$T_N(s) = \frac{1}{(s^2 + 0.7654s + 1)(s^2 + 1.8478s + 1)}, \qquad Q_1 = \frac{1}{1.8478} = 0.5412,\quad Q_2 = \frac{1}{0.7654} = 1.3066$$

Equal-component Sallen–Key biquad

                 +--------[C]----------+
                 |                     |
 Vi o--[R]-------+--[R]---+----(+)     |
                          |       A ---+---o Vo
                         [C]  +--(-)   |
                          |   |        |
                         GND  +--[RB]--+
                              |
                             [RA]
                              |
                             GND

Equal-component Sallen–Key lowpass ($R_1 = R_2 = R$, $C_1 = C_2 = C$, amplifier gain $K = 1 + R_B/R_A$):

$$T(s) = \frac{K\omega_0^2}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}, \qquad \omega_0 = \frac{1}{RC}, \qquad Q = \frac{1}{3 - K} \;\Rightarrow\; K = 3 - \frac{1}{Q}$$

Element values

$\omega_0 = 2\pi(10^4) = 62{,}832\ \text{rad/s}$, $C$ = 0.1 μF (all four capacitors):

$$R = \frac{1}{\omega_0 C} = 159.2\ \Omega$$

Overall passband gain $K_1K_2$ = 2.5748 (8.21 dB).

Gain compensation

To bring the DC gain back to 1, replace the input resistor $R$ of each section by a divider $R_{1a}$ (from $V_i$) and $R_{1b}$ (to ground) that attenuates by $1/K$ while keeping the Thevenin resistance equal to $R$:

$$R_{1a} = KR, \qquad R_{1b} = \frac{KR}{K-1}$$

(Check: $R_{1a} \parallel R_{1b} = R$ and $R_{1b}/(R_{1a}+R_{1b}) = 1/K$.)

 Vi -->[SK 1: K=1.1522]-->[SK 2: K=2.2346]--> Vo
       R=159.2, C=0.1uF     R=159.2, C=0.1uF
       RA=10k, RB=1.522k    RA=10k, RB=12.35k

With the forced 0.1 μF capacitors at 10 kHz the resistors are small (159.2 Ω); a good op-amp with low output resistance is needed.

Answer: all C = 0.1 μF, R = 159.2 Ω; section 1: K = 1.1522 ($R_B$ = 1.522 kΩ, $R_A$ = 10 kΩ); section 2: K = 2.2346 ($R_B$ = 12.35 kΩ); uncompensated gain 2.5748; for unity gain use input dividers 183.4 Ω / 1.205 kΩ (section 1) and 355.7 Ω / 288.1 Ω (section 2).