### butterworth spice netlist

Below shows the contents of: butterworth.spice

* Third order low pass filter, butterworth, with cutoff frequency (fc) = 10 MHz

* Create a third-order Butterworth filter, according to https://en.wikipedia.org/wiki/Butterworth_filter#Example
* The circuit diagram is:
*
*  ┌─L1─┬─L3─┬── +
*  V    C2   R4
*  └────┴────┴── -
*
* We take the simple example, with values:
*  L1 = 1.20 nH
*  C2 =  430 pF
*  L3 =  390 nH
*  R4 =   50 Ω
*
* This yields a transfer function of:
*   H(s) = 1/(R1 + (L1+L3)*s + (L1*C2*R4)*s^2 + (L1*C2*L3)*s^3)
*
* The magnitude of the steady-state response:
*   G(f) = abs(H(2π*f*im))
* so at f=10 MHz we should get 1/2 gain
*
* If we drive this system with a (one-sided) sinusoidal input of frequency 10 MHz, we obtain the following Laplace transform:
*   H(s) = ωₒ/(s^2 + ωₒ^2) * 1/(1 + s/ωₒ + (s/ωₒ)^2 + (s/ωₒ)^3)
*
* where ωₒ = 1e7*2π. This corresponds to a time-domain solution via the inverse laplace transform of:
*   vout(t) = (e^(-ωₒt) - sin(ωₒt) - cos(ωₒt))/2 + (2 * sin((sqrt(3) * ωₒt)/2))/(sqrt(3) * sqrt(e^(ωₒt)))
*
.param L1=1.2n
.param C2=430p
.param L3=390n
.param R4=50
.param freq=10MEG

V1 vin 0 AC=1 SIN(0 1 'freq')
LL1 vin n1 'L1'
CC2 n1 0 'C2'
LL3 n1 vout 'L3'
RR4 vout 0 'R4'