# Analogue Filter Calculator

Graph colours Red = Magnitude, Blue = Phase.
Complex frequency: s = σ + jω = σ + j·2π·fo.

## Passive Filters

### RC Circuits [a] [b] [c] Low-pass [a] $H$$s$$ = 1/{1 + s ⋅ R ⋅ C}$ $I_R = V/R \text" , " I_C = s ⋅ V ⋅ C$ $H$$s$$ = {s ⋅ R ⋅ C}/{1 + s ⋅ R ⋅ C}$ $f_\o = 1/{2 ⋅ \π ⋅ R ⋅ C}$ R = Ω C = F fo = Hz τ = s

### RL Circuits [a] [b] [c] High-pass [a] $H$$s$$ = {s ⋅ L}/{R + s ⋅ L}$ $I_R = V/R \text" , " I_L = V/{s ⋅ L}$ $H$$s$$ = R/{R + s ⋅ L}$ $f_\o = R/{2 ⋅ \π ⋅ L}$ R = Ω L = H fo = Hz

### LC Circuits [a] [b] [c] Resonant (peak) [a] $I_L = V/{s ⋅ L} \text" , " I_C = s ⋅ V ⋅ C$ $H$$s$$ = 1/{s^2 ⋅ L ⋅ C + 1}$ $H$$s$$ = {s^2 ⋅ L ⋅ C}/{s^2 ⋅ L ⋅ C + 1}$ $f_\o = 1/{2 ⋅ \π ⋅ √{L ⋅ C}}$ L = H C = F fo = Hz

## Active Filters

### OpAmp Low-pass/High-pass Circuits [a] [b] Low-pass [a] $H$$s$$ = {-R_f}/R ⋅ 1/{1 + s ⋅ R_f ⋅ C}$ $H$$s$$ = {-R_f}/R ⋅ {s ⋅ R_f ⋅ C}/{1 + s ⋅ R_f ⋅ C}$ $Gain = {-R_f}/R$ $f_\o = 1/{2 ⋅ \π ⋅ R_f ⋅ C}$ R = Ω C = F fo = Hz

### OpAmp Circuit with Complex Poles  Band-pass with Q-factorand gain, G $H(s) = {-(R_1 ⋅ C)^{-1} ⋅ s}/{s^2 + 2 ⋅ (R_3 ⋅ C)^{-1} ⋅ s + ({R_1 ⋅ R_2}/{R_1 + R_2} ⋅ R_3 ⋅ C^2)^{-1}}$ R1 = Q/G R2 = Q/(2 · Q2 - G) R3 = 2 · Q Resistor Scaling Factor = 1/(2 ·π· f0 · C) fo = Hz Q = G = C = F R1 = Ω R2 = Ω R3 = Ω