RC, LC & Sallen-Key Filter Designer with Bode Plot

The plotted transfer functions are derived from impedance division and Kirchhoff's laws. With $u=s/\omega_0$, a first-order low-pass is $1/(1+u)$ and its high-pass is $u/(1+u)$. For RC, $\omega_0=1/(RC)$; for RL, $\omega_0=R/L$.

A second-order denominator is $u^2+u/Q+1$. The numerator is 1 for low-pass and $u^2$ for high-pass. Let $a=2-Q^{-2}$ and $k=\sqrt{(a+\sqrt{a^2+4})/2}$. The half-power cutoff is $kf_0$ for low-pass and $f_0/k$ for high-pass. We solve this relation rather than treating pole frequency and cutoff as interchangeable.

The LC network has a series L and shunt C‖R for low-pass, or series C and shunt L‖R for high-pass. In either case, $\omega_0=1/\sqrt{LC}$ and $Q=R\sqrt{C/L}$. The resistor is the total specified shunt load.

For unity-gain Sallen-Key, $\omega_0=1/\sqrt{R_1R_2C_1C_2}$. Low-pass uses $Q=\sqrt{R_1R_2C_1C_2}/[C_1(R_1+R_2)]$, with C1 to ground and C2 feeding back from the output. High-pass uses $Q=\sqrt{R_1R_2C_1C_2}/[R_2(C_1+C_2)]$, with R1 to ground and R2 in the feedback branch. See TI SLOA024B, attached pages 8 and 13–14. Synthesis sets equal resistors for low-pass and equal capacitors for high-pass.

Prototype Q is $1/\sqrt2$ for Butterworth, $1/\sqrt3$ for Bessel, 0.9565 for 1 dB Chebyshev and 1.3049 for 3 dB Chebyshev (TI SLOA049D, attached pages 11–12). Only Q is taken from those tables; this tool uses its stated half-power normalization.

Preferred-value selection minimizes the absolute logarithmic ratio to the unrounded component. E6/E12/E24 use the established mantissas; E96 uses three-significant-digit geometric values, consistent with Vishay's decade table. Tolerance bounds use the exact monotone pole-frequency limits and extrema of $\sqrt{x}/(1+x)$ for Sallen-Key ratios. Combining separate pole and Q bounds conservatively encloses cutoff; these extrema need not occur together. The step response is the analytic inverse Laplace transform, including underdamped, critical and overdamped cases.

• All transfer equations are derived here from ideal circuit laws; published examples independently check the implementation.
• Ideal zero-impedance voltage drive. RC/RL outputs and amplifier inputs are unloaded; LC includes only the shown shunt resistor.
• Linear, lossless L and C; resistors are frequency independent. Initial capacitor voltage and inductor current are zero.
• Sallen-Key uses an ideal, unity-gain-stable voltage follower. Ground is the signal reference; single-supply bias circuitry is not synthesized.
• Tolerances are independent bounded variations about selected nominal values, with no assumed probability distribution or temperature dependence. The LC load resistor also varies by the resistor tolerance.
• First-order filters have no Q; the displayed Q entry says “Not applicable”. A 0.5 placeholder is used only by shared functions and does not change a first-order result.
• Preferred-series choice and tolerance are independent. Confirm actual product specifications and availability.

Use for one first- or second-order analog section in its linear small-signal range. For second-order analysis, keep Q between 0.05 and 10 so the plotted time window remains useful. A Bessel high-pass is a frequency-inverted prototype, not a guarantee of constant group delay.

The 100× gain-bandwidth-to-pole ratio is an initial engineering screen chosen for this tool, not a manufacturer guarantee. The ideal Bode curves are not bandwidth-limited. Use an actual op-amp model when the plotted frequency approaches its usable bandwidth; high-pass gain eventually rolls off, and real Sallen-Key low-pass rejection can deteriorate at high frequency.

Source resistance, additional loading, inductor DCR/ESR, self resonance, voltage-dependent capacitance, parasitics, op-amp noise, finite gain/phase, slew rate and clipping are excluded. Do not use ideal instantaneous step peaks to size amplifier slew rate. Tolerance cutoff bounds are conservative enclosures, not exact simultaneous extrema or a Monte Carlo confidence interval.