LC Low-Pass Filter Calculator

1. The Doubly Terminated LC Ladder

A passive low-pass filter for RF and signal work is almost always an LC ladder: series inductors and shunt capacitors in alternation, driven from a source resistance R_S and loaded by R_L. Inductors pass DC and block high frequencies; capacitors short high frequencies to ground. With N reactive elements the response falls by 20·N dB per decade (6·N dB per octave) far above the cutoff, so N is the order of the filter.

The ladder is lossless, so every bit of power that does not reach the load is reflected back to the source. Its performance is therefore described by the transducer gain |S21|², the fraction of the available source power delivered to the load, and the reflection |S11|², linked by the power balance of a lossless two-port. Doubly terminated ladders are also remarkably insensitive to component tolerances in the passband (Orchard's argument): at frequencies of full power transfer the loss is at a minimum, so its first-order sensitivity to any element is zero.

2. The Normalized Low-Pass Prototype and g-Values

Every LC low-pass design starts from a prototype normalized to a 1 Ω source and a cutoff of 1 rad/s. Its element values g₁ … g_N are listed from the source end; g_k is an inductance in henries for a series element and a capacitance in farads for a shunt element. Following Matthaei, Young and Jones, g₀ is the source resistance and g_N+1 the load: a resistance if g_N is a shunt capacitor, a conductance if g_N is a series inductor.

The two topologies are duals. A ladder that starts with a shunt capacitor and one that starts with a series inductor have the same g-values and exactly the same response; only the component count differs. For an odd order, shunt-C-first (a π network) needs (N−1)/2 inductors and series-L-first (a T network) needs (N+1)/2. All the values in this calculator are normalized so that g-values give a −3 dB point at exactly 1 rad/s, whatever the filter type.

3. Impedance and Frequency Scaling

The prototype becomes a real filter by two scalings. Multiplying all impedances by R₀ moves the terminations to R₀; replacing s by s/ω_c moves the cutoff to ω_c = 2πf₋₃dB. Each inductor and capacitor then follows from its g-value:

Because the scaling is linear, a filter's shape depends only on its g-values. Doubling the cutoff halves every component; doubling the impedance doubles the inductors and halves the capacitors.

4. Butterworth: the Maximally Flat Response

The Butterworth approximation makes the first 2N−1 derivatives of |S21|² zero at DC, giving the flattest possible passband with no ripple. The response is monotonic, its −3 dB point is the natural cutoff, and its element values have a closed form:

Setting the attenuation at Ω_s = f_s/f₋₃dB equal to the requirement A_s and solving for N gives the order directly; the calculator rounds up to the next integer:

5. Chebyshev: the Equiripple Response

A Chebyshev (type I) filter lets the passband loss ripple between 0 and A_r dB in exchange for the steepest transition any all-pole filter of the same order can achieve. The response uses the Chebyshev polynomial T_N, which oscillates between ±1 inside |x| ≤ 1 and grows as cosh(N·arccosh x) outside it:

The element values follow from the classic recursion of Matthaei, Young and Jones, written here for a ripple band edge of 1 rad/s:

6. Ripple Bandwidth Versus the −3 dB Cutoff

Published Chebyshev tables are normalized to the ripple band edge Ω_p, the last frequency where the loss equals A_r, not to the −3 dB point. Using them directly with your −3 dB frequency puts the real cutoff too high; for a 0.1 dB, 3rd-order filter the −3 dB point lands 39% too high. The −3 dB point lies where ε²T_N² = 1, which gives the ratio between the two edges:

The calculator applies this correction for every order and ripple, so the −3 dB point of the designed filter lands exactly on the frequency you enter, and step 1 reports the ripple band edge separately. Because Ω_p changes with N, the Chebyshev order cannot be solved in closed form against a −3 dB spec; the calculator evaluates the exact attenuation of each order in turn.

7. Even-Order Chebyshev Filters and Unequal Terminations

At DC, T_N(0) is 0 for odd N but ±1 for even N. An even-order Chebyshev response therefore starts at the bottom of a ripple, with A_r dB of loss at DC. A lossless ladder at DC is just a wire, so this loss can only come from a deliberate mismatch between the source and load resistances:

In a 50 Ω system this is usually unacceptable, so in Auto mode the calculator only uses odd Chebyshev orders. If you force an even order, it designs the filter for the required load, shows that load in the schematic and the netlist, and warns you. Exporting the S-parameters with both ports at R₀ then shows the mismatch you would see with an equal load.

8. Bessel–Thomson: Maximally Flat Group Delay

The Bessel (Thomson) filter optimizes the time domain instead of the magnitude. Its transfer function makes the group delay as flat as possible at DC, so all frequency components of a pulse arrive together and the waveform keeps its shape. The denominator is the reverse Bessel polynomial θ_N(s), whose integer coefficients have a closed form:

That normalization is to delay, not to the −3 dB point, which for N = 5 lies at 2.427 rad/s. The calculator finds each order's −3 dB frequency numerically and rescales it to 1 rad/s, so a Bessel design uses the same cutoff definition as the other two types. There is no closed-form order equation either; the calculator evaluates |θ_N(jω)| for each order.

There is also no closed form for the Bessel ladder elements, so they are found by Darlington synthesis. From |S11|² = 1 − |S21|², the reflection polynomial P(s) is obtained from the spectral factorization of θ_N(s)θ_N(−s) − θ_N(0)², taking its left-half-plane roots. The input impedance of the terminated ladder is then expanded as a continued fraction about s = ∞, which peels off one element at a time (Cauer's first form):

The continued-fraction expansion loses several digits per stage in ordinary 64-bit arithmetic, so the Bessel element values in this calculator were synthesized at 80-digit precision for every order from 1 to 12 and stored. They agree with Zverev's published tables to every printed digit, and the response of each ladder matches θ_N(0)/θ_N(s) to better than 10⁻⁹ dB.

9. Choosing the Filter Type and Order

The three types trade selectivity against time-domain behaviour. At twice the cutoff frequency they give these attenuations:

Chebyshev gains roughly 10 dB per added element close to the passband, Butterworth about 6 dB, and the Bessel attenuation near the passband barely changes with order. As N grows, the Bessel magnitude response approaches a Gaussian, A(f) ≈ 3.01·(f/f₋₃dB)² dB, so at twice the cutoff it saturates near 12–14 dB at any order. Far above the cutoff all three types roll off at the same 6N dB per octave; they differ only in how quickly they reach that asymptote.

• Butterworth for a flat passband and a moderate skirt, the default choice for anti-aliasing and harmonic filters.
• Chebyshev when components are expensive or the stopband is close to the passband, such as transmitter harmonic filters and IF filters.
• Bessel for pulses, data, video and measurement front-ends, where overshoot and ringing matter more than rejection.

10. Group Delay and Step Response

The LC ladder is an all-pole network: its transfer function is a constant over a polynomial whose roots p_i = σ_i + jω_i are the natural frequencies of the circuit. The calculator finds these poles from the ladder itself (with ideal or standard values), and the group delay and step response follow exactly:

Selectivity costs time-domain quality. A 5th-order Butterworth overshoots a step by 12.8%, a 0.5 dB Chebyshev by 13.2% and a 7th-order 0.1 dB Chebyshev by 17.7%, with group delay that peaks sharply near the band edge. A Bessel filter of any order overshoots by less than 1%, and its delay at the cutoff is within 7% of the DC value for N = 3 and within 1% for N ≥ 5.

11. Passband Ripple and Return Loss

Because the ladder is lossless, any passband loss is reflection. A ripple of A_r dB means a worst-case return loss of:

This is why RF filters between 50 Ω stages typically use 0.01–0.1 dB ripple: the match to neighbouring amplifiers and mixers matters as much as the extra selectivity. Butterworth and Bessel filters have a perfect match at DC and a return loss that degrades smoothly toward the cutoff, as step 3 shows.

12. Standard Values, Tolerances and Real Components

Ideal values rarely match catalogue parts. The calculator rounds each element to the nearest E12, E24 or E96 value on a logarithmic scale (E12 steps are about 20%, E24 about 10%, E96 about 2.4%) and re-analyses the rounded ladder, so you can see the shift in cutoff, stopband attenuation, delay and overshoot before you build anything. The passband of a doubly terminated ladder is insensitive to small errors, but the band edge and the ripple peaks of a high-order Chebyshev filter move noticeably.

• Inductors have finite Q, which rounds the passband edge and adds a little loss; Q above 50–100 at the cutoff is usually enough.
• Each inductor resonates with its winding capacitance at its self-resonant frequency (SRF); above the SRF it behaves as a capacitor, and the stopband attenuation can collapse. Choose inductors with an SRF well above the highest frequency you need to reject.
• Capacitors have series inductance and resonate too; at VHF and above, use small-case C0G/NP0 parts and short connections to ground.
• PCB pads and traces add capacitance and inductance of a few tenths of a picofarad and nanohenry. Above roughly 500 MHz to 1 GHz lumped components become impractical and distributed (stepped-impedance or stub) filters take over.

13. Accuracy and Validation

The element values are compared with the standard published tables below, and each ladder is re-analysed and compared with the ideal transfer function of its type from 0.05 to 20 times the cutoff. The table is computed live in your browser by the same code that designs your filter. Chebyshev values are shown normalized to the ripple band edge, as the tables print them; Bessel values are listed from the source end.

With ideal components the results are exact. In practice the accuracy of a built filter is set by component tolerances and parasitics; verify critical designs in a circuit simulator with vendor models, using the SPICE netlist or Touchstone file from step 2, and by measurement on a network analyser.