LC High-Pass Filter Calculator

Q: Why does a Bessel high-pass filter not have constant group delay?

The transformation s → ωc/s maps the low-pass prototype's flat delay near DC onto very high frequencies, and the delay of a high-pass filter is largest near and below its cutoff. The Bessel high-pass keeps the low-Q poles of its prototype, so it still rings the least, but it is not a delay-equalized filter. If you need constant delay across a high-pass band, add an all-pass equalizer.

Step 1

Specification

Set the −3 dB cutoff and the attenuation you need at a stopband frequency below it. Each filter type shows the lowest order that meets the spec; pick one to design it.

Requirements

−3 dB cutoff frequency

Stopband frequency (below the cutoff)

Attenuation at the stopband frequency

Source and load impedance

Filter type

Chebyshev passband ripple

Order

Topology

Chebyshev

Attenuation all three types at the orders above · dot = your spec

Step 2

Components

The high-pass ladder from step 1, scaled to your cutoff and impedance. The g column is the low-pass prototype value each part is transformed from. Standard values are the nearest parts in the chosen E-series; step 3 shows how much they move the response.

Schematic shows

Schematic

Component values Standard series

Part	Position	g	Ideal	Standard	Error

Paste into a spreadsheet or a bill of materials.

Export Values

AC analysis of the filter between its source and load resistors; S21 = 2·V(out)·√(R_S/R_L).

Show the netlist

Real/imaginary S-parameters, both ports referenced to R₀.

Show the Touchstone data

About this LC high-pass filter calculator

This free calculator from NextAntennas designs passive LC high-pass filters of the Butterworth, Chebyshev and Bessel types. You give it the −3 dB cutoff frequency, the attenuation you need at a stopband frequency below the cutoff and the system impedance; it finds the filter order, computes every series capacitor and shunt inductor, rounds them to standard E12, E24 or E96 parts and shows the insertion loss, return loss, group delay and step response of the actual ladder network. It works from audio frequencies to RF and microwave, for any source and load impedance such as 50 Ω or 75 Ω. Typical uses are rejecting AM and HF broadcast signals ahead of a VHF receiver, removing low-frequency noise and DC from a signal path, and blocking the lower bands in a diplexer.

How to use it

Specification: enter the −3 dB cutoff, a stopband frequency below it and the attenuation required there. The three filter types show the lowest order that meets the spec; choose one, set the Chebyshev ripple, and pick series-C-first or shunt-L-first topology. You can also force the order and see how far it is from the spec.
Components: a ladder schematic with the capacitor and inductor values, the low-pass prototype g-values they come from, the nearest E12, E24 or E96 parts and their rounding error, with a SPICE netlist and Touchstone (.s2p) S-parameters to copy into a circuit simulator.
Response: |S21| against your attenuation mask, return loss, group delay and step response, each with ideal and standard values overlaid, plus the shift in cutoff and stopband attenuation caused by rounding.

High-pass filter design equations

Each component is transformed from a normalized low-pass prototype g-value: C = 1/(2πf₋₃dB·g·R₀) for series capacitors and L = R₀/(2πf₋₃dB·g) for shunt inductors. The attenuation of the high-pass filter at f equals that of the prototype at f₋₃dB/f, so the Butterworth order is N ≥ log(10^A/10 − 1)/(2·log(f₋₃dB/f_s)); Chebyshev and Bessel orders are found by evaluating each order's exact attenuation. Chebyshev values are rescaled so that the −3 dB point, not the ripple band edge, falls at the frequency you enter.

Frequently asked questions

How do I calculate the components of an LC high-pass filter?

Start from a low-pass prototype g-value and apply the high-pass transformation: every series inductor becomes a series capacitor C = 1/(2π·f₋₃dB·g·R₀) and every shunt capacitor becomes a shunt inductor L = R₀/(2π·f₋₃dB·g). The calculator does this for Butterworth, Chebyshev and Bessel responses at any order, cutoff and impedance.

What order high-pass filter do I need?

The order is the number of capacitors and inductors in the ladder. Far below the cutoff each one adds about 6 dB per octave of attenuation. Enter the −3 dB cutoff, a stopband frequency below it and the attenuation you need there, and the calculator finds the lowest order of each type that meets it. For 40 dB at half the cutoff, a Butterworth needs 7 components and a 0.5 dB Chebyshev needs 5; a Bessel cannot reach it at any practical order.

Butterworth, Chebyshev or Bessel: which high-pass response should I use?

Butterworth has the flattest passband and is the usual default. Chebyshev accepts a small passband ripple in exchange for a steeper transition, so it meets a stopband requirement with fewer parts. A Bessel high-pass has the gentlest skirt and the least ringing of the three, but unlike a Bessel low-pass it does not have flat group delay in its passband, because the high-pass transformation inverts the frequency axis.

Why does a Bessel high-pass filter not have constant group delay?

The transformation s → ω_c/s maps the low-pass prototype's flat delay near DC onto very high frequencies, and the delay of a high-pass filter is largest near and below its cutoff. The Bessel high-pass keeps the low-Q poles of its prototype, so it still rings the least, but it is not a delay-equalized filter. If you need constant delay across a high-pass band, add an all-pass equalizer.

What passband ripple should I choose for a Chebyshev high-pass filter?

Ripple sets both the passband flatness and the worst-case return loss: 0.01 dB ripple gives 26.4 dB return loss, 0.1 dB gives 16.4 dB, 0.5 dB gives 9.6 dB and 1 dB gives 6.9 dB. For RF filters between 50 Ω stages, 0.01–0.1 dB is common because it keeps the match good. Larger ripple buys a steeper skirt and slightly fewer components.

Why can't an even-order Chebyshev high-pass filter be matched to 50 Ω at both ends?

An even-order Chebyshev response ends on a ripple trough at high frequency, where the capacitors are short circuits and the inductors open circuits. A lossless ladder there is just a wire, so the loss equal to the ripple can only come from a mismatch between source and load: the load must differ by coth²(β/4), about 1.98 for 0.5 dB ripple. Auto mode therefore uses odd orders; if you force an even order, the calculator reports the load it needs.

Series-C first or shunt-L first: which high-pass topology is better?

Both give the same response with ideal parts. Starting with a series capacitor (a T-section for odd orders) uses fewer inductors, which is usually preferred because capacitors are smaller, cheaper and have higher Q, and it blocks DC at both ports. Starting with a shunt inductor (a π-section for odd orders) gives the input a DC path to ground, which drains static charge from an antenna or protects a receiver input.

Can I use standard E12 or E24 component values?

Yes. Choose E12, E24 or E96 and step 3 overlays the response with the rounded values on the ideal one, with the shift in cutoff and stopband attenuation. Butterworth and Bessel filters tolerate rounding well; high-order Chebyshev filters with small ripple are the most sensitive and usually need E96 capacitors or a trimmed inductor.

Why does my high-pass filter lose performance at high frequencies?

An ideal high-pass filter passes everything above its cutoff, but real parts do not. Series capacitors have series inductance and become inductive above their self-resonant frequency, and shunt inductors have winding capacitance and become capacitive above theirs, so a real LC high-pass is a wide band-pass. Choose parts whose self-resonant frequency is well above the highest frequency you need to pass, and simulate with the exported SPICE netlist and vendor models.

How accurate is this filter calculator?

For ideal components the results are exact. The low-pass prototype values come from the closed-form Butterworth and Chebyshev equations and an 80-digit Darlington synthesis for Bessel, and agree with the tables of Matthaei and Zverev to every printed digit. Every plotted response is computed from the actual high-pass ladder, and the validation table on this page shows that it matches the ideal transfer function to better than 10⁻⁹ dB.

Is the calculator free?

Yes. It is free, needs no sign-up and runs entirely in your browser; nothing you enter is uploaded.

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Engineering Theory

Theory Behind the Equations

How the calculator turns a cutoff frequency and an attenuation requirement into capacitor and inductor values: the normalized low-pass prototype, the exact low-pass to high-pass transformation, the Butterworth, Chebyshev and Bessel approximations, impedance and frequency scaling, and the analysis of the finished network.

1. The Doubly Terminated High-Pass LC Ladder

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

Lossless two-port

{| S_{21} (j ω) |}^{2} + {| S_{11} (j ω) |}^{2} = 1

A_{dB} = - 10 \log_{10} {| S_{21} |}^{2}

A_dB = insertion loss (attenuation) · |S21|² = transducer power gain

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

High-pass design does not start from scratch. It starts from the same low-pass prototype used for low-pass filters, 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 element, a conductance if it is a series element.

All the prototype values in this calculator are normalized so that the g-values give a −3 dB point at exactly 1 rad/s, whatever the filter type. The g column in step 2 lists the prototype value each high-pass component is derived from.

3. The Low-Pass to High-Pass Transformation

Replacing the normalized frequency variable s by ω_c/s maps DC to infinity and infinity to DC, so the passband of the prototype becomes the band above ω_c = 2πf₋₃dB and its stopband the band below. Under this substitution a series inductor g, with impedance s·g, becomes ω_cg/s: the impedance of a series capacitor. A shunt capacitor g becomes a shunt inductor in the same way. Scaling all impedances by R₀ then gives every component directly:

High-pass transformation and denormalization

C_{k} = \frac{1}{2 π f_{- 3 dB} g_{k} R_{0}}

L_{k} = \frac{R_{0}}{2 π f_{- 3 dB} g_{k}}

R_{L} = g_{N + 1} R_{0} or \frac{R_{0}}{g_{N + 1}}

Series L (g) → series C · shunt C (g) → shunt L · example: g = 1 at 100 MHz and 50 Ω gives a 31.83 pF series capacitor or a 79.58 nH shunt inductor

Because the transformation is exact, the high-pass attenuation at any frequency equals the prototype attenuation at the reciprocal normalized frequency, A_HP(f) = A_LP(f₋₃dB/f). A stopband half an octave below the cutoff needs exactly the same order as a low-pass stopband half an octave above it. The two topologies are again duals: series-C-first and shunt-L-first ladders have the same response, and for an odd order a series-C-first T network needs (N−1)/2 inductors while a shunt-L-first π network needs (N+1)/2.

4. Butterworth: the Maximally Flat Response

The Butterworth approximation gives the flattest possible passband with no ripple; in high-pass form it is maximally flat at infinite frequency. The response is monotonic, its −3 dB point is the natural cutoff, and its prototype values have a closed form:

Butterworth high-pass response and prototype

{| S_{21} |}^{2} = \frac{1}{1 + {(f_{- 3 dB} / f)}^{2 N}}

g_{k} = 2 \sin \frac{(2 k - 1) π}{2 N}

k = 1 … N · g_N+1 = 1 (equal terminations for every order)

Setting the attenuation at the stopband frequency f_s equal to the requirement A_s and solving for N gives the order directly; the calculator rounds up to the next integer:

Butterworth order

N \geq \frac{\log_{10} (10^{A_{s} / 10} - 1)}{2 \log_{10} (f_{- 3 dB} / f_{s})}

40 dB at f_s = f₋₃dB/2 → N ≥ 6.64 → N = 7

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. In high-pass form the ripple lies above the ripple band edge f_p, and the response uses the Chebyshev polynomial T_N, which oscillates between ±1 inside |x| ≤ 1 and grows as cosh(N·arccosh x) outside it:

Chebyshev high-pass response

{| S_{21} |}^{2} = \frac{1}{1 + ε^{2} T_{N}^{2} (f_{p} / f)}

ε = \sqrt{10^{A_{r} / 10} - 1}

A_r = passband ripple in dB · f_p = ripple band edge, above the −3 dB cutoff

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

Chebyshev g-values

β = \ln (\coth \frac{A_{r}}{17.37})

γ = \sinh \frac{β}{2 N}

a_{k} = \sin \frac{(2 k - 1) π}{2 N}

b_{k} = γ^{2} + \sin^{2} \frac{k π}{N}

g_{1} = \frac{2 a_{1}}{γ}

g_{k} = \frac{4 a_{k - 1} a_{k}}{b_{k - 1} g_{k - 1}}

17.37 = 40/ln 10 (the calculator uses the exact constant) · k = 2 … N

6. Ripple Bandwidth Versus the −3 dB Cutoff

Published Chebyshev tables are normalized to the ripple band edge, not to the −3 dB point. In a high-pass filter the ripple edge f_p lies above the −3 dB cutoff, so using the tables directly with your −3 dB frequency puts the real cutoff too low. The −3 dB point lies where ε²T_N² = 1, which gives the ratio between the two edges:

Ripple edge for a given −3 dB cutoff

\frac{f_{p}}{f_{- 3 dB}} = \cosh (\frac{1}{N} arccosh \frac{1}{ε})

Prototype values from the ripple-normalized table are rescaled before the high-pass transformation

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 f_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

T_N(0) is 0 for odd N but ±1 for even N. An even-order Chebyshev high-pass response therefore ends at the bottom of a ripple, with A_r dB of loss at very high frequency. There the capacitors are short circuits and the inductors open circuits, so the ladder is just a wire and this loss can only come from a deliberate mismatch between the source and load resistances:

Load for even N

g_{N + 1} = \coth^{2} \frac{β}{4}

\frac{4 R_{S} R_{L}}{{(R_{S} + R_{L})}^{2}} = \frac{1}{1 + ε^{2}}

0.5 dB ripple → g_N+1 = 1.9841: a 50 Ω source needs a 99.2 Ω or 25.2 Ω load

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. The Bessel High-Pass Filter

The Bessel (Thomson) low-pass prototype makes the group delay as flat as possible at DC, from the reverse Bessel polynomial θ_N(s). There is no closed form for its ladder elements, so the calculator uses values found by Darlington synthesis at 80-digit precision for every order from 1 to 12, rescaled to a −3 dB point of 1 rad/s; they agree with Zverev's tables to every printed digit.

Bessel high-pass transfer function

H_{HP} (s) = \frac{θ_{N} (0)}{θ_{N} (ω_{0} / s)}

θ_{N} (s) = \sum_{k = 0}^{N} \frac{(2 N - k)!}{2^{N - k} k! (N - k)!} s^{k}

ω₀ is chosen so that the −3 dB point falls at f₋₃dB · θ₃(s) = s³ + 6s² + 15s + 15

The transformation keeps the magnitude shape, but it does not keep the flat delay. The prototype's flat delay near DC is mapped to very high frequencies, while the delay of the high-pass filter is largest near and below its cutoff and falls off well inside the passband. What the transformation does preserve is the angle, and therefore the Q, of every pole: the Bessel high-pass has the lowest-Q poles of the three types, and so the gentlest skirt and the least ringing. Use it when the waveform shape matters more than rejection, and add an all-pass equalizer if you need constant delay across the band.

9. Choosing the Filter Type and Order

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

Order N	Butterworth	Chebyshev 0.1 dB	Chebyshev 0.5 dB	Bessel
3	18.1 dB	21.5 dB	23.7 dB	12.0 dB
5	30.1 dB	41.1 dB	44.9 dB	14.1 dB
7	42.1 dB	62.3 dB	67.0 dB	14.0 dB
9	54.2 dB	84.2 dB	89.4 dB	13.4 dB

Chebyshev gains roughly 10 dB per added element close to the passband, Butterworth about 6 dB, and the Bessel attenuation an octave below the cutoff barely changes with order. As N grows, the Bessel response approaches a Gaussian in f₋₃dB/f, A(f) ≈ 3.01·(f₋₃dB/f)² dB, so at half the cutoff it saturates near 12–14 dB at any order. Far below the cutoff all three types fall 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 removing broadcast-band and low-frequency interference ahead of a receiver.
Chebyshev when components are expensive or the stopband is close to the passband, such as the high-pass half of a diplexer or a band-limiting filter in front of an amplifier.
Bessel for pulses and baseband signals with a low-frequency cut, where ringing matters more than rejection.

10. Group Delay and Step Response

The high-pass ladder has N transmission zeros at DC and N poles q_i = σ_i + jω_i, the reciprocals of the prototype poles scaled by ω_c. The zeros at DC add only a constant phase, so the group delay comes from the poles alone. The calculator finds the poles from the ladder itself (with ideal or standard values), and the delay and step response follow exactly:

Group delay and step response from the poles

H (s) = \frac{K s^{N}}{\prod_{i} (s - q_{i})}

τ_{g} (ω) = \sum_{i} \frac{- σ_{i}}{σ_{i}^{2} + {(ω - ω_{i})}^{2}}

y (t) = \sum_{i} \frac{Res [H, q_{i}]}{q_{i}} e^{q_{i} t}

Exact for any set of component values; no time stepping or FFT

A high-pass step response jumps to the passband gain and decays to zero. For N ≥ 2 the transfer function has at least a double zero at DC, so the area under the step response is zero and an undershoot below zero is unavoidable. A 3rd-order Butterworth undershoots by 30.9%, a 5th-order by 36.9%, and a 7th-order 0.1 dB Chebyshev by 38.7%; a Bessel undershoots least, by 24.9% for N = 3 and 31.7% for N = 5. The group delay of a high-pass filter is largest near and below the cutoff and falls roughly as 1/f² well inside the passband.

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:

Return loss at a ripple peak

RL = - 10 \log_{10} (1 - 10^{- A_{r} / 10})

0.01 dB → 26.4 dB · 0.1 dB → 16.4 dB · 0.5 dB → 9.6 dB · 1 dB → 6.9 dB

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 high-pass filters approach a perfect match at high frequency 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 undershoot 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.

An ideal high-pass filter passes everything above its cutoff; a real one does not. Every series capacitor has series inductance and turns inductive above its self-resonant frequency (SRF), and every shunt inductor has winding capacitance and turns capacitive above its own SRF. A real LC high-pass is therefore a wide band-pass; choose parts whose SRF is well above the highest frequency you need to pass.
Inductors have finite Q, which rounds the passband edge and adds a little loss near the cutoff; Q above 50–100 at the cutoff is usually enough.
Series capacitors carry the full signal current; at VHF and above use small-case C0G/NP0 parts, and check the voltage rating in transmitter filters.
The shunt-L-first topology gives a DC path to ground at the input, which drains static charge from an antenna; the series-C-first topology blocks DC at both ports.
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 (stub) filters take over.

13. Accuracy and Validation

The prototype values are compared with the standard published tables below, and each high-pass ladder is re-analysed and compared with the ideal high-pass 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.

Type	N	Computed prototype g₁ … g_N	Max \|Δg\| vs table	Max high-pass response error	Reference
Computing…

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.

References

Matthaei, G.L., Young, L. & Jones, E.M.T., Microwave Filters, Impedance-Matching Networks, and Coupling Structures, McGraw-Hill, 1964; Artech House reprint, 1980, ch. 4 and 7.
Zverev, A.I., Handbook of Filter Synthesis, Wiley, 1967.
Williams, A.B. & Taylor, F.J., Electronic Filter Design Handbook, 4th ed., McGraw-Hill, 2006.
Pozar, D.M., Microwave Engineering, 4th ed., Wiley, 2011, ch. 8.
Hunter, I.C., Theory and Design of Microwave Filters, IET, 2001.
Darlington, S., “Synthesis of Reactance 4-Poles Which Produce Prescribed Insertion Loss Characteristics,” Journal of Mathematics and Physics, vol. 18, 1939.
Thomson, W.E., “Delay Networks Having Maximally Flat Frequency Characteristics,” Proc. IEE, vol. 96, pt. III, 1949.
Weinberg, L., Network Analysis and Synthesis, McGraw-Hill, 1962.
Orchard, H.J., “Inductorless Filters,” Electronics Letters, vol. 2, no. 6, 1966.
Bowick, C., RF Circuit Design, 2nd ed., Newnes, 2008, ch. 3.