LC Band-Stop & Notch Filter Calculator

1. The Doubly Terminated LC Band-Stop Ladder

A lumped band-stop filter is an LC ladder of traps: parallel LC branches in the signal path alternating with series LC branches to ground, driven from a source resistance R_S and loaded by R_L. Every trap is tuned to the notch frequency f₀. There, the parallel branches in the line are open circuits and the series branches to ground are short circuits, so nothing reaches the load. Far below f₀ the in-line inductors and the capacitors to ground take over, far above it the in-line capacitors and the inductors to ground, and in both limits the ladder becomes a plain wire from source to load. With N traps (2N components) the response is an N-th order filter, so N is the order.

The ladder is lossless, so every bit of power that does not reach the load is reflected back to the source: inside the stopband a band-stop filter reflects the unwanted signal rather than absorbing it. Its performance is 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 insensitive to component tolerances in the passbands (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 band-stop design starts from a low-pass 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 prototype topologies are duals with the same g-values and exactly the same response. In the band-stop filter they become shunt-trap-first and series-trap-first ladders. All the values in this calculator are normalized so that the prototype's −3 dB point is at exactly 1 rad/s, whatever the filter type, which puts the band-stop −3 dB points exactly on f₁ and f₂.

3. The Low-Pass to Band-Stop Transformation

The band-stop transformation is the reciprocal of the band-pass one. It maps the prototype's passband around DC onto the two passbands of the band-stop filter, below f₁ and above f₂, its cutoff points ±1 rad/s onto the band edges, and the prototype's infinite frequency, where its attenuation is greatest, onto the notch f₀:

Under this substitution a series inductor g becomes the impedance of a parallel LC resonator and a shunt capacitor becomes a series LC resonator, each tuned to f₀. Because the transformation is exact, every property of the prototype carries over: ripple, return loss, equal or unequal terminations, and the attenuation at any frequency. A rejection edge at f_s inside the stopband maps to Ω_s = |Ω(f_s)| > 1, and since the attenuation keeps rising from there towards the notch, meeting the spec at the edge meets it across the whole rejection band. With edges on both sides, the calculator takes the larger of the two orders.

4. Impedance Scaling and the Trap Values

Multiplying all impedances by R₀ moves the terminations to R₀, and the transformation above sets the frequencies. Each prototype element then becomes one trap tuned to f₀:

These are the band-pass values with the two arms swapped: the in-line inductors scale as BW and the inductors to ground as 1/BW, so the ratio between them is about 1/FBW². That spread is what limits narrow multi-resonator band-stop ladders in practice; section 13 returns to it.

5. Geometric Symmetry and the Band Edges

Since Ω(f) depends on f/f₀ − f₀/f, two frequencies with the same ratio to f₀, such as f₀/k and k·f₀, have the same attenuation. The response is symmetrical on a logarithmic frequency axis around the geometric centre √(f₁f₂), which always lies below the arithmetic mean of the band edges. On a linear axis the upper wall of the notch is the steeper one: for a 3rd-order Butterworth band-stop centred at 100 MHz with a 20 MHz −3 dB width, 95 MHz is 17.5 dB down but 105 MHz is 18.8 dB down.

When you enter a notch frequency and width, the calculator treats the notch as the geometric centre and places the edges at f₁ = (√(BW² + 4f₀²) − BW)/2 and f₂ = f₁ + BW. When you enter the edges, it computes f₀ and BW from them. Either way the −3 dB points of the designed filter fall exactly on f₁ and f₂.

6. Butterworth: the Maximally Flat Response

The Butterworth approximation makes the first 2N−1 derivatives of the prototype's |S21|² zero at DC, which in the band-stop filter means both passbands are as flat as possible far from the notch, with no ripple. The attenuation rises monotonically towards f₀ from both sides, and the element values have a closed form:

Setting the attenuation at Ω_s equal to the requirement A_s and solving for N gives the order directly; the calculator rounds up to the next integer and repeats this for each rejection edge:

7. 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 walls any all-pole filter of the same order can achieve. In a band-stop filter the ripple appears in both passbands, increasingly closely spaced as they approach the stopband. 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:

8. Ripple Edges Versus the −3 dB Stopband

Published Chebyshev tables are normalized to the ripple band edge, the last frequency where the loss equals A_r, not to the −3 dB point. In a band-stop filter the ripple edges lie outside the −3 dB edges, further from the notch. Used directly with a −3 dB width, ripple-normalized values make the real stopband too narrow; for a 0.1 dB, 3rd-order filter the −3 dB stopband comes out 28% narrower than intended. The −3 dB point of the prototype lies where ε²T_N² = 1, which gives the ratio between the two widths:

The calculator applies this correction for every order and ripple, so the −3 dB edges of the designed filter land exactly on f₁ and f₂, and step 1 reports the ripple edges separately: below the lower one and above the upper one, the passband loss stays within the ripple. Because the ratio 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.

9. Even-Order Chebyshev Filters and Unequal Terminations

At Ω = 0, T_N(0) is 0 for odd N but ±1 for even N, so an even-order Chebyshev response has A_r dB of loss there. In a band-stop filter Ω = 0 is DC and infinite frequency, far out in both passbands, where the trap ladder is a plain wire from source to load. That 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.

10. Bessel–Thomson: Maximally Flat Group Delay

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

There is no closed form for the Bessel ladder elements, so they are found by Darlington synthesis: the reflection polynomial is obtained by spectral factorization of θ_N(s)θ_N(−s) − θ_N(0)², and the input impedance is expanded as a continued fraction that peels off one element at a time. That expansion is numerically ill-conditioned in double precision, so the element values were synthesized at 80 significant digits for every order from 1 to 12 and stored; they agree with Zverev's tables to every printed digit.

After the band-stop transformation the flat delay of the prototype belongs to the far passbands, where the delay is small anyway; close to the stopband edges the transformation stretches the frequency axis and the delay rises on both sides. A Bessel band-stop filter still has the smoothest delay and the least ringing of the three types, but its walls are so gentle that it rarely meets a tight rejection spec: at Ω_s = 2 its attenuation never exceeds about 14 dB whatever the order. Step 1 marks it as not reachable in that case.

11. Choosing the Filter Type and Order

The three types trade selectivity against time-domain behaviour. At a rejection edge where Ω_s = 2, which for a narrow notch is where the rejection band is about half as wide as the −3 dB stopband, they give these attenuations:

Chebyshev gains roughly 10 dB per added trap at the rejection edge, Butterworth about 6 dB, and the Bessel attenuation there barely changes with order. All three types reach infinite attenuation at f₀ with ideal parts; they differ in how wide the deep part of the notch is.

• Single trap (N = 1) for one narrow interferer: a mains-hum notch, a pilot tone, a single transmitter near a receiver. Its depth is set by the inductor Q (section 13).
• Butterworth for flat passbands and a moderately wide rejection band, the default for broadcast-band rejection filters.
• Chebyshev when wanted signals sit close to the band you are removing, or components are expensive.
• Bessel only when passband pulse fidelity matters more than rejection.

12. Group Delay and the Interferer Transient

The group delay is the derivative of the transmission phase, computed by the calculator from the actual ladder (with ideal or standard values) at every frequency. In a band-stop filter it is largest just outside the −3 dB edges and falls away in the far passbands; inside the stopband the signal is rejected and the delay has no practical meaning, so step 3 plots it only in the passbands.

The interferer transient answers the question that matters for a notch: when an unwanted carrier at f₀ switches on, how long does it take the filter to remove it? The leading edge of the carrier contains energy at all frequencies and passes at full amplitude; the leakage then dies away as the traps fill with energy. Each prototype pole p_k becomes a pair of band-stop poles next to ±j2πf₀, and the slowest one sets the decay:

Narrow notches are therefore slow: halving the width doubles the time an interferer leaks through. Higher-order Chebyshev filters have poles close to the jω axis and ring longest; for the 80–120 MHz FM rejection example, the 5th-order 0.5 dB Chebyshev takes about 215 ns to reach −40 dB, the 8th-order Butterworth about 184 ns. The calculator obtains the transient by simulating the actual ladder in the time domain (trapezoidal integration, pre-warped so that the notch sits exactly on the carrier), so it includes the effect of standard-value rounding: a detuned ladder settles to a residual leakage floor instead of zero.

13. Notch Depth, Inductor Q, Standard Values and Real Components

With lossless parts the notch is infinitely deep, but every real trap has loss. For a single trap between equal terminations, an inductor of unloaded Q limits the depth to:

So a narrow notch needs a high-Q resonator, often a helical, coaxial or crystal resonator rather than a chip inductor. Enter the inductor Q in step 1 and the calculator models every inductor with a series loss resistance 2πf·L/Q, then reports the notch depth and the rejection at your edges with lossy parts. Multi-resonator filters multiply their depth, but beyond 60–80 dB stray coupling from input to output around the filter usually sets the floor.

The calculator also 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%), shows how far each rounded trap is tuned from f₀, and re-analyses the rounded ladder. Detuned traps null at slightly different frequencies, so the single deep null splits into one null per trap with shallower dips between them, and the attenuation at f₀ itself becomes finite; step 3 reports the worst rejection across your rejection band and the attenuation left at f₀. A trap detuned by a fraction δ moves its null by roughly δ/FBW of the stopband width, so narrow notches are far more sensitive than wide ones.

• The ratio of the largest to the smallest inductor grows as about 1/FBW². Below roughly 5% fractional width, the inductors to ground become very large and their capacitors very small, comparable to stray capacitance, while the in-line traps need tiny inductors and large capacitors. One or two high-Q traps, or a quarter-wave stub, are then usually the better choice.
• Each inductor resonates with its winding capacitance at its self-resonant frequency (SRF), and capacitors resonate with their series inductance. A trap to ground built from a capacitor above its own resonance no longer behaves as designed, and the upper passband can develop spurious notches. Choose parts whose resonances lie well above the highest frequency you need to pass.
• PCB pads and traces add a few tenths of a picofarad and nanohenry. At VHF and above, use small-case C0G/NP0 capacitors, short ground connections, keep input and output apart, and include pad parasitics in a circuit simulator using the exported netlist.

14. 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 passband 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. Inside the stopband the return loss falls to 0 dB, because the filter reflects the rejected signal back to the source; make sure the stage driving it tolerates that reflection, or put an attenuator or isolator in front.

15. Accuracy and Validation

The prototype element values are compared with the standard published tables below. Each prototype is then transformed into band-stop ladders of 1%, 10% and 100% fractional width, which are re-analysed and compared with the transformed ideal response at 80 frequencies across both passbands and the stopband (|Ω| from 0.05 to 4). 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. The time-domain simulation reproduces the closed-form transient of a single trap, and the −40 dB suppression time of a 1% notch agrees with ln 100/(π·BW) to within 0.3%.

With ideal components the results are exact. In practice the accuracy of a built filter is set by component tolerances, finite Q 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.