Bandpass Filter Calculator calculates Butterworth and Chebyshev LC values, prototype g-values, passband edges, and Q for precise RF and microwave filter circuit design.
From Lowpass Prototype g-Values to a Physical LC Bandpass Ladder
This tool is a lumped-element filter synthesis calculator, not an approximation or a rule of thumb. It runs the same two-step process used in filter design references going back to Matthaei, Young, and Jones: generate a normalized lowpass prototype from the filter order and ripple, then scale that prototype into real inductor and capacitor values at your target center frequency, bandwidth, and impedance.
Step one produces a set of dimensionless prototype coefficients, $g_1$ through $g_n$, assuming a 1-ohm source and load. For a maximally flat (Butterworth) response, each coefficient has a closed-form solution:
$$g_k = 2\sin\left(\frac{(2k-1)\pi}{2n}\right)$$
For an equal-ripple (Chebyshev) response, the coefficients come from a recursion instead, seeded by the ripple value in dB:
$$\beta = \ln\left[\coth\left(\frac{Ripple_{dB}}{17.372}\right)\right], \quad \gamma = \sinh\left(\frac{\beta}{2n}\right)$$
$$g_1 = \frac{2\sin\left(\frac{\pi}{2n}\right)}{\gamma}, \quad g_k = \frac{4\sin\left(\frac{(2k-3)\pi}{2n}\right)\sin\left(\frac{(2k-1)\pi}{2n}\right)}{b_{k-1}\,g_{k-1}}$$
Step two converts each $g_k$ into an inductor and capacitor pair, using bandwidth and center frequency to set the resonance and $Z_0$ to set the impedance scale. Odd-numbered elements in the ladder alternate with even-numbered ones between series and shunt LC branches:
$$L_{series} = \frac{g_k Z_0}{2\pi \cdot BW}, \quad C_{series} = \frac{BW}{2\pi f_c^2 \, g_k Z_0}$$
$$L_{shunt} = \frac{BW \cdot Z_0}{2\pi f_c^2 \, g_k}, \quad C_{shunt} = \frac{g_k}{2\pi \cdot BW \cdot Z_0}$$
Each series branch is an LC pair that resonates at $f_c$ and drops in series with the signal path; each shunt branch resonates the same way but sits in parallel, tapped to ground.
Synthesizing a 100 MHz, 20 MHz-Wide, 3rd-Order 0.1 dB Chebyshev Filter
Take a 3-pole filter centered at 100 MHz with a 20 MHz bandwidth, a 50-ohm system, and 0.1 dB of passband ripple. Ripple above the 0.0001 dB threshold means this runs the Chebyshev branch, not Butterworth.
First, $\beta = \ln[\coth(0.1/17.372)] = 5.157$, and $\gamma = \sinh(5.157/6) = 0.9694$. Working through the recursion gives three prototype coefficients: $g_1 = 1.032$, $g_2 = 1.147$, $g_3 = 1.032$ — symmetric, as odd-order Chebyshev prototypes always are.
Center frequency and bandwidth in Hz are $1\times10^8$ and $2\times10^7$. Q, the calculator’s headline number, is just center over bandwidth: $100/20 = 5.00$. Band edges fall at $100 – 10 = 90.00$ MHz and $100 + 10 = 110.00$ MHz.
| Element | Type 1: series | Type 1: shunt (dual) |
|---|---|---|
| L1, C1 (position 1) | 410.44 nH, 6.17 pF | 15.43 nH, 164.18 pF |
| L2, C2 (position 2) | 13.87 nH, 182.61 pF | 456.53 nH, 5.55 pF |
| L3, C3 (position 3) | 410.44 nH, 6.17 pF | 15.43 nH, 164.18 pF |
Position 1 uses $g_1$ : $$L_{series} = (1.032 \times 50)/(2\pi \times 2\times10^7) = 410.44$$ nH, paired with $$C_{series} = 2\times10^7/(2\pi \times (10^8)^2 \times 1.032 \times 50) = 6.17$$ pF. Position 2 uses $$g_2 = 1.147$$ the same way, but in the opposite branch type. The two columns aren’t two different filters — they’re the same electrical response built as dual networks, one starting with a series branch, the other starting with a shunt branch.
Why the Two Component Sets Aren’t Two Different Filters
Both ladders in the table above produce the same passband, the same ripple, and the same rolloff. What differs is only which element sits at position 1 — series or shunt — and every position after it alternates from there. Pick whichever set gives you rounder, more buildable component values, or whichever topology fits your board layout better; there’s no electrical penalty either way.
Q of 5.00 corresponds to a 20% fractional bandwidth (20 MHz over 100 MHz). That’s on the wide side for this style of synthesis. The series/shunt scaling used here assumes a narrowband approximation — the passband response stays close to symmetric around $f_c$ only as long as bandwidth stays small relative to center frequency.
Filter design references generally treat fractional bandwidths above roughly 20 to 30% as the point where this symmetric approximation starts to visibly skew, and a full bandpass transform (rather than this simplified series/shunt scaling) becomes worth the extra complexity.
What Actually Changes the Component Values
Ripple is the single biggest switch in this calculator, not a minor tuning knob. Anything above 0.0001 dB routes through the Chebyshev recursion instead of the Butterworth closed form, and that changes every $g_k$, which cascades into every L and C in the ladder.
Butterworth gives the flattest passband and the gentlest rolloff for a given order; Chebyshev trades a small amount of passband ripple for a steeper transition into the stopband, and more ripple buys a steeper transition still.
Filter order does what you’d expect — more poles means a sharper skirt — but it also means more components, each one now depending on tighter tolerances to hit the target response. Nine poles is this calculator’s ceiling; ask for a 10th and it stops rather than silently rounding down.
Impedance and bandwidth move values in opposite directions by design. Doubling $Z_0$ doubles every inductor and halves every capacitor, since impedance scaling is linear in L and inverse in C.
Narrowing the bandwidth for a fixed center frequency does the opposite of what intuition suggests for the shunt capacitors and series inductors — a narrower band pushes $Q$ up, and the calculator’s series inductors and shunt capacitors grow while the series capacitors and shunt inductors shrink, since bandwidth sits in the numerator for one pair and the denominator for the other.
One edge case worth knowing: at $n = 1$, the Chebyshev recursion never runs past $g_1$, since the loop for $k = 2$ onward requires $k \leq n$. A single-pole filter is just one series or shunt LC resonator, full stop, regardless of which ripple value is entered.
Bandpass Filter Design Questions
What’s the difference between a Butterworth and Chebyshev bandpass filter?
Butterworth is maximally flat in the passband, with no ripple but a gentler rolloff into the stopband for a given order. Chebyshev allows a specified amount of passband ripple in exchange for a steeper transition band at the same order, which is why it’s the common choice when selectivity matters more than flatness.
How many poles does a bandpass filter need?
It depends on how fast the response has to roll off past the band edges relative to how far the interfering signal sits from those edges. Each added pole steepens the skirt but adds another component whose tolerance directly affects the realized response, so most practical designs land between 3 and 7 poles rather than pushing toward the extremes.
What does Q mean in a bandpass filter calculation?
Here it’s simply center frequency divided by bandwidth — a 100 MHz filter with a 20 MHz passband has a Q of 5. It’s a shorthand for how selective the filter is relative to where it sits in the spectrum, not a measure of any single component’s quality factor.
How do you convert lowpass prototype values to bandpass component values?
Each normalized prototype coefficient becomes a resonant LC pair tuned to the center frequency, with bandwidth and impedance setting the actual inductance and capacitance. Series prototype elements become series LC branches; shunt prototype elements become shunt LC branches, alternating down the ladder.
Why do LC bandpass filters have two possible component sets for the same order?
Because the ladder can start with either a series or a shunt branch and still produce an electrically identical response — the two options are dual networks, not competing designs. Which one you build usually comes down to which set of values is easier to source or fits your layout better.
What ripple value should I use for a Chebyshev filter?
Lower ripple, like 0.01 to 0.1 dB, stays closer to a Butterworth-like flat passband while still sharpening the rolloff slightly. Higher ripple, in the 0.5 to 1 dB range, buys a noticeably steeper transition band at the cost of more passband amplitude variation, which matters more in applications sensitive to gain flatness.
How does characteristic impedance affect the filter’s component values?
Impedance scales inductors up and capacitors down by the same factor, since $L$ carries $Z_0$ directly and $C$ carries $1/Z_0$. A 50-ohm design and a 75-ohm design for the same frequency and bandwidth will have the same response shape but noticeably different-sized components.
What’s the difference between series and shunt resonators in a filter ladder?
A series resonator sits directly in the signal path and presents low impedance at resonance, passing the center frequency through with minimal loss. A shunt resonator taps to ground and presents high impedance at resonance, so it stays out of the way at the center frequency while shorting out energy farther from it.
This synthesis assumes ideal, lossless inductors and capacitors with no parasitic resistance, self-resonance, or tolerance error, and it treats every component as a true lumped element — an assumption that gets shakier as frequency climbs and physical component size starts to matter relative to the signal’s wavelength.