5-Section Distributed Low-Pass Filter Calculator (Stepped Impedance)

A stepped-impedance (hi-lo) low-pass filter is made directly from PCB microstrip with no SMD capacitors or inductors: wide low-impedance sections act as shunt capacitors and narrow high-impedance sections act as series inductors. Enter the system impedance, the two line impedances and the ripple to get the electrical length of all five sections, for both maximally flat and Chebyshev responses.

5 microstrip sections: wide–narrow–wide–narrow–wideMaximally flat / ChebyshevElectrical length in degreesNo SMD parts needed

① Input parameters

System impedance ZoΩ Low impedance Z1 (wide)Ω High impedance Z2 (narrow)Ω Chebyshev rippledB

Z1 is limited to 5–30 Ω and Z2 to 80–300 Ω; values outside are clamped to the limit. A lower Z1 and higher Z2 behave more like ideal L and C, but line widths are limited by the PCB process. The ripple only affects the Chebyshev column.

② Results (electrical length at fc)

Section	Impedance	Maximally flat	Chebyshev
θ1	Z1 (wide)	°	°
θ2	Z2 (narrow)	°	°
θ3	Z1 (wide)	°	°

θ is the electrical length at the cutoff frequency fc. The layout is symmetrical: θ1–θ2–θ3–θ2–θ1. Convert to physical length using your substrate and line widths (see the formulas below).

Microstrip layout

The wide (low-impedance Z1) sections have lengths θ1, θ3, θ1; both narrow (high-impedance Z2) sections are θ2. Both ends connect to Zo (usually 50 Ω) feed lines.

Equivalent lumped circuit

5th-order low-pass prototype (shunt C first): each wide section corresponds to a shunt capacitor g₁, g₃, g₅, and each narrow section to a series inductor g₂, g₄.

How to use

Enter the system impedance Zo (usually 50 Ω).
Choose the low impedance Z1 (wide line, 5–30 Ω) and high impedance Z2 (narrow line, 80–300 Ω) that your substrate and process can make; use the Microstrip Line Calculator to find the widths.
For a Chebyshev response enter the passband ripple (dB), then click Calculate to get θ1, θ2 and θ3.
Look up the effective permittivity ε_eff of the Z1 and Z2 lines with the microstrip calculator, convert the angles to physical lengths with the formula below and draw the PCB.

Theory and formulas

Start from the 5th-order low-pass prototype values. Maximally flat: g₁ = 0.618, g₂ = 1.618, g₃ = 2.000 (g₄ = g₂, g₅ = g₁). Chebyshev values depend on the ripple L_r:

β = ln[ coth( L_r / 17.37 ) ]　(L_r = ripple in dB)
γ = sinh( β / 10 )
a_k = sin( (2k − 1)π / 10 )　　b_k = γ² + sin²( kπ / 5 )
g₁ = 2a₁ / γ　　g_k = 4a_k−1a_k / ( b_k−1 · g_k−1 ), k = 2 … 5

A short low-impedance line behaves as a shunt capacitor and a short high-impedance line as a series inductor. The electrical lengths are:

Low-Z section (equivalent shunt C): θ_C = sin⁻¹( g_k · Z₁ / Z_o )
High-Z section (equivalent series L): θ_L = sin⁻¹( g_k · Z_o / Z₂ )
θ1 ← g₁ (= g₅)　θ2 ← g₂ (= g₄)　θ3 ← g₃　(θ in degrees at fc)

Converting angles to physical length

λ_g = c / ( f_c · √ε_eff )　　ℓ = ( θ / 360° ) · λ_g　(c = 3×10⁸ m/s)

For example, with θ = 30°, fc = 1 GHz and ε_eff = 3.3, λ_g ≈ 165 mm and ℓ ≈ 13.8 mm. Z1 and Z2 lines have different ε_eff, so convert them separately.

Design tips

A larger impedance ratio Z2/Z1 gives a closer approximation to ideal L and C and better stopband rejection; Z1 = 15–20 Ω with Z2 = 100–130 Ω is a common choice.
Keep each section shorter than about 45° (λ/8) for the lumped-element approximation to hold; if the angles come out too large, lower Z1 or raise Z2.
Distributed structures show spurious passbands above roughly 2–3 × fc because transmission lines are periodic in frequency; cascade a lumped low-pass if you need a wide stopband.
Fringing capacitance and step discontinuities usually pull the cutoff slightly lower; correct the lengths with EM simulation or a measured prototype.

FAQ

How does a distributed LPF differ from an LC LPF?: A distributed LPF is made from PCB traces, so it is repeatable, low-cost and well suited to GHz frequencies. An LC LPF is smaller and has a wider stopband, which suits lower frequencies.
Why is there no cutoff frequency input?: The results are electrical lengths (degrees) at the cutoff frequency and do not depend on frequency. You only need fc and the substrate ε_eff when converting to physical length.
What if some fields cannot be calculated?: It means g·Z1/Zo or g·Zo/Z2 is greater than 1, so the required capacitance or inductance cannot be realised with these impedances. Lower Z1, raise Z2 or reduce the Chebyshev ripple.

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