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Butterworth Low-Pass Ladder Synthesis

Synthesize nominal order-2–5 low-pass ladder values from a cutoff frequency or stopband attenuation target using an equally terminated Butterworth prototype.

FORMULAv1.1.0
INPUT PARAMETERS

Select an Ideal Low-Pass Prototype and Design Target

Synthesizes nominal order-2–5 Butterworth low-pass ladder values under equal source/load resistance; no parasitic values are filled in automatically.

CALCULATION RESULT

Calculation result

Calculated locally
∿

Enter the parameters and run the calculation to see results and model assumptions here.

FORMULA & TRACEABILITY

Formulas and applicability

Formulas, variables, and model boundaries are published. References last reviewed on 2026-08-30.

4 formulas · 3 references
CORE EQUATIONS

Core equations

These equations directly produce the primary results and define the tool’s core model.

2
F01

Order-n Butterworth Low-Pass Magnitude Response

|H(jΩ)|²=1/(1+Ω²ⁿ)，A=10log₁₀(1+Ω²ⁿ)，Ω=f/fc

A maximally flat low-pass response normalized to cutoff frequency. At Ω=1, the power ratio is 1/2 and insertion loss is 3.0103 dB.

H
Normalized voltage transfer of the lossless, equally terminated prototype1
Ω
Normalized frequency f/fc1
n
Filter order1
A
Ideal insertion lossdB
Applicability
  • Lossless lumped-element Butterworth prototype
  • Source and load resistance are equal
F02

Normalized Butterworth Ladder Prototype Coefficients

gₖ=2sin[(2k−1)π/(2n)]，k=1…n

Generates a Butterworth low-pass ladder prototype with unit cutoff angular frequency and unit termination resistance; order 2 is [√2,√2] and order 3 is [1,2,1].

gₖ
Normalized value of passive element k1
k
Element index starting from the source1
n
Prototype order1
Applicability
  • Equal resistive terminations at both ports
  • This tool publishes only order-2–5 low-pass ladders; elements alternate from the selected source-end first element
DERIVED & CONVERSION

Derived and conversion equations

Derive units, levels, and supporting engineering quantities from the core values.

1
F03

Frequency and Impedance Scaling

L_series=R₀gₖ/(2πfc)，C_shunt=gₖ/(R₀2πfc)

Scales the normalized prototype to termination resistance R₀ and cutoff frequency fc; the result contains ideal nominal component values only.

R₀
Equal source and load resistanceΩ
fc
Butterworth cutoff frequencyHz
L_series
Nominal series inductanceH
C_shunt
Nominal shunt capacitanceF
Applicability
  • Both source and load are terminated in R₀
  • Not valid for arbitrary unequal-termination synthesis
BOUNDARIES & RULES

Boundary and rule equations

Check model applicability, measurement conditions, and regulatory rules.

1
F04

Cutoff Frequency from a Stopband Attenuation Target

fc=fs/[10^(A/10)−1]^(1/(2n))

An algebraic inversion of the Butterworth magnitude response; target A must exceed 3.0103 dB so fs lies above cutoff.

fs
Target stopband frequencyHz
A
Ideal target attenuation at that frequencydB
fc
Synthesized cutoff frequencyHz
Applicability
  • The target describes an ideal Butterworth prototype
  • The real circuit requires separate parasitic and termination verification
REFERENCES

References

01Texas Instruments / National Semiconductor · Application Report AN-779 (SNOA224A)Basic Introduction to Filters—Active, Passive, and Switched-Capacitor2010 revision · Provides the maximally flat Butterworth response, normalized-frequency definition, cutoff point, and second/third-order polynomials.
02Keysight Technologies · Genesys 2010.05 DocumentationSynthesis—Filter Synthesis2010 · Documents Butterworth low-pass prototypes, G-value tables, and passive ladder synthesis by termination impedance and cutoff frequency.
03Analog Devices · Application Note AN-1098AN-1098: Low-Cost Video Filter Using the ADA4430-1Official application material · Shows frequency and impedance scaling from a normalized passive filter prototype to a target cutoff and load impedance.
Engineering use notice

Results use the models and assumptions shown on this page for design estimates and pre-compliance risk review. Complex structures, dispersion, near-field coupling, and test setup can cause significant deviation.

FAQ

Frequently asked questions

How is the Butterworth order back-calculated?

From the stopband requirement (attenuation A_s at f_s): n ≥ log₁₀(10^(A_s/10) − 1) / (2·log₁₀(f_s / f_c)), rounded up.

How are normalized g-values used?

Look up the Butterworth normalized element table (e.g., 3rd order: 1.0 / 2.0 / 1.0), then denormalize by cutoff frequency and terminating impedance to get actual L and C values.

Butterworth or Chebyshev?

Butterworth offers maximally flat passband and smoother phase; Chebyshev rolls off faster with passband ripple. Mains EMI filters commonly use Butterworth for a flat passband.

PRE-COMPLIANCE SUPPORT

Need accredited testing or pre-compliance support?

These tools support design-stage estimates. For accredited EMC testing, compliance decisions, and troubleshooting, contact us.

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