Stepped Impedance Filter | Qucs-S S-parameter Viewer Documentation

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# Stepped Impedance Filter

## Overview

Lowpass filter topology using alternating high and low impedance transmission line sections to emulate series inductors and shunt capacitors. Compact implementation suitable for microstrip realization.

## Principle

Series inductors are replaced by high-impedance (narrow) TL sections, and shunt capacitors are replaced by low-impedance (wide) TL sections. The electrical lengths are chosen such that the TL sections approximate lumped elements at the design frequency.

## Design Equations

### Impedance Assignment

For a CLC topology (shunt-first):

- Shunt capacitor → Low-impedance line

- Series inductor → High-impedance line

For an LCL topology (series-first):

- Series inductor → High-impedance line

- Shunt capacitor → Low-impedance line

### Line Length Calculation

For low-impedance sections (capacitor replacement):

    l = (gₖ × Z_{low}) / (β × Z₀)

For high-impedance sections (inductor replacement):

    l = (gₖ × Z₀) / (β × Z_{high})

Where:

Parameter | Description
--- | ---
gₖ | normalized lowpass prototype coefficient
β = 2πfc / c | propagation constant at cutoff frequency
Z₀ | System impedance
c | speed of light

## Input Parameters

Parameter | Range | Default | Description
--- | --- | --- | ---
Minimum Z | 1 – 40 Ω | 10 Ω | Low-impedance line (Z_{low})
Maximum Z | 50 – 400 Ω | 200 Ω | High-impedance line (Z_{high})
Cutoff freq (fc) | — | 1 GHz | Lowpass cutoff
Order (N) | 2 – 10 | 3 | Number of sections

## References

Pozar, D. M. (2012). Microwave Engineering (4th ed.), Section 8.5, pp. 424–426. Wiley.

## See Also

- RF Circuit Synthesis Tools
- RF Filter Design Tool