Multisection Quarter-Wave Transformer | Qucs-S S-parameter Viewer Documentation

Skip to content

# Multisection Quarter-Wave Transformer

## Description

The multisection quarter-wave transformer uses multiple cascaded λ/4 transmission line sections with optimized characteristic impedances to achieve broadband impedance matching between real impedances.

## When to Use

- Broadband matching required (20-80% bandwidth)

- Matching between real impedances only

- Space available for multiple sections

## Design Theory

Each λ/4 section acts as an impedance inverter. The cascade of N sections creates a distributed low-pass prototype filter, with impedance values determined by weighting functions.

### Reflection Coefficient

For an N-section transformer:

\[\Gamma(\theta) = e^{-j(N+1)\theta} \sum_{n=0}^{N} \Gamma_n \cos(n\theta)\]

where θ is the electrical length and Γₙ are the reflection coefficients at each junction.

## Weighting Methods

### Binomial (Maximally Flat)

Provides maximum bandwidth with flat passband response. No ripple in the passband.

Impedance Formula

\[\ln(Z_n) = \ln(Z_0) + \frac{C_{N-1}^{n-1}}{2^{N-1}} \ln\left(\frac{R_L}{Z_0}\right)\]

where $C_{N-1}^{n-1}$ is the binomial coefficient.

### Chebyshev (Equal Ripple)

Provides sharper cutoff with equal-ripple passband response. Allows trade-off between bandwidth and ripple level.

Impedance Formula

\[Z_n = Z_{n-1} \exp(\pm \gamma \omega_n)\]

where γ is the ripple parameter and ωₙ are the Chebyshev polynomial weights.

Ripple Parameter

\[\gamma = \frac{1}{2} \ln\left(\frac{R_L}{Z_0}\right) / \Gamma_{max}\]

## Parameters

Parameter | Description
--- | ---
Z0 | Source impedance (Ω)
RL | Load resistance (Ω, real only)
Frequency | Center frequency (Hz)
Sections (N) | Number of λ/4 sections (2-10)
Weighting | Binomial or Chebyshev
Ripple (Chebyshev) | Maximum reflection coefficient (0.001-1.0)
Implementation | Ideal TL or microstrip

## Microstrip Considerations

### Impedance Range

Practical microstrip impedances: 20-120Ω

- Below 20Ω: Very wide lines, high loss

- Above 120Ω: Very narrow lines, fabrication difficult

### Discontinuities

Steps between sections modeled automatically:

- Capacitance at width increase

- Inductance at width decrease

- Affects higher-frequency response

## Example

Match 100Ω to 50Ω at 1 GHz (3-section Binomial)

Input data

Parameter | Value
--- | ---
Z0 | 50Ω
RL | 100Ω
frequency | 1GHz
Weighting | Binomial

Results

Parameter | Value
--- | ---
Z₁ | 54.5Ω
Z₂ | 70.7Ω
Z₃ | 91.7Ω
λ/4 @ 1 GHz | 74.9 mm
Total length | 224.7 mm

Circuit topology:

    Port ── Z₁(λ/4) ── Z₂(λ/4) ── Z₃(λ/4) ── Load(100Ω)
             54.5Ω      70.7Ω      91.7Ω

## Reference

- Pozar, D. M. “Microwave Engineering”, 4th Edition, Wiley, 2012

## See Also

- RF Circuit Synthesis Tools