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