# 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:

```{math}
\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**

```{math}
\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**

```{math}
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**

```{math}
\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

```{toctree}
:maxdepth: 1
/RFCircuitSynthesis/index
/RFCircuitSynthesis/ImpedanceMatching/index
```