# Quarter-Wave Transformers: Theory and Equations

### Key Takeaways

• A quarter-wavelength transmission line equals the load's impedance in a quarter-wave transformer.

• Quarter-wave transformers target a particular frequency, and the length of the transformer is equal to λ0/4 only at this designed frequency.

• The disadvantage of a quarter-wave transformer is that impedance matching is only possible if the load impedance is real-valued. The quarter-wave transformer adds length to a line for the necessary impedance matching.

Impedance matching is equivalent to circuit performance in transmission line theory. When there is a mismatch between the characteristic impedance of the circuit and the real-valued resistance of the load, some of the signal energy reflects to the source and reduces efficiency. One of the simplest matching networks available is the quarter-wave transformer. The network doesn’t require any additional discrete components for impedance matching; instead, the length of the transmission line imparts the necessary impedance.

When to Use a Quarter-Wave Transformer for Impedance Matching

 1 Matching impedances between a resistive load and transmission lines. 2 Matching impedances between two resistive loads. 3 Matching impedances between two transmission lines of differing characteristic impedances.

## The Parameters of the Quarter-Wave Transformer

The characteristic impedance of the quarter-wave transformer is the geometric average of the line impedance and the load resistance. This value is essential because full-power delivery only occurs when the line impedance equals this value. At the characteristic impedance of the quarter-wave transformer, the bandwidth for the transformer is at its widest. Importantly, matching occurs at only a single frequency, necessitating careful frequency-control methods to ensure the quarter-wave transformer performs as expected. Fortunately, multiple quarter-wavelength transmission lines can combine to increase the bandwidth and reduce the vulnerability of the transformer design.

Understanding the impact of reflections on a non-matched transmission line helps characterize some of the drawbacks of the transformer. At the boundaries of differing impedance mediums, some energy associated with a signal can pass through unabated while the remainder reflects toward the signal source. This process will repeat at the impedance boundary for the reflected wave, leading to a second-level reflection (and a third, fourth, etc.). However, damping reduces the power of each successive wave reflection until the power eventually runs out and the reflection subsides. It’s worth noting that reflections are zero at these boundaries at steady-state conditions, i.e., a sufficiently long time after any circuit perturbations that would alter the general circuit performance.

One disadvantage of the quarter-wave transformer is that impedance matching is only possible if the load impedance is real. For impedance matching using a quarter-wave impedance transformer, the complex load impedance must be converted to a real load impedance using shunt reactive elements or an appropriate length of transmission line between load and quarter-wave impedance. However, this approach affects the load's frequency dependence and reduces the match's bandwidth.

## Quarter-Wave Impedance Transformer Equations

In RF and microwave circuits, transmission lines must transfer maximum power to the load; impedance matching techniques ensure maximum power transfer without reflections. Short-circuit transmission lines of tunable lengths in stub matching can also fulfill impedance matching requirements. A quarter-wave impedance transformer helps match the real load impedance of the transmission lines with a transmission line length equal to one-quarter of the guided wavelength. The characteristic impedance associated with quarter-wave impedance transformers differs, and it minimizes the energy reflections in the transmission lines connected to the load.

### A Single-Section Quarter-Wave Impedance Transformer This equation provides a single-section quarter-wave transformer’s characteristic impedance for the matching section. This transformer targets a particular frequency, and the length of the transformer is equal to λ0/4 only at this designed frequency. The length is different at other frequencies, and simultaneous impedance matching is impossible.

The following equation gives the input impedance of the combination of the quarter-wave impedance transformer and load: The reflection coefficient: The magnitude of the reflection coefficient uses the equation below. The reflection coefficient is zero only at the desired frequency, where θ=𝜋/2. For narrow-band impedance matching, a single-section quarter-wave transformer is an option. The quarter-wave impedance transformer can be employed in multisection designs to provide broader bandwidth. Optimum matching characteristics are achievable in the desired frequency band with multi-section quarter-wave transformers.