Quarter-Wavelength Impedance Matching Networks

January 1, 2023

Introduction

This article describes a loss-less impedance matching technique that does not require the use of discrete components but instead uses cables or printed circuit board (PCB) traces, i.e., distributed elements or transmission lines. It is a matching circuit built using a piece of transmission line whose length is equal to a quarter of the signal’s wavelength, hence the name “quarter-wavelength impedance matching transformer” or “quarter-wavelength impedance matching network.” But before jumping into specifics of the quarter-wavelength impedance transformers/networks, let us first discuss the use of discrete components (inductors, capacitors, etc.) and their limitations in matching networks.

What’s Wrong with Using Discrete Components in Matching Networks?

Discrete components, or as used in this context, lumped elements, include non-useful traits (parasitics) that become dominant contributors of errors at higher frequencies. Capacitors act like inductors past their self-resonant frequency (SRF) due to parasitic series inductance of their leads. Inductors act like capacitors past their SRF due to parasitic parallel capacitance of the windings. Transformers have similar issues that crop up as the frequency of interest increases. As the frequency increases, it becomes nearly impossible to design matching networks using discrete components and their associated interconnections that contain parasitic effects. It is impossible to make parasitics effects negligible using these devices. When using discrete components, the parasitics dominate the circuit’s behavior, so we must utilize other solutions. These other options utilize distributed (not lumped) elements; one such device is called the quarter-wavelength matching network or quarter-wavelength matching transformer.

What is a quarter-wavelength matching transformer?

The quarter-wavelength matching transformer or network works by transforming or inverting the impedance of the source and load it is connected to. It is a transmission line (distributed element) that has a specific characteristic impedance and allows matching source and load impedances of the line using the following equation:

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Note that this equation is only valid at the frequency where the transmission line length is equal to a quarter of the signal’s wavelength (or odd multiples of a quarter-wavelength, which we will discuss later).

Theory of Operation

Recall that a wave traveling through a transmission line with a different impedance than the connecting element will partially reflect at both ends. Suppose the line has a quarter of the wavelength. In that case, the first reflection will occur after a quarter-wavelength, so the wave will return to the start of the transmission line after another quarter-wavelength, for a total of half a wavelength – one quarter-wavelength down the line and one quarter-wavelength back. This condition means that the wave arriving at the beginning of the line will be inverted or 180 degrees phase shifted in reference to the incident wave. Based on this phase inversion and the reflection coefficients at the ends of the line, the final impedance equation is determined. Impedance transformation with transmission lines will occur with any line length and with any type of impedance, but the simple formula only works with quarter-wavelengths and real impedances.

Example

Imagine you have a 100 MHz source of 100 Ω driving a load of 25 Ω and want to construct a quarter-wavelength matching network that minimizes reflections and losses.

Using the equation provided, you determine the line impedance should be 50 Ω.

To find the correct line length, you first determine the wavelength (λ) using the following equation:

The correct line length that will provide quarter-wavelength (λ/4) impedance matching for this example is 3 m divided by 4 or 0.75 m.

This matching network will provide correct matching at 100 MHz and some other frequencies, i.e., 300 MHz, 500 MHz, 700 MHz, and so on, which are all odd multiples of the fundamental 100 MHz frequency. Proper matching also occurs at multiple odd quarter-wavelengths because the line needs to invert the signal that is passing through it, for a quarter-wavelength line. But if the line is a half-wavelength (2 times λ/4), then the reflected signal ends up being phase shifted by 360 degrees and is in phase with the incident signal at the beginning of the line resulting in no inversion. The inversion process occurs again for a line that is three-quarters of a wavelength long (3λ/4). Continuing with this thought process, we can put together multiple combinations of inverting (λ/4) and non-inverting (λ/2) segments that result in impedance matching if they are odd multiples of a quarter-wavelength (3λ/4, 5λ/4, 7λ/4, 9λ/4, etc.). Together they form a line that generates a phase inversion. For any given length of transmission line, you can determine the multiple frequencies where the phase inversion occurs and see that these are odd multiples of a quarter-wavelength.

Summary

This brief article described the limitations of discrete components in matching networks. It provided a solution through the use of quarter-wavelength impedance matching networks that work at a frequency that corresponds to a quarter-wavelength and at multiple odd quarter-wavelengths.

References