# The Difference Between Lumped and Distributed Elements in Microwave Circuits

### Key Takeaways

• Lumped and distributed elements are important in microwave circuits, and their properties are related to the operating frequency or wavelength of the system.

• When compared to the operating wavelength, the main difference between lumped and distributed elements is their size.

• In microwave circuits, transmission lines can be considered lumped elements or distributed elements based on their dimensions. Microwave integrated circuit (MIC) technology used to use distributed elements, which limited the ability to reduce the size of MICs. However, today, the use of lumped elements makes it possible to reduce the size of MICs at the lower end of the microwave frequency band. When compared to the operating wavelength, the main difference between lumped and distributed elements is their size. Let’s take a closer look at lumped and distributed elements to better understand their differences.

## Generalizing the Difference Between Lumped and Distributed Elements

Lumped and distributed elements are important in microwave circuits, and their properties are related to the operating frequency or wavelength of the system.

### Lumped Elements

If the size of an element is smaller than the wavelength of the applied signals, then it is a lumped element. In lumped elements, the effect of wave propagation can be neglected. The physical dimensions of lumped elements make it so that signals do not vary over the interconnects interfacing them. There are only minimal phase differences between the input and output signals in lumped elements.

Generally, the size of lumped elements is less than 1/20 times the operating guided wavelength. This approximation is considered for lumped element designs at RF and microwave frequencies. Lumped inductor transformers and baluns are commonly used in RF and microwave applications. Other examples of lumped elements include:

### Distributed Elements

The physical dimension of distributed elements is comparable with the operating wavelength. They are distributed over lengths in an RF or microwave circuit.

When conventional lumped elements are difficult to implement at microwave frequencies, distributed-elements are used instead. They perform the same functions as lumped elements, but the signals vary along the lines and between the elements. These signals undergo considerable phase change across various points within the distributed elements. Signal transit time cannot be neglected in distributed elements.

Distributed element circuits can be built by forming the medium itself into specific patterns. The significant advantage of distributed element circuits is that they can be manufactured at a lower cost. Distributed elements can be found in printed circuit boards and coaxial and waveguide formats. Other examples of distributed elements used in microwave circuits such as filters, power dividers, and circulators include:

• Stubs

• Coupled lines

## Describing Lumped and Distributed Elements Mathematically

The difference between lumped and distributed elements can be described mathematically. In lumped elements such as resistors, capacitors, and inductors, physical quantities, such as voltage or current, are functions of time. The quantities are considered to be concentrated at singular points in the space. There is no variation in the quantities over the length of the lumped element. The mathematical equations governing lumped elements are ordinary differential equations.

In distributed elements, the physical quantities associated with the component are distributed in space. Quantities such as voltage and current are dependent on time and space in distributed elements. Therefore, it is essential to describe the physical quantities associated with a distributed element as the partial derivatives of time and length. Partial differential equations are used for describing distributed element quantities.

## Transmission Lines as Lumped and Distributed Elements

In microwave circuits, transmission lines can be considered lumped elements or distributed elements based on their dimensions. A transmission line is only considered a distributed element after it crosses a certain length. When the line length is less than this specific value, it is considered a singular lumped element.

Lumped element values are dependent on the geometry of the transmission line and the material with which it is made of. The transmission line can be modeled as series inductance, series resistance, shunt conductance, and shunt capacitance values. This combination of R, L, G, and C form the lumped element model of the transmission line.

Understanding the difference between lumped and distributed elements is important when working on transmission lines and filters in microwave circuits. Cadence’s software offers modeling and analysis of transmission lines and filters in RF or microwave circuits.