Designing with lumped elements at microwave and millimeter-wave frequencies can be a challenge. This is due to the fact that at higher frequencies the size of lumped element components becomes an appreciable portion of a wavelength. This means that distributed effects begin to become important. This article discusses some of the issues with designing and modeling lumped elements at microwave and millimeter-wave frequencies.

The issue with lumped elements is that above a certain frequency, lumped elements no longer perform electrically like lumped elements. The frequency at which a particular inductor, capacitor or resistor begins to deviate from the ideal or desired frequency response depends upon the type of lumped element, its size/shape, material, manufacturing technique and manufacturer.

*Figure 1. A comparison of measured data for a lumped inductor, and ideal inductor and electromagnetic simulation results for an embedded spiral inductor.*

As a result, at microwave and millimeter-wave frequencies, lumped elements are often integrated or “embedded” as part of the packaging. The performance benefit of integration is that the lumped elements can be designed to perform to higher frequencies. Also, integration usually leads to smaller size and, often, lower cost.

Figure 1 compares an ideal lumped element inductor with measured data for an inductor and electromagnetic simulation results for an embedded spiral inductor. Notice how the ideal lumped inductor insertion loss (S21) increases smoothly as a function of frequency. This is exactly what you would expect from a series inductive element. However, the measured data for the lumped inductor shows that the insertion loss begins to deviate from ideal at about 0.5 GHz. Observe from the graph how the behavior of the lumped inductor not only deviates from the ideal inductor slope, it also has a resonance at about 1.8 GHz. This type of response means that the lumped inductor is not very useful above about 0.5 GHz. The figure also shows the electromagnetic simulation results for an embedded spiral inductor. It follows the ideal inductor performance to about 1GHz or about double the frequency of the lumped element. Though the example is for an inductor, most capacitors and resistors embedded elements perform over a wider frequency range than their lumped equivalent value components.

Embedded lumped elements can be realized using most of the available fabrication technologies available including low temperature co-fired ceramic, high temperature co-fired ceramic, thick film, thin film and laminates.

*Figure 2. Spiral inductor model used in the electromagnetic modeling.*

The gigure shows the model used in the electromagnetic simulations of the embedded inductor. The spiral inductor is formed by the metal pattern on the dielectric substrate. The bandwidth of the spiral inductor is limited by the stray capacitance to ground plane and capacitance between the windings.