*This post authored by John Coonrod **originally appeared **on the **ROG Blog** hosted by **Microwave Journal**.*

Microwave circuit dimensions are related to their wavelengths / frequencies and to the **dielectric constant** (Dk) of their substrates. Quite simply, higher-frequency signals have smaller wavelengths and their electromagnetic (EM) energy of those smaller wavelengths will propagate through circuits with smaller dimensions. Phase velocity is related to wavelength, with slower EM waves having shorter wavelengths which propagate through circuit structures with smaller dimensions than faster waves with their longer wavelengths.

For a given frequency / wavelength, the circuit dimensions can also be reduced in size by fabricating the circuits on laminates with high dielectric constants, such as **RO3010**™ and **RT/duroid® 6010.2LM** circuit materials from Rogers Corp. The high Dk values of these circuit materials results in shorter wavelengths and smaller circuit dimensions, but also slows the propagation of the EM waves along the conductors of a printed circuit board (PCB).

Transmission-line conductors on PCBs for RF/microwave circuitry take on various forms, such as **microstrip**, **stripline**, **coplanar waveguide** (CPW), and even **substrate-integrated-waveguide** (SIW) transmission lines for circuits at millimeter-wave frequencies. For a conventional microstrip transmission line with solid conductors, for example, the phase velocity has relatively high speed. It is related to the current density, with most of the current running along the edges of the solid microstrip conductors in a circuit. If the solid conductors are modified into a conductor pattern that is not solid, such as a ladder pattern, the phase velocity will be slowed and the wavelength will be reduced.

The phase velocity is a function of the separation distance between the storage locations in a conductor for the EM energy’s electric and magnetic field components, resulting from capacitive loading (and storage capabilities) within a transmission line’s conductors. When the magnetic and electric energy storage locations are in close proximity, the phase velocity is higher or faster than when the magnetic and electric energy storage locations within the conductor are further apart. For the slower phase velocity, the wavelength of the transmission-line’s conductors is shorter than for the conductors with the faster phase velocity. Slowing the phase velocity of a microstrip transmission-line’s conductors is a way to miniaturize circuits fabricated on PCB materials with a given Dk value.

By modifying conventional microstrip conductors into unconventional patterns, such as a ladder configuration, distances between stored magnetic and electrical energy field components are increased beyond the separation distances for standard solid microstrip conductors. A microstrip ladder pattern effectively conducts EM energy, but with slower phase velocity than a solid microstrip transmission-line conductor because of its greater distance between magnetic and electric energy storage locations. Just how much slower the phase velocity is than a standard solid microstrip conductor will depend upon the geometry of the microstrip ladder pattern.

Such slow wave structures reduce the group velocity of a transmission line, effectively increasing its group delay compared to a conventional solid microstrip transmission line, and reducing its wavelength compared to the wavelength of a conventional microstrip transmission line. As an example, consider a half-wavelength gap-coupled resonator constructed from microstrip transmission lines. When fabricated with a standard solid conductor pattern on 20-mil-thick **RO4835**™ laminate from Rogers Corp., with typical Dk of 3.48 at 10 GHz in the z-direction of the material, the circuit structure will resonate at 4 GHz. When using the appropriate microstrip ladder conductor pattern on the same material, the resonant frequency of the gap-coupled resonator circuit can be reduced by 25%, to 3 GHz.

Coupled resonators are often used to realize various high-frequency filter functions, so this gap-coupled-resonator example with its microstrip ladder pattern of conductors can be applied to these types of filters to achieve a reduction in the size of the filter circuitry for a given design frequency. An even greater reduction in filter size can be gained by combining the ladder conductor pattern approach with high-Dk PCB material, such as **RO3010** ceramic-filled polytetrafluoroethylene (PTFE) laminate with a Dk of 10.2 measured in the z-direction of the material at 10 GHz and **RT/duroid 6010LM** ceramic-PTFE composite material with Dk of 10.2 in the z-direction at 10 GHz.

What is the impact on circuit size when using one of these high-Dk PCB materials? If a circuit material such as **RO3003**™ laminate with a Dk of 3.0 in the z-direction at 10 GHz is used as a starting point for a filter circuit with microstrip edge-coupled structure, including edge-coupled resonators, and the same filter is designed on a PCB material with much higher Dk, such as **RT/duroid 6010LM** laminate with its Dk of 10.2, a healthy reduction in size is possible. The size of the same microstrip edge-coupled filter function will be about 30% smaller on the higher-Dk material.

By combining slow-wave circuit patterns such as the microstrip ladder pattern with high-Dk circuit laminates such as the 10.2 Dk **RO3010** or **RT/duroid 6010LM** PCB materials, the circuit-shrinking effects of both approaches can be applied to an RF/microwave circuit design for a significant reduction in circuit size. Both of these low-loss circuit materials feature tightly controlled Dk values for excellent high-frequency performance and reduced circuit sizes.

Ladders are just one type of microstrip conductor pattern that can be used to create slow-wave structures, with many different patterns capable of achieving slow-wave propagation and contributing to a reduction of circuit size for a given design frequency and circuit laminate Dk value. The key to success for forming any slow-wave pattern (and shrinking the dimensions of a high-frequency circuit) lies in separating the electric and magnetic energy storage within the structure, thereby slowing the phase velocity and reducing the wavelength of the transmission line for a given circuit material and Dk value.

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