Slow-wave structures

Updated August 16, 2008

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Slow wave structures act to reduce the group velocity of a transmission line, or increase its group delay compared to a normal "fast-wave" structure.

Slow-wave structures are useful in shrinking distributed elements. In microwave engineering, cost is a linear function of circuit area, so any tricks we can employ to shrink designs are worth knowing about.

Let's start the topic by providing the following definition:

Slow-wave factor: the ratio of the wave velocity of a "native" transmission line to a slow-wave structure. The higher the slow-wave factor, the more the transmission line's wavelength is reduced compared to its "normal" wavelength in the chosen dielectric medium. By "normal" we are comparing it to CPW or microstrip or stripline.

Metal-insulator-semiconductor structures (MIS)

Silicon is usually a very poor insulator. By creating a microstrip line on top of silicon, with a dielectric layer such as silicon dioxide in between, a strange thing can happen. The electric field does not penetrate into the silicon (which tries to shorts it out) but the magnetic field does. The separation of electrical and magnetic energy slows the wave propagation. Substantial reduction in wavelength is possible. Here's the earliest reference to the topic that we could find:

IEEE Hasegawa and Furukawa, Slow-wave propagation along a microstrip line on Si-SiO2 system, IEEE Proceedings, 1979, pp 297-299.

GaAs can also be used to create a MIS slow-wave structure. In this case an epitaxial layer is grown on top of the wafer, because GaAs is such a good semi-insulating substrate. A layer of low-loss dielectric is added on top (silicon dioxide, silicon nitride, etc.) Here's the original reference on the subject:

Hasegawa and Okizaki, M.I.S. and Schottky Slow Wave Coplanar Striplines on GaAs Substrates, IEEE Electronics Letters, 27th October 1977, Vol.13, No 20.

Cross-tie slow-wave circuits

This class of circuits are sometimes called "artificial" transmission lines. Note that there are other implementations of slow-wave that are similar, all that is requires is a periodic structure of electrically short high-impedance and low-impedance lines.

The cross-tie slow-wave structure was first described by Seki and Hasegawa, in a paper titled Cross Tie Slow-Wave Coplanar Waveguide on Semi-Insulating GaAs Substrates published in Electronics Letters of the IEEE, 10 December 1981, Volume 17 No 25 [1]. It was further developed as a structure for IBM's silicon CMOS (or SiGe) circuits in the 2004 IEEE International Solid-State Circuits Conference paper titled On-Chip Interconnect for mm-Wave Applications Using an All-Copper Technology and Wavelength Reduction [2].

The cross-tie slow-wave structure replaces a continuous transmission line with an artificial transmission line made up of many electrically-small segments. The segment impedance alternates from Z1 to Z2 along the chain. One impedance is lower than Z0, the other is higher than Z0. The high impedance section can be created using a CPW structure with very wide gaps, the low impedance section is enabled by the cross-tie capacitance shunting from the center conductor to the dual grounds. We'll add a figure soon!

In order for the circuit to remain impedance matched, the impedances must obey the following according to reference [1]:

Z₁*Z₂=Z₀²

Thanks for the correction, Will! The above condition is valid only if E₁=E₂ (the strip lengths are equal). Thus there are four degrees of freedom you can play with.

The slow-wave factor "K" can be calculated as

(coming soon)

As a convention, Z₁<Z₀<Z₂ so that K is a number greater than unity. The plot (not shown) below shows the slow-wave factor as a function of Z₂/Z₁. Note that very high factors require extremely disparate line impedances; a slowing factor of 5 would require Z₁=~5 ohms, Z₂=~500 ohms (Z₂/Z₁=~100) for Z₀=50 ohms.

Another condition is that the length of the segments must be less than lambda/20 at the maximum frequency of operation.

Bragg frequency - what's that?

Coming soon