# Microstrip

David Jefferies

Transmission line notes.
Waveguides and Cavity Resonators

## What is microstrip transmission line?

Microstrip transmission line is a kind of "high grade" printed circuit construction, consisting of a track of copper or other conductor on an insulating substrate. There is a "backplane" on the other side of the insulating substrate, formed from similar conductor. A picture (37kB).

Looked at end on, there is a "hot" conductor which is the track on the top, and a "return" conductor which is the backplane on the bottom. Microstrip is therefore a variant of 2-wire transmission line.

If one solves the electromagnetic equations to find the field distributions, one finds very nearly a completely TEM (transverse electromagnetic) pattern. This means that there are only a few regions in which there is a component of electric or magnetic field in the direction of wave propagation. There is a picture of these field patterns (incomplete) in T C Edwards "Foundations for Microstrip Circuit Design" edition 2 page 45. See the booklist for further bibliographic details.

The field pattern is commonly referred to as a Quasi TEM pattern. Under some conditions one has to take account of the effects due to longitudinal fields. An example is geometrical dispersion, where different wave frequencies travel at different phase velocities, and the group and phase velocities are different.

The quasi TEM pattern arises because of the interface between the dielectric substrate and the surrounding air. The electric field lines have a discontinuity in direction at the interface. The boundary conditions for electric field are that the normal component (ie the component at right angles to the surface) of the electric field times the dielectric constant is continuous across the boundary; thus in the dielectric which may have dielectric constant 10, the electric field suddenly drops to 1/10 of its value in air. On the other hand, the tangential component (parallel to the interface) of the electric field is continuous across the boundary. In general then we observe a sudden change of direction of electric field lines at the interface, which gives rise to a longitudinal magnetic field component from the second Maxwell's equation, curl E = - dB/dt.

Since some of the electric energy is stored in the air and some in the dielectric, the effective dielectric constant for the waves on the transmission line will lie somewhere between that of the air and that of the dielectric. Typically the effective dielectric constant will be 50-85% of the substrate dielectric constant.

As an example, in (notionally) air spaced microstrip the velocity of waves would be c = 3 * 10^8 metres per second. We have to divide this figure by the square root of the effective dielectric constant to find the actual wave velocity for the real microstrip line. At 10 GHz the wavelength on notionally air spaced microstrip is therefore 3 cms; however on a substrate with effective dielectric constant of 7 the wavelength is 3/(sqrt{7}) = 1.13cms. Thus the maximum length for a stub to be used in stub matching, which is no more than half a wavelength, is about 5.6 mm.

A set of detailed design formulae and algorithms is presented in T C Edwards, Op Cit.

There is a rough and ready nomogram for calculating the impedance of microstrip from the dielectric properties and the geometry in this picture (37kB)

## Substrate materials.

Important qualities of the dielectric substrate include

• The microwave dielectric constant
• The frequency dependence of this dielectric constant which gives rise to "material dispersion" in which the wave velocity is frequency-dependent
• The surface finish and flatness
• The dielectric loss tangent, or imaginary part of the dielectric constant, which sets the dielectric loss
• The cost
• The thermal expansion and conductivity
• The dimensional stability with time
• The surface adhesion properties for the conductor coatings
• The manufacturability (ease of cutting, shaping, and drilling)
• The porosity (for high vacuum applications we don't want a substrate which continually "outgasses" when pumped)

Types of substrate include plastics, sintered ceramics, glasses, and single crystal substrates (single crystals may have anisotropic dielectric constants; "anisotropic" means they are different along the different crystal directions with respect to the crystalline axes.)

### Common substrate materials

• Plastics are cheap, easily manufacturable, have good surface adhesion, but have poor microwave dielectric properties when compared with other choices. They have poor dimensional stability, large thermal expansion coefficients, and poor thermal conductivity.

• Dielectric constant: 2.2 (fast substrate) or 10.4 (slow substrate)
• Loss tangent 1/1000 (fast substrate) 3/1000 (slow substrate)
• Surface roughness about 6 microns (electroplated)
• Low themal conductivity, 3/1000 watts per cm sq per degree

• Ceramics are rigid and hard; they are difficult to shape, cut, and drill; they come in various purity grades and prices each having domains of application; they have low microwave loss and are reasonably non-dispersive; they have excellent thermal properties, including good dimensional stability and high thermal conductivity; they also have very high dielectric strength. They cost more than plastics. In principle the size is not limited.

• Dielectric constant 8-10 (depending on purity) so slow substrate
• Loss tangent 1/10,000 to 1/1,000 depending on purity
• Surface roughness at best 1/20 micron
• High thermal conductivity, 0.3 watts per sq cm per degree K

• Single crystal sapphire is used for demanding applications; it is very hard, needs orientation for the desired dielectric properties which are anisotropic; is very expensive, can only be made in small sheets; has high dielectric constant so is used for very compact circuits at high frequencies; has low dielectric loss; has excellent thermal properties and surface polish.

• Dielectric constant 9.4 to 11.6 depending on crystal orientation (slow substrate)
• Loss tangent 5/100,000
• Surface roughness 1/100 micron
• High thermal conductivity 0.4 watts per sq cm per degree K

• Single crystal Gallium Arsenide (GaAs) and Silicon (Si) are both used for monolithic microwave integrated circuits (MMICs).

• Dealing with GaAs first we have.....

• Dielectric constant 13 (slow substrate)
• Loss tangent 6/10,000 (high resistivity GaAs)
• Surface roughness 1/40 micron
• Thermal conductivity 0.3 watts per sq cm per degree K (high)

GaAs is expensive and piezoelectric; acoustic modes can propagate in the substrate and can couple to the electromagnetic waves on the conductors.

• Now dealing with Silicon we have.....

• Dielectric constant 12 (slow substrate)
• Loss tangent 5/1000 (high resistivity)
• Surface roughness 1/40 micron
• Thermal conductivity 0.9 watts per sq cm per degree K (high)

The dielectric strength of ceramics and of single crystals far exceeds the strength of plastics, and so the power handling abilities are correspondingly higher, and the breakdown of high Q filter structures correspondingly less of a problem.

It is also a good idea to have a high dielectric constant substrate and a slow wave propagation velocity; this reduces the radiation loss from the circuits. However at the higher frequencies the circuits get impossible small, which restricts the power handling capability. For these applications one often choses fused quartz (dielectric constant 3.8).