Microstrip

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Cross-section of microstrip geometry. Conductor (A) is separated from ground plane (D) by lower dielectric (C). Upper dielectric (B) is typically air.

A microstrip is a thin, flat electrical conductor separated from a ground plane by a layer of insulation or an air gap. Microstrips are used in printed circuit designs where high frequency signals need to be routed from one part of the assembly to another with high efficiency and minimal signal loss due to radiation. They are of a class of electrical conductors called transmission lines, having specific electrical properties that are determined by conductor width and resistivity, spacing from the ground plane, and dielectric properties of the insulating layer. A microstrip transmission line is similar to a stripline, except that the stripline is sandwiched between two ground planes and respective insulating layers.

Microstrips can also be designed to launch electromagnetic waves into space, in which case they are called microstrip antennas.

1 Inhomogeneity
2 Characteristic Impedance
3 References
4 External links
5 See also

[edit] Inhomogeneity

The electromagnetic wave carried by a microstrip line, exists partly in the dielectric substrate, and partly in the air above it. In general, the dielectric constant of the substrate will be greater than that of the air, so that the wave is travelling in an inhomogeneous medium. In consequence, the propagation velocity is somewhere between the speed of radio waves in the substrate, and the speed of radio waves in air. This behaviour is commonly described by stating the effective dielectric constant (or effective relative permittivity) of the microstrip; this being the dielectric constant of an equivalent homogeneous medium (i.e. one resulting in the same propagation velocity).

Further consequences of an inhomogeneous medium include:

The line will not support a true TEM wave; at non-zero frequencies, both the E and H fields will have longitudinal components (a hybrid mode).^[1] The longitudinal components are small however, and so the dominant mode is referred to as quasi-TEM.

The line is dispersive. With increasing frequency, the effective dielectric constant gradually climbs towards that of the substrate, so that the phase velocity gradually decreases.^[1]^[2] This is true even with a non-dispersive substrate material (the substrate dielectric constant will usually fall with increasing frequency).

The characteristic impedance of the line changes slightly with frequency (again, even with a non-dispersive substrate material). The characteristic impedance of non-TEM modes is not uniquely defined, and depending on the precise definition used, the impedance of microstrip either rises, falls, or falls then rises with increasing frequency.^[3] The low-frequency limit of the characteristic impedance is sometimes referred to as the quasi-static characteristic impedance.

The wave impedance varies over the cross-section of the line.

[edit] Characteristic Impedance

A closed-form approximate expression for the characteristic impedance of a microstrip line was developed by Wheeler:^[4]^[5]^[6]

$Z_\textrm{microstrip} = \frac{Z_{0}}{2 \pi \sqrt{2 (1 + \varepsilon_{r})}} \mathrm{ln}\left( 1 + \frac{4 h}{w_\textrm{eff}} \left( \frac{14 + \frac{8}{\varepsilon_{r}}}{11} \frac{4 h}{w_\textrm{eff}} + \sqrt{\left( \frac{14 + \frac{8}{\varepsilon_{r}}}{11} \frac{4 h}{w_\textrm{eff}}\right)^{2} + \pi^{2} \frac{1 + \frac{1}{\varepsilon_{r}}}{2}}\right)\right)$

where $w eff$ is the effective width, which is the actual width of the strip, plus a correction to account for the non-zero thickness of the metallization. The effective width is given by

$w_\textrm{eff} = w + t \frac{1 + \frac{1}{\varepsilon_{r}}}{2 \pi} \mathrm{ln}\left( \frac{4 e}{\sqrt{\left( \frac{t}{h}\right)^{2} + \left( \frac{1}{\pi} \frac{1}{\frac{w}{t} + \frac{11}{10}}\right)^{2}}}\right)$

with

Z 0 =

impedance of free space,

$\varepsilon_{r} =$ dielectric constant of substrate,

w =

width of strip,

h =

thickness ('height') of substrate and

t =

thickness of strip metallization.

This formula is asymptotic to an exact solution in three different cases

$w \gg h$ , any $\varepsilon_{r}$ (parallel plate transmission line),
$w \ll h$ , $\varepsilon_{r} = 1$ (wire above a ground-plane) and
$w \ll h$ , $\varepsilon_{r} \gg 1.$

It is claimed that for most other cases, the error in impedance is less than 1%, and is always less than 2%.^[6] By covering all aspect-ratios in one formula, Wheeler 1977 improves on Wheeler 1965^[5] which gives one formula for $w / h > 3.3$ and another for $w / h \le 3.3$ (thus introducing a discontinuity in the result at $w / h = 3.3$ ). Nevertheless, the 1965 paper is perhaps the more often cited.

Curiously, Harold Wheeler disliked both the terms 'microstrip' and 'characteristic impedance', and avoided using them in his papers.

[edit] References

^ ^a ^b E. J. Derdinger, “A frequency dependent solution for microstrip transmission lines”; IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 30-39, Jan. 1971.
^ H. Cory, “Dispersion characteristics of microstrip lines”; IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 59-61, Jan. 1981.
^ B. Bianco, L. Panini, M. Parodi, and S. Ridetlaj “Some considerations about the frequency dependence of the characteristic impedance of uniform microstrips”: IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 182-185, March 1978.
^ H. A. Wheeler, “Transmission-line properties of parallel wide strips by a conformal-mapping approximation”, IEEE Trans. Microwave Theory Tech., vol. MTT-12, pp. 280-289, May 1964.
^ ^a ^b H. A. Wheeler, “Transmission-line properties of parallel strips separated by a dielectric sheet”, IEEE Tran. Microwave Theory Tech., vol. MTT-13, pp. 172-185, Mar. 1965.
^ ^a ^b H. A. Wheeler, “Transmission-line properties of a strip on a dielectric sheet on a plane”, IEEE Tran. Microwave Theory Tech., vol. MTT-25, pp. 631-647, Aug. 1977.