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Microwave Engineering 3eAuthor - D. Pozar

Presented by Alex Higgins

Sections 3.6 – 3.8

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Outline

● Section 3.6 – Surface Waves on a Grounded Dielectric Slab

● Section 3.7 – Stripline● Section 3.8 – Microstrip● An Investigation of Microstrip

Characteristic Impedance Formulae

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Surface Waves on a Grounded Dielectric Slab

Surface waves are typified by a field that decays exponentially away from the dielectric surface, with most of the field tightly bound near the surface of the dielectric.

Also known as forced surface waves or tightly bound slow surface waves.

Geometry of a grounded dielectric slab[4].

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Transverse Magnetic ( TM ) mode

For any nonzero thickness slab, with a permittivity greater than unity, there is at least one propagating TM mode.

Equation (3.167) – Cutoff frequency.

Equation (3.168) – Field expressions for surface wave in a grounded dielectric slab.

TM 0Dominant mode

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Transverse Electric ( TE ) mode

TE modes can also be supported by the grounded dielectric slab, where the Hz field satisfies the Helmholtz equation.

Equation (3.174) – Cutoff frequency.

Equation (3.175) – Field expressions for surface wave in a grounded dielectric slab.

TE1Dominant mode

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Dominant Mode Comparison

n TM (GHz) TE (GHz)

0 0 -

1 4.3 2.2

2 8.7 6.5

3 13.0 10.8

Equation (3.174) – TE Cutoff frequency.

Equation (3.167) – TM Cutoff frequency.

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Stripline

● Supports TEM waves● Like the parallel plate waveguide

and coaxial lines, can also support higher order TM and TE modes

● Since TEM mode is dominant an electrostatic analysis sufficiently determines the propagation and characteristic impedance

Stripline transmission line (a) geometry. (b) E and H field lines[4].

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Characteristic Equations for stripline

● Since the stripline supports TEM mode the attenuation due to dielectric loss is of the same for as that for other TEM lines, i.e. coaxial.

● The attenuation due to conductor loss can be found by the perturbation method or Wheeler's incremental inductance rule (outlined in Section 2.7).

Equation (3.179) – Characteristic impedance of a stripline.

Equation (3.180) – Inverse design formula for a stripline of a given characteristic impedance.

(3.179) and (3.180) assume a zero strip thickness, and are quoted as being accurate to about 1% of their exact values.

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Microstrip

● One of the most popular types of planar transmission lines due to the ease in which they are fabricated and their easy integration with other microwave devices.

● Exact fields of a microstrip line constitute a hybryid TM-TE wave, however when the dielectric substrate is electrically thin (d >> λ) the fields are quasi-TEM.

p=c

e=k 0 e

eWhere is the effective dielectric constant of the microstrip line.

Cross-sectional geometry for a microstrip transmission line.

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Pozar – Characteristic Equations for Microstrip

Equation (3.195) – Effective dielectric constant

Cross-sectional E-field lines for a microstrip transmission line[2].

Equation (3.196) – Characteristic impedance of a microstrip line

Equation (3.197) – Inverse design formula for a microstrip line of a given characteristic impedance.

● Considering the microstrip as a quasi-TEM line, attenuation is due to both dielectric and conductor losses.

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An Investigation into Microstrip Characteristic Impedance Formulae

● Motivation● Large number of papers and books with a wide variety of

equations. Which one to use?● Qualitative comparison of three different formulae

● Pozar – From “A Designer's Guide to Microstrip Line” by Bahl and Trivedi

● Ulaby – From “Microstrip Circuit Analysis” by Schrader● Lee – From “High Frequency Amplifiers” by Carson

● MATLAB ™ functions written for each formula.● Representative physical widths for microstrip lines varied using 1/16”

and 1/32” thick FR4 substrate and 1 oz Cu traces.

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Ulaby Lee

Cross-sectional geometry for a microstrip transmission line.

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Conclusions

● Agreement is seen between Pozar and Ulaby when varying the relative dielectric constant.

● All three models show agreement in the 1/16” substrate thickness, while they all deviate quite a bit for thicknesses beyond 1/14”.

● Note also that Pozar and Ulaby show agreement in the 1/32” substrate thickness.

● All three models show agreement for trace widths in the range of 0.055 … 0.1 in. while, at trace widths less than 0.055 in. Ulaby and Lee approach infinity.

● When designing a microstrip transmission line, know the limitations of the formula that you are working with.

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References

[1] C. A. Balanis, Advanced Engineering Electromagnetics. Wiley, 1989.[2] S. M. Wentworth, Applied Electromagentics - Early Transmission Lines

Approach. Wiley, 2007.[3] F. Ulaby, E. Michielssen, and U. Ravaioli, Fundamentals of Applied

Electromagnetics. Prentice Hall, 6 ed., 2010.[4] D. M. Pozar, Microwave Engineering. Wiley, 3 ed., 2005.[5] Thomas H. Lee, Planar Microwave Engineering, Cambridge, 2004

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Questions ?