Meander lines

Triple-track resistor and conductor coupons are made by deposition of Ta2N and Ti-Pd-Au metallization, respectively, on the AI2O3 substrate. This test pattern consists of three parallel meandering lines with 3-mil spaces between lines. Each line is approximately 3-mil wide and has 2.86 X ]0 squares, with an overall length of 8.5 in. The number of squares of insulator between adjacent lines is approximately 3.5 X 10". ... 

A polarizer is a device that transforms a linear polarized wave into a circular polarized wave, or vice versa. The common principle is simply to decompose the incident field into two components where the phase of one is advanced and the other is delayed such that their difference is 90° while their amplitudes are the same. It appears that Pakan [128] was the first to utilize this principle. Later improvements were introduced by Lemer [129]. These devices were not of the meander-line type, as will be discussed here. These seem to appear first in a paper by Young et al. [130] and were subsequently unproved by Epis [131]. Later, a paper by Terret et al. [132] discussed how to calculate the susceptance of a meander line. All of these contributions were primarily focused on normal angle of incidence while Chu and Lee [133] extended the calculation to include oblique angle of incidence. A recent contribution was supplied by Marino [134], It was apparent that meander-line polarizers gradually deteriorate for higher angles of incidence. The present appendix will demonstrate that introduction of a dielectric profile can greatly improve this calamity. 

The general principle for a meander-line polarizer is illustrated in Fig. C.l, top. It consists of one (or more) sheet(s) of meander-line-shaped conductors usually printed on a circuit board. The E vector of an incident plane wave is tilted 45° with respect to the principal axis of the meander line. It is decomposed into a vertical component E, and a horizontal component E. The equivalent circuit for the vertical component is a transmission line with a shunt inductance... 

Fig. C. 1 The workings of a singe meander-iine polarizer. Top A meander-line sheet acts as a shunt inductance for the vertical components and as a shunt capacitance for the horizontal components of an incident signal. Bottom The vertical componmt of the incident field is reflected with reflection coefficient and transmitted with the transmission coefficient Ty= 1 + Ty as shown in the Smith chart. Similarly, the horizontal component is reflected and transmitted with reflection coefficient T/, and transmission coefficient r/, = 7 + T/, respectively.

There will also be a reflected field. The vertical and horizontal components have amplitudes and phases given simply by F and F , respectively. In general this reflected field will be elliptic polarized. What is of concern, however, is the fact that when a single meander-line sheet transmits a perfect circular polarized field, we readily see from the Smith chart that F = t = F = Tft. In other words A single meander-line sheet will produce a reflected field of the same amplitude as the transmitted field that is, the efficiency is only 50%. The remedy for this calamity is to use several meander-line sheets cascaded after each other as will be discussed in the next section. See also Problem C.l, where you are asked to consider a two-sheet meander-line polarizer. 

Fig. C.2 By cascading two or more meander-line sheets, we may obtain broader bandwidth and tower refiecVon. The Smith charts show the input impedances at the various locations as indicated in the schematic.

We next rotate point 3 into point 4, corresponding to Ihe sheet separation d2 as shown in the Smith chart, bottom. Finally, adding sheet reactance 7X1 in parallel brings us to point 5 located at the center of the Smith chart that is, we should now have practically nothing reflected from this three-layered design, in contrast to the single meander-line case discussed above. 

APPENDIX C MEANDER-LINE POLARIZERS FOR OBLIQUE INCIDENCE... 

The impedance of the outer meander-line sheet including the free space Zq behind the individual sheets is shown in the Smith chart in Fig. C.3 for the vertical as well as the horizontal components. Similarly, we show the impedances for the inner meander-line sheets including Zq in the Smith chart shown in Fig. C.4. We observe that both the vertical and horizontal components vary with frequency however, the angular difference between the transmission coefficients for the two components remains reasonably constant with ffeqnency. This feature is quite essential for producing polarizer designs with a large frequency range. 

We also note that the angles of the transmission coefficient for the inner meander lines are approximately twice that for the outer meander lines, as they should be (see discussion above). The meander-line dimensions are given in the respective Smith charts. We observe that both of these designs have the same dimensions and D, and so on. This is done so as not to violate Floquet s Theorem (see reference 26). The differences in impedances are obtained by using different line widths w in the two cases. 

To assist in designing the proper meander-line sheets, we show in Fig. C.5 the typical impedance variance as a function of the strip width w and meander-line spacings Note that the variation is one-sided as a function of the strip width w and symmetric with meander-Une spacing... 

Fig. C.3 The input impedance of a single encapsulated meander line (outer) including free space Zo behind. Vertical as well as horizontal cases.

This design is comprised of three meander-line sheets separated by air of thickness 0.8657 cm, as shown at the bottom of Fig. C.6. At / = 8.7 GHz this corresponds to a separation of A.o/4. However, when incorporating the dielectric substrate around the meander lines, an effective quarter-wave separation is obtained at... 

Fig. C.5 Typical movement of the meander-line impedance including free space Zo for Top Variation of line width w Bottom Variation of line spacing Dx.

The reason for this dilemma is simply that the electrical separation between the meander-line sheets typically is given by Po droy (d is the physical separation). Thus, at 60° the electrical separation will be reduced by a factor of 1/ cos 60° = 2, making the electrical separation approximately correct at the higher frequencies... 

Fig. C.6 Design 1. Three encapsulated meander-line sheets without dielectric. Top The magnitude ratio Ev/Eh as a function of frequency. Bottom The phase difference Qy -9f,asa function of frequency.

We just observed how Design 1 fails at the lower frequencies at higher angles of incidence because the factor roy becomes less than one. In Design 2 we shall attempt to correct for this discrepancy simply by increasing the meander-line sheet separation. 

More specifically, the meander-line sheet separation is increased from 0.8657 cm to 0.9996 cm. It does indeed lead to some improvement in phase response at the lower frequencies but also to some degradation at the higher frequencies for normal angle of incidence. Thus, it is clear that a more radical treatment is needed. Since space is limited in this book, we have chosen simply not to show the results for this design. 

The reason for this shortage of phase difference is that when we use dielectric spacers, we lower the intrinsic impedance by This simply implies that the meander-line impedances must also be lowered accordingly as will be demonstrated in the next design. 

Meander-Line Sheet Separation A/4 in Dielectric Dielectric Constant of Spacers e i = 6 2 = 2.0 Meander-Line Impedances Lowered... 

Fig. C.9 Design 3. Three encapsulated meander-line sheets with dielectric separation. Input impedances for the vertical as well as the horizontal field components.

This design has the same dielectric profile as Design 3, but the meander-line impedances have been reduced. This has been accomplished for the outer meander lines by simply increasing the line width w of the meander lines from 0.02081 cm to 0.03954 cm while aU other dimensions remain the same. A plot of the outer meander-line sheet incorporating a dielectric slab is shown in Fig. C.IO. It should be compared with the original meander-line sheet in Fig. C.3 except that the dielectric slab rotates the impedance approximately 90° clockwise (on the... 

Fig. C.11 Design 4. input impedances (verticai and horizontal) of two meander lines separated by a dielectric slab.

Fig. C. 12 Design 4. Input impedances (vertical and horizontal) for two cascaded meander-line sheets and two dielectric slabs as shown.

Fig. C.13 Design 4. Input impedances Vertical and horizontal) for three cascaded meander-line sheets separated by dielectric spacers as shown.

Fig. C. 15 Design 5. Three meander-line sheets separated by dielectric slabs and further provided with outer matching transformers. The input impedances (vertical and horizontal).

---