Introduction
In a uniform, zero-thickness isolated stripline, the normalized current density J(x) has the form
where W is the width of the strip and x is the position, measured from the center of the strip (Fig. 1(a)), as given in Appendix A for W/2 = 1. It has the same form as the charge density [Reference Collin1]. Power loss depends on the integral of this function, which has singularities at x = ±W/2, because the finite thickness t of the strip has been ignored. For a thick strip, currents flow on the top, bottom, and side surfaces because of the skin effect, and the 90° corners do not have this singularity.
Fig. 1. Fitting parameters: integration limits Δand d for (respectively): (a) strip with finite thickness, and (b) MOM calculation with a zero-thickness conductor.
The singularities are avoided by integrating only within a range −(W/2) + Δ < x < W/2 − Δ, where fitting parameter Δ is chosen as follows. In Fig. 1(a), s = x + W/2, that is, s is measured from the left-hand edge and s<<W. At a 90° corner, the thick-metal current density squared $J_m^2$
(s) increases only as s −2/3 instead of s− 1, as s decreases. When the two curves shown have equal integrals, the power dissipation per unit length for the finite-thickness strip with surface resistance Rs can be represented by the thin-metal case, $P_\Delta = 4R_s\int_0^{W/2-\Delta } {J^{2\;}\lpar x \rpar dx}$
, considering the right-hand half of the top surface, and then multiplying by 4 for the whole strip. The value of Δ, sometimes called the “stopping point” [Reference Holloway and Kuester2] or “stopping distance” [Reference Booth and Holloway3], is t/290 for perfect conductors [Reference Lewin4], typically t/150 to t/250 for finite conductor skin depths as confirmed in experiments [Reference Holloway and Kuester2], and t/130 for superconductors [Reference Booth and Holloway3] in one example [Reference Huang, Bolli, Cresci, Mariotti, Panella, Lopez-Perez and Garcia5]. Losses can also be obtained from thick-layer simulations [Reference Rautio and Demir6, Reference Rautio7] and are also considered in [Reference Horton, Easter and Gopinath8–Reference Hamham, Mesa, Medina and Khalladi11].
One version of the Method-of-Moments (MOM) [Reference Harrington12] uses thin conductors as an approximation and approximates J 2(s) as rectangular basis functions, shown in Fig. 1(b). The combination of rectangles does not account for the singularity of 1/s and furthermore, for the second rectangle onwards, the discrepancies between the rectangle functions and 1/s may not exactly cancel. The aim of this work is to establish an integration limit (W/2-d) such that $P_d = 4R_s\int_0^{W/2-d} {J^{2\;}\lpar x \rpar dx}$
is the MOM estimate of power loss. The ratio $P_\Delta$
/P d becomes the correction factor for simulations. It is useful because the power loss, which is proportional to J 2 (x), is particularly sensitive to errors in J(x). The derivation for d is straightforward, but the assumptions are not very obvious, so the bulk of the effort is to confirm that P d is a fair estimate of $P_{sim} = 4R_s\sum\nolimits_n {J_n^2 X_n}$
from actual MOM calculations (with notation from Fig. 1(b)).
The expressions for Δ are already experimentally confirmed in [Reference Holloway and Kuester2]. The present work proposes and confirms the value of d, a numerical exercise. Both a manual MOM procedure and commercially available software Sonnet® [13] are used, for several configurations: isolated stripline; uniformly spaced parallel strips; microstrips and striplines close to ground planes; and strips in a uniform, externally applied magnetic field. Isolated stripline is the simplest case, relevant because the edge singularities are not strongly affected when ground planes are added, as confirmed by comparison with a microstrip where the substrate thickness is only half the strip width. Coplanar line, in which currents in the adjacent ground planes are in the opposite direction to the center line, are similar to the parallel strips with currents in adjacent lines in the opposite direction. The thick metal calculations and experiments [Reference Holloway and Kuester2] are supplemented by some additional simulations and experimental results to illustrate the calculation process. The comparisons also reveal some features of microstrips which might not be expected, such as a very significant loss under-estimate for coarse meshing.
The MOM calculation (existing software) already gives the overall current distribution, taking into account any dielectrics in the layout. The additional adjustment for the current crowding (this work, characterised by d) is assumed to be independent of the dielectrics, so free-space configurations are considered. This is because, for localized currents, Ampere's law (Appendix B) can be used instead of Maxwell's equations. Supplementary simulations on spirals and meander lines, and experiments with microstrips confirm that the results are still applicable when dielectrics with high relative permittivity (ɛr ~ 10) are present.
For comparison, existing papers [Reference Holloway and Kuester2, Reference Horton, Easter and Gopinath8–Reference Hamham, Mesa, Medina and Khalladi11] give stand-alone calculations for $P_\Delta$
, using extensive computations with software which does not appear to be intended for arbitrary filter shapes, that is, not simulation software. For the present work, the overall current distribution is found by existing MOM software, requiring only the very localized adjustment to find P d. Despite being defined by similar equations, $\;P_\Delta \;$
and P d are not the same quantity and are not compared to find the error level in one or the other. Instead, previous work [Reference Holloway and Kuester2] is accepted and used in the correction factor$\;P_\Delta$
/P d, which can be inserted into the software by entering a modified value of metal resistance. The differences and relative merits between [Reference Holloway and Kuester2,Reference Horton, Easter and Gopinath8–Reference Hamham, Mesa, Medina and Khalladi11] and the MOM itself [Reference Harrington12,13] are not part of the current study. The main advantage is that the improvement in the MOM can be used for complex shapes in filters such as step-impedance hairpins, with very little additional effort.
Evaluation of the integration limit
Using Fig. 1(b), in which the current density (omitting a constant, for clarity) is $J\lpar s \rpar = 1/\sqrt s$
so the mean value of J(s) (not J 2(s) as illustrated) between s = 0 and s = X 1 is given by
For the rectangular function J 12 = 4/X 1 and the curve J 2(s) = 1/s (for s = d to X 1) to have equal areas and therefore to represent the same power loss,
where Rs is the surface resistance, taken to be very small so that it does not affect the current distribution, so
This simple equation is the centerpiece of the present work. The second rectangular function with height J 22 underestimates 1/s for s slightly greater than X 1, causing the MOM equations to over-estimate J 1 and the crowding factor, as will be investigated numerically.
The estimate for the MOM value of power dissipated per unit length of a strip is approximately
For a uniformly distributed current, it is
where I is the current in the top surface of the right-hand half of the strip, which has width W/2. Hence an approximate equation for the current crowding ratio included in the MOM is
The actual MOM current crowding in simulations is (with notation given in Fig. 1(b))
Isolated strips
Using J 2(x) for an isolated strip given in Appendix A, with width adjusted from 2 units to W,
for small d. The actual current crowding factor $\eta _\Delta$
for thick conductors based on [Reference Lewin4] is the same equation but with d replaced by Δ and the correction factor $\int_0^{W/2-\Delta } {J^{2\;}\lpar x\rpar dx\;} /\;\int_0^{W/2-d} {J^{2\;}\lpar x\rpar dx}$
is equal to $\eta _\Delta /\;\eta _d\;$
(much simpler than $\eta _\Delta /\;\eta _{sim}$
). Although η d and $\eta _\Delta \;$
share the same equation and the same red line, simulations are likely to have X 1/W between 0.05 and 0.25, on the right-hand half of the graph, while real filters will usually have Δ/W on the left-hand side, so $\eta _\Delta /\eta _d\;$
can be substantial, typically 1.5–2.0.
Figure 2 also compares η d with η sim from manual MOM calculations as given in Appendix B, revealing only a slight variation, believed to be due to the variable-ratio X 2:X 1. Here, η d is a function of d, while η sim is a function of X 1, and the agreement shows that the relation (5) is valid. Commercially available software Sonnet® [13] was also used. A uniform, straight strip of width 40 μm is placed in a 2 × 2 × 2 mm3 enclosure, and excited at 1 GHz. Exceptions are for X 1/W = 0.002, where W = 100 μm, which illustrate that scaling does not affect the results. Current density plotted by the software is manually read off, and (9) evaluated. Results are also given in Fig. 2(a), with good agreement with the manual MOM and (10). The horizontal discrepancy between the η d and η simlines is about 20%, but with the logarithmic relationship, the vertical discrepancy is only about 3%, slightly more for large X1/W where the approximation $J\lpar s\rpar = 1/\surd s$
breaks down. It is the vertical separation that is relevant in calculations. Transmission line loss also depends on R/Z 0, where R is the resistance per unit length and Z 0 the characteristic impedance. Because the MOM slightly overestimates the characteristic impedance, there is a small additional error, ignored here but found in some example calculations to be of the order of 2%.
Fig. 2. Current crowding factor: approximate equation (10) compared with MOM calculations (9). For strips carrying a known current. (a) For sub-sectioning X 1:X 2:X 3:X 4 = 1:2:4:8 : typical values of X 1/W for copper and superconductor are given. Red and dashed black lines correspond to the similar lines in Fig. 1. (b) For various values of X 1:X 2:X 3:X 4; with the approximate equation subtracted. For 1:1:1:1, m is irrelevant, so the value shown on the graph is arbitrary. (c) Comparison of sub-sectioning: X 1:X 2:X 3:X 4 = 1:2:4:8 and x 1:x 2:x 3:x 4 = 1:6:12:24.
Narrow subsections are usually used near the strip edges where the current density varies rapidly, and larger subsections in the middle. In Sonnet, a default ratio X 1:X 2: X 3: X 4, … is 1:2:4:8…, working inwards, except for the subsections near the center, which are chosen so that the total width is the strip width. This ratio is used in Fig. 2(a), with two values using 1:6:12 … for comparison. Other ratios are shown in Fig. 2(b) where (10) is subtracted so the discrepancy can be greatly magnified. The values of η sim- η d for the ratios 1:1:1:1 and 1:2:4:8, are very close, implying that the sizes of inner subsections are virtually immaterial. However, comparing 1:2:4:… with 1:6:12…, there is a significant difference. The widths of the inner subsections in corresponding positions are similar; for example, x 3 has a magnitude between X 4 and X 5 (not X 3), as shown in Fig. 2(c). Thus, only the widths of the first and possibly the second subsections are significant. This is re-confirmed with the plots of η sim- η d for 1:1:m:2m, which also show little change when X 1 and X 2 are not varied.
Parallel striplines
Equations for current density are given in Appendix A for parallel strips, with the currents either all in the same direction or “anti-parallel”, alternating in the direction in neighboring strips. The widths and the spaces are both 40 μm. For the anti-parallel strips, the neighbors were represented by the image currents in the walls of a very narrow box. For the parallel strips, the strip under test was next to a symmetry plane with 32 neighbors on one side and 33 images in the symmetry plane, on the other side. The results, in Fig. 2(a), also show very good agreement.
To observe non-idealized situations instead of straight transmission lines in air, simulations were also done for spirals and meander lines. One case was a spiral similar to Fig. 3, with input and output lines, omitting the other resonator, a meander line combined with an interdigital capacitor (IDC). In the second case, the meander/IDC was provided with nearby input and output lines, and the spiral was omitted. In the fundamental resonance of the spiral [Reference Huang, Bolli, Cresci, Mariotti, Panella, Lopez-Perez and Garcia5], currents in adjacent turns are in the same direction, while for the meander/IDC, the current is largest in the meander line, where currents in adjacent sections are anti-parallel.
Fig. 3. Resonators used in simulations: one spiral and the other including a meander line, used separately or together.
The substrate has a relative permittivity 9.65 and thickness 0.5 mm, representing MgO (Magnesium Oxide). Line width is 50 μm. The spiral is a 2 mm square with a 3.7 mm tail. The original study was for superconductors [Reference Huang, Bolli, Cresci, Mariotti, Panella, Lopez-Perez and Garcia5], but for easier evaluation of bandwidth and quality factor, the surface resistance was entered as 4.6 × 10−4 ohms per square, approximately 100 times that of a typical YBCO superconductor at 2.3 GHz and 20 K with top and bottom surfaces in parallel. For the meander line, it was 8.1 × 10−4 ohms/square, approximately 20 times that of YBCO at the 10 GHz resonance. Losses are estimated from the 3 dB bandwidths of the S 21 response, giving allowance for the input and output loading found from a re-simulation with zero resonator loss. They are scaled to fit the η d curve at X 1/W = 0.05; there is a close match, except near X 1/W = 0.5, where η d = 0.881 and 0.928 for the parallel and anti-parallel strips, respectively. Had the values been scaled to η d = 1 at X 1/W = 0.5, the curves would have been higher. The cause is probably the skewed current distribution and transverse currents near the bends, which introduce more loss in both the simulations and in the thick films. Since only the ratio $\eta _\Delta /\;\eta _d\;$
is required, the curves are confirmed except for extremely coarse mesh.
Strips in a uniform field
When an external uniform perpendicular magnetic field induces a current in the isolated strip, the induced current is also concentrated at the edges (insets, Fig. 4). The conformal mappings, MOM calculation and the simulation layout with a nearly-uniform field are given in Appendices A–C. A non-idealized configuration for simulation is shown in Fig. 3. The lossless spiral, resonating at 2.3 GHz, creates a magnetic field which induces current in the meander/IDC resonator, which has surface resistance 4.6 × 10−4 ohms per square, as before. The second resonator is not resonant at this frequency but loads the spiral. The current density is approximately anti-symmetric, as sketched in the inset of Fig. 4 and in Fig. 13(c). Resonance 3 dB bandwidths are very narrow, but the surface resistance cannot be increased much further without perturbing the current distribution. Very good agreement between the approximate equation, the MOM calculations, and the meander line simulations is shown in Fig. 4. As before, the meander line data are scaled to fit at X 1/W = 0.05.
The resonator combination is intended for other work, where the meander resonator attenuates the third-order resonance of the spiral, as in [Reference Huang14–Reference Huang17].
Microstrips and striplines with ground planes
At microwave frequencies, the current is confined to the upper and lower surfaces of the microstrip. With a nearby ground plane underneath, the current density is greater on the microstrip lower surface because of the larger magnetic field, studied by modeling the top and bottom surfaces as two separate layers with a very thin gap. However, in most simulations, they are combined as one layer, since halving the number of sub-sections may result in an eightfold reduction in computation time. These two cases are given in Fig. 5, where W = 40 μm. Ground plane losses [Reference Collin1,Reference Holloway and Kuester18,Reference Holloway and Kuester19], are omitted. The value of η d is assumed to be close to η sim for separate layers, being determined by the 1/s dependence of J 2(s) at the edges, as before, and confirmed in the next section.
Fig. 5. Current crowding factor for very thin microstrips and striplines, with nearby ground planes, based on MOM (Sonnet) calculations.
Errors arise from conflicting requirements for the separation t = 0.02 μm between the upper and lower surfaces of the strip. It should be small to avoid obscuring the variation of η sim with X 1/W. For the isolated strip, which has equal top and bottom currents, the difference between using a single layer or separate layers is due solely to t. The worst error is about 5%. To check if t is too small for other values of h, some strips were re-simulated with three layers enclosing two gaps of 0.02 μm. The middle layer does not have zero current density (as it should) because the rectangle function estimates of currents in the top and bottom layers shield the center layer incompletely. In the absence of the middle layer, the field under the strip would influence the current in the top layer. The center current is greatly reduced with t = 0.04 μm, even though η sim changed by much less than 0.1, implying very little effect.
The curves should approach the isolated strip for large h/W. This applies even for h/W = 0.5; furthermore, the transition to microstrip occurs only at approximately h/W = 0.2. For very small h/W it should tend towards η sim = 2 for the double layer model since current is concentrated on the lower layer, while for a single layer, η sim = 1. However, for h/W = 0.05, X 1/W~0.125, the current crowding factor is only about 1.65, showing that a significant current still flows in the top surface. Meanwhile, for X 1/W~0.005, all the two-layer curves are near 1.8, giving a reasonable rule of thumb, because current crowding at the edges compensates for the incomplete current crowding between top and bottom.
Turning to striplines, crowding is significant even for h/W = 0.2, but less than for the microstrip because more current flows near the center of the top surface, in addition to the bottom surface.
Comparison with thick strips
To supplement $\eta _\Delta$
and experimental data in [Reference Holloway and Kuester2], and the additional experimental data in the section “Comparisons with measurements”, new simulations for η thk using the thick-layer MOM model was generated for comparison with η sim, the above thin-metal model of 1 or 2 layers. The data are shown in Fig. 6. X 1/W is sufficiently small so that the current crowding is determined almost exclusively by the thickness. Thus, the horizontal axis for the thin metal is X 1/W, but for the thick metal it is Δ/W, and agreement between the two cases confirms the relation (5). Data are given for h/W→∞ and h/W = 0.2; t = 32 μm, resistivity is 4.31 × 10−9 Ωm (a quarter of room-temperature copper), and frequency is 1.092 GHz, to make the skin depth δ = 1.0 μm, that is, much thinner than t. Nevertheless, the skin depth is represented approximately using the appropriate skin resistance and reactance in the thin layer model for each of the top and bottom surfaces, while the side walls were 1 μm wide stacks of thin layers (that is, the software's thick-layer model) with 1 μm gaps. This stack has a rectangular function for a current density of width δ, instead of the exponential profile, but the skin resistance and field penetration are approximately correct. Box length is 100 μm, while other dimensions are much smaller than a wavelength. In this section, the crowding factor was found by comparing S 21 with the corresponding value of a thin-film reference line with X 1 = W/2, instead of using (9), to avoid manual summation of very many terms. Comparing S 21 values includes the difference arising from different estimates of characteristic impedance (as in the section “Isolated strips”), not considered in (9). The software estimates of characteristic impedance suggest that the difference is particularly large when h/W = 0.2, typically 5%.
Fig. 6. Current crowding factor for very thick microstrips with nearby ground planes, based on MOM (Sonnet) calculations, t/2δ = 16, compared with the thin-layer simulations.
Figure 6 shows that (5) is a good approximation, except for large t/W, as expected [Reference Lewin4] because the conformal mapping there assumes t << W. Two values of η thk for t/2δ = 5.5 from [Reference Horton, Easter and Gopinath8] are also given. Their graphed data have been adjusted by removing dielectric and ground plane losses as given in [Reference Collin1,Reference Edwards20]. The point at X 1/W = 0.00245 is a consensus of several works which are compared in [Reference Stracca9]. For the point at X 1/W = 0.0098, h/W is 2 but taken to be infinite since even h/W = 0.5 is not very different from h/W→∞.
In Fig. 7, results are compared with η thk from medium-thickness (t ~ δ) simulations consisting of a stack of thin layers with t 0 = 0.2 μm spacing and W = 40 to 400 μm, but (because of computer limitations) t 0 = 0.33 μm for W = 640 μm. Again, h/W→∞ and h/W = 0.2. For resistivity 17.24 × 10−9 Ωm (room temperature copper) and frequency 1 GHz, skin depth is δ = 2.09 μm. Because of the unusually small subsections, several of them are required to resolve the variation of current density near the edge. Eight 1 μm subsections were therefore added, produced in the software by manually drawing narrow touching rectangles instead of one larger rectangle. Maximum t/Δ occurs when t/2δ≈1.44 [Reference Holloway and Kuester2], which minimizes Δ. Thus, the series of open circles represents progressively decreasing t, starting with the pink-filled circle; Δ initially decreases and then increases. The same applies to the series of upright triangles.
Fig. 7. Current crowding factor for t/2δ of the order of 1, compared with thin layers.
To find $\eta _\Delta$
, S 21 loss was compared with a single-layer reference line with X 1 = W/2 and sheet resistance
based on [Reference Holloway and Kuester2], where R s is the skin resistance and k is the (complex) wave number within the metal. For t/2δ ≫ 1, top and bottom layers are separate and in parallel, so R~R s/2, while when t/2δ~1, the skin layers overlap, R is larger, and the reference line is itself affected by the overlapping top and bottom skin layers, but not the current crowding at the edges.
Example series of points with constant t/W, t, and W are given. For h/W→∞, t/2δ > 0.5, and t/W < 1/16, three calculations agree closely:
• ηsim, thin-metal simulations together with (5) (broad black line, long dashes);
• η Δ, where [Reference Holloway and Kuester2,Reference Booth and Holloway3] give data on t/Δ, which with (10) gives the crowding factor (thin red line); and
• ηthk, using the same data on t/Δ but with new thick-metal simulations (black circles).
Admittedly, because of the logarithmic relationship in (10), the horizontal discrepancy, that is, the error in Δ, is larger, but it is ηsim which is required in calculations. If ηthk is accurate, the horizontal shift between it and η Δ may be due to the approximation that t/Δ in [2] is independent of W. In particular, for the very small and little-used range t/2δ < 0.5 (not shown), ηthk and η Δ agree only for t/W~1/16, but here, [Reference Holloway and Kuester2] can be ignored and the thin-metal simulations can be used on their own.
For h/W = 0.2, (1) does not apply, but nevertheless, the estimates of ηsim are a fair match to ηthk for t/2δ > 1. For smaller t, the values of ηsim approach the single-layer simulations, as expected, with t/2δ = 0.72 near the center of the transition. For h/W = 0.05, the agreement is fair for t/2δ > 1.2.
Thin strips with resistance
Introducing resistance to the thin conductor simulations modifies the current distribution and hence ηd (as it does with the thick model via the parameter t/2δ, previously discussed). The inset in Fig. 8 shows a diagonal line which represents any of the three curves of ηd, the black dashed lines in Fig. 2(a). Strip resistance limits the current crowding factor; moving leftwards, the line reaches a limiting value depending on the resistance. This limit is given in the main figure, obtained in MOM computations. For easier scaling, strip width was changed to 100 μm. The excitation frequency remains 1 GHz. For the parallel strips, there are 16 near neighbor strips included in the calculation on each side of the main strip to reduce computation time. (In two runs with 32 neighbors, the difference is not visible to the scale of the graph). The thin film resistor model was used; for t/2δ > 1, the surface resistance of the top and bottom surfaces in parallel is approximately Rs 0/2, where Rs 0 is the surface resistance for the given 100 μm strip at 1 GHz.
Fig. 8. Current crowding factor for a thin resistive isolated 100 μm wide stripline, based on MOM (Sonnet) calculations.
For the same overall resistance, a wider strip has a proportionately larger resistance per unit width. At a higher frequency, the reactance increases, so for a proportionate contribution by resistive and reactive components, the resistance is proportionately larger. Thus, the scaling rule is
Alternatively, the horizontal axis Rs 0/2 can be replaced by (0.1256 /μ0fW) (Rs/2) where the permeability of free space μ0 makes the variable dimensionless and removes the very large constants. To test the scaling, simulations with f and W varied by a factor of 10 made no difference, to within the drawing accuracy. The crowding factor is then the smaller of ηR and ηd, except in the small transition region, where both the finite subsection width and the resistance contribute.
In Fig. 8, X 1/W = 0.002 limits ηR as Rs is reduced. Otherwise, the curves would continue to rise to the left, with approximately constant gradient.
Different curves apply when the metal is modeled with a surface impedance of R + jX, where R = X, as when the skin effect is included, and these are also shown. The appropriate curve depends on the model in the simulation, not on the actual properties of the metal when the aim is to appraise and adjust the MOM loss estimate.
Comparisons with measurements
Two similar hairpin filters, described in [Reference Huang14] for harmonics suppression, were revisited to investigate current crowding; one is shown in Fig. 9(a). They are made with 1.27 mm thick RT Duroid 6010.2 with 18 μm copper thickness. The tracks are 0.5 mm wide, and each resonator is approximately 7 × 28 mm2. The manufacturer's latest dielectric data are given in [21]. The external circuits in the simulation and measurement are straightforward: “in” and “out” are connected to source and load, or to ports 1 and 2 of a network analyzer. For one of the filters, simulation results progressively including more and more of the loss mechanisms (dielectric, microstrip resistance, and ground plane) are shown in Fig. 9(b) and compared with the measurement; some of the curves almost coincide. In Fig. 10, the mid-band losses are normalized, dividing by the stated reference value, which is the simulation S 21 loss for copper only, without the adjustment $\eta _\Delta$
/η d for current crowding.
Fig. 9. Hairpin filter with 22% bandwidth and 1 GHz center frequency. (a) Layout, (b) simulated frequency response for various loss mechanisms, and measured response.
Fig. 10. Losses: measurements compared with computations for microstrip (h ≫ W) and strips with near neighbors; scaled so that the un-corrected resistive loss (X 1/W = 0.25) is 1 unit.
Despite the relatively coarse mesh of X 1/W = 0.25 in [Reference Huang14], predictions of filter characteristics had been excellent, except for the losses, which required the unrealistic assumption that current only flows on the bottom surface (“current ratio” = 0 in Sonnet) even though current actually flows on both sides when the ground plane is not very close (h > W). This is similar to the 1.8 rule of thumb given earlier. The present work shows that η d is only 1.09, far short of $\eta _\Delta = 1.73$
, as estimated from [Reference Holloway and Kuester2]. Taking current to flow on both sides, and including the adjustment $\eta _\Delta$
/η d, as shown in Fig. 10, substantially improves the loss estimate.
Two further resonators with the same dielectric but no ground plane each comprised two intertwined spirals as in Fig. 11(a) with the inner ends joined as in Fig. 11(b), giving a “spiral-in-spiral-out” resonator with a total of seven turns. The overall dimensions are 10.5 × 10.5 mm2 and 7.5 × 7.5 mm2, with line widths 0.25 and 0.125 mm, respectively, and X 1/W = 0.25. In the fundamental resonance, adjacent turns have anti-parallel currents, while in the second-order resonance, they are parallel. (Meanwhile, the electric fields of the first two resonances probe the dielectric to different depths, so a resonator can also be used to investigate a possible variation of dielectric constant with depth. The double spiral (Fig. 11(a)) is a complementary resonator because the shallow field occurs at the higher resonance instead of the lower frequency.)
Fig. 11. Resonator to measure losses: (a) precursor: dual spiral resonator. (b) spiral-in-spiral-out resonator. (c) Response.
Losses are estimated from the bandwidths of the resonances. Because of a significant skew in the peaks, the 1 dB bandwidth (where B 1dB/f 0 = 1/2Q) as in Fig. 11(c), instead of 3 dB (B 3dB/f 0 = 1/Q) was taken, but even so, the losses are not exactly proportional to bandwidth. To make a reasonable comparison (Fig. 10), all losses including the measured loss were based on these bandwidths. The adjusted estimate was obtained by re-simulating with bulk resistance increased by a factor of $\lpar \eta _\Delta /\;\eta _d\rpar ^2$
, which increases the surface resistance by $\eta _\Delta /\;\eta _d$
because of the skin effect. For a better comparison, the bandwidth for a lossless resonator is subtracted, as it corresponds to a loss in the external circuit. The values are then divided by the bandwidth for the un-adjusted copper loss. It depends on R/ωL, where L and R are the inductance and resistance per unit length of the transmission line; because the magnetic fields in the parallel and anti-parallel cases differ markedly, the normalization factors are also very different.
The bar graph (Fig. 10) shows that this correction again leads to a considerable improvement. The least accurate is for the anti-parallel currents in the strips with 0.125 mm gaps. It may be because the proximity of near neighbors affects the s −1/3 dependence of current density near the corners [Reference Holloway and Kuester2], and not only the general distribution described by (1). The strong skew in the frequency response may also have contributed.
Superconductor losses have historically been rather unrepeatable, even with measurements by the same worker at about the same time [Reference Farber, Deutcher, Contour and Jerby22–Reference Avenhaus, Porch and Lancaster24]. The same applies to the filter in [Reference Huang, Bolli, Cresci, Mariotti, Panella, Lopez-Perez and Garcia5], compared with [Reference Semerad25,Reference Irgmaier, Semerad, Prusseit, Ludsteck, Sigl, Kinder, Dzick, Sievers, Freyhardt and Peters26], or perhaps the surface resistance has improved by about 25% over the past 20 years.
Conclusion
This work evaluates the error in power loss calculations based on the standard MOM with rectangular basis functions and thin metal strips, resulting from the incomplete consideration of the concentration of current near the edges. It uses a simple approximation for current density near an edge and compares the result with accepted thick-metal calculations from the literature which are already supported by measurements. The resulting adjustment makes the loss estimates commensurate with the excellence of other MOM results, even with relatively coarse meshing, allowing more complex circuits to be analyzed. Isolated strips and strips with near-neighbors are considered, as is a strip in an externally-applied uniform field. New simulation data with thick metal are also presented. Non-ideal cases with spirals and meander lines, including dielectric layers, are simulated for comparison. For isolated strips (h/W→∞), extending down as far as h/W = 0.5, an equation is given to estimate the current crowding factor given by MOM calculations of microstrips and striplines. For smaller values of h/W, numerical data are provided. In example calculations with relatively coarse mesh X 1/W = 0.25, significant underestimates of η d were corrected and compared with experiments, showing that the correction factor is valid, while other calculations show that the ground plane is significant only when it is very close to the strip.
Appendix A: Conformal mappings for stripline
Conformal mappings [Reference Cohn27] for striplines with width W = 2 are given in Table 1 and Fig. 12. The complex variable w in the conformal mappings is not to be confused with W. For unity current density in the t-plane, the normalized power dissipated in the right-hand half of a strip is
Fig. 12. Conformal mappings for stripline. (a) An isolated strip, or parallel strips with all currents in the same direction. (b) Parallel strips with current directions alternating. (c) An isolated strip in an externally applied uniform field. (d) Parallel strips in an external field.
Table 1. Conformal mappings for current crowding factor η d
Results are given for d <<1. The strips are either isolated or have 1:1 mark-to-space ratio. The “parallel” strips have currents in the same direction while for the “antiparallel” strips, currents in adjacent strips are in opposite directions.
The vertical walls in the z-planes do not exist in the actual structures but are inserted so that the fields in the region |Re(z)| < 2, Im(z) > 0 in Figs 12(b) and (d) or |Re(z)| > 0 in Fig. 12(c) are appropriate.
Transform 2 for the anti-parallel strips arises from the Schwartz–Christoffel transformation [Reference Cohn27], and since the total current on the right-hand half of the strip is 1,
found in tables of the complete elliptic integral to be 1.854.
For the strips in an external field, a factor of 22 is added to allow for current in both sides of the strip, while in Fig. 12(d), there is a further factor of 22 to re-normalize the flux density in the z-plane at large Im(z).
For the parallel strips in an external field, the limits of integration in (A.1) are 1 + d to 2.
Appendix B: The method of moments
By Ampere's law, a long, straight wire at point x carrying a current I results in a magnetic flux density
where r is the distance from the wire (Fig. 13). Integrating between points P and Q,
so in Fig. 13(b), integrating between xa and xb where the constant current density is J 1, it is
Fig. 13. Method of moments calculation using piecewise constant approximations (rectangular functions). (a) Magnetic flux φpqin a zone PQ due to a current I. (b) Current distribution in a stripline carrying a current. (c) Current distribution in a stripline in an externally applied uniform field.
Similar terms for the other five rectangular current distributions are omitted for clarity.
In this version of the MOM, P and Q are at the centers of the rectangular functions on the right-hand half of the strip. By taking various points for P and Q, N-1 independent equations can be formulated for N unknown values of Jn; attempting to take more combinations leads to non-independent equations. Flux penetration φpq within the strip is zero. The total current on the right-hand half of the strip is
There are enough equations to solve for J 1, J 2, and J 3 in terms of Itot. This is easily adapted for fewer rectangular functions, but with more than three unknowns, using a pocket calculator is very tedious and general-purpose mathematical software such as MathCad, Matlab, Mathematica or “C” would be required.
For an externally applied field (Fig. 13c), φpqis equal but opposite to that field, making the total zero. Taking xp = 0 and two symmetrically placed rectangular functions,
Again, similar terms for J 2 and J 3 should be included, but there is no equation for Itot since it is always zero with the anti-symmetric current density.
Appendix C: Uniform applied magnetic field
A nearly uniform field can be generated in a simulation using Helmholtz coils [28], consisting of two co-axial circular coils of radius and separation both r. The two-dimensional equivalent is an infinitely long rectangle with short side 2l, shown in Fig. 14 as the two upper strips. Because only the vertical component By has to be uniform, the lower coil is omitted. The strip is in a plane 0.55l below the plane of the coil. (In a plane l/√3 below the coil, By is maximally flat up to the x 3 term, while for 0.55l, the ripple is only ± 0.07% and the region of nearly constant By is larger.) The short 2000 μm strip length saves computation time, but it could have been even shorter. It can be shown by Ampere's law that the flux density is given by
where I is the current in the upper two strips.
Fig. 14. Approximating a uniform field in simulations: l = 80 μm, W = 40 μm, and the cubical cavity has side 2000 μm.
Frederick Huang received the B.A. and D. Phil degrees in Engineering Science from the University of Oxford in 1980 and 1984. Since 1989 he has been a lecturer and then an honorary research fellow with the University of Birmingham, UK. Previous research interests are surface acoustic wave (SAW) dot array pulse compressors, analogue voice scramblers, Langmuir-Blodgett films, SAW and superconducting linear phase, and chirp filter synthesis using inverse scattering, slow-wave structures, superconducting quasi-lumped element filters, switched filters and delay lines, together with microstrip and waveguide discontinuities. The main current interests are spiral band-pass filters and filter harmonic suppression.
