Losses in Planar Waveguides

Maik Holzhey and Richard Schulz

w Abstract—A comprehensive study to calculate losses in planar waveguide structures is presented theoretically with approximate δn formulas and numerical simulation, respectively. Results are t critically reviewed by comparison of the two approaches. Approx- imate formulas to calculate line impedance, ohmic and dielectric attenuation are given. CST MICROWAVE STUDIO R [1] is used h k = εr to perform the numerical calculations and validate the formu- δn las’ approximate results. Furthermore, a summary of relevant research, referring to losses in planar waveguides, is provided.

I.INTRODUCTION Fig. 1. Illustration of the example geometry with specified parameters according to the listed notation. In the 1960th and 1970th has been a strong urge to determine important electrical properties of a planar waveguide, consisting of two metal planes separated by a dielectric sheet. Line w0 = w + ∆w w Impedance and attenuation constants, resulting from a specific ε0ε2 geometry, are particularly relevant to the application engineer. ε ε ⇒ 0 1 ε Back in the time, engineers faced the challenge of little eff computation power and limited accuracy in measurement technology. In order to build a sufficiently accurate theory, an Fig. 2. Transformation of the geometry and applying an effective dielectric analytical approach was necessary. However, today’s engineers constant to calculate line impedance Z0. Note the assumption t = 0 to calculate line impedance. have a large amount of computational power at their disposal. A numerical simulation which can handle complex shapes and an inhomogeneous dielectric filling much easier may offer µ magnetic permeability of the metal strips better approximations to reality. Comparing these approaches ρ electrical resistivity of the metal strips shows limitations of the analytic theory and builds trust towards f frequency in Hz simulating planar waveguides. | MH | Rs effective skin resistance of the metal strips L inductance GLOSSARY q dielectric filling fraction αc ohmic attenuation in dB/m αd dielectric attenuation in dB/m k relative dieelectric constant ε of the dielectric sheet, r II. THEORY seperating the two metal strips ka (k + 1)/2 average of dielectric constants of sheet and Analysing losses in planar waveguide structures similar to free space [8] the presented structure in Fig. 1, imposes various challenges. kFR4 dielectric constant of FR-4 material The geometry divides into two half spaces introducing an kPTFE dielectric constant of Teflon material inhomogeneous dielectric filling as is shown in Fig. 1. This εr equal to relative dieelectric constant k complicates the analysis due to continuity conditions of the w width of the conductor electromagnetic fields. In [7], Wheeler proposes a solution, 0 w0 w = w + ∆w - edge correction using a conformal mapping approach to transform the ge- t thickness of the conductor ometry to only one dielectric filling. Using this conformal h height of the dielectric sheet mapping and by applying an effective dielectric constant to the Z0 line impedance transformed dielectric sheet, Wheeler [8] was able to calculate e Naperian base line impedance based on geometry information. The geometry λ wavelength is hereby altered to a slightly different shape to consider fringe λ0 wavelength in free space fields at the edges (edge correction). However, this technique is c0 light velocity in free space based on some simplifications such as an approximate formula Rc = 377Ω - wave resistance in free space for the conformal mapping or conductor thickness being set to zero. Figure 2 illustrates this procedure. The original formula This work was supported by Technische Universität Berlin (Berlin Institute of Technology), Faculty of Electrical Engineering and Computer Science, [8] yields only little accuracy for narrow strips but is sufficiently Department Theoretical Electromagnetics accurate for wide strips. t 2h w/h 2 : w0 = w + ∆w , with ∆w = ln + 1 ≥ π t   t Rs 8.68 w0 w0/(πh) h h 2h 1 + h αc = 2 1 + + ln + 1 t (1) Z0h · w0 2 w0 h − w0/(2h) + 0.94 w0 πw0 · t − 1 + h + π ln 2πe 2h + 0.94       2h  


[Wheeler65] including conductor thickness, requires lengthy, yet straight [Heinrich] forward calculations that result into (1), given by Pucel et al. [Schneider] 102 [5]. It gives the conductor attenuation αd in [dB/m] for shape ratios w/h 2. Pucel et al. give formulas for all shape ratios in their publication≥ which is not reproduced at this point, due in [Ω] 0

Z to space limitations. Finally an equation for dielectric attenuation αd in [dB/m] is w/h 3: [Wheeler65] 1 ≥ given by Schneider [6] with the equations (7)-(9). The superior 10 R 1 1 4h 1 w 2 1 k 1 π 1 4 = ln + − ln + ln Rc √ka · π w 8 h − 2 k+1 2 k π accuracy of this empirically found formula given in (9) has 

 1 0 1 10− 10 10 been praised in many subsequent publications, such as [9]. w/h Using the effective dielectric constant (7)

1 Fig. 3. Comparison of line impedance calculation by various authors  + 1  1 10h − 2  = r + r − 1 + (7) eff 2 2 w   For that reason, it underwent some empirical corrections over and the true dielectric filling fraction (8), a decade later, made possible by increased calculation power 1 1 and more accurate measurement data [9]. Such a corrected q = eff − tanδ (8) 1 1 · formula is given by Heinrich [3] in (2) and (3) for different r − shape ratios w/h (compare Fig. 3). the dielectric losses (9) αd can be calculated.

120Ω 20π 1 λ0 w/h ≤ 3.3 : Z0 = p · αd = q using λ = (9) 2(k + 1) ln10 · · λ √eff     s 2 (2) 4h  h  1 k − 1  0.242  The attenuation constants αd and αc are given in [dB/m] and ln  + 16 + 2 − · 0.452 +  w w 2 k + 1 k require no additional post-processing. | MH |

188.5Ω III. SIMULATION w/h ≥ 3.3 : Z0 = √ · R k CST MICROWAVE STUDIO [1] and its time domain  w k + 1 h  w i k − 1 −1 solver is used to evaluate the losses in a microstrip model. Line 0.44 + + 1.452 + ln + 0.94 + 0.082 2h 2πk 2h k2 impedance, dielectric losses, and ohmic losses are determined (3) for different geometry and material properties. The used CAD-model, which is shown in Fig. 4, consists Having calculated line impedance makes a prediction of of a rectangularly shaped microstrip, that is separated from ohmic losses feasible. Pucel et al. [5] use Wheeler’s incremental the ground-plane by dielectric material. The ground-plane is inductance rule (4) [8],[7] to estimate ohmic attenuation. modeled using a rectangular shape of the same length and 1 ∂L width like the dielectric material. The isolators tangent-loss R = R (4) µ sj ∂n and permittivity are assumed to be constant in the simulated 0 j j X frequency range. Multiple permittivity values are used, the 4 Using effective skin resistance to model the conductor tangent-loss is chosen to be 10− . Lossy metal with different resistivity (5) leaves the challenge to determine the conductor conductivity values is used for metallization. inductance L (6) For utilization of the model symmetry a magnetic symmetry boundary is used. The ground plane is simulated as electric 1 Rsj = (πfµρs)− 2 (5) boundary. The excitation is performed by one waveguide port. The free space boundaries are chosen to be simulated as open boundaries. Few mesh elements are added between geometry w0 t L = √µ  Z f , , k (6) and open boundaries. Figure 4 gives an overview of the used 0 0 0 · h h   boundary conditions. Giving accurate expressions for the n deviations of the For reduction of simulation errors resulting from the edges inductance, using Wheeler’s conformal mapping approach, on the microstrip geometry, edge refinement is applied. | RS | 0.35 k = 1 (simulation) 0.3 k = 2 (simulation) k = 4 (simulation) 0.25 k = 8 (simulation) k = 16 (simulation) k = 1 (formula) m] 0.2 / k = 2 (formula)

[dB 0.15 k = 4 (formula)

α k = 8 (formula) 0.1 k = 16 (formula)

2 5 10− · 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f [Hz] 1010 ·

Fig. 6. Dielectric attenuation for different k - Schneider [6] formula in comparison to simulation results Fig. 4. CST MICROWAVE STUDIO R [1] - simulation assembly and boundary 0.25 conditions. Colors: green - electrical boundary, purple - open boundary, blue - w/h = 0.1 (simulation) magnetic symmetry plane w/h = 6.4 (simulation) 0.2 wh = 30 (simulation) w/h = 0.1 (formula) w/h = 6.4 (formula) k = 1 (simulation) 0.15 w/h = 30 (formula) m]

k = 2 (simulation) /

k = 4 (simulation) [dB

102 k = 8 (simulation) α 0.1 k = 16 (simulation)

k = 1 (formula) 2 5 10−

[Ω] k = 2 (formula) · 0 k = 4 (formula) Z k = 8 (formula) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 101 k = 16 (formula) f [Hz] 1010 ·

Fig. 7. Dielectric attenuation for different w/h - Schneider [6] formula in 1 0 1 10− 10 10 comparison to simulation results w/h

Fig. 5. Line impedance for different k - Heinrich [3] formula in comparison to simulation results for commonly used k (kPTFE 2 and kFR4 4) Schneider’s estimations show very good agreement≈ to the simulations≈ results as shown in Fig. 6. In general Schneider’s formula is slightly IV. RESULTS lower than the simulation results for k > 8 and slightly higher The shown diagrams compare the simulation results to the for k = 1, which might be neglectable. related approximate formulas. Simulation results are expressed Comparing Schneider’s estimation [6] to simulation results as dashed lines with circles for actually simulated values reveals, that best agreement is achieved especially for high whereas the approximate formulas are expressed with solid (w/h = 30) and low values (w/h = 0.1) of w/h as shown in lines. Fig. 7. Overall Schneider’s formula shows high agreement with the A. Line Impedance simulation results for common k and high as well as low The simulated line impedance is compared to estimations w/h. This leads to the assumption that Schneider’s formula from Wheeler [8], Schneider [6] and Heinrich [3]. Simulations is especially developed for commonly used k as well as very are performed for different values of w/h as well as different narrow and wide microstrips. As the disagreement is small values of k. For w/h 10 all three formulas converge to both methods seem to be equally appropriate for dielectric loss the simulation results. For small values of w/h Heinrich’s evaluation. estimation seems to be best, as it is very close to the simulation results as shown in Fig. 5), whereas Schneider’s estimation is C. Ohmic Losses too low and Wheeler’s estimation is too high. The simulated ohmic losses are compared to formulas by Heinrich’s estimations are more recent than Wheeler’s and Pucel et al. [5]. Different ρ, w/h and k are used for the Schneider’s estimations. Earlier formulas were mainly based simulations. on theoretical simplifications (chapter II) with high errors The disagreement between simulated and estimated ohmic for uncommonly used geometry properties, whereas recent losses for different k increases when using higher values. For formulas are fitted using simulation and measurement data. high values of k the frequency dependent disagreement is Therefore it can be assumed, that the simulation, as well as not neglectable as shown in Fig. 8. Nevertheless, the general Heinrich’s formula, deliver the most accurate results for the qualitative behavior is similar in simulation and formula. line impedance. Smaller disagreement occurs when using different values of ρ. The disagreement increases with higher values of ρ as shown B. Dielectric Losses in Fig. 9. The influence of ρ on the disagreement between The simulated dielectric losses are compared to estimations simulation and formula is small in comparison to the influence from Schneider [6]. Different k are simulated. Specially of k. 2.5 k = 1 (simulation) very accurately according to the simulation. Line impedance k = 2 (simulation) 2 k = 4 (simulation) calculation, despite optimized formulas, still shows small k = 8 (simulation) k = 16 (simulation) tolerances, at delivering close approximations for very narrow 1.5 k = 1 (formula) m] / k = 2 (formula) strips. This is due to approximate formulas for the conformal

[dB k = 4 (formula) mapping and the incorrect assumption of considering transver- α 1 k = 8 (formula) k = 16 (formula) sal electromagnetic waves propagating inside the waveguide. 0.5 These aspects introduce an error to calculate ohmic losses

0 accurately. The electrical properties resulting from the geometry 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f [Hz] 1010 can be calculated, using numerical simulation when edges · R are properly treated. CST MICROWAVE STUDIO uses an Fig. 8. Ohmic attenuation for different k - Pucel [5] formula in comparison analytical model to help the numerical analysis in avoiding to simulation results incorrectly large values for the electromagnetic fields around

1.4 the edges. Using a mesh grid refinement around the shape ρ = 8.33 µΩ cm (simulation) 1.2 ρ = 4.17 µΩ cm (simulation) edges makes the simulation more accurate. | MH | ρ = 1.72 µΩ cm (simulation) 1 ρ = 0.79 µΩ cm (simulation) ρ = 0.40 µΩ cm (simulation) VI.OUTLOOK ρ = 8.33 µΩ cm (formula) m] 0.8 / ρ = 4.17 µΩ cm (formula) Computer-aided design can also be used to fit and find [dB 0.6 ρ = 1.72 µΩ cm (formula) α ρ = 0.79 µΩ cm (formula) accurate formulas to predict losses in microstrips. Hammerstad 0.4 ρ = 0.40 µΩ cm (formula) et al. [2] used this approach to find accurate formulas for 0.2 attenuation constants derived from Q factor estimation. How- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f [Hz] 1010 ever for numerical simulation this approach holds challenges · to calculate correct eigenmodes in resonant problems while Fig. 9. Ohmic attenuation for different ρ - Pucel [5] formula in comparison considering losses. Kirschning et al. [4] used an empirical to simulation results approach to find an accurate, frequency dependent formula to determine the effective dielectric constant which is crucial to Also different w/h are simulated. Especially for small values predict dielectric losses. Over a very large frequency range, of w/h increased disagreement occurs as shown in Fig.10. the linear, empirical approximation proposed by Schneider [6] Overall the ohmic losses show the same qualitative behavior might not be sufficient. | MH | comparing simulations and formula. However, a not neglectable REFERENCES frequency dependent disagreement occurs. The formula is con- [1] CST. Computer simulation technology. microwave studio - https://www. sistently lower than the simulation results. The same behavior cst.com/products/cstmws. has also been shown through measurement conducted, by Pucel [2] HAMMERSTAD,E., AND JENSEN,O. Accurate models for microstrip [5]. As Pucel et al. [5] use Wheeler’s conformal mapping [7], computer-aided design. In 1980 IEEE MTT-S International Microwave symposium Digest (May 1980), pp. 407–409. ohmic losses are wrongly predicted. Through measurement [3] HEINRICH, W. Passive Elemente der Hochfrequenztechnik - passive conducted, by Pucel [5], it is known that theory consistently components in high frequency design. lecture notes (TU Berlin) (2016). predicts lower ohmic losses. The numerical simulation yields [4] KIRSCHNING,M., AND JANSEN,R.H. Accurate model for effective dielectric constant of microstrip with validity up to millimetre-wave larger ohmic losses, so a better accuracy can be assumed. This frequencies. Electronics Letters 18, 6 (March 1982), 272–273. is why the simulation might deliver a more accurate ohmic [5] PUCEL,R.A.,MASSE,D.J., AND HARTWIG, C. P. Losses in microstrip. loss prediction. | RS | IEEE Transactions on Microwave Theory and Techniques 16, 6 (Jun 1968), 342–350. [6] SCHNEIDER, M. V., GLANCE,B., AND BODTMANN, W. F. Microwave V. SUMMARY and millimeter wave hybrid integrated circuits for radio systems. The Bell System Technical Journal 48, 6 (July 1969), 1703–1726. The theoretical analysis shows good agreement with the nu- [7] WHEELER,H.A. Transmission-line properties of parallel wide strips by merical solutions. Especially dielectric losses can be predicted a conformal-mapping approximation. IEEE Transactions on Microwave Theory and Techniques 12, 3 (May 1964), 280–289. [8] WHEELER,H.A. Transmission-line properties of parallel strips separated by a dielectric sheet. IEEE Transactions on Microwave Theory and w/h = 3.2474 (simulation) 0.4 w/h = 30.00 (simulation) Techniques 13, 2 (Mar 1965), 172–185. w/h = 3.2474 (formula) [9] WHEELER,H.A. Transmission-line properties of a strip on a dielectric 0.35 w/h = 30.00 (formula) sheet on a plane. IEEE Transactions on Microwave Theory and Techniques

m] 0.3 25, 8 (Aug 1977), 631–647. /

[dB 0.25 α

0.2

0.15

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f [Hz] 1010 ·

Fig. 10. Ohmic attenuation for different w/h - Pucel [5] formula in comparison to simulation results