Modeling of Transmission Lines with Multiple Coated Conductors*

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1 Modeling of Transmission Lines with Multiple Coated Conductors* PlanarCal Konstantin Lomakin Institute of Microwaves and Photonics Friedrich-Alexander-Universität Erlangen-Nürnberg 23 June 2016 *accepted for EuMW 2016

2 Contents Introduction Modeling Approach Skin Effect in Layered Conductors Application in Transmission Line Models Comparison to Measurement, Discussion Conclusion 2

3 Introduction Coated surfaces in microwave applications Ensure functionality: solderability, bondability Protection: oxidation Galvanic or chemical deposition Layer thickness in the order of µm Often intermediate plating with low conductivity required (e.g. Au & Ni) Transmission lines applications Ranging from DC to microwaves At low frequencies skin depth δ easily exceeds layer thickness δ GHz EM-field penetrates materials beneath top coating Multiple coatings have to be considered for precise modeling 3

85 mm: inner lead diameter di = 0.8 mm skin depth 14MS/m, 1 GHz: δ = 4 µm 0.

4 Is Plane Skin Effect valid for Coaxial Cable? m k 4 6 Mont Blanc 64 km Matterhorn Coaxial cable 1.85 mm: inner lead diameter di = 0.8 mm skin depth 14MS/m, 1 GHz: δ = 4 µm 0.8 mm / 4 µm = 200 δ << di Earth: diameter: 12,756 km distance Mont Blanc-Matterhorn: 64 km 12,756 km / 64 km = 200 Warpage negligible, roughness visible! 4

5 Gradient Model Gradient Model is deduced from Maxwell s equations with location dependent conductivity σ(x) and time harmonic fields Assuming a surface in the y-z-plane: w By is the dominant component Translation invariance parallel to the surface x t y The third terms originates from the location dependent conductivity σ(x) Without the third term, i.e. σ = const.: Helmholtz equation describing the classical skin effect in ideally smooth surfaces Solution: with 5

6 Modeling Approach Single layer Surface roughness modeled as σ transition σ proportional to cumulative distribution function (CDF) of the surface profile: σ(x) = σdc CDF(x) CDF Multiple layers Determine (or estimate) RMS-roughnesses Rq for inner layers Calculate respective PDFs with correct position xm according to layer boundary Weigh PDFs with difference between bulk conductivities of current layer and layer above x PDF1 Obtain CDF by integrating sum of layer PDFs PDF3 PDF2 6

7 precision air-line setup. DueNi)to 8,6 the goldauplating, !m Au (15% MS/m (15% Ni) additional 8.6 MS/m layers, especially nickel are required to achieve adhesion. The basethe material is roughness CuBe2 andof itsthe layer stack summarized in dethe surface inner andis outer lead was following termined table: measuring the Firestone-Abbott curve with a tactile profilometer as an estimate for the CDF of the surface profile T[4]. ECHNOLOGICAL DATA (the CONDUCTIVITY AND LAYER THICKNESS From these curves RMS-roughness of the inner lead is) leadand Rq=90 nm outer roughlylayer evaluated to be Rqinner =60 nm forlead the outer material CuBe 21 MS/m CuZn 14 identims/m 2 3Pb2 are leadbase respectively. It was assumed, that these values cal 1to those0.5of-1the layers, which!m inner Cu 58 MS/mcan hardly Cu be measured 58 MS/m directly. The conductivity profile is used to 1.8 calculate resulting -3!m Ni 1.8 MS/m Ni MS/m the magnetic field profile over the depth x, as shown in Fig !m Au (15% Ni) 8,6 MS/m Au (15% Ni) 8.6 MS/m for the inner lead at a frequency of 10 GHz. Skin Effect in Layered Conductors 1.85mm precision air line assembly Inner lead: CuBe2 base material Au plating requires additional Ni layer σdc of Au layer heavily reduced due to Ni Exponential decline in ideal, single layer case Multiple conductive layers increase inner lead ity and magmetic field total loss and phase delay normalized conductivity and magmetic field The surface roughness of the inner and outer lead was determined measuring the Firestone-Abbott curve with a tactile inner lead: Rq=60 nm / outer lead: Rq=90 nm profilometer as an estimate for the CDF of the surface profile [4]. From these curves the RMS-roughness of the inner lead is roughly evaluated to be Rq=60 nm and Rq=90 nm for the outer Gradient 10 GHz lead respectively. It was assumed, that these values are identical to those of the inner layers, which can hardly be measured Magnetic field obtained from CDF directly. The resulting conductivity profile is used to calculate the magnetic field profile over the depth x, as shown in Fig 2 Magnetic field declines depending on σ(x) for the inner lead at a frequency of 10 GHz. depth x (µm) Fig. 2: Conductivity profile of layered conductors and resulting mag 7 netic field profile at 10 GHz

8 III. SKIN EFFECT IN LAYERED CONDUCTORS surface. The classical skin effect delivers the well known exponential decline of the magnetic field strength. IV.A T L M Surface Impedance of Layered Conductors The suggested method is now applied to model a 1.85mm precision air-line setup. Due to the gold plating, additional ayers, especially nickel are required to achieve adhesion. The base material is CuBe2 and its layer stack is summarized in the ollowing table: PPLICATION IN RANSMISSION INE ODELS From the magnetic field response, a complex surface impedance reflecting both increased loss and propagation delay of the rough surface can be calculated [3]. For the air-line setup with layered conductivity, the real and imaginary part is Surface impedance Z can be obtained from fields TECHNOLOGICAL DATA (CONDUCTIVITY AND LAYER THICKNESS) shown in Fig 3. The knee at low frequencies is due to the fact layer inner lead outer lead that the skin depth is still large enough, so that the magnetic Z is complex, thus capable of reflecting both loss and delay base material CuBe2 21 MS/m CuZn3Pb2 14 MS/m field penetrates into the nickel layer which has a very low conductivity. -1!mof Z Curepresents 58 MS/m loss Cu 58 MS/m 1 Real0.5 part !m Ni 1.8 MS/m Ni 1.8 MS/m Imaginary part of Z corresponds to 1-1.5!m Au (15% Ni) 8,6 MS/m Au (15% Ni) 8.6 MS/m inner surface inductance L by X = ωl gmetic field The surface roughness of the inner and outer lead was deermined measuring the Firestone-Abbott curve with a tactile profilometer as an estimate for the CDF of the surface profile 3D field solvers arbitrary crossections 4]. From these curves for the RMS-roughness of the inner lead is oughly evaluated to be Rq=60 nm and Rq=90 nm for the outer Z applicable to any surface ead respectively. It was assumed, that these values are identialto those the inner layers,the which can hardly are be measured Fieldofeffects inside conductor directly. The resulting conductivity profile is used to calculate projected to its surface he magnetic field profile over the depth x, as shown in Fig 2 ortheno innermesh lead atinside a frequency of 10 GHz. the conductor surface impedance (Ohm/square) 2 No increase in simulation time! frequency (GHz) Fig. 3: Real and imaginary part of the surface impedance caused by the penetration into layers with higher resistivity at low frequencies 8

9 ! tance L!: Z! = R! +j!l! New Concept of Equivalent Circumference ζeq (4) These quantities are related to the per unit length paramet R and Li of a transmission line by a dimension depende factor!eq: Analytical two wire transmission line models (5) Propagation coefficient γ described by per unit length parameters R, L, G, C (6) The equivalent circumference!eq combines circumferenc Inductance L consists of inner and outer inductance: L Li + Lassembly a of a=two-wire to that of equivalent one-wire asse bly, where!eq is smaller than every single circumference itse Current I in the inner equals the current in the outer lead: du(end)=0 In the example of a coaxial transmission line like sketch in Fig 4, the current I in the inner lead equals the current in t outer lead. Voltage drop along dz is du(dz) = dr I with I Resistance per unit length R : I Equivalent circumference ζeq: D d! Ci! Co dz Fig. 4: Coaxial transmission line with dimensions and equivale circumferences "i and "o 9

10 Relation between Surface Impedance and Per Unit Length Parameter Comparing per unit length parameters R and L to ideal surface impedance Z Advantages of equivalent circumference ζeq ζeq, ZL and material paramters describe transmission line properties completely ζeq can be calculated from lossless 2D-fields, simulated for arbitrary crossections ζeq can be found for all types of transmission lines: Not limited to coaxial lines! Since warpage is negligible ζeq is applicable for planar transmission lines e.g. CPWs ζeq,1 ζeq,2 ζeq,3 10

11 air-line, the technological data from Tab 1 was used to calculate R and Li. The other per unit length parameters result from the dimensions of the line. To obtain comparable responses, the same adaptions as stated in [2], i.e. #3=4.7 MS/m and #2=3.5 MS/m were made. The deviation in conductivity of the outer layers could be the consequence of diffusion from gold atoms into the nickel layer and vice versa. The calculated Measured insertion loss and phase delay of multiple coated 1.85mm coaxial responses then show good agreement with the responses shown [2],frequencies in which a steady state model for the skin effectdeviate from non-coated case At in low both measured responses in a multilayer-conductor [6] was utilized. In Fig 5, the inseri.e. f-dependence for loss tion loss in db/m calculated with the Gradient Model is compared to the measurement. Additionally Gradient Model precisly predicts measured phase delay Comparison to Measurement, Discussion s21 (db/m) phase delay (ns/m) air-line frequency (GHz) frequency (GHz) Fig. 5: Measured and simulated insertion loss 1.85mm precision Fig. 6: Measured and simulated phase delay of 1.85mm precision airk. Lomakin, G. Gold, K. of Helmreich 11 air-line up to 67 GHz line up to 67 GHz

12 Conclusion Gradient Model Gradient Model not only precisly predicts EM interaction with rough conductor surfaces but also with stacked conducting layers Completely describes impact of rough, coated conductors on transmission line properties, i.e. on both loss and delay Application Easily applicable in 3D field simulators by utilizing a complex surface impedance Z as BC No increase of simulation time! New concept of equivalent circumference ζeq allows for easy adaption in analytical transmission line models 12

Effective Conductivity Concept for Modeling Conductor Surface Roughness G. Gold, K. Helmreich Institute of Microwaves and Photonics

Effective Conductivity Concept for Modeling Conductor Surface Roughness Institute of Microwaves and Photonics Friedrich-Alexander-University Erlangen-Nuremberg gerald.gold@fau.de klaus.helmreich@fau.de