A New, Accurate Quasi-Static Model for Conductor Loss in Coplanar Waveguide M. S. Islam, E. Tuncer, D. P. Neikirk Dept. of Electrical & Computer Engineering Austin, Texas
Conductor loss calculation techniques Simple dc+skin depth resistance model Incremental inductance rule - Numerical evaluation (Wheeler) - Analytical evaluation (Gupta) Conformal mapping (Collin) Closed form expression (e.g., W. einrich, Trans. MTT, Jan.'9) Fullwave calculations
New Quasi-Static Technique Conformal mapping used to find frequency dependent series resistance and inductance, in addition to normal capacitance per unit length Requires the use of a "scaled" conductivity in mapped domain - surface impedance can be calculated using scaled conductivity - surface impedance can be found in real space, "scaled" in mapped domain
Length Scaling Under the Influence of a Conformal Map Conformal maps "preserve" local shape j y - differential lengths in w(u, v) plane are "scaled" by an amount M(u, v) z(x, y) plane dx dy w(u, v) = f(z) j v w(u, v) plane du = M dx dv = M dy x Mu,v ( ) = df dz u,v u 4
Example of scaling for a simple cylindrical wire conformal map σ rdθ σ= dv du σ Μ σ M = + 5
Shunt capacitance depends on the ratio of side lengths : C = r θ ( v M) = = v r um u ( ) scale factor cancels! Series resistance depends on the area : R= σ r r θ = σ ( u M) vm ( ) = M σ u v σ M ( scaled)= σ M scale factor enters as M!
Conductor Surface Impedance of a Circular Cylindrical Wire Can solve elmholtz equation exactly Z s = ( ) ( ) T J o Tr o πr o σ J o Tr o where, T = - jωµσ In mapped domain requires solution for conducting rectangular slab with non-uniform conductivity - can be solved using non-uniform transmission line equations For both real & mapped planes, analytic expressions for Z surf are identical 7
Conformal mapping & "scaling" in Co-planar waveguide Mapping is performed by evaluating elliptic integral of first kind Surface impedance is "scaled" in the mapped domain to include effect of current crowding Surface resistance is approximated as R s ( ω) = Re tanh ωµ o σ ωµ o σ ( + j ) t ( + j) 8
conformal map Z-plane y v du(u, v t ) W-plane dx b dx t x M u,v ( ) = dw dz u,v v t v b dr t = R s Mu,v ( t) du ( ) dl = µ o v t v b du dr b = ( ) du R s M u,v b u o du(u, vb ) u conformal map Diagram illustrating use of a conformal map to find the series impedance of a transmission line including the effect of finite resistance
Series impedance Z due to differential width du is the series combination of mapped surface resistance and parallel plate inductance : Z = R S Mu,v ( t)+mu,v b ( ) { } du + jωµ o v t v b du Therefore, total series impedance per unit length is Z( ω) = u o du jωµ o v t v b + R s Mu,v ( t )+Mu,v b ( ) { }
At low frequency total series impedance reduces to R Z = R dc + jωµ o v t v dc b R s u o du Mu,v ( t )+Mu,v b ( ) [ ] At high frequency Z = R s u o u o o ( ) du M( u,v t )+Mu,v b [ ] + jωl ext
b a w Low-loss substrate Cross-sectional view of CPW. Dimensions used for SI GaAs and Pyrex samples are a = 5 µm, b = µm, w = 5 µm
Comparison with our model: SI GaAs substrate 7 9 Attenuation (d/cm) 6 5 4 GaAs substrate Effective index of refraction 8 7 6 5 4 GaAs substrate.. Frequency (Gz).. Frequency (Gz)
Equivalent series resistance and inductance: GaAs substrate 7 Series Resistance (ohms/cm) 8 6 4 resistance inductance GaAs substrate 6 5 4 Series Inductance (n/cm).4. Frequency (Gz)
Comparison with our model: Pyrex substrate 8 8 Attenuation (d/cm) 7 6 5 4 Pyrex substrate Effective index of refraction 7 6 5 4 Pyrex substrate.. Frequency (Gz).. Frequency (Gz) 5
Equivalent series resistance and inductance: Pyrex substrate 6 7 Series Resistance (Ohms/cm) 4 8 6 4 resistance inductance Pyrex substrate 6 5 4 Series Inductance (n/cm).4. Frequency (Gz)
Comparison with other techniques (GaAs substrate) 7 7 6 6 Attenuation (d/cm) 5 4 Measured Rdc+skin effect.. Frequency (Gz) Attenuation (d/cm) 5 4 Measured Wheeler.. Frequency (Gz) 7
Comparison with other techniques (GaAs substrate) 7 7 6 6 Attenuation (d/cm) 5 4 Gupta Measured Attenuation (d/cm) 5 4 Measured Collin.. Frequency (Gz).. Frequency (Gz) 8
Comparison with aydl's experimental data b a w t InP substrate 9
attenuation, d/mm F F F J InP : a = µm, b = 5 µm, t =.5 µm : a = µm, b = 5 µm, t =.5 µm F F F F F : a = 6 µm, b = 45 µm, t =.5 µm J J J J J J J : a = 6 µm, b = 45 µm, t =.5 µm 4 5 6 7 8 frequency, Gz
attenuation, d/mm F J InP F J : a = µm, b = 5 µm, t =.5 µm : a = µm, b = 5 µm, t =.5 µm F F F F : a = µm, b = 45 µm, t =.5 µm F J J J J J J : a = µm, b = 45 µm, t =.5 µm 4 5 6 7 8 frequency, Gz
Conclusions Conformal mapping can be used to accurately predict conductor loss in quasi-tem transmission lines - can be applied to coplanar strips, microstrip, strip-line Numerically efficient Closed form Easy to implement in CAD software Useful for interconnect loss calculation or timing analysis in Multi Chip Modules (MCMs) - generates equivalent circuit model that is very efficient for time domain simulation