ω ω ω ω ω ω ω ω ω ωLj

2. The line admittance matrix is[ Y( ω )]= [ G(ω )]+ jω[C(ω )]3. Find a similarity transformation matrix such that⎡zm(1,1)0 L 0 ⎤ ⎡Z(ω )1,1Z⎢0 z⎥ ⎢(2,2)Z()2,1ZZm ⎢mO M⎥ tT ⎢ ω==⎢ M O O 0 ⎥ ⎢ M M⎢⎥ ⎢⎢⎣0 L 0 zm(n,n)⎥⎦⎢⎣Z(ω ) n,1Zand( ω )1,2( ω )2,2( ω )2, n[ ] [ ] ⎥[ T ] (6)⎡y⎢0m(1,1)y0LO0( ω )1,1( ω ) n,2( ω )1,2(5)LLOLZZZ( ω )1, nM( ω ) n,nm(2,2)1 ( )2,1 ( )2,2( )2, n t[ Ym] ⎢⎥ −[ T ] ⎢ ω ωω ⎥ −==[ T ] (7)⎡Y⎢Y⎢ M O O 0 ⎥ ⎢ M M⎢⎥ ⎢⎢⎣0 L 0 ym(n,n)⎥⎦⎢⎣Y(ω ) n,1Y4. Find the complex propagation factor matrix2⎡γ10 L 0 ⎤⎢ 2 ⎥2 00[ ] ⎢ γ LΓ1m=⎥ = [ Zm][ Ym] (8)⎢ M M O M ⎥⎢2⎥⎢⎣0 0 L γ1 ⎥⎦5. Finally, the characteristic impedance can be calculated from:−1−1Zc = [ Z ][ T ] Γ T (9M⎤⎥[ ] [ ] [ ] )( ω )mYY( ω ) n,2LLOLYYY( ω )1, nM( ω ) n,nThe accuracy of the transmission line impedance so calculated depends on very much the RLCGmatrices extracted in the BEM2D field solver. This field solver, basically is under staticassumption and the skin effect is included under Quasi-tem Approximation of a single line modelin a closed form[2].4. References1. K. Oh, D. Kuznetsov, and J. Schutt-Aine, “Capacitance Computation in a MutilayeredDielectric Medium Using Closed-Form Spatial Green’s Functions,” IEEE, MTT, Vol.42, No. 8, pp. 1443-1453, August, 1994.2. A. Djordjevic and T. Sarkar, “Closed-Form Formulas for Frequency-DependentResistance and Inductance per Unit Length of Microstrip and Strip TransmissionLines,” IEEE, MTT, Vol. 42, No. 2, pp. 241-248, Feb., 1994.⎤⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎦