Notes#11, ECE594I, Fall 2009, E.R. Brown 169 Quick Overview of THz Transmission Lines Three Types 1). TEM - e.g., coax structures by micromachining 2). TE or TM - Waveguide: rectangular and circular 3). Quasi-TEM (best approach for integrated circuits on semiconductor substrates - Coplanar strips, coplanar slots (i.e., coplanar waveguide) - Microstrip Uniform transmission line & TEM mode • Two separate conductors with translational symmetry AND homogeneous cross-section • If conductors are assumed to extend to ± infinity along translational axis, then electric and magnetic fields must be perpendicular to coordinates • The electric field between conductors can be associated with a specific capacitance C’ • The magnetic field can be associated with a specific inductance L’ .EndView.SideViewH fields by RH ruleConstructive interferencei××iH FieldE Field..EndView..SideViewH fields by RH ruleConstructive interferencei××iH FieldE FieldNotes#11, ECE594I, Fall 2009, E.R. Brown 170 Lumped-Element Model (O..Heaviside) Can solve by phasor technique of circuit theory })(~Re{),(tjezVtzvω= })(~Re{),(tjezItziω= Application of circuit rules (KCL & KVL) along with Faraday and Maxwell laws for lumped elements (v = L di/dt , i = C dv/dt) yields: jzjzeVeVzV−−++=)(~ wave moving wave moving toward toward positive z negative z γ → propagation constant ( analogous to k in free space) γ2 = ( R’ + jωL’ ) ( G’ + jωC’ ) Also zzeIeIzIγγ−−++=)(~ Key result: oZCjGLjRIVIV≡++=−=−−++''''ωω { Characteristic Impedance Special (and very useful) case: Lossless LineNotes#11, ECE594I, Fall 2009, E.R. Brown 171 Ö R’ = G’ = 0 LCCjLj22)')('(ωωωγ−== ''CLjωγ= zzCLjCLjeVeVzV'''')(~ωω−−++→ has the form e-jkz iff k = v/'' ω≡ω CL ⇒ ''1CL=v phase velocity Zo → ''CjLjωω = ''CL purely real But this is not dissipative ! It’s just a ratio of fields Example: Coaxial line From electrostatics: )ln(2'abCπε= )ln(2'abLπµ=  orεεε= orµµµ= ⇒ Zo = )ln(60)ln(2120)ln(2)ln(2ababababrrrrεµεµπππεπµ== [Ω] rroorrcCLµεµεµεµευρ====1.11''1 [m/s] Bandwidth is determined by presence of higher-order mode 1st higher mode is TE11 mode In air filled coax, turn-on of this mode is determined by ccbakλπ22=+≈ abEHεr,µrabEHεr,µrNotes#11, ECE594I, Fall 2009, E.R. Brown 172 ⇒ )( baccfcc+≈=πλ Absorption in TEM Transmission Line (example of coaxial line) • In THz region absorption in the metal walls of transmission lines establishes the lower limit on attenuation per unit length. And the fundamental absorption mechanism is caused by the skin effect. • The skin effect is a classical consequence of the tendency in metals and other good conductors for the ac electric field to penetrate less in the material as the frequency increases (recall that at dc or, say, 60 Hz, the current flows uniformly through a good conductor, such as the copper power lines). The penetration depth, called the “skin depth” is given by δ = (πfµσ)-1/2 , where µ is the magnetic permeability and σ is the (ac) electrical conductivity. Its value in various common metals is shown in the plot below. • From analysis of, total current in center conductor I0 = δπσθπ⋅⋅≈∫∫∫⋅=⋅ aErdrdrJdSJSa2)(0200 if δ << a But the small but non-zero longitudinal component of E field must also generate a voltage drop V0 = LEdzEL∫=⋅00, so the series resistance is R ≡ V0/I0 ≈ L /( 2πaσδ ) and the specific series resistance is R’ ≡ R/L = 1/( 2πaσδ ) ≡ Rs/(2πa) Where Rs is the “surface” resistance = (σδ)-1 EoRecall: tangential component of E field must be continuous; soE0is small, but not zeroCoaxAxis (z)ErE(r) = E0e(r-a)/δJ(r) ≈σE0e(r-a)/δ(Ohm’s Law)acenter conductor analysisrEoRecall: tangential component of E field must be continuous; soE0is small, but not zeroCoaxAxis (z)ErE(r) = E0e(r-a)/δJ(r) ≈σE0e(r-a)/δ(Ohm’s Law)E(r) = E0e(r-a)/δJ(r) ≈σE0e(r-a)/δ(Ohm’s Law)acenter conductor analysisr Lines of electric force for TE11