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1 EE Planer Micrwave Circuit Design Ntes Returning t the incremental sectin, we will nw slve fr V and I using circuit laws. We will assume time-harmnic excitatin. v( z,t ) = v(z)cs( ωt ) jωt { s } v z,t = Re V z e v z,t t =jωv s ( z) Applying KVL: Applying KCL: i z,t -v( z,t ) + i( z,t) RΔz + LΔz + v( z+δz,t ) = 0 t v z+δz,t i( z+ Δz,t) i( z,t ) + v( z+δz,t) GΔz + CΔz = 0 t Rearranging each equatin, dividing by Δz, and taking the limit as Δz 0, we find lim Δ z 0 v z+δz,t -v z,t Δz i z,t = -i ( z,t) R-L t Δ z 0 i z+δz,t -i z,t lim Δz v z+δz,t = -Gv ( z+δz,t) - C t Recgnizing that the left hand side f the equatins define a derivative, we have the Telegrapher f Transmissin Line equatins: v z,t i z,t i z,t = Ri( z,t) - L t v z,t = Gv( z,t) - C t Under steady-state cnditins (suppressing the time factr) 16

2 EE Planer Micrwave Circuit Design Ntes v z =- ( R+jωL ) iz i( z ) =- G+jωC v z (1) () Rewritting equatin (), v z = 1 ( G+jωC) i z and substituting this result back int equatin (1): 1 i z ( G + jωc) =- R+jωL iz i z = R+jωL G + jωc iz and similarly, v z = R+jωL G + jωc v z Let γ = α +jβ ( R+jωL)( G+jωC) γ cmplex prpagatin cnstant α attenuatin cnstant [Np/m] β phasecnstant [rad/m] Using the prpagatin cnstant we uncver wave equatins. i z v z = γ i z 0 = γ v z 0 The slutins t the wave equatins cme in the frm f traveling waves. 17

3 EE Planer Micrwave Circuit Design Ntes + -γz - +γz v z = V e + Ve (3) + -γz - +γz i z = I e + I e (4) -γz +γz e +z directed wave e -z directed wave Substituting equatin (3) int equatin (1) + -γz - +γz -γv e +γve = R+jωL iz γ iz = Ve Ve (5) ( + -γz - +γz ) R+jωL In cmparing equatins (4) and (5), we find that we can define a characteristic impedance. R+jωL Z = R + jωl + - = = = + - γ G+jωC I I V V Z R+jωL = (6) G+jωC Thus ur traveling wave expressins are as fllws. + -γz - +γz v z = V e + Ve + - V -γz V i( z ) = e e Z Z The prpagatin cnstant (γ) and characteristic impedance (Z ) are tw fundamental prperties that define the behavir f transmissin lines. Bth depend n R, L, G and C f ur lumped element mdel that in turn depend n the gemetry f the transmissin line and the relevant material prperties. +γz 18

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5 EE Planer Micrwave Circuit Design Ntes Material Prperties Relevant T Transmissin Lines Permittivity When an applied electric field interacts with a material, the field tends t plarize (align) the atms r mlecules f the material. The permittivity f a material defines the extent t which this plarizatin ccurs. In general, a material s permittivity is a cmplex quantity, the imaginary prtin (ε ) f which is a measure f a material s lss behavir. That is, ε ' '' permittivity = ε -jε In the field f micrwaves, a dielectric s lss is mre cmmnly given by its lss tangent (tanδ) which includes bth damping f the vibrating diple mments and cnductive lss. ' ε = ε ( 1-jtanδ ) The permittivity f free space (cnsidered lssless) is: -1 ε permittivity f free space = F/m. T simplify matters, when quting the real part f the permittivity f a material (its dielectric cnstant ), we nrmalize the value t that f free space and thus qute the material s relative dielectric cnstant (ε r ). ε r ε' relative dielectric cnstant = ε Example values arund 10 GHz Material ε r tanδ Frequency Tefln GHz AlO GHz Silicn GHz GaAs GHz Styrfam GHz Distilled water GHz Nte: These values are frequency dependent; tanδ values depend n resisitivity. Fr example the tanδ value quted fr Si is based n high-resistivity material, nt CMOS Si! Permeability In a similar fashin t the permittivity, we need a measure f the extent t which a material is influenced by a magnetic field. The permeability f a material describes the ability f a magnetic field t align the magnetic diples within the material. The permeability f free space is 0

6 EE Planer Micrwave Circuit Design Ntes -7 μ permeability f free space = 4π 10 H/m. While magnetic materials find applicatin in micrwave systems (phase shifters and circulatrs are example micrwave cmpnents expliting magnetic behavir), we will cnfine urselves in EE 433 t nn-magnetic materials and take the relative permeability (the analg t relative dielectric cnstant) t be unity. That is, μ r = 1. Cnductivity While permittivity and permeability are used t describe materials that are pr electrical cnductrs (i.e. insulatrs), a material that readily cnducts electrical current is characterized by its cnductivity (σ) which is ften given in 1/(Ω-m). The cnductivity values f several metals are given belw. Resistivity Example Cnductivity Values Material Cnductivity (S/m) Aluminum 3.816x10 7 Gld 4.098x10 7 Cpper 5.618x10 7 Silver 6.173x10 7 Slder 7.0x10 6 Nte: These are typical values fr the given materials. Resistivity is simply the inverse f the cnductivity. The units fr resistivity are thus resistance-length, mst ften quted in (Ω-cm). Skin Depth High frequency EM fields d nt penetrate cnductrs very far beneath the cnductr surface. The skin depth indicates the depth at which the amplitude f the field decays t 1/e (t ~ 37% f its riginal value). The skin depth is given by: δ skin depth = s = = α πfμσ 0 π f σ [ GHz ] Fr example, the skin depth f gld at 6 GHz is apprximately 1 μm. As a rule f thumb, ne strives t have a metal thickness in a planar circuit that is a few skin depths deep. Therefre, a gld thickness f a few micrns at 6 GHz shuld be sufficient! <end lecture 4> 1

7 EE Planer Micrwave Circuit Design Ntes Transmissin Line Parameters fr a Simple Caxial Line Cnsider the fllwing crss sectin f a caxial cable, recalling that the EM field is cnfined t the dielectric regin. b a cnductrs dielectric (ε r ) The lumped element values fr the distributed line parameters f a cax may be derived frm electrstatic cnsideratins (see standard EM texts) and are given as fllws. ' ' '' R s ω μ b πε 1 1 π ε L= ln C= R= + G= b π a π a b b ln ln a a where R s 1 surface resistivity = σδ s [ Ω ] A few things t nte Bth gemetry (a and b) and materials prperties (μ, ε) play a rle in defining the line parameters. L and C are lssless, whereas R and G intrduce lss. In general, we will deal with practical TLs, that is, TLs that exhibit lw lss. If we assume that the metal used in the transmissin line has infinite cnductivity (σ, Perfect Electrical Cnductr -- PEC) and that the dielectric used is lssless (ε = 0), frm equatin 6 we find that the characteristic impedance f a cax is given by: R+jωL L 1 b μ' Z = = = ln. G+jωC C π a ε' Again in the lssless case, the prpagatin cnstant f a cax is purely imaginary, resulting in a phase cnstant slely dependent n material parameters as given by: μ' b πε' β= ω LC = ω ln = ω μ'ε' π a b ln a

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11 EE Planer Micrwave Circuit Design Ntes <end lecture 5> 6

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16 EE Planer Micrwave Circuit Design Ntes <end lecture 6> 31