Topic 5: Transmission Lines Profs. Javier Ramos & Eduardo Morgado Academic year.13-.14
Concepts in this Chapter Mathematical Propagation Model for a guided transmission line Primary Parameters Secondary Parameters Bandwidth and Attenuation Common Transmission Lines Copper pair Coaxial Microstrip Theory classes: 1.5 sessions (3 hours) Problems resolution:.5 session (1 hour)
Bibliography Líneas de Transmisión. Vicente E. Boria. Universidad Politécnica de Valencia. Sistemas de Telecomunicación. Transmisión por Línea. J. Hernando Rábanos. Servicio de Publicaciones de la ETSI Telecomunicación UPM 3
Problem Statement: a practical example An engineer tells you the measured clock is non-monotonic and because of this the flip flop internally may double clock the data. Dev a data Dev b Signal Measured here Clk Switch Threshold 4
Transmission Line Concept Voltage and current on a transmission line is a function of both time and position Must think in terms of position and time to understand transmission line behavior This positional dependence is added when the assumption of the size of the circuit being small compared to the signaling wavelength I 1 I V I = = f f ( z, t) ( z, t) V 1 V dz 5
Examples of Transmission Lines Cables and wires Coax cable Wire over ground Tri-lead wire Twisted pair (two-wire line) Long distance interconnects - - - - - 6
Transmission Line Definition General transmission line: a closed system in which power is transmitted from a source to a destination In this course we will study only TEM (transversal Electro- Magnetic) mode transmission lines A two conductor wire system with the wires in close proximity, providing relative impedance, velocity and closed current return path to the source Characteristic impedance is the ratio of the voltage and current waves at any one position on the transmission line Propagation velocity is the speed with which signals are transmitted through the transmission line in its surrounding medium Z = v = V I c εr 7
Transmission Lines as a Complex Electromagnetic Problem Both Electric and Magnetic fields are present in the transmission lines These fields are perpendicular to each other and to the direction of wave propagation for TEM mode waves, which is the simplest mode Electric field is established by a potential difference between two conductors. Implies equivalent circuit model must contain capacitor Magnetic field induced by current flowing on the line Implies equivalent circuit model must contain inductor I I I I H I I I V - - E - - V V I I I V H V V I I 8
Transmission Line Equivalent Circuit General Characteristics (Primary Parameters) Per-unit-length Capacitance (C ) [pf/m] Per-unit-length Inductance (L ) [nf/m] Per-unit-length (Series) Resistance (R ) [Ω/m] Per-unit-length (Parallel) Conductance (G ) [1/Ωm] dzr dzl dzg dzc dz 9
Ideal Transmission Line Ideal (lossless) Characteristics of Transmission Line Ideal TL assumes: Uniform line Perfect (lossless) conductor (R ) Perfect (lossless) dielectric (G ) dzl dzc 1
Transmission Line Equivalent Circuit B i ( z, t ) x x x v ( z, t ) - - - - - - - - - - z i(z,t) R z L z i(z z,t) v(z,t) - G z C z v(z z,t) - z i( z, t) v( z, t) = v( z z, t) i( z, t) R z L z t v( z z, t) i( z, t) = i( z z, t) v( z z, t) G z C z t 11
Transmission Line Signal Model Hence Now let z v( z z, t) v( z, t) i( z, t) = Ri( z, t) L z t i( z z, t) i( z, t) v( z z, t) = Gv( z z, t) C z t v i = Ri L z t i v = Gv C z t which are known as Telegrapher s Equations 1
Transmission Line Signal Model To combine these, take the derivative of the first one with respect to z: v i i R L z z z t = i i = R L z t z v = R Gv C t v v L G C t t Switch the order of the derivatives 13
Transmission Line Signal Model v v v v R Gv C L G C = z t t t Hence, we have v v v ( RG) v ( RC LG) LC = z t t The same equation also holds for i d V ( RG) V ( RC LG) jωv LC( ω ) V = dz 14
Transmission Line Signal Model d V ( ) ω ( ω ) = RG V j ( RC LG) V LC V dz RG jω ( RC LG) ω LC = ( R jω L)( G jωc) Z = R jωl = series impedance/length Y = G j ω C = parallel admittance/length We can re-write d V = ( ZY ) V dz 15
Transmission Line Signal Model Defining γ = ZY The differential equation governing the signal is d V = ( γ ) V dz Which solution is Where γ z V ( z) = Ae Be γ z ( R jωl)( G jωc) γ = z = j / z e θ γ = α jβ α, β α = β = attenuation contant phase constant π < θ < π 16
Transmission Line Signal Model Forward travelling wave (a wave traveling in the positive z direction): t = λ g V ( z) = V e = V e e γ z α z j z ( ) { α z jβ z jωt} v ( z, t) = Re V e e e = Re {( ) } jφ α z jβ z j ω t V e e e e α z ( ω β φ ) = V e cos t z V z e β = α z The wave repeats when Hence: βλ = g π π λ g 17
Transmission Line Secondary Parameters Attenuation V ( z) = V e = V e e γ z α z j z V ( z) γz az jβz = V e = V e e = V e az az e = attenuation λ g V z e α α Re 1 [( R jωl)( G jωc) ] = z t = 18
Transmission Line Secondary Parameters Phase Velocity v p (phase velocity) guided wavelength λ g π λg = [ m] β z Hence Set ωt β z = z v ( z, t) = α V e cos( ωt β z φ) dz ω β = dt dz = dt vp = ω β v p = In expanded form: ω {[ R jωl G jωc ] } 1/ Im ( )( ) constant ω β 19
Transmission Line Secondary Parameters Characteristic Impedance Z v z dv dz I (z) V (z) - V ( z) = V e γ z z V = Z = I I ( z) I e γ = Ri = RI = ZI i L t jωli γv e z A wave is traveling in the positive z direction. V ( z) Z I ( z) = ZI e γ z γ z (Z is a number, not a function of position) Z V Z Z = = = I γ Y Z 1/ R jωl = G jωc 1/ Z = R jωl Y = G jωc
Power-wise Power is proportional to the voltage squared P( d) e α d So, attenuation dependence with distance is At = K e α d When expressing it on dbs At( db) = K1 Kx (constants vary with frequency) Javier Ramos. Comunicaciones de Banda Ancha. 1
Transmission Line Model When forward and backward traveling waves Terminating impedance (load) V ( z) = V e V e 1 I ( z) = V e V e Z l V γ z γ z = γz γz γz V e 1 e V e L V = 1 Γ γ z γ z Γ = L Z Z L L ( γz e ) Γ L Load reflection coefficient Z Z
Impedance Matching in Transmission Lines Case 1: Matched load: (Z L =Z ) Z, β Z ( ) Z l No reflection from the load For any l 3
Impedance Matching in Transmission Lines Case : Short circuit load: (Z L = ) Z, β β Note: l = l π λ λ g Always imaginary! l / λ g S.C. can become an O.C. with a λ g /4 trans. line 4
Impedance Matching in Transmission Lines Case 3: Open Circuit (Z L = ) Homework 5
Impedance Matching in Transmission Lines Voltage Standing Wave Ratio Z, γ 1- ΓL z 1 Γ L 1 V ( z) V z = λ / z = Vmax Voltage Standing Wave Ratio ( VSWR ) = 1 Γ Vmin VSWR = 1 Γ L L 6
Transmission Lines Transmission lines commonly met on printed-circuit boards w ε r ε r h w h Microstrip Stripline w w w ε r h ε r h Coplanar strips Coplanar waveguide (CPW) 7
Transmission Lines ε is electric permittivity ε = 8.85 X 1-1 F/m (free space) ε r is relative dielectric constant Tipicalε r : 3.9 paper; 1 air; 3 to 4 polietilene and 4 to 6 PVC µ = µ r µ ; ε = r ε ε µ is magnetic permeability ρ µ -7 = 4p X 1 H/m (free space) µ r is relative permeability For most of dielectric materials, µ r is close to one is resistivity: Cooper: 17.4; Bronze 19; Iron 1 (x 1-8 Ω/m) 8
Microstrip Transmission Line w t ε r h ( f ) ( ) ( ) ( f ) eff eff εr 1 εr Z ( f ) = Z ( ) eff eff εr 1 εr Z ( ) = ε eff r eff eff εr ε εr ( f ) = εr () 1 4F ε ( ) 1π ( ) ( w h) ( w h) ( ) / 1.393.667 ln / 1.444 eff r 1.5 () ε 1 ε 1 1 ε 1 t / h = 1 1 ( h / w) 4.6 w / h eff r r r r ( w / h 1) t h w = w 1 ln π t 9
Copper Pair / Twisted Pair Primary parameters of a Copper Pair µ L =.3 log π ρ R = π 4 d πε C =,43 D log d D d D: Distance between cables d: diameter of each cable G: conductance depends on the dialectric, usually negligible but depending on the frequency Javier Ramos. Comunicaciones de Banda Ancha. 3
Coaxial Cable a b ε r, σ L π ε ε r C = [ F/m] b ln a µ µ r = π b a ln [ H/m] R G = = ρ 1 πa πσ b ln a 1 πb [S/m] 31
Summary of Concepts in this Chapter What are the primary parameters of the transmission lines Calculation of attenuation, and propagation velocity for transmission lines Calculate above parameters for common transmisión lines 3