ECE 497 JS Lecture -03 Transmission Lines Spring 2004 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jose@emlab.uiuc.edu 1
MAXWELL S EQUATIONS B E = t Faraday s Law of Induction H = J D + t Ampère s Law D = ρ Gauss Law for electric field B = 0 Gauss Law for magnetic field Constitutive Relations B = µ H D = ε E 2
WAVE PROPAGATION λ λ z Wavelength : λ λ = propagation velocity frequency 3
Why Transmission Lines? In Free Space At 10 KHz : λ= 30 km At 10 GHz : λ = 3 cm Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength. 4
Transmission Line Model Let d be the largest dimension of a circuit circuit z If d << λ, a lumped model for the circuit can be used λ 5
Transmission Line Model circuit z If d λ, or d > λ then use transmission line model λ 6
Modeling Interconnections Low Frequency Mid-range Frequency L T High Frequency C T /2 C T /2 Z o Short L T /2 or L T /2 Transmission Line C T Lumped Reactive CKT 7
Single wire near ground h d + for d << h, Z = o 1 µ 4h ln 2π ε d -1 Z o = 120 cosh (D/d) C = 2πε 4h ln d L = µ ln 2π 4h d 8
Single wire between grounded parallel planes ground return h/2 + d h/2 Z = o 1 µ 4h ln 2π ε πd 9
Wires in parallel near ground D + + d h for d << D, h { } ( 69/ ε ) r log10 ( 4 / ) 1 ( 2 / ) Z = h d + h D O 2 10
Balanced, near ground D + d h for d << D, h Z O ( ε ) r 276/ log ( 2 D/ d) = 10 2 ( D h) 1 + /2 11
Open 2-wire line in air d D -1 Z o = 120 cosh (D/d) 12
Parallel-plate Transmission Line w a µ, ε L = µa w C = εw a 13
Types of Transmission Lines ε r Coaxial line Air Waveguide e r Coplanar line e r Microstrip e r Stripline e r Slot line 14
Coaxial Transmission Line µ ε a b TEM Mode of Propagation L = µ ln b a C = 2πε ln(b/a) 15
Coaxial Air Lines µ ε a b Infinite Conductivity Zo = µ / ε ln / 2π ( b a) Finite Conductivity ( 1/ a+ 1/ b) µ / ε Zo = ln ( b/ a) 1 + 1 j 2π 4 π fµσ ln( b/ a) ( ) 16
Coaxial Connector Standards µ ε a b Connector Frequency Range 14 mm DC - 8.5 GHz GPC-7 DC - 18 GHz Type N DC - 18 GHz 3.5 mm DC - 33 GHz 2.92 mm DC - 40 GHz 2.4 mm DC - 50 GHz 1.85 mm DC - 65 GHz 1.0 mm DC - 110 GHz 17
Coaxial Connectors Source: Allied Electronics, Inc. 18
Microstrip ε 19
Microstrip Microstrip Characteristic Impedance 100 90 Zo (ohms) 80 70 60 h = 21 mils h = 14 mils h = 7 mils 50 40 30 0 1 2 3 W/h dielectric constant : 4.3. 20
Electric Field Configuration Microstrip Stripline Consequence: Wave propagation in stripline is closer to the TEM mode of propagation and the propagation of velocity is approximately c/ ε r. 21
TEM PROPAGATION I L + V C - z z Z 1 Z o β Z 2 V s 22
Reflection in Transmission Lines 1. 2. 3. 23
I Telegrapher s Equations L + V C - z V = z I L t I = C z V t L: Inductance per unit length. C: Capacitance per unit length. 24
Transmission Line Solutions z (Frequency Domain) Z o β forward wave backward wave β = ω LC V( z) j z = Ae β + j z + Be β Z o = L C I( z) 1 jβ z j z = Ae Z Be + β o 25
Transmission Line Equations z (Frequency Domain) V s Z 1 Z o β Z 2 l A = TVs ( ω) 1 ΓΓ e 1 2 2 jβl T = Z 1 Zo + Z o 2 jβl B =Γ2e A ωl β = v o Γ Γ 2 1 2 o = = Z Z Z Z 1 1 2 Z + Z Z + Z o o o 26
Transmission Line Equations (Frequency Domain) A= TV Γ Γ e k k 2 jω kl / vo 1( ω) 1 2 k= 0 B = TV Γ Γ e k k+ 1 2 jω ( k+ 1) l/ vo 1( ω) 1 2 k= 0 k k 2 / / ( ) j ω kl v o j ω z v o Vf z = Γ1Γ2e e TV1( ω) k= 0 k k+ 1 2 ( 1) / / ( ) j ω k + l v o + j ω z v o Vb z = Γ1Γ2 e e TV1( ω) k= 0 V( z) = V ( z) + V ( z) f b 27
Time-Domain Solutions k k Vf (,) z t = T Γ1Γ2V t k= 0 z + 2kl v o = Γ Γ k k 1 Vb (,) z t T + 1 2 V t k= 0 z 2( k+ 1) l v o V( z, t) = V ( z, t) + V ( z, t) f b 28
Wave Shifting Solution* v f1 v f2 Z 1 Z o τ Z 2 v b1 V g1 v () t =Γv ( t τ ) + TV () t f1 1 b2 1 g1 v () t =Γ v ( t τ ) + TV () t b2 2 f1 2 g2 v () t = v ( t τ ) f 2 f1 v () t = v ( t τ ) b1 b2 v () t = v () t + v () t 1 f1 b1 v () t = v () t + v () t 2 f 2 b2 line length τ = delay= velocity v b2 Γ = 1 T 1 2 = Γ = T 2 V g2 = Z Z Z 1 1 1 Z Z Z Z + Z Zo + Z 2 2 2 o o o Z + Z Zo + Z o o o *Schutt-Aine & Mittra, Trans. Microwave Theory Tech., pp. 529-536, vol.36 March 1988. 29
Frequency Dependence of Lumped Circuit Models At higher frequencies, a lumped circuit model is no longer accurate for interconnects and one must use a distributed model Transition frequency depends on the dimensions and relative magnitude of the interconnect parameters. f 0.3 109 t r 0.35 10d ε r f 30
Lumped Circuit or Transmission Line? Determine frequency or bandwidth of signal RF/Microwave: f= operating frequency Digital: f=0.35/t r Determine the propagation velocity and wavelength Material medium v=c/(ε r ) 1/2 Obtain wavelength λ=v/f Compare wavelenth with feature size If λ>> d, use lumped circuit: L tot = L* length, C tot = C* length If λ 10d or λ<10d, use transmission-line model 31
Frequency Dependence of Lumped Circuit Models Level Dimension Frequency Edge rate PCB line 10 in > 55 MHz < 7ns Package 1 in > 400 MHz < 0.9 ns VLSI int* 100 um > 8 GHz < 50 ps * Using RC criterion for distributed effect 32
Metallic Conductors Area σ Length Re sist an ce : R Package level: W=3 mils R=0.0045 Ω/mm R = Le ng th σ Area Submicron level: W=0.25 microns R=422 Ω/mm 33
Metal Metallic Conductors Conductivity σ(ω -1 m 1 10-7 ) Silver 6.1 Copper 5.8 Gold 3.5 Aluminum 1.8 Tungsten 1.8 Brass 1.5 Solder 0.7 Lead 0.5 Mercury 0.1 34
Loss in Transmission Lines RF SOURCE 35
Skin Effect in Transmission Lines δ Low Frequency High Frequency Very High Frequency 36
.. Skin Effect in Microstrip D t d w σ y J = J o e - y /d e - jy/ d V Magnitude of current density e ε r 37
Skin Effect The electric field in a material medium propagates as E o e γz = E o e αz e jβz where γ = α + jβ. We also have z γ = ω µε(1+j σ ωε ). W int H int δ s CURRENT AREAS δ s 38
Lossy Transmission Line I + R L V - C G z Telegraphers Equation V z = (R+jωL)I = ZI I z = (G+jωC)V = YV 39
Lossy Transmission Line z R, L, G, C, forward wave backward wave 40