Nguyễn Công Phương Engineering Electromagnetics Transmission Lines Contents I II III IV V VI VII VIII IX X XI XII XIII XIV XV Introduction Vector Analysis Coulomb’s Law & Electric Field Intensity Electric Flux Density, Gauss’ Law & Divergence Energy & Potential Current & Conductors Dielectrics & Capacitance Poisson’s & Laplace’s Equations The Steady Magnetic Field Magnetic Forces & Inductance Time – Varying Fields & Maxwell’s Equations Transmission Lines The Uniform Plane Wave Plane Wave Reflection & Dispersion Guided Waves & Radiation Transmission Lines - sites.google.com/site/ncpdhbkhn Transmission Lines 10 Introduction The Transmission Line Equations Lossless Propagation Transmission Line Equations & Their Solutions in Phasor Form Wave Reflection at Discontinuities Voltage Standing Wave Ratio Transmission Lines of Finite Length Some Transmission Line Examples Graphical Method Transients Analysis Transmission Lines - sites.google.com/site/ncpdhbkhn Introduction I1 R1 R2 + + + V2 V1 – E − I2 D − N Ida Engineering Electromagnetics Springer 2015 Transmission Lines - sites.google.com/site/ncpdhbkhn The Transmission Line Equations (1) I1 I1 I2 I2 + + D – – dz I + dI I ( z,t ) + Rdz + Ldz V + dV v Cdz − Transmission Lines - sites.google.com/site/ncpdhbkhn Gdz − The Transmission Line Equations (2) I + dI I ( z,t ) + Rdz + Ldz V Cdz − V + dV Gdz − I − ( I + dI ) − (Gdz )(V + dV ) − (Cdz )(V + dV )′ = −V + ( Rdz ) I + ( Ldz ) I ′ + V + dV = dI dV + ( Rdz ) i + ( Ldz ) =0 dt → dI + (Gdz )v + (Cdz ) dV = dt ∂I ∂V − = RI + L ∂z ∂t → − ∂I = GV + C ∂V ∂z ∂t Transmission Lines - sites.google.com/site/ncpdhbkhn The Transmission Line Equations (3) I + dI I ( z,t ) + Rdz Ldz v Cdz − + ∂I ∂V − = RI + L ∂z ∂t − ∂I = GV + C ∂V ∂z ∂t V + dV Gdz − ∂ 2V ∂ 2V ∂V = LC + ( LG + RC ) + RGV ∂z ∂t ∂t → 2 ∂ I ∂ I ∂V = LC + ( LG + RC ) + RGI ∂z ∂t ∂t Transmission Lines - sites.google.com/site/ncpdhbkhn Transmission Lines 10 Introduction The Transmission Line Equations Lossless Propagation Transmission Line Equations & Their Solutions in Phasor Form Wave Reflection at Discontinuities Voltage Standing Wave Ratio Transmission Lines of Finite Length Some Transmission Line Examples Graphical Method Transients Analysis Transmission Lines - sites.google.com/site/ncpdhbkhn Lossless Propagation (1) ∂I ∂V − ∂z = RI + L ∂t , − ∂I = GV + C ∂V ∂t ∂z ∂2V ∂ 2V ∂V = LC + ( LG + RC ) + RGV ∂z ∂t ∂t 2 ∂ I ∂ I ∂V = LC + ( LG + RC ) + RGI ∂z ∂t ∂t ∂ 2V ∂I ∂ 2V ∂V − ∂z = L ∂t ∂z = LC ∂t R = 0, G = → , 2 ∂ I ∂ V ∂ I ∂ I − = C = LC ∂z ∂z ∂t ∂t z z → V ( z , t ) = f1 t − + f t + = V + + V − v v Transmission Lines - sites.google.com/site/ncpdhbkhn Lossless Propagation (2) z V ( z , t ) = f1 t − + f t + v z + − = V + V v ∂f1 ∂f1 ∂(t − z / v) = = − f1′ ∂z ∂(t − z / v) dz v ∂f1 ∂f1 ∂(t − z / v) = = f1′ ∂t ∂(t − z / v) ∂t ∂2 f1 = f1′′, ∂t v ∂2 f1 = f1′′ ∂t → v= ∂2V ∂ 2V = LC ∂z ∂t Transmission Lines - sites.google.com/site/ncpdhbkhn LC 10 Graphical Method (1) 1+ Γ ZL − Z0 → Z L = Z0 Γ= ZL + Z0 1− Γ ZL = z L (normalized load impedance) Z0 1+ Γ → zL = 1− Γ + [ Re{Γ} + j Im{Γ}] → Re{z L } + j Im{ zL } = − [ Re{Γ} − j Im{Γ}] = − Re 2{Γ} − Im2{Γ} + j Im{Γ} [1 − Re{Γ}] + Im 2{Γ} Transmission Lines - sites.google.com/site/ncpdhbkhn 39 Graphical Method (2) Re{ z L} + j Im{z L } = − Re 2{Γ} − Im2{Γ} + j Im{Γ} [1 − Re{Γ}] + Im 2{Γ} Re{ zL } = − Re 2{Γ} − Im2{Γ} [1 − Re{Γ}] + Im2{Γ} → Re{ zL }[ Re{Γ} − 1] + Re ({Γ} − 1 + (= 0) 1 − + Re{Γ}Im {Γ} + Im {Γ} + + Re{ zL } + Re{ zL } 2 Re{z L } Re{z L } + Im {Γ} = → Re{Γ} − + Re{z L } + Re{z L } Transmission Lines - sites.google.com/site/ncpdhbkhn =0 40 Graphical Method (3) Re{ z L} + j Im{z L } = − Re 2{Γ} − Im2{Γ} + j Im{Γ} [1 − Re{Γ}] + Im 2{Γ} 2 Re{ z L } Re{ z L } Re{Γ} − + Im {Γ} = + Re{ z L } + Re{ z L } (Re{Γ} − 1)2 + Im{Γ} − 2 = Im{ z L } Im { z L } Transmission Lines - sites.google.com/site/ncpdhbkhn 41 Graphical Method (4) Re{ z L } Re{ zL } Re{Γ} − + Im 2{Γ} = + Re{ z L } + Re{ zL } ,0 & a radius of Equation of a circle, centered at + Re{z L } + Re{ z } L r = Re{z L } Im{Γ} Re{Γ} Transmission Lines - sites.google.com/site/ncpdhbkhn 42 Graphical Method (5) (Re{Γ} − 1) + Im{Γ} − = 21 Im{ zL } Im {z L } 1 Equation of a circle, centered at 1, & a radius of Im{ z } L Im{ z L } s = Im{ z L } Im{Γ} Re{Γ} Transmission Lines - sites.google.com/site/ncpdhbkhn 43 Graphical Method (6) Find the normalized load impedance ZL zL = = Re{z L } + j Im{ z L } Z0 Find the circle corresponding to Re{zL} Find the arc corresponding to Im{zL} The intersection of the circle & the arc is Γ Transmission Lines - sites.google.com/site/ncpdhbkhn 44 Graphical Method (7) Ex.: ZL = 25 + j100 Ω, Z0 = 50 Ω; Γ = ? Normalization: zL = (25 + j100)/50 = 0.5 + j2 The circle corresponds to 0.5 The arc corresponds to Γ is the intersection of the circle & the arc Γ = 0.52 + j0.64 Transmission Lines - sites.google.com/site/ncpdhbkhn 45 Transmission Lines - sites.google.com/site/ncpdhbkhn 46 Transmission Lines 10 Introduction The Transmission Line Equations Lossless Propagation Transmission Line Equations & Their Solutions in Phasor Form Wave Reflection at Discontinuities Voltage Standing Wave Ratio Transmission Lines of Finite Length Some Transmission Line Examples Graphical Method Transients Analysis Transmission Lines - sites.google.com/site/ncpdhbkhn 47 Transient Analysis (1) Γ= V+ V0 Z L − Z RL − Z = Z L + Z RL + Z I+ t=0 0V + Z0 RL = Z → Γ = V0 RL = → Γ = −1 Rg RL − z=0 z=L RL = ∞ → Γ = ΓL = Γg = RL − Z RL + Z Rg − Z Rg + Z VL V0 L/v Transmission Lines - sites.google.com/site/ncpdhbkhn t 48 L Transient Analysis (2) z (m) V1+ Γg t=0 L/v Z0 V0 RL Rg z=0 z=L VL = V1+ + V1− + V2+ + V2− + V3+ + V3− + 5L/v 6L/v = V1+ (1 + Γ L )(1 + Γg Γ L + Γ2g Γ L2 + ) 7L/v 1− Γ gΓ L 8L/v V1+ = V0 Z Rg + Z V2+ = Γ g Γ LV1+ V2− = Γ g Γ2LV1+ 4L/v = V1+ (1 + ΓL + Γg Γ L + Γ g Γ2L + Γ2g Γ L2 + ) = V1+ (1 + ΓL ) V1− = Γ LV1+ 2L/v 3L/v 9L/v 10L/v ΓL V3+ = Γ 2g Γ2LV1+ V3− = Γ 2g Γ3LV1+ V4+ = Γ 3g Γ3LV1+ V4− = Γ3g Γ4LV1+ V5+ = Γ 4g Γ4LV1+ V5− = Γ 4g Γ5LV1+ t (s) Transmission Lines - sites.google.com/site/ncpdhbkhn t (s) 49 L Transient Analysis (3) z (m) V1+ Γg t=0 L/v Z0 V0 RL Rg z=0 3L / z=L + − + − V +V +V +V V1+ +V1− 9L/v 3L 4v V3+ = Γ 2g Γ2LV1+ V3− = Γ 2g Γ3LV1+ V4+ = Γ 3g Γ3LV1+ V4− = Γ3g Γ4LV1+ V5+ = Γ 4g Γ4LV1+ 8L/v 0 V2− = Γ g Γ2LV1+ 6L/v 7L/v V1+ +V1− +V2+ V1+ V2+ = Γ g Γ LV1+ 4L/v 5L/v V3/4 V1− = Γ LV1+ 2L/v 3L/v 5L 4v 11L 4v 13L 4v t 10L/v ΓL 3L / V5− = Γ 4g Γ5LV1+ t (s) Transmission Lines - sites.google.com/site/ncpdhbkhn t (s) 50 L Transient Analysis (4) z (m) Γg t=0 L/v Z0 V0 RL Rg z=0 z=L I L = I1+ + I1− + I 2+ + I 2− + I 3+ + I3− + I1+ = V1+ / Z I1− = −V1− / Z0 I 2+ = V2+ / Z0 2L/v 3L/v I 2− = −V2− / Z0 4L/v 5L/v I 3+ = V3+ / Z0 I 3− = −V3+ / Z0 I 4+ = V4+ / Z0 6L/v 7L/v I 4− = −V4− / Z0 I 5+ = V5+ / Z0 8L/v 9L/v 10L/v ΓL I 5− = −V5− / Z0 t (s) Transmission Lines - sites.google.com/site/ncpdhbkhn t (s) 51 L Transient Analysis (5) z (m) Γg t=0 L/v Z0 V0 RL Rg z=0 z=L 3L / I1+ + − + I 2− = −V2− / Z0 I 3+ = V3+ / Z0 I 3− = −V3+ / Z0 I 4+ = V4+ / Z0 6L/v + I +I +I I1+ + I1− I 2+ = V2+ / Z0 4L/v 5L/v I3/4 I1− = −V1− / Z0 2L/v 3L/v − + − I +I + I +I 7L/v I 4− = −V4− / Z0 I 5+ = V5+ / Z0 8L/v 9L/v 0 3L 4v 5L 4v 11L 4v 13L 4v t 10L/v ΓL 3L / I1+ = V1+ / Z I 5− = −V5− / Z0 t (s) Transmission Lines - sites.google.com/site/ncpdhbkhn t (s) 52 Transient Analysis (6) Ex Rg = Z0 = 50 Ω, RL = 25 Ω, V0 = 10 V The switch is closed at time t = Determine the voltage at the load resistor and the current in the battery as functions of time ΓL = RL − Z0 25 − 50 = = −0.33 RL + Z0 25 + 50 Rg − Z 50 − 50 Γg = = =0 Rg + Z 50 + 50 V1+ = V0 10 Z0 = 50 = 5V Rg + Z0 50 + 50 t=0 Z0 V0 z (m) V1+ t (s) t (s) z (m) Γg V1+ I = = = 0.1A Z0 50 + 2L/v ΓL I1+ L/v V1− (−1.67) I =− =− = 0.033A Z0 50 ΓL V1− = Γ LV1+ 2L/v V1− = Γ LV1+ = (−0.33)5 = −1.67V − Rg Γg L/v RL I1− t (s) Transmission Lines - sites.google.com/site/ncpdhbkhn t (s) ... Wave Ratio Transmission Lines of Finite Length Some Transmission Line Examples Graphical Method Transients Analysis Transmission Lines - sites.google.com/site/ncpdhbkhn 28 Transmission Lines of... + RC ) + RGI ∂z ∂t ∂t Transmission Lines - sites.google.com/site/ncpdhbkhn Transmission Lines 10 Introduction The Transmission Line Equations Lossless Propagation Transmission Line Equations... = Re{Vs ( z) e } = jω t Transmission Lines - sites.google.com/site/ncpdhbkhn 13 Transmission Lines 10 Introduction The Transmission Line Equations Lossless Propagation Transmission Line Equations