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Lecture 3 Homework 1. The fields of a traveling TEM wave inside the coaxial line shown left y can be expressed as a ρ V0 r -g z I 0 f -g z E= e ; H= e r ln b / a 2pr b μ, where is the propagation constant of the line. The conductors are assumed to have a surface resistivity Rs, and the material filling the space between the conductors is assumed to have a complex permittivity = ’ - j" and a permeability μ = μ0μr. Determine the transmission line parameters (L,C,R,G). y x w 2. For the parallel plate line shown left, derive the R, L, G, and C parameters. Assume w >> d. d r x z 1.4 Field analysis of transmission lines The Telegrapher equations Derived from Field Analysis y The fields inside the coaxial line will satisfy Maxwell's curl equations E j H a b H j E ρ x μ, Expanding the above equations in cylindrical coordinates and then gives the following vector equations (TEM waves,Ez = Hz = 0, no ψdependence) 0 0 ¶Er ¶E0 ¶ f -r +f +z (r Ef ) = - jwm (rH r + f Hf ) ¶z ¶z0 r¶r 0 0 ¶H r ¶Hf ¶ -r +f +z ( r Hf ) = - jwe ( rEr + f Ef ) ¶z ¶z r¶r Since the z components of these two equations must vanish and E = 0 at ρ = a and b, the components must have the forms Ej = H r = 0. Er = h(z) r , Hj = g(z) r (p: 55-56) Using the above equations, we obtain h( z ) jg ( z ) z g ( z ) jh( z ) z Eliminate h(z) and g(z) ¶V (z) wm ln b / a =-j I(z) ¶z 2p ¶I(z) 2p V (z) = - jw (e '- je '') ¶z ln b / a Voltage between the two conductors: V(z) = b b a a ò Er (r, z)d r = h(z) ò dr r = h(z)ln b a Current on the inner conductor: I(z) = 2p Substitute the coefficients by L, G, C ò Hj (a, z)adj = 2p g(z) j =0 The same telegraphic equations as derived from distributed theory. ¶V (z) = - jw LI(z) ¶z ¶I(z) = -(G + jeC)V (z) ¶z Waves in lossless coaxial waveguides h( z ) jg ( z ) z g ( z ) jh( z ) z Helmholtz equation ¶2 h(z) 2 + w me h(z) = 0 2 ¶z ¶2 g(z) 2 + w me g(z) = 0 2 ¶z k = w me A+ k m = = B+ wm e For one-way propagation, eg, along +z axis h(z) 1 + -ik×z Er = = Ae r r g(z) 1 + -ik×z Hf = = Be r r Electric and magnetic field (TEM): Er = Hf = h(z) 1 + -ik×z - ik×z = (A e + A e ) r r g(z) 1 + -ik×z = (B e + B-eik×z ) r r Waves in lossless coaxial waveguides Voltage between the two conductors at z = 0 V(0) = b b ò Er (r, 0)d r = A ò + a a dr Er = V0 A = ln b / a b = A ln r a + + h(z) V0 = e-ik×z r r ln b / a Current on the inner conductor (Ampere’s circuit law) at z = 0: I(0) = 2p ò Hj (a, z)adj = 2p B j =0 + B+ = I0 2p Hf = g(z) I 0 -ik×z = e r 2pr Propagation Constant, impedance and Power Flow for the Lossless coaxial Line Propagation constant: b = k = w me = w LC (for a lossless medium) TEM transmission lines have the same form of propagation constant as that for plane waves in a lossless medium. Wave impedance: Zw E H Characteristic impedance: Z0 ( ¶Er ¶z = - jwm H f ) V0 E ln b / a ln b / a ln b / a I0 2H 2 2 Power flow (computed from the Poynting vector): 2 b V0 I 0 1 1 1 P E H ds d d V I 0 0 2S 2 0 a 22 ln b / a 2 (agreement with circuit theory) 1.5 The terminated lossless transmission line What is a voltage reflection coefficient? V(z) = V + (z)+V - (z) I(z) = I + (z)+ I - (z) Assume an incident wave ( V0 e jz ) generated from a source at z < 0. We have seen that the ratio of voltage to current for such a traveling wave is Z0, the characteristic impedance. But when the line is terminated in an arbitrary load ZL Z0, the ratio of voltage to current at the load must be ZL. Thus, a reflected wave must be excited with the appropriate amplitude to satisfy this condition. What is a voltage reflection coefficient? Total voltage and current on the line (superposition of incident and reflected waves): V ( z ) V0 e jz V0 e jz V0 jz V0 jz I ( z) e e Z0 Z0 (V0+: incident; V0-: reflected) The total voltage and current at the load are related by the load impedance, so at z = 0, we must have V (0) V0 V0 ZL Z0 I (0) V0 V0 Voltage reflection coefficient Γ: V0 Z L Z 0 0 V0 Z L Z0 (Phase difference: π) What is a voltage reflection coefficient? The total voltage and current waves on the line : V (z) = V0+e- jb z +V0-e jb z = V0+ [e- jb z + Ge jb z ] V0+ - jb z V0- jb z V0+ - jb z I(z) = e e = [e - Ge jb z ] Z0 Z0 Z0 V0G= + V0 Consider the time-average power flow along the line at the point z: 2 V 1 1 2 0 Pav Re[V ( z ) I ( z )* ] Re{1 *e 2 jz e 2 jz } 2 2 Z0 which can be simplified: 2 1 V0 2 Pav (1 ) 2 Z0 • Constant average power flow at any point on the line; • Total power delivered to the load = incident power – reflected power What is a voltage reflection coefficient? (a). Standing wave ( = -1) (b). Voltage standing wave ratio ( < 1) V ( z ) V0 1 e 2 jz V0 1 e 2 jl SWR Vmax 1 Vmin 1 RL 20 log (l = -z) (1 SWR < , where SWR=1 implied a match load.) (dB) (return loss) What is a voltage reflection coefficient? (l) The reflection coefficient at z = -l: V0 (0) Zin ZL Z0 V0 e jl (l ) jl (0)e 2 jl V0 e V0 z l 0 At a distance l = -z from the load, the input impedance seen looking toward the load is jl jl 1 e 2 jl V (l ) V0 (e e ) Z in Z0 Z0 I (l ) V0 (e jl e jl ) 1 e 2 jl A more usable form of input impedance: ( Z L Z 0 ) e j l ( Z L Z 0 ) e j l ) Z in Z 0 ( Z L Z 0 ) e j l ( Z L Z 0 ) e j l ) Z0 Z L cos l jZ0 sin l Z 0 cos l jZ L sin l Z0 Z L jZ0 tan l Z 0 jZ L tan l • Input impedance of oe portion of transmission line with an arbitrary load impedance. • Transmission line impedance Equation. 1.5The terminated lossless transmission line Voltage and current in the line: V ( z ) V0 [e jz e jz ] V0 jz I ( z) [e e jz ] Z0 (1). Short circuit transmission line (ZL = 0) voltage Zin d=l 0 current impedance (2). Open circuit transmission line ZL = voltage Zin current impedance (3). Quarter-wave transmission line (l / 4 n / 2, n 1,2,3,...) (4). Interface of two transmission lines Reflection coefficient: Transmission coefficient: Insertion loss: Grad, Div and Curl in Cylindrical Coordinates