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1 MIT OpenCourseWare /ESD.013J Electromagnetics and Applications, Fall 2005 Please use the following citation format: Markus Zahn, 6.013/ESD.013J Electromagnetics and Applications, Fall (Massachusetts Institute of Technology: MIT OpenCourseWare). (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit:

2 Electromagnetics and Applications Fall 2005 Lecture 10 - Transmission Lines Prof. Markus Zahn October 13, 2005 I. Transmission Line Equations A. Parallel Plate Transmission Line E must be perpendicular to the electrodes and H must be tangential, so E = Ex (z, t)ī x H = Hy (z, t)ī y H E x H y E = µ = µ t z t E H y E x H = ɛ = ɛ t z t From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, Used with permission. 1

3 2 v(z, t) = E d l = E x (z, t)d 1 z = constant i(z, t) = K z (z, t)w = H y (z, t)w v i µd = L L = henries / meter z t w Inductance per unit length i v ɛw = C C = farads / meter z t d Capacitance per unit length B. Transmission Line Structures µd ɛ w 1 LC = = µɛ = w d 2 c 2

4 C. Distributed Circuit Representation with Losses i(z, t) i(z + Δz, t) = CΔz v(z, t) + GΔz v(z, t) t v(z, t) v(z + Δz, t) = LΔz i(z + Δz, t) + i(z + Δz, t)rδz t i(z + Δz, t) i(z, t) i v lim = = C Gv Δz 0 Δz z t v(z + Δz, t) v(z, t) v i lim = = L ir Δz 0 Δz z t R is the series resistance per unit length, measured in ohms/meter, and G is the shunt conductance per unit length, measured in siemens/meter. 3

5 If the line is lossless (R = G = 0), we have the Telegrapher s equations: i z v z v = C t i = L t Including loss, Poynting s theorem for the circuit equivalent form is: i v v z = C t Gv v i i = L ir z t ] i v (vi) [ 1 1 Add: v + i = = Cv 2 + Li 2 Gv 2 i 2 R z z z t 2 2 D. Wave Equation (Lossless, R = 0, G = 0) i v 2 i 2 v = C t z t z t = C t 2 v i 2 i 1 2 v z z = L t z t = L z v 2 v 2 v 2 v 1 2 v = C = LC = L z 2 t } z 2 t {{ 2 c t } II. Sinusoidal Steady State Wave equation A. Complex Amplitude Notation jωt v(z, t) = Re vˆ(z)e i(z, t) = Re î(z)e jωt Substitute into the wave equation: 2 v 1 2 v d 2 vˆ ω 2 ω z 2 = c 2 t 2 dz 2 = c 2 vˆ(z), let k = c d 2 vˆ dz 2 + k2 vˆ = 0 vˆ(z) = Vˆ+e jkz + Vˆ e +jkz dvˆ 1 ( ) jk Vˆ+e jkz +jkz = Ljωî î(z) = + jk Vˆ e dz Ljω k ω k LC = = C LC = = = Y0 is the Line Admittance ω c ω ωl L L 1 L Z 0 = = is the Line Impedance Y 0 C ( ) î(z) = Y Vˆ+e jkz +jkz 0 Vˆ e 4

6 vˆ(z) = Vˆ+e jkz + Vˆ e +jkz v(z, t) = Re Vˆ+e j(ωt kz) + Vˆ e j(ωt+kz) i(z, t) = Re Y 0 Vˆ+e j(ωt kz) Vˆ e j(ωt+kz) k = ω = ω LC = ω ɛµ c B. Short Circuited Line (v(z = 0, t) = 0, v(z = l, t) = V 0 cos(ωt)) vˆ(z) = Vˆ+e jkz + Vˆ e +jkz vˆ(z = 0) = 0 = Vˆ+ + Vˆ Vˆ+ = Vˆ ( ) vˆ(z = l) = V 0 = Vˆ+e +jkl + Vˆ e jkl = Vˆ+ e jkl e jkl 5

7 = 2j Vˆ+ V 0 V ˆ + = Vˆ = 2j V ( ) 0 vˆ(z) = e jkz +jkz = V 0( 2j) sin(kz) e 2j 2j V 0 sin(kz) = ( ) ( ) î(z) = Y 0 Vˆ+e jkz Vˆ e jkz Y 0 V 0 = e jkz + e +jkz 2j 2 Y 0 V 0 cos(kz) = 2 j jy 0 V 0 cos(kz) = V 0 sin(kz) V 0 sin(kz) cos(ωt) v(z, t) = Re vˆ(z)e jωt = Re e jωt = [ ] jy 0 V 0 cos(kz) Y 0 V 0 cos(kz) sin(ωt) i(z, t) = Re î(z)e jωt = Re e jωt = We have resonance when = 0 kl = nπ = ωl ω = ω n nπc c l, n = 1, 2, 3,... Complex impedance: Z(z) = vˆ(z) = jz 0 tan(kz) î(z) In the following, take n = 1, 2, 3,...: Z(z = l) = +jz 0 tan(kl) kl = nπ Z(z = l) = 0 short circuit π kl = (2n 1) 2 Z(z = l) = open circuit π (n 1)π < kl < (2n 1) 2 Z(z = l) = +jx, X > 0 (positive reactance, inductive) 1 (n 2 )π < kl < nπ Z(z = l) = jx, X > 0 (negative reactance, capacitive) kl 1 Z(z) = jz 0 k L = j ω L C C = jlz Z(z = l) = j(ll) inductive V 0 z kz 1 v(z, t) = l cos(ωt) v(z = l, t) = V 0 cos(ωt) V 0 Y 0 i(z, t) = kl 6 di = (Ll) (z = l, t) dt V 0 sin(ωt) sin(ωt) i(z = l, t) = (Ll)ω

8 C. Open Circuited Line (i(z = 0, t) = 0) v(z = l, t) = V 0 sin(ωt) î(z) = Y 0 Vˆ+e jkz Vˆ e +jkz î(z = 0) = 0 = Y 0 Vˆ+ Vˆ Vˆ+ = Vˆ ( ) vˆ(z = l) = jv 0 = Vˆ+e +jkl + Vˆ e jkl = Vˆ+ e jkl + e jkl = 2 Vˆ+ cos(kl) jv 0 Vˆ+ = Vˆ = 2 cos(kl) vˆ(z) = jv 0 ( e jkz + e +jkz ) 2 cos(kl) jv 0 2 cos(kz) = 2 cos(kl) jv 0 cos(kz) = cos(kl) î(z) = jy ( 0V ) 0 e jkz e +jkz 2 cos(kl) ( jy 0 V 0 )( 2j) sin(kz) = 2 cos(kl) Y 0 V 0 sin(kz) = cos(kl) Complex Impedance v(z, t) = Re [ vˆ(z)e jωt] = V 0 cos(kz) sin(ωt) cos(kl) i(z, t) = Re [ î(z)e jωt ] = V 0Y 0 sin(kz) cos(ωt) cos(kl) π Resonance: cos(kl) = 0 (kl) = (2n 1) 2, n = 1, 2, 3,... (2n 1) π ω n = 2 2l vˆ(z) Z(z) = = Z 0 j cot(kz) î(z) Z(z = l) = jz 0 cot(kl) kl 1 v(z, t) = V 0 sin(ωt) i(z, t) = V 0 Y 0 kz cos(ωt) dv i(z = l, t) = (Cl)ωV 0 cos(ωt) = (Cl) (z = l, t) dt 7

9 Open circuited line Impedance for short and open circuited wires 8