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Telegrapher ’ s Equations

Telegrapher ’ s Equations. dV/dz = -Z ’ I. dI/dz = -Y ’ V. d 2 V/dz 2 = g 2 V V = V 0 + (e - g z + G e g z ) I = V 0 + (e - g z - G e g z )/Z 0. = Z ’ Y ’ = (R ’ +j w L ’ )(G ’ +j w C ’ ) = a + j b : Decay constant

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Telegrapher ’ s Equations

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  1. Telegrapher’s Equations dV/dz = -Z’I dI/dz = -Y’V d2V/dz2 = g2V V = V0+(e-gz + Gegz) I = V0+(e-gz - Gegz)/Z0 • = Z’Y’ = (R’+jwL’)(G’+jwC’) = a + jb : Decay constant Z0 = Z’/Y’ = (R’+jwL’)/(G’+jwC’): Characteristic Impedance

  2. Lossless Line R’ = G’ = 0 g =jb = jwL’C’ Z0 = L’/C’ = 377 W in free space (impedance of free space) a=0 (lossless) v = w/b: indep. of frequency (dispersionless) • Dispersionless: undistorted signal • Genl. Condition: R’/G’ = L’/C’

  3. Transmission Line LOAD Reflection at a load Z0 ZL V(z) = V+0(e-jbz + Gejbz) I(z) = V+0(e-jbz - Gejbz)/Z0 V(0)/I(0) = ZL  G = (ZL-Z0)/(ZL+Z0) = |G|ejqr ZL=Z0 [(1+G)/(1-G)]

  4. Work backwards to input impedance Zin Z0 ZL Transmission Line LOAD V(z) = V+0(e-jbz + Gejbz) I(z) = V+0(e-jbz - Gejbz)/Z0 V(-l)/I(-l) = Zin Zin=Z0 [(1+Gejbl)/(1-Gejbl)] ie, to get Z(d), G Ge-jbd in ZL formula

  5. Zin Z0 ZL Vin Transmission Line LOAD Set Vin = IinZin = VSZin/(ZS+Zin) = V(-l) = V0+(ejbl + Ge-jbl)  gives V0+ Nailing down V0+ Iin ZS VS V(z) = V+0(e-jbz + Gejbz) Why is V(-l)/I(-l) ≠ Zs? Because we haven’t included the source current Iin Iin = VS/(ZS + Zin)

  6. Special Cases: Impedance match. (ZL = Z0) G = 0 No reflection

  7. Special Cases: Short ckt. (ZL = 0) G = -1 Refln. at a hard wall (phase p)

  8. Special Cases: Open Ckt. (ZL = ∞) G = 1 Refln. at a soft wall (phase 0)

  9. Special Cases: Reactive load(ZL: imaginary) • = ejf |G| = 1 Fully reflected, but with intermediate phase (reactive component picked up)

  10. V(z)s for diff cases V(z) = V+0(e-jbz + ejbz) = 2V+0cos(bz) for open ckt V(z) = V+0(e-jbz - ejbz) = 2jV+0sin(bz) for short ckt V(z) = V+0e-jbz for matched load 2|V+0| |V+0| http://www.bessernet.com/Ereflecto/tutorialFrameset.htm

  11. Standing Wave Ratio V(z) = V+0(e-jbz + Gejbz) |V(z)| = V+01 + |G|2 + 2|G|cos(2bz+qr) S = VSWR = |Vmax|/|Vmin| = (1+|G|)/(1-|G|) = 1 for matched load, ∞ for open/short/reactive load CSWR = |Imax|/|Imin| http://www.bessernet.com/Ereflecto/tutorialFrameset.htm

  12. ZS Z0 ZL Transmission Line Applications (Antireflection coating) GL = (ZL-Z0)/(ZL+Z0) ZL=Z0 [(1+GL)/(1-GL)] Zin=Z0 [(1+GLe-2jbl)/(1-GLe-2jbl)]

  13. ZS Z0 ZL Transmission Line • Qr. wave plate (Phase p for reflection to cancel incident) 2. “Bridge” impedance is geometric mean (Depends on product + correct dimensions) Zin=ZS  (Zin/Z0) =  (ZS/Z0) • ZS = ZL • bl=p/2, l=l/4, Z0 = ZSZL Applications (Antireflection coating) Trick: Looking from Source side, effective impedance of line should be matched with ZS

  14. ZS Z0 ZL Transmission Line Applications (Antireflection coating) Explanation: Consider incident and reflected waves To cancel, reflected wave must • have opposite phase to incident in order to cancel it • be of equal strength as incident wave for complete cancellation Reflected wave has extra phase 2bL relative to incident For condition (a), need 2bL = p, implying 2(2p/l)L=p L = l/4 For condition (b), need ZL/Z0 = Z0/Zs  Z0 = ZL.Zs

  15. V = Vacos(wt) I = Iacos(wt+f) T 1 P = ∫ P(t)dt = VaIacos(f) T 0 Power Transfer V(z) = V0(e-jbz + Gejbz) I(z) = V0(e-jbz - Gejbz)/Z0 P(z) = ½ Re[V(z)I*(z)] P(t) = IV = VaIa[cos(2wt+f) + cos(f)] ½ ½ = ½ Re[VI*]

  16. Pi = |V0|2/2Z0 Power Transfer V(z) = V0(e-jbz + Gejbz) I(z) = V0(e-jbz - Gejbz)/Z0 P(z) = ½ Re[V(z)I*(z)] Pt = P(0) = Re[V0(1+|G|ejq)V0*(1-|G|e-jq)/2Z0] = |V0|2(1-|G|2)/2Z0 Power transfer Pt/Pi = 1 - |G|2 (No reflection  all power transferred)

  17. Impedance Matching d Z0 Z0 ZL Matching stub Zs Matching network Adjust d so Yin = Y0 + jB cos(bd) = -G, B =[(ZL2-Z02)/(2ZLZ0)]1-G2 Adjust the reactance of the stub so Ys = -jB bd =cot-1(B/Y0)

  18. Summary • Learned key concepts – wave propagation, reflection, phase and impedance matching, power transmission etc • Note that we did not worry about how the waves were created, or what determines the parameters such as C, Z0, L etc. We will learn that in chapters 4-7. • We will encounter these wave properties again when we talk about real EM waves that also reflect, transmit etc at boundaries. But they do so in 3-D, which needs the language of vector algebra.

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