Telegrapher’s Equations

Presentation on theme: "Telegrapher’s Equations"— Presentation transcript:

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 Reflection at a load Z0 ZL G = (ZL-Z0)/(ZL+Z0) = |G|ejqr
Transmission Line 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 Nailing down V0+ Iin ZS Zin Z0 ZL VS Vin V(z) = V+0(e-jbz + Gejbz)
Set Vin = IinZin = VSZin/(ZS+Zin) = V(-l) = V0+(ejbl + Ge-jbl)  gives V0+ Vin Zin Z0 Transmission Line LOAD ZL 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|

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|

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

13 (Antireflection coating)
Applications (Antireflection coating) Transmission Line Z0 ZL ZS Trick: Looking from Source side, effective impedance of line should be matched with ZS 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

14 (Antireflection coating)
Applications (Antireflection coating) Transmission Line Z0 ZL ZS 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 P(t) = IV = VaIa[cos(2wt+f) + cos(f)]
Power Transfer V(z) = V0(e-jbz + Gejbz) I(z) = V0(e-jbz - Gejbz)/Z0 P(z) = ½ Re[V(z)I*(z)] V = Vacos(wt) I = Iacos(wt+f) P(t) = IV = VaIa[cos(2wt+f) + cos(f)] P = ∫ P(t)dt = VaIacos(f) T 1 = ½ Re[VI*]

16 Power Transfer V(z) = V0(e-jbz + Gejbz) I(z) = V0(e-jbz - Gejbz)/Z0
Pi = |V0|2/2Z0 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 Zs Adjust d so Yin = Y0 + jB
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.