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Transmission Lines ® Transformation of voltage, current and impedance ® Impedance ® Application of transmission lines 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 The Telegraphist Equations ® We can rewrite the above equations as (Telegraphist Equations) ∂V iωZo = I ∂z v ∂I iω = V ∂z Zo v ® See the equivalent web brick derivation in terms of the inductance and capacitance per unit length along the line. ® The Telegraphist Equations become ∂V Zo µo d = iωLI, L = = ∂z v w ∂I 1 ǫoǫr w = iωCV, C = = ∂z Zo v d 2 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 The Telegraphist Equations ® The velocity and characteristic impedance of the line can be expressed in terms of L and C. 1 L v= Zo = LC C ® L and C are the inductance and capacitance per unit length along the line. ® For coaxial cable the formula is quite different (a,b inner, outer radii). 2πǫoǫr 2πln(b/a) C= L= ln(b/a) µo µo ln(b/a) 1 Zo = v= ǫoǫr 2π µoǫoǫr 3 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Proof of the Coaxial Cable Relations ® L and C are the inductance and capacitance per unit length along the line. ® For coaxial cable C and L are given by, 2πǫoǫr 2πln(b/a) C= L= ln(b/a) µo µo ln(b/a) 1 Zo = v= ǫoǫr 2π µoǫoǫr ® On board. 4 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Reﬂection Coefﬁcient ® Consider a wave propagating toward a load ® In general there is a wave reﬂected at the load. The total voltage and current at the load are given by, Vload = Vf + Vr Iload = If + Ir where Vf = ZoIf Vr = −ZoIr 5 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Reﬂection Coefﬁcient ® At the load, ZL Vload = ZLIload = ZL If + Ir = Vf + Vr = Vf − Vr Zo ® Solving for ρ = Vr /Vf , we obtain, ZL − Z o ρ= ZL + Z o ® ρ is the reﬂection coefﬁcient. ® If ZL = Zo there is no reﬂected wave. ® A line terminated in a pure reactance always has |ρ| = 1 6 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Impedance Transformation Along a Line ® Consider a transmission line terminated in an arbitrary impedance ZL. ® The impedance Zin seen at the input to the line is given by ZL + jZo tan kL Zin = Zo Zo + jZL tan kL ® If ZL = Zo, then Zin = Zo. ® If ZL = 0, then Zin = jZo tan kL ® If ZL = ∞, then Zin = Zo/(j tan kL) 7 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Voltage and Current Transformation Along a Line ® Consider a transmission line terminated in an arbitrary impedance ZL. ® The voltage Vin and current Iin at the input to the line are given by Vin = Vend cos kL + jZoIend sin kL V Iin = Iend cos kL + j end sin kL Zo ® If a line is unterminated then the voltage and current vary along line. 8 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Proof of Voltage and Current Transformation ® Consider a transmission line terminated in an arbitrary impedance ZL. ® The voltage and current waves propagating on the line are, V (z, t) = exp j(ωt − kz) + ρ exp j(ωt + kz) 1 ρ I(z, t) = exp j(ωt − kz) − exp j(ωt + kz) Zo Zo 9 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Proof of Voltage and Current Transformation ® At z = 0, V (0, t) = exp j(ωt) + ρ exp j(ωt) 1 ρ I(0, t) = exp j(ωt) − exp j(ωt) Zo Zo ® At z = +L, V (L, t) = exp j(ωt − kL) + ρ exp j(ωt + kL) 1 ρ I(L, t) = exp j(ωt − kL) − exp j(ωt + kL) Zo Zo 10 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Proof of Voltage and Current Transformation ® Compute V (0, t) in terms of V (L, t) and I(L, t) ... V (0, t) = exp j(ωt) + ρ exp j(ωt) V (L, t) = exp j(ωt − kL) + ρ exp j(ωt + kL) jZoI(L, t) = j exp j(ωt − kL) − jρ exp j(ωt + kL) 11 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Proof of Voltage and Current Transformation ® Multiply the equations for V (L, t) and I(L, t) by cos kL and sin kL, V (0, t) = exp j(ωt) + ρ exp j(ωt) V (L, t) cos kL = exp j(ωt − kL) cos kL + ρ exp j(ωt + kL) cos kL jZoI(L, t) sin kL = j exp j(ωt − kL) sin kL − jρ exp j(ωt + kL) sin kL V (L, t) cos kL + jZoI(L, t) sin kL = exp j(ωt − kL) (cos kL + j sin kL) + ρ exp j(ωt + kL) (cos kL − j sin kL) V (L, t) cos kL + jZoI(L, t) sin kL = V (0, t) 12 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Impedance Transformation Along a Line ® The voltage Vin and current Iin at the input to the line are given by V (0, t) = V (L, t) cos kL + jZoI(L, t) sin kL V (L, t) I(0, t) = I(L, t) cos kL + j sin kL Zo ® Divide these V (0, t) V (L, t) cos kL + jZoI(L, t) sin kL = I(0, t) I(L, t) cos kL + j V (L,t) sin kL Zo ZL cos kL + jZo sin kL Z + jZo tan kL Zin = = Zo L cos kL + j ZL sin kL Zo Zo + jZL tan kL 13 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Applications of Transmission Lines ® Hybrids and baluns 14 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Applications of Transmission Lines ® Filters 15 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 16 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 17 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Matching Networks ® Use a matching network to match a source to a load for maximum transferred power. ∗ ® ZL = ZS ® Two different types we consider: L-networks and Pi/T networks ® Consist entirely of Ls and Cs. ® How to deal with reactive source and load impedances? Either treat by absorption or resonance ® Dont forget that if there is a transmission line in between the source and the load network then there are two matching networks: one to match the source impedance to Zo and one to match Zo to the load impedance. 18 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 L-Networks ® Consist of two matching elements. ® Choose shunt arrangement at ZL (resp. ZS ) if RS < RL (resp. RL < RS ). Use series arrangement on the other side. ® Try to absorb source and load reactances into the matching impedance reactances. ® Since the impedances seen in either direction through the green line must be complex conjugates of each other, then the Q is the same for the circuits on either side of the green line. 19 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 L-Networks ® The relationships between the rS , xS and rL, xL are given by rL 2 xL 1 + Q2 =1+ Q , = 2 . RL shunt. RS series. rS xS Q 2 rS =1+ Q2, xS = 1 + Q . R shunt. R series. S L rL xL Q2 ® Q obtained from, rL Q= − 1. rS rS Q= − 1. rL 20 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 L-Networks: Summary ® Place the shunt of the L-network across the highest resistance and the series of the L-network in series with the lower resistance. ® Compute the Q required to match the source and load resistances. ® Use the Q to ﬁnd xS and xL from rS and rL. ® Remember to place inductors in series with capacitors and vice versa in order to allow for complex conjugates. ® Absorb or resonate the source and load stray reactances XS and XL of the matching network with xS and xL. ® Whether we absorb or resonate depends on how large the strays are. 21 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 L-Networks: Limitations ® The value of Q arises from the calculation. ® But what if we need to specify Q? ® Solution: T and Pi networks. 22 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 T and Pi: High Q Networks ® Can allow us to choose Q. ® Q however is always higher than for an L-network. Why? 23 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Analysis of T and Pi Networks ® Choose Q. ® Consider the T or Pi network to be a pair of back to back L networks. ® The virtual resistance in a Pi network must be smaller that those on the source and load. 24 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 T Networks ® The virtual resistance in a T network must be larger that those on the source and load. 25 ENGN4545/ENGN6545: Radiofrequency Engineering L#21 Low Q Networks ® Q however is always lower than for an L-network. ® OK for broadband match. 26 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

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Transmission Lines ® Transformation of voltage, current and ...

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