Transmission Lines. Transformation of voltage, current and impedance. Impedance. Application of transmission lines

Transcription

1 Transmission Lines Transformation of voltage, current and impedance Impedance Application of transmission lines 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

2 The Telegraphist Equations We can rewrite the above equations as (Telegraphist Equations) ( ) V iωzo z = I v ( ) I iω z = V Z o 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 z = iωli, L = Z o v I z = iωcv, C = 1 Z o v = µ od w = ǫ oǫ r w d 2 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

3 The Telegraphist Equations The velocity and characteristic impedance of the line can be expressed in terms of L and C. v = 1 LC Z o = L 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). C = 2πǫ oǫ r ln(b/a) L = 2πln(b/a) µ o Z o = µo ln(b/a) ǫ o ǫ r 2π v = 1 µ o ǫ o ǫ r 3 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

4 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, C = 2πǫ oǫ r ln(b/a) L = 2πln(b/a) µ o Z o = µo ln(b/a) ǫ o ǫ r 2π v = 1 µ o ǫ o ǫ r On board. 4 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

5 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, V load = V f + V r I load = I f + I r where V f = Z o I f V r = Z o I r 5 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

6 Reflection Coefficient At the load, V load = Z L I load = Z L ( If + I r ) = Vf + V r = Z L Z o ( Vf V r ) Solving for ρ = V r /V f, we obtain, ρ = Z L Z o Z L + Z o ρ is the reflection coefficient. If Z L = Z o there is no reflected wave. A line terminated in a pure reactance always has ρ = 1 6 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

7 Impedance Transformation Along a Line Consider a transmission line terminated in an arbitrary impedance Z L. The impedance Z in seen at the input to the line is given by Z in = Z o Z L + jz o tan kl Z o + jz L tan kl If Z L = Z o, then Z in = Z o. If Z L = 0, then Z in = jz o tan kl If Z L =, then Z in = Z o /(j tan kl) 7 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

8 Voltage and Current Transformation Along a Line Consider a transmission line terminated in an arbitrary impedance Z L. The voltage V in and current I in at the input to the line are given by V in = V end cos kl + jz o I end sin kl I in = I end cos kl + j V end Z o sin kl If a line is unterminated then the voltage and current vary along line. 8 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

9 Proof of Voltage and Current Transformation Consider a transmission line terminated in an arbitrary impedance Z L. The voltage and current waves propagating on the line are, V (z, t) = exp j(ωt kz) + ρexp j(ωt + kz) I(z, t) = 1 Z o exp j(ωt kz) ρ Z o exp j(ωt + kz) 9 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

10 Proof of Voltage and Current Transformation At z = 0, V (0, t) = exp j(ωt) + ρexp j(ωt) I(0, t) = 1 Z o exp j(ωt) ρ Z o exp j(ωt) At z = +L, V (L, t) = exp j(ωt kl) + ρexp j(ωt + kl) I(L, t) = 1 Z o exp j(ωt kl) ρ Z o exp j(ωt + kl) 10 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

11 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) + ρexp j(ωt) V (L, t) = exp j(ωt kl) + ρexp j(ωt + kl) jz o I(L, t) = j exp j(ωt kl) jρexp j(ωt + kl) 11 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

12 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) + ρexp j(ωt) V (L, t)cos kl = exp j(ωt kl)cos kl + ρexp j(ωt + kl)cos kl jz o I(L, t)sin kl = j exp j(ωt kl)sin kl jρexp j(ωt + kl)sin kl V (L, t)cos kl + jz o I(L, t)sin kl = exp j(ωt kl)(cos kl + j sin kl) + ρexp j(ωt + kl)(cos kl j sin kl) V (L, t)cos kl + jz o I(L, t)sin kl = V (0, t) 12 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

13 Impedance Transformation Along a Line The voltage V in and current I in at the input to the line are given by V (0, t) = V (L, t)cos kl + jz o I(L, t)sin kl V (L, t) I(0, t) = I(L, t)cos kl + j sin kl Divide these Z o V (0, t) I(0, t) = V (L, t)cos kl + jz oi(l, t)sin kl I(L, t)cos kl + j V (L,t) sin kl Z o Z in = Z Lcos kl + jz o sin kl cos kl + j Z L Z o sin kl = Z Z L + jz o tan kl o Z o + jz L tan kl 13 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

14 Applications of Transmission Lines Hybrids and baluns 14 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

15 Applications of Transmission Lines Filters 15 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

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18 Matching Networks Use a matching network to match a source to a load for maximum transferred power. Z L = Z S 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 Z o and one to match Z o to the load impedance. 18 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

19 L-Networks Consist of two matching elements. Choose shunt arrangement at Z L (resp. Z S ) if R S < R L (resp. R L < R S ). 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

20 L-Networks The relationships between the r S, x S and r L, x L are given by r L r S = 1 + Q 2, r S r L = 1 + Q 2, Q obtained from, x L = 1 + Q2 x S Q 2 x S = 1 + Q2 x L Q 2. R L shunt. R S series.. R S shunt. R L series. Q = rl r S 1. Q = rs r L ENGN4545/ENGN6545: Radiofrequency Engineering L#21

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 x S and x L from r S and r L. 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 X S and X L of the matching network with x S and x L. Whether we absorb or resonate depends on how large the strays are. 21 ENGN4545/ENGN6545: Radiofrequency Engineering L#21

22 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

23 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

24 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

25 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

26 Low Q Networks Q however is always lower than for an L-network. OK for broadband match. 26 ENGN4545/ENGN6545: Radiofrequency Engineering L#21