Transmission Lines. Transmission lines. Telegraphist Equations. Reflection Coefficient. Transformation of voltage, current and impedance

Transcription

1 Transmission Lines Transmission lines Telegraphist Equations Reflection Coefficient Transformation of voltage, current and impedance Application of trasnmission lines 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

2 Definition If stray inductance and capacitance causes all wires to be dispersive (frequency dependent). Then how can we transport any power at radiofrequency? A trasnmission line is a waveguide in which stray capacitance and inductance lead to non dispersive propagation. (discrete picture) A trasnmission line is a waveguide that allows plane electromagnetic waves to propagate. (wave picture) 2 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

3 Properties of Transmission Lines Many different varieties.. balanced lines such as twin wire and unbalanced lines such as coaxial cable. Transmisssion lines must always have two parallel conductors (a single wire is always an antenna at radiofrequency. At low powers coax is often flexible and the conductors are separated by a dielectric. Example dielectrics are polythene (RG-58 Coax), foam PFE high quality sem-rigid (Aluminium jacket) coax and PTFE (teflon). On PCBs you need to manufacture your own transmission lines... PCB tracks are designed as striplines. In this case the dielectric is FR4 (fibreglass). 3 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

4 Properties of Transmission Lines No net current flows along the line. In coax, current flows along the outside of the centre conductor and in the opposite direction on the inside of the outer conductor. Why? Coaxial cables are shielded. Electric fields point radially between the inner and outer conductors. The magnetic fieldlines circle the inner conductor. Transmission lines are versatile. Use them in impedance transforming elements (baluns), filters, antennas, DC blocks. 4 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

5 Properties of Transmission Lines 5 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

6 The Planar Transmission Line: Identical to the plane EM wave Same derivation as for plane waves (but include a dielectric)... E z = iωb B z = iω v 2E Integral over γ Integral over ζ 6 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

7 The Planar Transmission Line From the definition of potential difference and Ampere s law we obtain V = Ed and B = µ oi w Substitute these in the differential equations for E and B. E z = iωb => V z = B z = iω I v2e => z = ( iωµo d w ( iωw µ o dv 2 For sinusoidal dependence V, I = exp i (±kz ωt) ω k = ±v = ± 1 µ o ɛ o ɛ r V I = ±µ ovd µo w = ± d ɛ o ɛ r w ) ) I V 7 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

8 The Planar Transmission Line For a wave travelling in one direction only, the ratio of V to I is a constant For a given wave, the ratio V/I is referred to as the Characteristic Impedance. The planar transmission line is also called a stripline. For the stripline: µo d Z o = ɛ o ɛ r w Note that the characteristic impedance can be complex if the dielectric material separating the conductors is lossy (has finite loss tangent). In the latter case the propagation speed is also complex. 8 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

9 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 = ɛ oɛ r w d 9 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

10 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 = 2πɛ oɛ r ln(b/a) L = µ oln(b/a) 2π Z o = µo ln(b/a) ɛ o ɛ r 2π v = 1 µ o ɛ o ɛ r 10 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

11 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 11 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

12 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 12 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

13 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) 13 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

14 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. 14 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

15 Applications of Transmission Lines Hybrids and baluns 15 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

16 Applications of Transmission Lines Filters 16 ENGN4545/ENGN6545: Radiofrequency Engineering L#9

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