Lecture Outline. Attenuation Coefficient and Phase Constant Characteristic Impedance, Z 0 Special Cases of Transmission Lines

Transcription

1 Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 4b Transmission Line Parameters Transmission These Line notes Parameters may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide 1 Lecture Outline Attenuation Coefficient and Phase Constant Characteristic Impedance, Z Special Cases of Transmission Lines General transmission lines Lossless lines Weakly absorbing lines Distortionless lines Why is 5 a Standard Impedance? Transmission Line Parameters Slide 1

2 Fundamental Vs. Intuitive Parameters Fundamental Parameters Intuitive Parameters Electromagnetics Electromagnetics,, n,,,, tan Transmission Lines R, L, G, C Transmission Lines Z,,, VSWR The fundamental parameters are the most basic parameters needed to solve a transmission line problem. However, it is difficult to be intuitive about how they affect signals on the line. An electromagnetic analysis is needed to determine R, L, G, and C from the geometry of the transmission line. The intuitive parameters provide intuitive insight about how signals behave on a transmission line. They isolate specific information to a single parameter. The intuitive parameters are calculated from R, L, G, and C. Transmission Line Parameters Slide 3 Attenuation Coefficient and Phase Constant Transmission Line Parameters Slide 4

3 Derivation and (1 of 7) Step 1 Start with the expression for. j G jc R jl Square this expression to get rid of square root on right hand side. j G jcr jl Expand this expression. j RG jrc jlg LC Collect real and imaginary parts on the left hand and right hand sides. j RG LC j RC LG Transmission Line Parameters Slide 5 Derivation and ( of 7) Step Generate two equations by equating real and imaginary parts. j RG LC j RC LG RCLG RG LC There are now two equations and two unknowns. RC LG RG LC Transmission Line Parameters Slide 6 3

4 Derivation and (3 of 7) Step 3 Derive a quadratic equation for. RC LG Eq. (1a) RG LC Eq. (1b) Solve Eq. (1a) for. RC LG Eq. () Substitute Eq. () into Eq. (1b) and simplify. RCLG RG LC RC LG RG 4 LC RG RC LG 4 Transmission Line Parameters Slide 7 LC 4 4 RCLG 4 RG4 LC Derivation and (4 of 7) Step 4 Solve for using the quadratic equation. b b 4ac Recall the quadratic formula: ax bx c x a The equation for is in the form of the quadratic equation where a 1 LC RG RC LG 4 x The solution is 4 LC RG LC RG RC LG RG LC R L G C b LCRG c RCLG Transmission Line Parameters Slide 8 4

5 Derivation and (5 of 7) Step 5 Resolve the sign of the square root. RG LC R L G C The final expression is In order for this expression to always give a real value for, the sign of the square root must be positive. RG LC R L G C Transmission Line Parameters Slide 9 Derivation and (6 of 7) Step 6 Solve for using the expression for. Recall Eq. (1b): RG LC RG LC R L G C Derive an equation for by substituting the expression for into Eq. (1b). RG LC R L G C RG LC R L G C RG LC Transmission Line Parameters Slide 1 5

6 Derivation and (7 of 7) Step 7 The final expressions for and are derived in terms of the fundamental parameters R, L, G, and C by taking the square root of the latest expressions for and. RG LC R L G C RG LC R L G C Both and must be positive quantities for passive materials. This means the positive sign is taken for the square roots. RG LC R L G C RG LC R L G C Transmission Line Parameters Slide 11 Characteristic Impedance Z Transmission Line Parameters Slide 1 6

7 Characteristic Impedance, Z () The characteristic impedance Z of a transmission line is defined as the ratio of the voltage to the current at any point of a forward travelling wave. V V Z I I Definition for a forward travelling wave. Definition for a backward travelling wave. Notice the negative sign! Most characteristic impedance values fall in the 5 to 1 range. The specific value of impedance is not usually of importance. What is important is when the impedance changes because this causes reflections, standing waves, and more. Transmission Line Parameters Slide 13 Derivation of Z (1 of 5) Step 1 Substitute the solutions back into the transmission line equations. V z V e V e I z I e I e z z z z dv z R jl I z dz d V e dz V e z z z z R jlie Ie di z dz d I e dz z z G jcv z I e z z G jcv e V e Transmission Line Parameters Slide 14 7

8 Derivation of Z ( of 5) Step Expand the equations and calculate the derivatives. d V e V e dz z z z z R jlie Ie d I e I e dz z z z z G jcv e V e V e V e z z z z z R jli e R jli e I e I e z z G j C V e G j C V e z Transmission Line Parameters Slide 15 Derivation of Z (3 of 5) Step 3 Equate the expressions multiplying the common exponential terms. V R jl I V e V e R jl I e R jl I e z z z z V R jl I I G jc V I e I e G jc V e G jc V e z z z z I G jc V Transmission Line Parameters Slide 16 8

9 Derivation of Z (4 of 5) Step 4 Solve each of our four equations for V /I to derive expressions for Z. V R jl I V R jl I I G jc V I G jc V V R jl Z I V R jl Z I V Z I G jc V Z I G jc Transmission Line Parameters Slide 17 Derivation of Z (5 of 5) Step 5 Put Z in terms of just R, L, G, and C. Recall our expression for : j G jcr jl We can substitute this into either of our expressions for Z. R jl Z G jc Proceed with the first expression. R jl Z R jl G jcr jl R jl G jcr jl R jl G jc Transmission Line Parameters Slide 18 9

10 Final Expression for Z () We have derived a general expression for the characteristic impedance Z of a transmission line in terms of the fundamental parameters R, L, G, and C. Definition: Z V I V I Expressions: Z R jl R jl G jc G jc Transmission Line Parameters Slide 19 Dissecting the Characteristic Impedance, Z The characteristic impedance describes the amplitude and phase relation between voltage and current along a transmission line. With this picture in mind, the characteristic impedance can be written as Z Z Z z V z V e V V I z I e e e e Z Z z z z jz The characteristic impedance can also be written in terms of its real and imaginary parts. Z R jx Reactive part of Z. This is not equal to jl or 1/jC. Resistive part of Z. This is not equal to R or G. Transmission Line Parameters Slide 1

11 Special Cases of Transmission Lines: General Transmission Line Transmission Line Parameters Slide 1 Parameters for General TLs Propagation Constant, j G jc R jl Attenuation Coefficient, RG LC R L G C Phase Constant, RG LC R L G C Characteristic Impedance, Z Z R jx R jl G jc Transmission Line Parameters Slide 11

12 Special Cases of Transmission Lines: Lossless Lines Transmission Line Parameters Slide 3 Definition of Lossless TL When we think about transmission lines, we tend to think of the special case of the lossless line because the equations simplify considerably. For a transmission line to be lossless, it must have R G Transmission Line Parameters Slide 4 1

13 Parameters for Lossless TLs Propagation Constant, j j LC Attenuation Coefficient, Phase Constant, LC Characteristic Impedance, Z Z R jx R L C L C X Transmission Line Parameters Slide 5 Special Cases of Transmission Lines: Weakly Absorbing Line Transmission Line Parameters Slide 6 13

14 Definition of Weakly Absorbing TL Most practical transmission lines have loss, but very low loss making them weakly absorbing. We will define a weakly absorbing line as R L and G C Ensures low ohmic loss for signals propagating through the line. Ensures very little conduction between the lines through the dielectric. Transmission Line Parameters Slide 7 Parameters for Weakly Absorbing TLs Attenuation Coefficient, 1 R GZ Z Conductance through the dielectric dominates attenuation in high impedance transmission lines. Resistivity in the conductors dominates attenuation in low impedance transmission lines. In weakly absorbing transmission lines, there usually exists a sweet spot for the impedance where attenuation is minimized. Transmission Line Parameters Slide 8 14

15 Special Cases of Transmission Lines: Distortionless Lines Transmission Line Parameters Slide 9 Definition of Distortionless TL In a real transmission line, different frequencies will be attenuated differently because is a function of. This causes distortion in the signals carried by the line. RG LC R L G C To be distortionless, there must be a choice of R, L, G, and C that eliminates from the expression of, effectively making independent of frequency. The necessary condition to be distortionless is R G L C Transmission Line Parameters Slide 3 15

16 Parameters for Distortionless TLs Propagation Constant, j RG j LC Attenuation Coefficient, RG Phase Constant, LC To be distortionless, we must have. is a measure of how quickly a signal accumulates phase. Different frequencies have different wavelengths and therefore must accumulate different phase through the same length of line. Characteristic Impedance, Z R L Z R jx G C R L R X G C Transmission Line Parameters Slide 31 Why 5? Transmission Line Parameters Slide 3 16

17 Cable Loss Vs. Characteristic Impedance As we adjust the cable dimensions (i.e. b/a), we change both its impedance and its loss characteristics. This let s us plot the cable loss vs. characteristic impedance for a coax with different dielectric fills. For the air filled coax, we observe minimum loss at around 77, where b/a 3.5. A coaxial cable filled with polyethelene ( r =.), the minimum loss occurs at 51. (b/a = 3.6). fifty ohms Transmission Line Parameters Slide 33 Power Handling Vs. Characteristic Impedance As we adjust the cable dimensions (i.e. b/a), we affect the peak voltage handling capability (breakdown) and its power handling capability (heat). We observe the lowest peak voltage at just over 5 which we interpret as the point of best voltage handling capability. We observe the lowest peak current at around 3 which we interpret as the point of best power handling capability. fifty ohms Transmission Line Parameters Slide 34 17

18 Why 5 Impedance is Best? Two researchers, Lloyd Espenscheid and Herman Affel, working at Bell Labs produced this graph in 199. They needed to send 4 MHz signals hundreds of miles. Transmission lines capable of handling high voltage and high power were needed in order to accomplish this. The data shown at right was generated for an air filled coaxial cable. Best for High Voltage: Z = 6 Best for High Power: Z = 3 Best for Attenuation: Z = 75 5 seemed like the best compromise. Transmission Line Parameters Slide 35 Why 75 Impedance Standard for Coax? Nobody really knows!! The ideal impedance is closer to 5, however this requires a thicker center conductor. Maybe 75 is a compromise between low loss and mechanical flexibility? Transmission Line Parameters Slide 36 18