Lecture Outline. Scattering at an Impedance Discontinuity Power on a Transmission Line Voltage Standing Wave Ratio (VSWR) 8/10/2018

Transcription

1 Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (95) E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 4d Scattering on a Transmission Line Scattering These on a notes Transmission may contain Line copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide Lecture Outline Scattering at an Impedance Discontinuity Power on a Transmission Line Voltage Standing Wave Ratio (VSWR) Scattering on a Transmission Line Slide 2

2 Scattering at an Impedance Discontinuity Scattering on a Transmission Line Slide 3 Problem Setup Transmission Line Transmission Line 2, Z 2, Z2 z? z We will get a reflection Scattering on a Transmission Line Slide 4 2

3 Incorporate Reflected Wave Transmission Line Transmission Line 2, Z 2, Z2 z z V z V e V e z z V V I z e e Z Z z z V z V e 2 2 V 2z 2 2z I2 z e Z2 Scattering on a Transmission Line Slide 5 Enforce Boundary Conditions ( of 2) Transmission Line Transmission Line 2, Z 2, Z2 z 2 I z I z V V V e e e Z Z Z 2 z z 2 2z 2 Scattering on a Transmission Line Slide 6 V z V z V e V e V e z z 2z 2 z Boundary conditions require the voltage and current on either side of the interface to be equal. 3

4 Enforce Boundary Conditions (2 of 2) Transmission Line Transmission Line 2, Z 2, Z2 z Scattering on a Transmission Line Slide 7 V V V V V I I2 V V V2 Z Z Z z The interface occurs at z = so the exponential terms all equal. Reflection Coefficient, Enforcing the boundary conditions at z = gave us V V V Eq. 2 V V V2 Z Z Z2 Substitute Eq. () into Eq. (2) to eliminate V 2. V V V V Z Z Z 2 Eq. 2 Solve this new expression for V V. V V V V Z Z Z Z 2 2 V V V V Z Z Z Z 2 2 V V Z Z Z Z Z Z V Z Z V V Z Z V Z Z 2 2 V Z Z V Z Z 2 2 Scattering on a Transmission Line Slide 8 4

5 Revised Equations for V(z) and I(z) The total voltage and current in any section of line was written as z z V z V z V zv e V e I z e e Z Z V e e z z V z z e e Z Using the concept of the reflection coefficient, these equations can now be written as z V z V e V e z V z V I z e e Z Z z V Z Z V Z Z 2 2 Reflection coefficient at the load Scattering on a Transmission Line Slide 9 Power on a Transmission Line Scattering on a Transmission Line Slide 5

6 Power Flowing Along Length of Line The RMS power flowing at a distance z from the load is * Pavg z Re VzI z 2 * is complex conjugate This equation is valid for any line, even those with loss. For lossless lines (not lossless loads), we have jz jz V jz jz V zv e Le Iz e Le Z Substituting these equations into our expression for P avg (z) gives * V Pavg z Re V e e e e 2 Z P avg jz jz jz jz L L V 2Z L 2 Notice that the z dependence vanished. This is because power flows uniformly without decay in lossless lines. * Scattering on a Transmission Line Slide Voltage Standing Wave Ratio (VSWR) Scattering on a Transmission Line Slide 2 6

7 Voltage Standing Wave Ratio (VSWR) The VSWR is essentially the same concept as the standing wave ratio (SWR) discussed along with waves. The only difference is that it describes voltage and current instead of electromagnetic fields. max V z max I z VSWR min V z min I z Scattering on a Transmission Line Slide 3 Derivation of VSWR ( of 2) We start with our expression for waves travelling in opposite directions on a transmission line. We will assume a lossless line. jz jz V jz jz V zv e Le Iz e Le Z The magnitude of the voltage signal V(z) is 2 V z V e e V e j z j z j z L L By inspection of this equation, we determine the maximum and minimum values of this function. max min L L V max V z V V min V z V Scattering on a Transmission Line Slide 4 7

8 Derivation of VSWR (2 of 2) The VSWR is therefore max V z L VSWR min V z V L V L VSWR L The VSWR is an easily measured quantity and we can calculate the magnitude of the reflection coefficient from the VSWR. L VSWR VSWR Scattering on a Transmission Line Slide 5 Animation of VSWR ( of 6) Case : 5 transmission line terminated with a short circuit load. L Scattering on a Transmission Line Slide 6 8

9 Animation of VSWR (2 of 6) Case 2: 5 transmission line terminated with an open circuit load. L Scattering on a Transmission Line Slide 7 Animation of VSWR (3 of 6) Case 3: 5 transmission line terminated with a 6.5 load. Z Z L L.5 Scattering on a Transmission Line Slide 8 9

10 Animation of VSWR (4 of 6) Case 4: 5 transmission line terminated with a 5 load. Z Z L L.5 Scattering on a Transmission Line Slide 9 Animation of VSWR (5 of 6) Case 5: 5 transmission line terminated with an RL load. Scattering on a Transmission Line Slide 2

11 Animation of VSWR (6 of 6) Case 6: 5 transmission line terminated with an RC load. Scattering on a Transmission Line Slide 2