9/28/2017 Course Instructor Dr. Raymond C. Rumpf Office: A‐337 Phone: (915) 747‐6958 E‐Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 4b Transmission Line Behavior TheseLine notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Transmission Behavior Slide 1 Lecture Outline • • • • • • Scattering at an Impedance Discontinuity Power on a Transmission Line Voltage Standing Wave Ratio (VSWR) Input Impedance, Zin Parameter Relations Special Cases of Terminated Transmission Lines – Shorted line (ZL = 0) – Open‐circuit line (ZL = ) – Matched line (ZL = Z0) Transmission Line Behavior Slide 2 1 9/28/2017 Scattering at an Impedance Discontinuity Transmission Line Behavior Slide 3 Problem Setup Transmission Line 2 Transmission Line 1 1 , Z1 2 , Z2 z0 z ? We will get a reflection Transmission Line Behavior Slide 4 2 9/28/2017 Incorporate Reflected Wave Transmission Line 2 Transmission Line 1 1 , Z1 2 , Z2 z z0 V1 z V1 e1z V1 e1z I1 z V2 z V2 e 2 z V1 1z V1 1z e e Z1 Z1 I2 z V2 2 z e Z2 Transmission Line Behavior Slide 5 Enforce Boundary Conditions (1 of 2) Transmission Line 2 Transmission Line 1 1 , Z1 2 , Z2 z0 z V1 z V2 z 1z 1 V e V1 e1z V2 e 2 z I1 z I 2 z Boundary conditions require the voltage and current on either side of the interface to be equal. 1 V 1z V1 1z V2 2 z e e e Z1 Z1 Z2 Transmission Line Behavior Slide 6 3 9/28/2017 Enforce Boundary Conditions (2 of 2) Transmission Line 2 Transmission Line 1 1 , Z1 2 , Z2 z z0 V1 0 V2 0 1 V V1 V2 The interface occurs at z = 0. I1 0 I 2 0 1 V V V 1 2 Z1 Z1 Z 2 Transmission Line Behavior Slide 7 Reflection Coefficient, Enforcing the boundary conditions at z = 0 gave us V1 V1 V2 Eq. 1 V1 V1 V2 Z1 Z1 Z 2 Eq. 2 Substitute Eq. (1) into Eq. (2) to eliminate V2. V1 V1 V1 V1 Z1 Z1 Z2 Solve this new expression for V1 V1. 1 1 1 1 V1 V1 V1 V1 Z1 Z1 Z2 Z2 1 1 1 1 V1 V1 V1 V1 Z1 Z2 Z1 Z2 1 1 1 1 V1 V1 Z1 Z 2 Z1 Z 2 Z 2 Z1 V1 Z 2 Z1 V1 V1 Z 2 Z1 V1 Z 2 Z1 Transmission Line Behavior V1 Z 2 Z1 V1 Z 2 Z1 Slide 8 4 9/28/2017 Revised Equations for V(z) and I(z) The total voltage and current in any section of line was written as V z V0 e z V0 e z I z V0 z V0 z e e Z0 Z0 Using the concept of the reflection coefficient , these equations can now be written as V z V0 e z V0 e z V0 e z e z I z V0 z V0 z V0 z e e e e z Z0 Z0 Z0 Reflection coefficient at the load V0 Z 2 Z1 V0 Z 2 Z1 Transmission Line Behavior Slide 9 Power on a Transmission Line Transmission Line Behavior Slide 10 5 9/28/2017 Power Flowing Along Length of Line The RMS power flowing at a distance z from the load is Pavg z 1 Re V z I * z 2 * is complex conjugate This equation is valid for any line, even those with loss. For lossless lines (not lossless loads), we have V z V0 e j z L e j z I z V0 j z e L e j z Z0 Substituting these equations into our expression for Pavg(z) gives * V0 j z * 1 j z j z Le L e j z Pavg z Re V0 e e 2 Z0 Pavg V0 2Z 0 1 2 L Notice that the z dependence vanished. This is because power flows uniformly without decay in lossless lines. Transmission Line Behavior Slide 11 Voltage Standing Wave Ratio (VSWR) Transmission Line Behavior Slide 12 6 9/28/2017 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. VSWR max V z min V z max I z min I z Transmission Line Behavior Slide 13 Derivation of VSWR (1 of 2) We start with our expression for waves travelling in opposite directions on a transmission line. We will assume a lossless line. V z V0 e j z L e j z I z V0 j z L e j z e Z0 The magnitude of the voltage signal V(z) is V z V0 e j z L e j z V0 1 L e j 2 z By inspection of this equation, we determine the maximum and minimum values of this function. Vmax max V z V0 1 L Vmin min V z V0 1 L Transmission Line Behavior Slide 14 7 9/28/2017 Derivation of VSWR (2 of 2) The VSWR is therefore VSWR max V z min V z V0 1 L 0 V 1 L VSWR 1 L 1 L The VSWR is an easily measured quantity and we can calculate the magnitude of the reflection coefficient || from the VSWR. L VSWR 1 VSWR 1 Transmission Line Behavior Slide 15 Animation of VSWR (1 of 6) Case 1: 50 transmission line terminated with a short‐circuit load. L 1 Transmission Line Behavior Slide 16 8 9/28/2017 Animation of VSWR (2 of 6) Case 2: 50 transmission line terminated with an open‐circuit load. L 1 Transmission Line Behavior Slide 17 Animation of VSWR (3 of 6) Case 3: 50 transmission line terminated with a 16.5 load. Z 0 Z L L 0.5 Transmission Line Behavior Slide 18 9 9/28/2017 Animation of VSWR (4 of 6) Case 4: 50 transmission line terminated with a 150 load. Z 0 Z L L 0.5 Transmission Line Behavior Slide 19 Animation of VSWR (5 of 6) Case 5: 50 transmission line terminated with an RL load. Transmission Line Behavior Slide 20 10 9/28/2017 Animation of VSWR (6 of 6) Case 6: 50 transmission line terminated with an RC load. Transmission Line Behavior Slide 21 Input Impedance, Zin Transmission Line Behavior Slide 22 11 9/28/2017 Problem Setup Generator Transmission Line Load Zg Vg , Z0 Z in ZL z0 z Generator Input Impedance The input impedance Zin is the impedance observed by the generator. Zg Vg z Z in z Z in z The input impedance Zin is NOT necessarily the line’s characteristic impedance Z0 or the load impedance ZL. Transmission Line Behavior Slide 23 Animation of Impedance Transformation Z2 Z in m 0 2 ZL 4 Input impedance inverts Z in 125 j125 Transmission Line Behavior Z in m Z L 2 Input impedance repeats Z in 10 j10 Slide 24 12 9/28/2017 Derivation of Input Impedance, Zin (1 of 2) The reflection coefficient at any point z from the load is Backward Wave z V0 e z V0 2 z e V0 e z V0 Forward Wave This means that from the perspective of the generator, the reflection going into the transmission line will change depending on the length of the transmission line. This can only happen of the input impedance to the transmission line is changing. Transmission Line Behavior Slide 25 Derivation of Input Impedance, Zin (2 of 2) We define the impedance of the line at position z to be Z z V z I z We previously wrote V(z) and I(z) as V z V0 e z L e z I z 0 V e z L e z Z0 L V0 Z L Z 0 V0 Z L Z 0 Substituting in our expressions for V(z) and I(z) gives Z z V0 e z L e z V0 z e L e z Z0 Z0 e z L e z e z L e z It makes sense that the impedance is not a function of voltage in a linear system. Transmission Line Behavior Slide 26 13 9/28/2017 Sanity Check: Input Impedance at Load The input impedance at the load can be determined by setting z = 0 in our previous equation. Z in 0 Z 0 Z0 e0 L e0 e 0 L e0 1 L 1 L ZL Z0 ZL Z0 Z0 Z Z0 1 L ZL Z0 1 Z0 ZL Z0 ZL Z0 ZL Z0 ZL Z0 Z0 2Z L 2Z 0 ZL Transmission Line Behavior Slide 27 Input Impedance at z The input impedance at location z is Z in Z 0 e Le e L e Z 0 e L e e Le A Note About Sign: Backing away from the load, z becomes negative. However, we defined z so stays positive in this equation and for equations that follow. Transmission Line Behavior Slide 28 14 9/28/2017 Impedance Transformation Formula (1 of 2) Recall that Z Z 0 e L e e L e L Z L Z0 Z L Z0 We can eliminate L from the input impedance equation by substituting in our expression for L. Z L Z 0 e Z L e e Z 0 e e Z L Z0 Z in Z 0 Z0 Z Z 0 Z L e e Z 0 e e e L e Z L Z0 e Transmission Line Behavior Slide 29 Impedance Transformation Formula (2 of 2) Now recall the definitions of hyperbolic sine and cosine functions. sinh z e z e z 2 cosh z e z e z 2 This lets us write the input impedance expression as sinh Z 2 cosh Z 0 2sinh cosh Z in Z 0 L Z0 sinh Z L 2sinh Z 0 2 cosh Z ZL 0 cosh Z L Z0 Recognizing that tanh(z) = sinh(z)/cosh(z), our expression reduces to Z in Z 0 Transmission Line Behavior Z L Z 0 tanh Z 0 Z L tanh Slide 30 15 9/28/2017 Input Transformation for Lossless Line The lossless line has 0 j Putting these values into our impedance transformation formula gives Z in Z 0 Z L Z 0 tanh j Z 0 Z L tanh j Recognizing that tanh(jz) = jtan(z), our expression for lossless lines becomes Z in Z 0 Z L jZ 0 tan Z 0 jZ L tan Transmission Line Behavior Slide 31 Input Impedance Repeats for Lossless Lines For lossless lines, the tan function in the impedance transformation equation tells us that the function is periodic and repeats. The function repeats every integer multiple of . m m , , 3, 2, 1, 0,1, 2,3, , Recognizing that = 2/, the above expression leads to m 2 Note: is the wavelength in the transmission line, not the free space wavelength 0. This means the input impedance repeats for every half‐wavelength long the transmission line is. We will revisit this when we cover Smith charts, which will give you a way to visualize the impedance transformation phenomenon. Transmission Line Behavior Slide 32 16 9/28/2017 Example: Impedance Transformation (1 of 3) A transmission line with 50 characteristic impedance is connected to a 10 nF capacitor as the load. If the phase constant of the transmission line is = 60 m-1, what is the input impedance Zin of a 1 inch section of line operating at 4.0 GHz? What equivalent circuit would the source see? Transmission Line Load 50 Z in 10 nF z0 z z 1 inch Transmission Line Behavior Slide 33 Example: Impedance Transformation (2 of 3) Loss was not specified so we assume a lossless transmission line. Our impedance transformation equation is therefore Zin Z 0 Z L jZ 0 tan Z 0 jZ L tan The variables in this equation are Z 0 50 2.54 cm 1 m 1.524 1 inch 100 cm 1 1 1 ZL j 0.004 9 1 jC j 2 fC j 2 4.0 10 s 10 109 F 60 m 1 1 inch Transmission Line Behavior Slide 34 17 9/28/2017 Example: Impedance Transformation (3 of 3) Substituting in the values of our variables gives j 0.004 j 50 tan 60 0.0254 3 Z in 50 j1.07 10 50 j j 0.004 tan 60 0.0254 The input impedance is purely imaginary and positive. Thus, the input impedance looks like an inductor to the generator. Z in j Leq Leq Z in Z in j1.07 103 4.24 108 H 42.4 nH 9 1 j j 2 f j 2 4.0 10 s Transmission Line Behavior Slide 35 Parameter Relations Transmission Line Behavior Slide 36 18 9/28/2017 Vmax, Vmin, Imax & Imin in Terms of VSWR Vmax and Vmin 2 VSWR VSWR 1 2 V0 1 L V0 VSWR 1 Vmax V0 1 L V0 Vmin Imax and Imin I max V0 V0 2 VSWR 1 L Z VSWR 1 Z0 0 I min V0 Z0 1 L V0 2 Z 0 VSWR 1 Transmission Line Behavior Slide 37 Z0 in Terms of VSWR The characteristic impedance Z0 can be calculated from Vmax and Imax or Vmin and Imin. Z0 Vmax Vmin I max I min The input impedance Zin repeats as you back away from the load. We can calculate the maximum and minimum impedance as max Z in Vmax Z 0 VSWR I min Z0 V min Z in min I max VSWR Transmission Line Behavior min Z in Z in max Z in Slide 38 19 9/28/2017 Example (1 of 3) A 50 impedance transmission line is connected to an antenna with a 72 input impedance. A source provides an input signal of 24 V peak‐to‐peak. What is the reflection coefficient at the antenna? Z Z 0 72 50 In this case, the antenna is the load. L L 0.1803 Z L Z 0 72 50 What fraction of the input power is delivered to the antenna? 2 2 R L 0.1803 0.0325 T 1 R 1 0.0325 0.9675 96.7% Despite the mismatch, almost all power is still delivered to the antenna. This still does not mean the antenna will radiate! What is the VSWR on the line feeding the antenna? VSWR 1 L 1 0.1803 1.44 1 L 1 0.1803 VSWR dB 20 log10 VSWR 20 log10 1.44 3.17 dB Transmission Line Behavior Slide 39 Example (2 of 3) What is the minimum and maximum voltage on the line? First, we need to convert voltage peak‐to‐peak Vp-p to voltage magnitude V0. V0 Vp-p 2 24 V 12 V 2 Now we are in a position to calculate Vmin and Vmax. 0 Vmin V 0 Vmax V 1 12 V 1 0.1803 9.84 V 1 12 V 1 0.1803 14.16 V L L When we are utilizing high voltages, we want to be sure Vmax will not cause arcing or any other breakdown problems. What is the minimum and maximum current on the line? V 9.84 V 0.1967 A I min min Z0 50 At high power, we want to be sure Imax will not Vmax 14.16 V 0.2833 A I max cause heating problems. Z0 50 Transmission Line Behavior Slide 40 20 9/28/2017 Example (3 of 3) What is the total range of input impedances a source could see? min Zin Vmin 9.84 V 34.72 I max 0.2833 A max Z in Vmax 14.16 V 72 I min 0.1967 A min Z in Z in max Z in 34.72 Z in 72 Transmission Line Behavior Slide 41 Special Cases of Terminated Transmission Lines Transmission Line Behavior Slide 42 21 9/28/2017 Shorted Line, ZL = 0 Reflection from Load Voltage Standing Wave Ratio L 1 VSWR Vmin and Vmax There exists V(z) = 0. Imin and Imax Vmin 0 I min 0 0 Vmax 2 V I max Z0 min[Zin] and max[Zin] Input Impedance Z tanh Z in 0 jZ 0 tan 2 V0 lossy lossless min Z in 0 short circuit max Z in open circuit Note 1: Zin for the lossless line is purely imaginary. This means it is purely reactive and no dissipation occurs in the line. The input impedance alternates between being capacitive and inductive as you back away from the load. Note 2: The shorted line behaves much the same way as the open‐circuit line. We also observe that Z in,short Z in,open Z 02 Transmission Line Behavior Slide 43 Open‐Circuit Line, ZL = Reflection from Load Voltage Standing Wave Ratio L 1 VSWR Vmin and Vmax Imin and Imax Vmin 0 I min 0 Vmax 2 V0 I max lossy lossless Note 1: Zin for the lossless line is purely imaginary. This means it is purely reactive and no dissipation occurs in the line. The input impedance alternates between being capacitive and inductive as you back away from the load. Transmission Line Behavior 2 V0 Z0 min[Zin] and max[Zin] Input Impedance Z coth Z in 0 jZ 0 cot There exists V(z) = 0. min Z in 0 short circuit max Z in open circuit Note 2: The open‐circuit line behaves much the same way as the shorted line. We also observe that Z in,short Z in,open Z 02 Slide 44 22 9/28/2017 Matched Line, ZL = Z0 Reflection from Load L 0 Vmin and Vmax Vmin Vmax V0 Input Impedance Z in Z 0 Voltage Standing Wave Ratio VSWR 1 because Vmax = Vmin Imin and Imax I min I max V0 Z 0 min[Zin] and max[Zin] min Z in max Z in Z 0 Note: F the matched line, there are no reflections and all of the power is delivered to the load. Transmission Line Behavior Slide 45 23