9/8/17 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 ine Behavior Transmission These ine notes Behavior may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide 1 ecture Outline Scattering at an Impedance Discontinuity Power on a Transmission ine oltage Standing Wave Ratio (SWR) Input Impedance, in Parameter Relations Special Cases of Terated Transmission ines Shorted line ( = ) Open circuit line ( = ) Matched line ( = ) Transmission ine Behavior Slide 1
9/8/17 Scattering at an Impedance Discontinuity Transmission ine Behavior Slide 3 Problem Setup Transmission ine 1 Transmission ine 1, 1,? We will get a reflection Transmission ine Behavior Slide 4
9/8/17 Incorporate Reflected Wave Transmission ine 1 Transmission ine 1, 1, e e 1 1 1 1 1 I1 e e 1 1 1 1 1 1 e I e Transmission ine Behavior Slide 5 Enforce Boundary Conditions (1 of ) Transmission ine 1 Transmission ine 1, 1, 1 1 Transmission ine Behavior Slide 6 1 e e e 1 1 1 1 I1 I e e e 1 1 1 1 Boundary conditions require the voltage and current on either side of the interface to be equal. 3
9/8/17 Enforce Boundary Conditions ( of ) Transmission ine 1 Transmission ine 1, 1, 1 1 1 1 1 1 1 1 Transmission ine Behavior Slide 7 I I The interface occurs at =. Reflection Coefficient, Enforcing the boundary conditions at = gave us Eq. 1 1 1 1 1 1 1 Substitute Eq. (1) into Eq. () to eliate. 1 1 1 1 1 1 Eq. Solve this new expression for. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Transmission ine Behavior Slide 8 4
9/8/17 Revised Equations for () and I() The total voltage and current in any section of line was written as e e I e e Using the concept of the reflection coefficient, these equations can now be written as e e e e I e e e e Reflection coefficient at the load 1 1 Transmission ine Behavior Slide 9 Power on a Transmission ine Transmission ine Behavior Slide 1 5
9/8/17 Power Flowing Along ength of ine The RMS power flowing at a distance from the load is 1 * Pavg Re I * is complex conjugate This equation is valid for any line, even those with loss. For lossless lines (not lossless loads), we have j j j j e e I e e Substituting these equations into our expression for P avg () gives * 1 Pavg Re e e e e P avg j j j j 1 Notice that the dependence vanished. This is because power flows uniformly without decay in lossless lines. * Transmission ine Behavior Slide 11 oltage Standing Wave Ratio (SWR) Transmission ine Behavior Slide 1 6
9/8/17 oltage Standing Wave Ratio (SWR) The SWR 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 max I SWR I Transmission ine Behavior Slide 13 Derivation of SWR (1 of ) We start with our expression for waves travelling in opposite directions on a transmission line. We will assume a lossless line. j j j j e e I e e The magnitude of the voltage signal () is e e e j j j 1 By inspection of this equation, we detere the maximum and imum values of this function. max max 1 1 Transmission ine Behavior Slide 14 7
9/8/17 Derivation of SWR ( of ) The SWR is therefore max SWR 1 1 1 SWR 1 The SWR is an easily measured quantity and we can calculate the magnitude of the reflection coefficient from the SWR. SWR1 SWR1 Transmission ine Behavior Slide 15 Animation of SWR (1 of 6) Case 1: 5 transmission line terated with a short circuit load. 1 Transmission ine Behavior Slide 16 8
9/8/17 Animation of SWR ( of 6) Case : 5 transmission line terated with an open circuit load. 1 Transmission ine Behavior Slide 17 Animation of SWR (3 of 6) Case 3: 5 transmission line terated with a 16.5 load..5 Transmission ine Behavior Slide 18 9
9/8/17 Animation of SWR (4 of 6) Case 4: 5 transmission line terated with a 15 load..5 Transmission ine Behavior Slide 19 Animation of SWR (5 of 6) Case 5: 5 transmission line terated with an R load. Transmission ine Behavior Slide 1
9/8/17 Animation of SWR (6 of 6) Case 6: 5 transmission line terated with an RC load. Transmission ine Behavior Slide 1 Input Impedance, in Transmission ine Behavior Slide 11
9/8/17 Problem Setup Generator Transmission ine oad g g in, Generator g Input Impedance The input impedance in is the impedance observed by the generator. g in in The input impedance in is NOT necessarily the line s characteristic impedance or the load impedance. Transmission ine Behavior Slide 3 Animation of Impedance Transformation in m 4 Input impedance inverts in 15 j15 in m Input impedance repeats in 1 j1 Transmission ine Behavior Slide 4 1
9/8/17 Derivation of Input Impedance, in (1 of ) The reflection coefficient at any point from the load is Backward Wave e e e 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 ine Behavior Slide 5 Derivation of Input Impedance, in ( of ) We define the impedance of the line at position to be We previously wrote () and I() as I e e I e e Substituting in our expressions for () and I() gives e e e e e e e e It makes sense that the impedance is not a function of voltage in a linear system. Transmission ine Behavior Slide 6 13
9/8/17 Sanity Check: Input Impedance at oad The input impedance at the load can be detered by setting = in our previous equation. e e in e e 1 1 1 1 Transmission ine Behavior Slide 7 Input Impedance at The input impedance at location is e e e e in e e e e A Note About Sign: Backing away from the load, becomes negative. However, we defined so stays positive in this equation and for equations that follow. Transmission ine Behavior Slide 8 14
9/8/17 Impedance Transformation Formula (1 of ) Recall that e e e e We can eliate from the input impedance equation by substituting in our expression for. e e e e e e in e e e e e e Transmission ine Behavior Slide 9 Impedance Transformation Formula ( of ) Now recall the definitions of hyperbolic sine and cosine functions. sinh e e cosh e e This lets us write the input impedance expression as sinh cosh sinh cosh in sinh cosh sinh cosh Recogniing that tanh() = sinh()/cosh(), our expression reduces to tanh in tanh Transmission ine Behavior Slide 3 15
9/8/17 Input Transformation for ossless ine The lossless line has j Putting these values into our impedance transformation formula gives tanh j in tanh j Recogniing that tanh(j) = jtan(), our expression for lossless lines becomes j tan in j tan Transmission ine Behavior Slide 31 Input Impedance Repeats for ossless ines 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,, 1,,1,, 3,, Recogniing that = /, the above expression leads to m Note: is the wavelength in the transmission line, not the free space wavelength. 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 visualie the impedance transformation phenomenon. Transmission ine Behavior Slide 3 16
9/8/17 Example: Impedance Transformation (1 of 3) A transmission line with 5 characteristic impedance is connected to a 1 nf capacitor as the load. If the phase constant of the transmission line is = 6 m -1, what is the input impedance in of a 1 inch section of line operating at 4. GH? What equivalent circuit would the source see? Transmission ine oad in 5 1 nf 1 inch Transmission ine Behavior Slide 33 Example: Impedance Transformation ( of 3) oss was not specified so we assume a lossless transmission line. Our impedance transformation equation is therefore j tan in j tan The variables in this equation are 5 1.54 cm 1 m 6 m 1 inch 1.54 1 inch 1 cm 1 1 1 j.4 jc j fc j 4. 1 s 1 1 F 9 1 9 Transmission ine Behavior Slide 34 17
9/8/17 Example: Impedance Transformation (3 of 3) Substituting in the values of our variables gives j.4 j 5 tan 6.54 5 jj.4 tan 6.54 3 in 5 j1.71 The input impedance is purely imaginary and positive. Thus, the input impedance looks like an inductor to the generator. in j eq j1.71 j j f j 4.1 s 3 in in eq 9 1 8 4.4 1 H 4.4 nh Transmission ine Behavior Slide 35 Parameter Relations Transmission ine Behavior Slide 36 18
9/8/17 max,, I max & I in Terms of SWR max and 1 max 1 SWR SWR1 SWR1 I max and I I I max 1 1 SWR SWR1 SWR1 Transmission ine Behavior Slide 37 in Terms of SWR The characteristic impedance can be calculated from max and I max or and I. I max max I The input impedance in repeats as you back away from the load. We can calculate the maximum and imum impedance as max max in I in SWR I max SWR max in in in Transmission ine Behavior Slide 38 19
9/8/17 Example (1 of 3) A 5 impedance transmission line is connected to an antenna with a 7 input impedance. A source provides an input signal of 4 peak to peak. What is the reflection coefficient at the antenna? 7 5.183 In this case, the antenna is the load. 7 5 What fraction of the input power is delivered to the antenna? R.183.35 Despite the mismatch, almost all power is still delivered to the antenna. This still T 1R1.35.9675 96.7% does not mean the antenna will radiate! What is the SWR on the line feeding the antenna? 1 1.183 SWR 1.44 1 1.183 SWR log SWR log 1.44 3.17 db db 1 1 Transmission ine Behavior Slide 39 Example ( of 3) What is the imum and maximum voltage on the line? First, we need to convert voltage peak to peak p-p to voltage magnitude. p-p 4 1 Now we are in a position to calculate and max. When we are utiliing high voltages, we want to 1 1 1.183 9.84 be sure max will not cause arcing or any other 1 1 1.183 14.16 breakdown problems. max What is the imum and maximum current on the line? 9.84 I.1967 A 5 At high power, we want max 14.16 to be sure I max will not Imax.833 A cause heating problems. 5 Transmission ine Behavior Slide 4
9/8/17 Example (3 of 3) What is the total range of input impedances a source could see? 9.84 in 34.7 Imax.833 A 14.16 max max in 7 I.1967 A max in in in 34.7 7 in Transmission ine Behavior Slide 41 Special Cases of Terated Transmission ines Transmission ine Behavior Slide 4 1
9/8/17 Shorted ine, = Reflection from oad 1 oltage Standing Wave Ratio SWR There exists () =. and max max I and I max I I max Input Impedance in j tanh lossy tan lossless Note 1: in 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. [ in ] and max[ in ] in short circuit max open circuit in Note : The shorted line behaves much the same way as the open circuit line. We also observe that in,short in,open Transmission ine Behavior Slide 43 Open Circuit ine, = Reflection from oad 1 oltage Standing Wave Ratio SWR There exists () =. and max max I and I max I I max Input Impedance in coth lossy j cot lossless Note 1: in 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. [ in ] and max[ in ] in short circuit max open circuit in Note : The open circuit line behaves much the same way as the shorted line. We also observe that in,short in,open Transmission ine Behavior Slide 44
9/8/17 Matched ine, = Reflection from oad oltage Standing Wave Ratio SWR 1 because max = and max max I and I max I I max Input Impedance in [ in ] and max[ in ] max in in Note: F the matched line, there are no reflections and all of the power is delivered to the load. Transmission ine Behavior Slide 45 3