Reflection Coefficient Equations for Tapered Transmission Lines

29 June – 1 July 2006, Sofia, Bulgaria Reflection Coefficient Equations for Tapered Transmission Lines

Zlata Cvetkovi ć1, Slavoljub Aleksi ć, Bojana Nikoli ć

Abstract −−− An approximation theory, based on the theory frequencies. Modern network analyzers measure the incident of small reflections, to predict the reflection coefficient and reflected waves directly during a frequency sweep, and response as a function of the impedance taper is applied in impedance results can be displayed in any number of formats. the paper. Obtained result is applied to a few common types of tapers. Transmission lines with exponential, Reflection loss is away to express the reflection coefficient triangular and Hermite taper are considered. All results in logarithmic terms (dB). The reflection coefficient is the are plotted using program package Mathematica 3.0. The ratio of the reflected signal voltage level to the incident signal presented results should be useful in solving reflection voltage level. Reflection loss is the number of decibels that the coefficient problems. reflected signal is below the incident signal. Reflection loss is always expressed as a positive number and varies between Keywords −−− Reflection coefficient, Voltage standing wave ratio (VSWR), Exponential, triangular and Hermite taper. infinity for a load at the characteristic impedance and 0 dB for an open or short circuit.

I. INTRODUCTION

The analysis of nonuniform transmission lines has been a subject of interest of many authors. Uniform transmission lines ( x ) Z p can be used as impedance transformers depending on the Z C 0 Z C frequency and length of the line [1]. The nonuniform lines have the advantage of wide-band impedance matching when x ∆ x they are used as impedance transformers and larger rejection x= 0 bandwidths when they are used as filters [1, 2]. Some results d for nonuniform exponential loss transmission line used as impedance transformer are presented in papers [3-5]. The paper [6] gives a solution in closed-form of the equation for Fig. 1. A tapered transmission line matching section. value of arbitrary complex impedance transformed through a length of lossless, nonuniform transmission line with Let the transmission line shown in Fig. 1 be considered. The exponential, cosine-squared and parabolic taper. continuously tapered line can be modelled by large number of incremental sections of length ∆ x . One of these sections, In this paper we will derive an approximation theory, based on the theory of small reflections, to determine the reflection connected at, has a characteristic impedance of Z C (x) + ∆ Z C coefficient response as a function of the impedance taper. and one before has a characteristic impedance of Z C (x) , as it Obtained result is applied to transmission lines with exponen- is shown in Fig. 2. These impedance values are conveniently tial, triangular and Hermite taper [7]. For all examples the normalized by Z . Then the incremental reflection voltage standing wave ratio is plotted using program package C0 coefficient from the step at distance x is given by Mathematica 3.0.

(Z C (x) + ∆ Z C )− Z C (x) ∆ Z C II. REFLECTION COEFFICIENT EQUATION FOR ∆Γ = ≈ . (1) ()Z C (x) + ∆ Z C + Z C (x) 2 Z C NONUNIFORM TRANSMISSION LINE The traditional way of determining RF impedance was to In the limit as ∆ x → 0 , it can be written as measure voltage standing wave ratio (VSWR) using an RF detector, a length of slotted transmission line and a VSWR d Z 1 d(ln Z (x)) d Γ = C = C d x (2) meter. VSWR is defined as the maximum value of the RF 2 Z 2 d x envelope over the minimum value of the RF envelope. As the C probe detector was moved along the transmission line, the since relative position and values of the peaks and valleys were d(ln f (x)) 1 d( f (x)) = . noted on the meter. From these measurements, impedance d x f (x) d x could be derived. The procedure was repeated at different

1 Zlata Z. Cvetkovi ć is with the Faculty of Electronic Engineering, University of Nis, Aleksandra Medvedeva 14, 18 000 Niš, Serbia and Montenegro, E-mail: {zlata, as, bojananik}@elfak.ni.ac.yu

45 Reflection Coefficient Equations for Tapered Transmission Lines The corresponding incremental reflection coefficient at the Transmission line attenuation improves the VSWR of a load input end can be written as follows or antenna. Therefore, if you are interested in determining the performance of antennas, the VSWR should always be − j2β x d Γin ≈ e d Γ . (3) measured at the antenna connector itself rather than at the output of the transmitter. Transmission lines should have their By using formula (2) of small reflections, the total reflection insertion loss (attenuation) measured in lieu of VSWR, but coefficient at the input end of the tapered section can be VSWR measurements of transmission lines are still important determined by summing all the partial reflections with their because connection problems usually show up as VSWR appropriate phase angles spikes. d d 1 d  Z (x)  Γ = d Γ = e − 2j β x ln C  d x = Γ e − jϕ . (4) III. TAPERED TRANSMISSION LINES in ∫ in ∫   in 2 d x  Z  0 0 C0 In this paper, three types of nonuniform transmission lines

with exponential, triangular and Hermite function taper are So if Z C (x) is known, reflection coefficient at x = 0 can considered. be found as a function of frequency. On the other hand, if Γ in A. Exponential taper is specified, then in principle Z C (x) can be determined. Along an exponential taper, the impedance is changing exponentially with distance, ∆Γ Z (x) = Z e k x , for 0 < x < d (8) Z c ( x )+ ∆Z c C C0 Z c ( x ) as indicated in Fig. 3. At x we have Z )0( = Z . At x ∆x = 0 C C0 k d x = d we wish to have Z C (d) = Z Cd = Z C0e , what Fig. 2. Model for an incremental step change in determines the constant impedance of the tapered line. 1 k = ln M , (9) When a transmission line is terminated with an impedance, d

Z P , that is not equal to the characteristic impedance of the where k is a taper coefficient and transmission line, Z , not all of the incident power is C Z absorbed by the termination. One part of the power is reflected M = Cd (10) back so that phase addition and subtraction of the incident and Z C0 reflected waves creates a voltage standing wave pattern on the is a taper ratio. Z and Z are the characteristic impedan- transmission line. The ratio of the maximum to minimum C0 Cd voltage is known as the voltage standing wave ratio (VSWR) ces of the transmission line at the left (source) and right (load) and successive maxima and minima are spaced by 180°. sides, respectively. If the equation for reflection coefficient is solved for the From (4), the total reflection coefficient at the input end is VSWR, it is found that found as

d 1+ Γin 1 − 2j β x d k x Γin = e ()ln e d x = VSWR = . (5) 2 ∫ d x 1− Γin 0

The reflection or return loss is related through the following 1 Z sin β d = ln Cd e − jβd = Γ e − jϕ . (11) equation in 2 Z C0 β d

Reflection loss = 20 log 10 Γin dB . (6) B. Triangular taper

Reflection loss is a measure in dB of the ratio of power in Characteristic impedance along triangular taper changes as the incident wave to that in the reflected wave, and as defined above always has a positive value.  x 2 Z 2  ln Cd  d  ZC 0 Also of considerable interest is the mismatch or insertion Z C ()x = Z C0 e , for 0 < x < d / 2 loss. This is a measure of how much the transmitted power is (12) attenuated due to reflection. It is given by the following  x  x 2  Z 4 −2  −1 ln Cd equation:  d  d   Z Z x = Z e   C 0 , for d / 2 < x < d , C () C0 2 Insertion loss = −10 log 1− Γ dB . (7) 10 ( in ) as it is presented in Fig. 3.

46 Zlata Cvetkovi ć, Slavoljub Aleksi ć, Bojana Nikoli ć Evaluating reflection coefficient from (4) gives

2 1 − j β d sin β d  − jϕ Γin = ln M e   = Γin e (13) 2  β d 

C. Hermite taper

In the case of Hermite taper, impedance of the transmission line varies with distance x as follow

()k x 2 Z C ()x = Z C0 e , for 0 < x < d (14)

The resulting reflection coefficient response is given by (4)

− 2j βd 2j βd Fig. 5. Resulting VSWR magnitude response e (1+ 2j βd − e ) − jϕ Γin = ln M = Γin e . (15) on exponential line for different taper ratio. ()2βd 2

Fig. 3 shows the impedance variations for the exponential, triangular and Hermite tapers of the same taper ratio, M = 3 .

Fig. 6. Resulting reflection coefficient magnitude versus frequency for the exponential, triangular Fig. 3. Impedance variations for the exponential, triangular and Hermite taper. and Hermite tapers of the same taper ratio, M = 3 .

IV. NUMERICAL RESULTS

According to the analysis presented above for exponential, triangular and Hermite transmission line, different calculations are done.

Fig. 7. Resulting reflection coefficient magnitude versus frequency for the exponential, triangular and Hermite taper.

Fig. 4 shows resulting reflection coefficient magnitude Fig. 4. Resulting reflection coefficient magnitude response on exponential line for different taper ratio response on exponential line for different taper ratio. M = 4,3,2 . For the same taper ratios on exponential line, Fig. 5 presents VSWR magnitude response.

47 Reflection Coefficient Equations for Tapered Transmission Lines In order to compare obtained results, we consider an resulting reflection coefficient and VSWR are shown for the exponential, triangular and Hermite taper transmission line, same taper ratio M = Z Cd / Z C0 = 3. which are used at the same taper ratio M = 3 , as impedance Obtained results are plotted using program package transformers from Z =100 Ω to Z = 300 Ω . C0 Cd Mathematica 3.0.

Resulting reflection coefficient magnitude versus frequency REFERENCES for the exponential, triangular and Hermite transmission line taper is shown in Fig. 6. [1] Devendra K. M., Radio-Frequency and Microwave Commu- nication Circuits: Analysis and Design, John Wiley & Sons, Inc. In Fig. 7, the voltage standing wave ratio (VSWR) of these 2001. nonuniform transmission lines is shown. [2] Ghausi, S. M., and Kelly, J. J., Introduction to Distributed-Pa- rameter Networks , Holt, Rinehart and Winston, Inc., New York, All the figures are plotted using program package 1968. Mathematica 3.0. [3] Cvetkovic Z.Z., Aleksic S.R., Javor V.: "Lossy Exponential Transmission Line as Impedance Transformer," Proceedings of V. CONCLUSION papers , XIII International Symposium on Theoretical Electrical Engineering PSC 2005 , pp. 167-170, Timisoara, Romania, The modern network analyzer system sweeps very large November 2005. frequency bandwidths and measures the incident power, P i, [4] Cvetkovic Z.Z. Petkovic R.A."Characteristics of Loss and the reflected power, P r . Because of the considerable th Proc. of Electronic Devices and computing power in the network analyzer, the reflection loss is Exponential Line Network", 4 Systems Conference EDS'96, pp. 34-38, Brno, Czech Republic, calculated from the equation given previously, and displayed 1996 . in real time. Optionally, the VSWR can also be calculated [5] Cvetkovic Z.Z., "Eksponencijalni vod kao sirokopojasni transfor- from the reflection loss and displayed real time. mator impedanse ", II Telekomunikacioni forum TELFOR'94, str. In this paper, by using small reflections, the total reflection 183-186. Beograd, 1994. coefficient at the input end of the tapered section is [6] Ahmed M. J.: Impedance Transformation Equations for Exponential, Cosine −Squared, and Parabolic Tapered Line determined by summing all the partial reflections up these Transmission Lines , IEEE Trans. on Microwave Theory and incremental reflections with their appropriate phase angles. Tech., Vol. MTT-29, No. 1, pp. 67-68, 1981. For exponential transmission line, resulting reflection [7] Kobayashi K., Nemoto Y., Sato R., “Time Equivalent coefficient and VSWR magnitude response for different taper Representation of Nonuniform Transmission lines Based on the ratio are presented. In order to compare obtained results for Extended Kuroda’s Identity”, IEEE Trans. on Microwave Theory exponential, triangular and Hermite transmission line, and Tech ., Vol. 30, No.2, pp. 140-146, 1982.