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Electronic Science Electrodynamics and Microwaves 17. Stub Matching

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Electronic Science Electrodynamics and Microwaves 17. Stub Matching 1 Module 17 Stub Matching Technique in Transmission Lines 1. Introduction 2. Concept of matching stub 3. Mathematical Basis for Single shunt stub matching 4 .Designing of single stub using Smith chart 5. Series single stub 6. Concept of Double stub matching 7. Mathematical Basis for Double Stub Matching 8. Design of Double stub using Smith Chart: 9. Summary Objectives After completing this module, you will be able to 1. Understand the concept of matching stub and different types of it. 2. Understand the mathematical principle behind the stub matching technique. 3. Design single stub and double stub using Smith chart. 4. Know about advantages and limitations of stub matching technique. Electrodynamics and Microwaves Electronic Science 17. Stub Matching Technique in Transmission Lines 2 1. Introduction In the earlier modules , we have discussed about how matching networks and QWT can be used for matching the load impedance with the lossless transmission line. Both these techniques suffer from some drawbacks .Especially the losses like, insertion loss, mismatch loss are of much more prominent losses. There is another technique known as “Stub Matching Technique” to match the load impedance with the Characteristic impedance of the given transmission line so as to have zero reflection coefficient. The technique is marvelous and the designing can be done using Smith chart in a very simple way. Let us discuss it with adequate details rather qualitatively avoiding mathematical rigor in it. 2. Concept of Matching Stub A section of a transmission line having small length is called as a stub. Two types’ namely single stub and double stub comprising of one or two stubs are in common use. A section of a two wire transmission line of small length l connected in parallel to the line to be matched at a distance d from the load as shown is known as a single stub. The stub may or may not have its characteristic impedance the same as that of the given transmission line. Generally the stub is chosen so as to have the same characteristic impedance as that of the given transmission line. Electrodynamics and Microwaves Electronic Science 17. Stub Matching Technique in Transmission Lines 3 Similarly the stub may be short circuited or open circuited at its open end. However, short circuited stub is preferred because practically it is easier to achieve a reliable perfect sho rt circuit rather than a perfect open circuit. Thus, a short circuited stub of proper length l connected at proper distance d serves the purpose of a perfect match. Determining precise values of I and d are the main computations involved in stub matching technique. The values of distance d and the length l of the stub depend upon whether the stub is desired to be inductive or capacitive. Different types of arrangements of stubs such as Single shunt stub ,Single series stub ,double stub, Triple stub etc. can be found to be documented in literature. 3. Mathematical Basis for Single shunt stub matching A simple concept of impedances in parallel is used to derive the necessary equation needed to compute the value of normalized stub impedance ys. Once the normalized stub impedance ys is known, the values of d and l can be calculated easily using Smith Chart. So, let us see how to derive the equation needed to compute the value of normalized stub impedance ys. Consider the circuit with a loss less transmission line of characteristic impedance Z0, terminated in a load with impedance ZL as shown. Let Vg and Zg be the voltage and impedance of the generator respectively. Assume that the line is matched with the generator .Hence we have, Zg = Z0. A single stub of length l and characteristic impedance Z0s is connected across the points P and P’ in parallel to the transmission line at a distance d from the load. Let Zd be the impedance of the line at a distance d from the load and Zs - the impedance of the stub. Now for matching of the impedances at the connection of the stub with the line, the impedance between P and P’ seen from both sides must be equal to each other. It can be easily understood that the impedance between P and P’ seen from the load towards the generator is equal to the characteristic impedance Z0 of the transmission line. Let Zpp’, be the impedance across the points P and P’ seen from the generator side towards the load. Let us determine Zpp’. Electrodynamics and Microwaves Electronic Science 17. Stub Matching Technique in Transmission Lines 4 At the point P there is a division of signal. A part of the signal enters the stub and the remaining signal passes towards the load. Thus the two branches namely the stub and the line of length d between the stub and load are in parallel. It is easy to understand that, when seen from generator side, Zpp’ equals the impedance of the parallel combination of Zd and Zs as shown in diagram. Thus we have, Zpp’ = Zd // Zs = (Zd x Zs) / (Zd + Zs) Hence, 1/ Zpp’ = Ypp’ = (Zd + Zs)/ (Zd x Zs) = (1 / Zs) + (1 / Zd) = Ys +Yd Where Ys ,Yd and Ypp’ represent the corresponding admittances . If we divide by Y0, we obtain the relation in normalized form as follows. Ypp’/ Y0 = (Ys / Y0) + (Yd/ Y0) ypp’ = (yos ys)/y0 + yd Note the notations used in the derivation. i) Y0 represents the characteristic admittance of the transmission line. ii) Ys = yos ys is the admittance, Y0s is the characteristic admittance and ys is the normalized admittance of the single stub. iii) Yd = yd y0 is the admittance of the line and yd is normalized admittance of the line at a distance d from the load. Thus we have two results for admittance: i) Admittance across P and P’ = Ypp’= Ys +Yd as seen from generator towards load ii) Admittance across P and P’ = Y0 = 1/ Z0 as seen from the load towards generator . For satisfying the condition of matching, the two impedances must be equal. Thus we have, Ypp’ = Ys +Yd = Y0 ------- (1) Dividing both sides of this equation by Y0, we obtain the equation in normalized form as shown. -------------- (2) Electrodynamics and Microwaves Electronic Science 17. Stub Matching Technique in Transmission Lines 5 + = 1 ------ (3) Equation (2) can be rearranged to obtain expression for as follows. = (1- ) ---------- (4) Equation (4) plays an important role in determining stub position and its distance from the load using Smith chart. If the characteristic impedance of the stub is equal to the characteristic impedance of the line, that is if Yo = Yos, then equation (4) reduces to = (1- ) ----------- (4’) This much background is sufficient for designing a single stub. 4. Designing of single stub using Smith chart Let us try to understand how the two quantities namely stub length and its distance d from the load can be determined using Smith chart. Instead of discussing theoretically, let us discuss with the help of a numerical example. The statement of the problem is as shown. Problem: - A lossless transmission line of characteristic impedance Z0 =50 Ω is terminated in a load of impedance ZL=100 + j50 Ω. Determine the smallest length and the distance d from the load of a single capacitive stub having characteristic impedance Zos = 75 Ω shorted at its ends, which when connected, will serve the purpose of a perfect match of load with the transmission line. The step wise procedure for the solution of this problem is as follows. Step (i) : Compute and plot the normalized load admittance = 1/ on smith chart .It be remembered that while using Smith chart for calculations involving impedances, constant r circles become constant g circles while , constant x circles become constant s circles .Let point P represent . Here, = ZL/Z0 = 2+j , =1/ = 0.4 -0.2 j . Thus point P has coordinates g= 0.4, s = -0.2j. We may describe the same fact as P (0.4,-0.2j). Electrodynamics and Microwaves Electronic Science 17. Stub Matching Technique in Transmission Lines 6 Step (ii): Draw a circle of radius OP with O as its center. As we know the circle drawn in this way is known as the VSWR circle. Note the points Q and Q’ in which the circle drawn intersects the r=0 circle now called as g= 0 of the Smith chart. Note the coordinates ( g , s ) of point Q and Q’. In this problem we have Q ( 1 , - j ) and Q’(1 , j ).The point Q and Q’ represent the value of yd in equation (4). If we want to have inductive stub, the point Q’ is to be used for yd while for capacitive stub, yd is to be taken from point Q. As per the requirement in the problem, we need a capacitive stub. Hence we will select point Q. Accordingly, we have yd = 1 - j . Step (iii) : Calculate the value of normalized admittance using equation (4) and plot on the smith chart . Let S represent . Note that point S will always lie on the r= 0 i.e. g = 0 circle. = (1- ) ---------- (4) = (1- (1-j)) = 1.5 j Step (iv): Join the center O with points P, Q and S .Extend OP, OQ and OS to meet the radial distance scale marked with “Towards Generator” to meet in points K,M and N respectively. and note the readings k , m and n respectively. In the problem under consideration, we have k= 0.464, m = 0.338, and n = 0.157λ. Step (v): For determination of distance of the stub from the load proceed as follows.
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