Impedance Matching and Transformation

1 Impedance Matching and TransformationMatching the source and load to the transmission line or waveguidein a general microwave network is necessary to deliver maximum powerfrom the source to the load. In many cases, it is not possible to choose allimpedances such that overall matched conditions result. These situationsrequire that Matching networks be used to eliminate or waveguide to termination Matching networkT-line or waveguide to t-line or waveguide Matching networkDepending on the application, Matching may be required over a band offrequencies such that the bandwidth of the Matching network is animportant design parameter. If the load Impedance varies over a givenrange, a Matching network which can be adjusted or tuned as necessary.

2 Ingeneral, Matching networks are constructed with reactive components onlyso that no loss is added to the overall Element NetworksFor frequencies up to approximately 1 GHz, Matching networkscontaining lumped elements (L-networks) may be used. The circuitelements (capacitors and inductors) must be small enough relative towavelength so that the normal circuit equations for voltage and current arevalid. L-networks are easily analyzed using either circuit equations or theSmith configuration of the Matching L-network will depend on the sizeoof the load Impedance relative to the characteristic Impedance Z. TheLLgeneral load Impedance Z and its corresponding normalized value zdefined byLoLLIf R < Z, then r < 1 (z is outside the r =1 circle on the Smith chart).

3 LoLLIf R > Z, then r > 1 (z is inside the r =1 circle on the Smith chart).LoLumped Element Matching Network for R > ZThe reactance jX and susceptance jB may be any combination of capacitors(X < 0, B > 0) or inductors (X > 0, B < 0). For a matched network, the inputinoimpedance Z must be equal to Z which givesMultiplying out this equation and equating the real and imaginary terms onboth sides of the equation yields two equations for the unknowns X and we solve Equation (1) for X, and insert the result into (2), we find aquadratic equation for B with a solution ofLoThe requirement that R > Z ensures that the term under the square root inthe numerator of the expression for B is real. Note that two solutions forB are possible and both solutions are physically realizable given that B canbe positive or negative.

4 Once B is determined, X can be found usingEquation (2):With two pairs of solutions for both B and X, there are two differentmatching network Element Matching Network for R < ZinoFor a matched network, the input admittance Y must be equal to 1/Zwhich givesMultiplying out this equation and equating the real and imaginary terms onboth sides of the equation yields two equations for the unknowns X and these equations for B and X yieldsLoThe requirement that R < Z ensures that the terms under the square rootsin the expressions for B and X and are real. Again, with two solution pairsfor B and X, there are two different Matching network (Lumped element Matching networks)Design two lumped element Matching networks to match a 50 lineLLoto a load Impedance of Z = (70 + j 100) [R > Z] at 700 Using the Matching network design equations, we findThese solutions correspond to lumped elements ofThe Smith chart can also be used to design the Matching networks.

5 Wefirst locate the load Impedance on the Smith chart. Given the parallelconnection of the rightmost Matching network element (jB) with the load,we add the admittance of the these two elements together. Since theparallel Matching network element is purely susceptive, we move along theconstant conductance circle from the load admittance in the properdirection for the given Matching network element (smaller admittance if jBrepresents an inductor or larger admittance if jB represents a capacitor).oUsing admittances, we rotate until we intersect the 1/Z + j0 admittanceocircle (or the Z Impedance circle using impedances). The change in thesusceptance of these two points represents the Matching element jB. Theimpedance of the parallel combination of the load and jB is then added toothe reactance of the Matching element jX by rotating along the Z + j0impedance circle until we reach the center of the Smith chart (matchedcondition).

6 Data Point #1 ( + ) Data Point #2 ( - ) Data Point #3 ( + ) Data Point #1 ( + ) Data Point #2 ( + ) Data Point #3 ( + ) Single Stub TunersGiven that we can obtain any value of reactance or susceptance withthe proper length of short-circuited or open-circuited transmission line, wemay use these transmission line stubs as Matching StubtloY = Y + jB[Input admittance of the terminated t-line section]sY = !jB[Input admittance of the stub (short or open circuit)]intlsoY = Y + Y = Y[Overall input admittance]Series StubtloZ = Z + jX[Input Impedance of the terminated t-line section]sZ = !jX[Input Impedance of the stub (short or open circuit)]intlsoZ = Z + Z = Z[Overall input Impedance ]Single Shunt Stub Tuner Design normalized load Impedance and draw VSWR circle(normalized load admittance point is 180 from the normalizedoimpedance point).

7 The normalized load admittance point, rotate CW (towardgenerator) on the VSWR circle until it intersects the r = 1 circle. Thisrotation distance is the length d of the terminated section of nomalized admittance at this point is 1 + at the stub end (rightmost Smith chart point is theadmittance of a short-circuit, leftmost Smith chart point is theadmittance of an open-circuit), rotate CW (toward generator) until thepoint at 0 ! jb is reached. This rotation distance is the stub length Series Stub Tuner Design normalized load Impedance and draw VSWR the normalized load Impedance point, rotate CW (towardgenerator) on the VSWR circle until it intersects the r = 1 circle. Thisrotation distance is the length d of the terminated section of nomalized Impedance at this point is 1 + at the stub end (leftmost Smith chart point is theimpedance of a short-circuit, rightmost Smith chart point is theimpedance of an open-circuit), rotate CW (toward generator) until thepoint at 0 !

8 Jx is reached. This rotation distance is the stub length (shunt stub tuner)Design a short-circuited shunt stub tuner to to match a loadLimpedance of Z = (25!j50) to a 50 transmission Point #1 ( - ) Data Point #2 ( - ) Data Point #3 ( - ) Example (open-circuited shunt stub tuner)Design two open-circuited shunt stub tuners to match the loadimpedance in the previous lumped element Matching network example,L[Z = (70+j100) , 50 transmission line at 700 MHz]. The twodesigns should represent the two shortest stub distances from the load. Data Point #1( + ) Data Point #1( + ) Data Point #2( - ) Data Point #2( + ) Data Point #3( + ) Data Point #3( + ) 12d = 101 mmd = 160 mm12l = 143 mml = 71 mmFrequency Response of Matching NetworksIdeal lumped element and single stub Matching networks provide perfectmatching ( =0) at only one frequency.

9 In our lumped element and singlestub Matching network examples, we showed two solutions that yieldedperfect Matching at the design frequency. However, the componentconfiguration in a lumped element Matching network and the stub positionin a stub Matching network will affect the frequency response of thenetwork away from the design frequency. We may plot the frequencyresponse of the reflection coefficient to illustrate the different either type of Matching network, the reflection coefficient lookinginto the Matching network may be written as inIn order to determine the variation of Z with respect to frequency, we needto know the variation of the load Impedance with respect to frequency. WeLassume that the load Impedance used in our examples (Z = 70 + j100 at700 MHz) consists of a series combination of a resistor (R = 70 ) and aninductor (L = nH) [ L = 100 at f = 700 MHz].

10 The resulting twocircuits for the lumped element Matching networks are shown to the design equations, the input impedances for the twonetworks looking into the Matching network input ports areFor the case of the shunt stub networks, the input admittance lookinginto the Matching network isin,111in,222We find Z by inserting (l, d) and find Z by inserting (l, d).Comparing the frequency responses of the lumped element matchingnetworks and the stub tuners shows that, in general, the lumped elementsyield a slightly broader Wave TransformerThe quarter wave transformer is a simple quarter wavelength section of1transmission line with characteristic Impedance Z that when placedobetween a transmission line of characteristic Impedance Z and a real loadL1impedance R yields a matched system.