ece4333notes5

Impe danc e Matching a nd Tr ansformation Matching the source and load to the transmission line or waveguide in a g eneral microwave net work i s neces sary to deliver max imum po wer from the source to the load. In many cases, it is not possible to choose all impedances such that overall matched condi tions result. The se situations require that matching networks be used t o eliminate reflections. T-line or waveguide to termin ation matching network T-line or waveguide to t-line or waveguide matching network Depending o n the application, matching m ay b e required ov er a band o f frequencies such t hat the b andwidth of the mat ching network i s an important design parameter. If the load impedance varies over a gi ven range, a matching network which can be adjusted or tuned as necessary. In general, matching networks are constructed with reactive components only so that no l oss is added to the overall network. Lumped Element Networks For frequencies up t o approximately 1 GH z, matching n etworks containing lumped element s (L-net works) may be used. The circuit elements (capacitors and inductors) must be small enough relative to wavelength so that the normal circuit equations for voltage and current are valid. L-networks are easily analyzed using either circuit equations or the Smith chart. The con figuration of the mat ching L-network will depend on the si ze oof the load impedan ce relative to the char acteristic impedan ce Z. The LLgeneral load impedan ce Z and its correspon ding normalized val ue z defined by Lo L LIf R < Z, then r < 1 (z is outside the r =1 circle o n the Smith chart). Lo L LIf R > Z, then r > 1 (z is inside the r =1 circle o n the Smith chart). LoLumped Element Matching Network for R > Z The reactance jX and suscept ance jB may be any combination of capacitors (X < 0, B > 0) or inductors (X > 0, B < 0). For a matched network, the input in oimpedan ce Z must be equal to Z which gives Multiplying out this equation and equating the real and imaginary terms on both sides of the equation yields two equations for the unknowns X and B. If we solve Equation (1) for X, and insert the result into (2), we find a quadratic equation for B with a solution o f LoThe requirement that R > Z ensures that the term under the square root in the numerator of the expression for B is real. Note that two solutions for B are possible and both solutions are physically realizable given that B can be p ositive o r neg ative. Once B is determined, X can be found using Equation (2): With two pa irs of solutions for both B and X, there are two different matching network sol utions. LoLumped Element Matching Network for R < Z in oFor a matched network, the input admi ttance Y must be equal to 1/Z which gives Multiplying out this equation and equating the real and imaginary terms on both sides of the equation yields two equations for t he unknowns X and B. Solving these equations for B and X yields LoThe requirement that R < Z ensures that the terms under the square roots in the expressions for B and X and are real. Again, with two solution pairs for B and X, there are two different matching network sol utions. Examp le (Lumped element matching networks) Design two lumped element matching networks to match a 50 Ù line LL oto a load impedan ce of Z = (70 + j 100) Ù [R > Z] at 700 MHz.. Using the matching n etwork design equations, we find These solutions correspond to lumped elements of The Smith chart can also be used to design the matching networks. W e first locate the load impedance on t he S mith ch art. Given the p arallel connection of the rightmost matching network element (jB) with the load, we add the admittance of the these t wo elements together. Since the parallel matching ne twork element is purely susceptive, we move along the constant con ductance ci rcle from the l oad admi ttance i n the p roper direction for the gi ven matching network el ement (smal ler admi ttance i f jB represents an inductor or larger admi ttance i f jB represents a capacitor). oUsing a dmittances, we rotate until we intersect the 1/Z + j0 ad mittance ocircle (or the Z impedance circle using impedances). The change in the susceptance of these two points represents the matching element jB. The impedance of the parallel combination o f the load and jB is th en added to othe reactance of the matching element jX by r otating along the Z + j0 impedan ce circle until we reach the center of the Smith chart (matched condition). Data P oint #1 (70.0 + j100.0) Ù Data P oint #2 (50.2 - j90.4) Ù Data P oint #3 (50.2 + j0.0) Ù Data P oint #1 (70.0 + j100.0) Ù Data P oint #2 (49.6 + j90.0) Ù Data P oint #3 (49.6 + j0.0) Ù Singl e Stub Tuners Given that we can obtain any value of reactance or susceptance with the proper length of short-circuited or open-circuited transmission line, we may use t hese t ransmi ssion line st ubs as mat ching networks. Shun t Stub tl oY = Y + jB [Input admittance of the terminated t-line section] sY = !jB [Input admittance of the stub (short or open circu it)] int l soY = Y + Y = Y [Overall input admi ttance] Series Stub tl oZ = Z + jX [Input impedance of the terminated t-line section] sZ = !jX [Input impedance of the stub (short or open circu it)] int l soZ = Z + Z = Z [Overall input impedan ce] Single Shunt Stub Tuner Design Procedure 1.Locate normalized load impedance and draw VSWR circle (normalized l oad admi ttance po int is 180 from the n orma lizedo impedance point). 2.From the normalized load admittance point, rotate C W (toward generator) on the V SWR circle until it intersects the r = 1 circle. This rotation d istance is the length d of the terminated sect ion of t-tline. The nom alized admittance at this point is 1 + jb. 3. Beginning a t the stub e nd ( rightmost Sm ith chart poi nt is the admittance of a short-circuit, leftmost Sm ith chart poi nt is the admittance of an ope n-circuit), rotate CW (toward generator) until the point at 0 ! jb is reached. This rotation di stance is the stub length l. Single Series Stub Tuner Design Procedure 1. Locate normalized l oad impedan ce and draw VSWR circle. 2.From the normalized load imp edance point, ro tate C W (toward generator) on the V SWR circle until it intersects the r = 1 circle. This rotation d istance is the length d of the terminated sect ion of t-tline. The nom alized impedance at this point is 1 + jx. 3. Beginning a t the stub e nd ( leftmost Sm ith chart poi nt is the impedance of a short-circuit, rightmost Sm ith char t point is the impedance of an open-circuit), rotate CW (toward generator) until the point at 0 ! jx is reached. This rotation distance is the stub length l. Example (sh unt stub tuner) Design a s hort-circuited shunt stub tuner to to mat ch a l oad Limpedan ce of Z = (25!j50) Ù to a 50 Ù transmi ssion line. Data P oint #1 (25.0 - j50.0) Ù Data P oint #2 (14.3 - j22.5) Ù Data P oint #3 (49.8 - j0.0) Ù Example (o pen-circu ited shunt stub tuner) Design two open-circuited shu nt stub tuners to mat ch the load impedan ce in the previous lumped element matching network examp le, L[Z = (70+j100) Ù, 50 Ù transmission line at 700 MHz ]. The two designs shoul d represent the two shortest stub d istances from the load. Data P oint #1(70.0 + j100.0) Ù Data P oint #1(70.0 + j100.0) Ù Data P oint #2(12.6 - j21.6) Ù Data P oint #2(12.6 + j21.8) Ù Data P oint #3(49.8 + j0.0) Ù Data P oint #3(50.3 + j0.0) Ù 12 d = 101 mm d = 160 mm 12 l = 143 mm l = 71 mm Frequ ency Respon se of Matching N etworks Ideal lumped element and single stub matching networks pr ovide perfect matching (Ã=0) at only one frequency. In our lumped element and single stub mat ching network examples, we showed two solutions that yielded perfect matching a t the design fr equency. However, the compone nt configuration in a lumped element matching network and the stub position in a stub m atching n etwork wi ll affect the frequency respons e of the network away from the d esign frequency. We may plot the fr equency respon se of the reflection coeff icient to illustrate the different respon ses. Given either type of matching n etwork, the reflection coefficient looki ng into the mat ching network may be w ritten as inIn order to det ermine the variation of Z with respect to frequency, we need to know the variation of the load impedance with respect to frequency. We Lassume that the load impedan ce used i n our examp les (Z = 70 + j100 Ù at 700 MHz) consists o f a series combination of a resisto r (R = 70 Ù) and an inductor (L = 22.74 nH) [ùL = 100 at f = 700 MHz ]. The resulting two circuits for the lumped element matching networks are shown below. According to the design equations, the input impedances for the two networks looking into the matching network input ports are For the case of the shunt stub networks, the input admittance looki ng into the matching network is in,1 1 1 in,2 2 2We find Z by inserting (l, d) and find Z by inserting (l, d). Comparing the frequency respons es of the lumped element matching networks and the stub tuners shows that, in general, the lumped elements yield a slightly broader bandwi dth. Quarter W ave Transform er The quarter wave transformer is a simple quarter wavelength section of 1transmi ssion line with ch aracteristic impedan ce Z that when placed obetween a t ransmi ssion line of characteristic impedan ce Z and a real load L1impedan ce R yields a mat ched system. The val ue of Z is determined by using the equation for the input impedance of a terminated transmission line. The input impedan ce is pu rely real since t he line length is on e qu arter wavelength: The tangent terms become unb ound ed so that, in the limit, we find oFor a matched system, the input impedance must equal Z which gives Example (Quarter wave transformer, Smith chart illustration) Determine the char acteristic impedan ce of t he quarter wave t ransfor mer required to match a 300 Ù load to a 50 Ù transmi ssion line. The fr equency r espon se of the q uarter-wave t ransfor mer is defi ned according to the frequency response of the reflection coefficient: The fr actional bandwi dth of the qua rter wave transformer may be approximated by where Äf is the bandwi dth for the matching n etwork at the maximum mallowable reflection coeff icient magn itude Ã. It shoul d be not ed that the frequency analysis of the qua rter wave transfor mer is only val id for waveg uiding structures that carry TEM waves since the characteristic impedance of guiding structures carrying non-TEM waves is frequency dependent. Double Stub T uner The single stub tuner is very flexible at matching any load impedance to a given transmi ssion line. However, if the lo ad impedance var ies so that an adjustable tuner is necessary, the single stub tuner requires that the position of the st ub tuner also varies. A double stub tuner allows for an adjustable mat ching network t hat utilizes adjustab le length stubs at fixed positions. The distance bet ween t he st ubs is defi ned as d while the stub 12 1lengths ar e l and l where l defines the length of the stub closest to the load. oThe distance d between t he load impedan ce and the first stub is somewhat tl,1arbitrary in that we may always determine the input admi ttance Y given Lothe load impedan ce Z and the length of l ine d. However, for a given value oof d, we may enco unter some load impedan ces which cann ot be mat ched for a given stub separation distance d. tl,1The first stub m ust add the proper amount of susceptance to Y that, tl,2when transfor med t hroug h the distance d to yield Y, places this input admi ttance on the g = 1 circle. The secon d stub is chosen so as t o cancel tl,2the suscept ance of Y. in,1The input admi ttance Y may be w ritten as This admittance represents the equivalent load on the end o f the tl,2transmission line segment of length d used t o find the input admi ttance Y. in,2 oThe real part of Y must equal Y in order to match the transmi ssion line, which yields the susceptance of the first stub. The resulting equation f or s1B is s1Note that there are two possible so lutions for B. The susceptance of the in,2 s1second stub must cancel that of Y. Setting B equal to the negative of the in,2imagi nary par t of Y yields The upper and lower signs on the stub susceptance equations correspond to the given pair o f susceptance solutions. T he len gths of the stubs are found using the susceptance equations for an open-circuited or short- circuited stub. or Theoretically, the stub spacing d can be any value. However, stub spacings near 0 a nd ë/2 yield matching networks that are highly sensitive to frequency. In practice, stub spaci ngs of ë/8 and 3ë/8 are commonly used. Example (D ouble stu b tuner) Design a double stub tuner using short circuited st ubs separated by ë/8 Lto mat ch a l oad impedan ce of Z = (25 + j 40) Ù to a 50 Ù line. Assume the value of the given load impedance is that seen at the conne ction point of the first stub. ss12Inserting into the equations for B and B yields s1B = 0. 0200 , 0.0560 S s2B = !0.0120 , 0.0520 S The corr espon ding stub lengths ar e s1l = 0. 375 ë, 0.445 ë s2l = 0. 164 ë, 0.442 ë ss12 B = 0. 0200 S B = !0.0120 S ss12 l = 0. 375 ël = 0. 164 ë DP #1 (86.5 - j15.5) Ù DP #2 (36.8 - j22.2) Ù DP #3 (50.2 - j0.0) Ù eq L s eqLs 11DP #1 Z = Z 2 ZY = Y + Y Frequ ency Respon se of the D ouble Stub T uner in,2The input admittance looking into the double stu b tuner (Y) is given by where the susceptances of the two stubs are The corresponding reflection coefficient is In order to plot the frequency response of the reflection coefficient, we assume that the load impedance used in our double stu b matching network Lexamp le (Z = 25 + j40 Ù at 1 GHz) consists of a s eries comb ination of a resisto r (R = 25 Ù) and an inductor (L = 6.37 nH) [ùL = 40 at f = 1 GHz]. The previously obtained solutions for the dou ble stub m atching n etworks are designated as dou ble stub tuner #1 a nd d oubl e stub tuner #2. Doubl e stub tuner #1 Doubl e stub tuner #2 ss ss 11 11B = 0. 0200 S (l = 0. 375 ë) B = 0. 0560 S (l = 0. 445 ë) sss s 222 2B = !0.0120 S(l = 0. 164 ë)B = 0. 0520 S (l = 0. 442 ë) Note that the fr equency r espon se of double stub tuner #1 is much l ess frequency se nsitive that of do uble stub tuner #2. This is due to the fact that the stub lengths for tuner #2 a re close to ë/2 while those of tuner #1 are not. Both designs are useful depending on the app lication. Double stub tuner #1 is preferable when a wider bandwi dth is required. However, double stub tuner #2 m ay be useful in a narrowband application whe re the matching network performs the function of a bandpass filter. The Theo ry of Smal l Reflections One way of designing broadband matching networks is to use multiple sections of transmission line rather than just one section as in the case of the q uarter wave t ransfor mer. In order to simplify the analysis of t hese multiple section mat ching networks, the theory of small reflections is utilized. LConsider a load impedan ce Z connected to a transmission line of 1characteristic impedan ce Z through a section o f transmission line of 2characteristic impedan ce Z as shown below. The length of the conne cting transmission line is l. We can easily write equations for the local (or partial) reflection and t ransmission c oefficients at the two po ints on t he transmissio n line network where reflec tions may occur. We assume a wave of unit amplitude is traveling toward the load on t he 1Z section of transmi ssion line (e). Using the conne ction between the!jâz two transmi ssion lines as reference ( z = 0), the wave amplitude incident on the transmi ssion line con nection point is 1p0. The amplitude of theo 11reflected w ave i s à while the amp litude of the transmi tted w ave i s T. The transmi tted w ave t ravels a distance l to the load connection (with a t otal phase shift of e = e). The reflected w ave at this point is !jâl !jè This reflected w ave no w travels a distance l in the opposite direction to the transmission line connection. The wave reflected back to the load at this point is (a ccounting for the additional e phase shift)!jè 1while the w ave t ransmi tted to the Z transmission line is This process is then repeated for the multiple reflections at each end of the conne cting transmission line section whi ch leads to an infinite sum of reflected and transmi tted si gnals to describe the total reflected w ave. The total reflected wave at the transmission line connection is The summation in the above equation can be written in closed form using the fol lowing geometric series. This gives so that the total reflected w ave may be w ritten as This expression can be fur ther simplified by noting that which yields 3If *Ã1 Ã*<<1, then the exp ression for the total reflection coeff icient may be app roximated by This expression for the total reflection coeffi cient shows that the reflected wave is dom inated by the initial reflection a t the transmission l ine conne ction and the first reflection from the load conne ction. Multisection Matching T ransform er A multisection m atching transformer may be formed by conne cting N transmission line sections in series between the feeder transmission line of oLcharacteristic impedan ce Z and the load impedan ce of Z. 012 NWe may defi ne local reflection coeff icients Ã, Ã, Ã, ... , à at the connection points of the multisection matching transformer. If we assume that each transmi ssion line section is of equal length, the local nreflection c oefficients are of the same sign ( Z increases or d ecreases monotonically along the length of the transformer), an d the change in impedance from section to sect ion m akes the theory of small reflections applicable, then the ove rall reflection c oefficient of the system may be written as If we further assume t hat the reflection coeffi cients along the transfor mer o N 1N !1are symmetric (i.e., à = Ã, à = Ã, etc.), then The last term in the reflection coeff icient equation is if N is odd and if N is even. The sums of complex expon entials in the reflection coefficient equation may be written as cosines to yield These exp ressions are similar in form to a F ourier series. This implies that we may synthesize any fr equency respon se given a large eno ugh number of sect ions with the proper reflection coeff icients. Binomial Multisection Matching T ransform er If we may choos e the parameters of the multisection m atching transfor mer such t hat then the reflection coeff icient has t he fol lowing property: This property defi nes a maximally flat respons e for the refection coefficient oat a ce nter frequency f defi ned by owhere ë defines the wavelength of the center frequency. Thus, the length of each sec tion of the mat ching transfor mer would be on e-quarter wavelength at th e designed center frequency. The characteristic impedan ces of the individual transfor mer sections mu st be determined such that the maximally flat respons e it obtained. The general form of the reflection coeffi cient approximation for the N- section matching transfor mer can eas ily be w ritten in terms of a bi nomial series according to nwhere A is the general amplitude coefficient and C is the b inomial coefficient given by Equa ting coefficients for the binom ial series and the reflection coefficient approximation yields The amplitude coefficient A may be determined by taking the limit of the reflection coeff icient equations as the fr equency decr eases to zero. In that case, and each sect ion of the mat ching transfor mer is zero length. This gives or The char acteristic impedan ces of t he adja cent sections of t he b inomial transformer are related by which yields The angle è for the binomial multisection matching transfor mer is related to the frequency according to where each section of the transfor mer is ë/4 in length. Thus, the bandwidth of the transformer may be defined in terms of the angle è. If we defi ne a mmaximum al lowable reflection coeff icient magn itude of Ã, then the bandwi dth Äf can be w ritten as 12where f and f define the lower and u pper frequencies of the bandwi dth, 12 1respect ively. The angle s è and è correspon ding to the fr equencies f and 2f are found by solving the equation for the overall reflection coefficient evaluated at the max imum allowable val ue. Solving this eq uation for è yields 11The angle è associ ated w ith the lower frequency f is the solution to the oabove equation whi ch is less than è = ð/2. Du e to symmetry, the two 12 oangles è and è are equally spaced about è. Thus , the bandwi dth of the transfor mer may be w ritten as In de termining t he individual characteristic impedances of the multisection binomial matching transformer, if o ne uses the previously derived formula significant errors may be introduced which bu ild from the input of the matching section to the load. Thi s is caused by the fact that the theory of smal l reflections was utilized t o determine the binomial constants but an exact formulation was used to determine the amplitude constant A. A more accurate determination of t he characteristic impedances results when the constant A is also determined using the theory of smal l reflections. The local reflection coeffi cients for the connections of the ind ividual transmi ssion line sect ions are defi ned by According to the theory of small reflections, the discontinuities in the characteristic impedance at the junctions of the multisection m atching transformer are small so that The local reflection coeffi cients may then be w ritten in terms of t he natural log of this quotient since so that nn+1Solving this eq uation for Z in terms of Z gives N+1We can show that, using this formula, the impedan ce Z is exactly equ al Lto Z as it shoul d. The binomial series has t he speci al property that so that Example (B inomial mu ltisection matching transformer) Design a 4-section bi nomial matching transformer to match a 300 Ù load to a 50 Ù line and de termine the resulting percentage bandwi dth for mà = 0. 1. The binomial coeff icients are given by The impedances of the 4 transformer sec tions are Data P oint #1 (300.0 + j0.0) Ù Data P oint #2 (239.8 + j0.0) Ù Data P oint #3 (122.5 + j0.0) Ù Data P oint #4 (62.5 + j0.0) Ù Data P oint #5 (50.0 + j0.0) Ù The percentage bandwidth of the 4-section binomial matching network is given by Chebyshev M ultisection Matching T ransform er A Cheb yshev multisection matching transfor mer can provi de even larger bandwidths than a binomial multisection matching transfor mer for a given num ber of transmission line sections. The increased ba ndwi dth of the Chebyshev transformer comes at the cost of increased ripple ove r the passband of the m atching net work. However, we may still designate some maxi mum allowable reflection coeff icient for the desi gn of the C hebyshev transformer. The Cheb yshev transformer exploits the characteristics of the Chebyshev p olynomials. Properties of Chebyshev Polynomials 1.Even ordered C hebyshev p olynomials are even fun ctions. 2.Odd ordered C hebyshev p olynomials are odd functions. 3.The magnitude of any Chebyshev po lynom ial is unity or less in the range of !1 # x #1. n4.T (1) = 1 for all Chebyshev p olynomials. 5.All zeros (roots) of the C hebshev po lynomials lie within the range of !1 # x #1. Using the properties of Chebyshev pol ynom ials, we may design m atching networks w ith a r eflection coeff icient at or below some prescribed level over a wide bandwi dth. Throug h the transfor mation of x = cos è, the C hebyshev p olynomials may be w ritten w ith an argument of cos è. Thus, the C hebyshev p olynomials may be w ritten com pactly as The C hebyshev p olynomials for all argument s x may be w ritten as In order to implement the Chebyshev matching transformer, the endpoints mmof the required passband (è, ð !è) [with a center frequency at ð/2] must be map ped onto the range where the C hebyshev p olynomials satisfy This mapping is defined by The resulting Chebyshev polynomials fo r the given mapping are We have previously shown us ing the theory of small reflections that the reflection coeff icient of an N section mul tisection mat ching transfor mer may be w ritten as Given that the maximum magnitude of each Chebyshev polynomial should be unity within the passband, the maximum magnitude of the reflection coefficient in the passband is The constant A is det ermined by t aking the limit as è approaches 0. The reflection coeff icient then becom es mIn order to determine the ang le è, we solve the following equation: The characteristic impedances are found after the local reflection coefficients are determined. The resulting f ractional bandwi dth for the Chebyshev m ultisection transformer is Example (C hebyshev multisection matching transformer) Design a 4-section Che byshev matching transformer to match a 300 Ù mload to a 50 Ù line and for à = 0. 1. The equation for the reflection coefficient with N = 4 gives mThe ang le è is found by mwhich yi elds è = 38 .2. Insering the fourth order Chebyshev p olynomialo given by into the reflection coeff icient formula gives Equating the coeffi cients associated w ith the cosi ne terms gi ves The remai ning local reflection coeff icients are found by symmet ry: The resulting characteristi c impedances are