Tài liệu RF và mạch lạc lò vi sóng P6 ppt

6 IMPEDANCE TRANSFORMERS In the preceding chapter, several techniques were considered to match a given load impedance at a ®xed frequency. These techniques included transmission line stubs as well as lumped elements. Note that lumped-element circuits may not be practical at higher frequencies. Further, it may be necessary in certain cases to keep the re¯ection coef®cient below a speci®ed value over a given frequency band. This chapter presents transmission line impedance transformers that can meet such requirements. The chapter begins with the single-section impedance transformer that provides perfect matching at a single frequency. Matching bandwidth can be increased at the cost of a higher re¯ection coef®cient. This concept is used to design multisection transformers. The characteristic impedance of each section is controlled to obtain the desired pass-band response. Multisection binomial transformers exhibit almost ¯at re¯ection coef®cient about the center frequency and increase gradually on either side. A wider bandwidth is achieved with an increased number of quarter-wave sections. Chebyshev transfor- mers can provide even wider bandwidth with the same number of sections but the re¯ection coef®cient exhibits ripples in its pass-band. This chapter includes a procedure to design these multisection transformers as well as transmission line tapers. The chapter concludes with a brief discussion on the Bode-Fano constraints, which provide an insight into the trade-off between the bandwidth and allowed re¯ection coef®cient. 189 Radio-Frequency and Microwave Communication Circuits: Analysis and Design Devendra K. Misra Copyright # 2001 John Wiley & Sons, Inc. ISBNs: 0-471-41253-8 (Hardback); 0-471-22435-9 (Electronic) 6.1 SINGLE SECTION QUARTER-WAVE TRANSFORMER We considered a single-section quarter-wavelength transformer design problem earlier in Example 3.5. This section presents a detailed analysis of such circuits. Consider the load resistance R L that is to be matched with a transmission line of characteristic impedance Z o . Assume that a transmission line of length ` and characteristic impedance Z 1 is connected between the two, as shown in Figure 6.1. Its input impedance Z in is found as follows. Z in  Z 1 R L  jZ 1 tanb` Z 1  jR L tanb` 6:1:1 For b`  90  (i.e., `  l=4 ) and Z 1   Z o R L p ; Z in is equal to Z o and, hence, there is no re¯ected wave beyond this point toward the generator. However, it reappears at other frequencies when b` T 90  . The corresponding re¯ection coef®cient G in can be determined as follows. G in  Z in À Z o Z in  Z o  Z 1 R L  jZ 1 tanb` Z 1  jR L tanb` À Z o Z 1 R L  jZ i tanb` Z 1  jR L tanb`  Z o  R L À Z o R L  Z o  j2  Z o R L p tanb`  r in exp jj ; r in  R L À Z o fR L  Z o  2  4Z o R L tan 2 b`g 1=2  1 1  2  Z o R L p R L À Z o secb` ! 2 8 < : 9 = ; 1=2 6:1:2 Figure 6.1 A single-section quarter-wave transformer. 190 IMPEDANCE TRANSFORMERS Variation in r in with frequency is illustrated in Figure 6.2. For b` near 90  , it can be approximated as follows: r in % jR L À z o j 2  Z o R L p tanb` % jR L À Z o j 2  Z o R L p cosb`6:1:3 If r M is the maximum allowable re¯ection coef®cient at the input, then cosy 1  2r M  Z o R L p R L À Z o   1 À r 2 M p           ; y 1 < p=2 6:1:4 In the case of a TEM wave propagating on the transmission line, b`  p 2  f f o , where f o is the frequency at which b`  p 2 . In this case, the bandwidth ( f 2 À f 1 Df is given by Df f 2 À f 1 2 f o À f 1 2 f o À 2f o p y 1  6:1:5 and the fractional bandwidth is Df f o  2 À 4 p cos À1 2r M  Z o R L p R L À Z o   1 À r 2 M p           6:1:6 Figure 6.2 Re¯ection coef®cient characteristics of a single section impedance transformer used to match a 100-O load to a 50-O line. SINGLE SECTION QUARTER-WAVE TRANSFORMER 191 Example 6.1: Design a single-section quarter-wave impedance transformer to match a 100-O load to a 50-O air-®lled coaxial line at 900 MHz. Determine the range of frequencies over which the re¯ection coef®cient remains below 0.05. For R L  100 O and Z o  50 O Z 1   100  50 p  70:7106781 O and, `  l 4  3  10 8 4  900  10 6 m  8:33 cm Magnitude of the re¯ection coef®cient increases as b` changes from p=2 (i.e., the signal frequency changes from 900 MHz). If the maximum allowed r is r M  0:05VSWR  1:1053, then fractional bandwidth is found to be Df f o  2 À 4 p cos À1 2r M  Z o R L p R L À Z o   1 À r 2 M p            0:180897 Therefore, 818.5964 MHz f 981.4037 MHz or 1.4287 b` 1.7129. 6.2MULTI-SECTION QUARTER-WAVE TRANSFORMERS Consider an N-section impedance transformer connected between a transmission line of characteristic impedance of Z o and load R L , as shown in Figure 6.3. As indicated, the length of every section is the same while their characteristic impedances are different. Impedance at the input of Nth section can be found as follows: Z N in  Z N exp jb`G N expÀjb` exp jb`ÀG N expÀjb` 6:2:1 where G N  R L À Z N R L  Z N 6:2:2 192 IMPEDANCE TRANSFORMERS The re¯ection coef®cient seen by the (N-1)st section is G H NÀ1  Z N in À Z NÀ1 Z N in  Z NÀ1  Z N e jb`  G N e Àjb` ÀZ NÀ1 e jb` À G N e Àjb`  Z N e jb`  G N e Àjb` Z NÀ1 e jb` À G N e Àjb`  or, G H NÀ1  Z N À Z NÀ1 e jb`  G N Z N  Z NÀ1 e Àjb` Z N  Z NÀ1 e jb`  G N Z N À Z NÀ1 e Àjb` Therefore, G H NÀ1  G NÀ1  G N e Àj2b` 1  G N G NÀ1 e Àj2b` 6:2:3 where G NÀ1  Z N À Z NÀ1 Z N  Z NÀ1 6:2:4 If Z N is close to R L and Z NÀ1 is close to Z N , then G N and G NÀ1 are small quantities, and a ®rst-order approximation can be assumed. Hence, G H NÀ1 % G NÀ1  G N e Àj2b` 6:2:5 Similarly, G H NÀ2 % G NÀ2  G H N e Àj2b`  G NÀ2  G NÀ1 e Àj2b`  G N e Àj4b` Figure 6.3 An N-section impedance transformer. MULTI-SECTION QUARTER-WAVE TRANSFORMERS 193 Therefore, by induction, the re¯ection coef®cient seen by the feeding line is G % G 0  G 1 e Àj2b`  G 2 e Àj4b` ÁÁÁG NÀ1 e Àj2NÀ1b`  G N e Àj2N b` 6:2:6 or, G  P N n0 G n e Àj2nb` 6:2:7 where G n  Z n1 À Z n Z n1  Z n 6:2:8 Thus, we need a procedure to select G n so that G is minimized over the desired frequency range. To this end, we recast the above equation as follows: G  G 0  G 1 w  G 2 w 2 ÁÁÁG N w N  G N Q N n1 w À w n 6:2:9 where j À2b` 6:2:10 and, w  e jj 6:2:11 Note that for b`  0 (i.e., l 3I), individual transformer sections in effect have no electrical length and load R L appears to be directly connected to the main line. Therefore, G  P N n0 G n  R L À Z o R L  Z o ; , w  16:2:12 and, only N of the N  1 section re¯ection coef®cients can be selected indepen- dently. 194 IMPEDANCE TRANSFORMERS 6.3 TRANSFORMER WITH UNIFORMLY DISTRIBUTED SECTION REFLECTION COEFFICIENTS If all of the section re¯ection coef®cients are equal then (6.2.9) can be simpli®ed as follows: G G N  1  w  w 2  w 3 ÁÁÁw NÀ1  w N  w N1 À 1 w À 1 6:3:1 or, G G N  e jN1j À 1 e jj À 1  e jNj=2 sin N  1 2 j  sin j 2  Hence, jGjrjr N sin N  1 2 j  sin j 2                  N  1r N sin N  1 2 j  N  1 sin j 2                  6:3:2 and, from (6.2.12), P N n0 G n N  1r N  R L À Z o R L  Z o 6:3:3 Therefore, equation (6.3.2) can be written as follows: rb` R L À Z o R L  z o         Á sinfN  1b`g N  1 sinb`         6:3:4 This can be viewed as an equation that describes magnitude r of the re¯ection coef®cient as a function of frequency. As (6.3.4) indicates, a pattern of rb` repeats periodically with an interval of p. It peaks at np, where n is an integer including zero. Further, there are N À 1 minor lobes between two consecutive main peaks. The number of zeros between the two main peaks of rb` is equal to the number of quarter-wave sections, N. Consider that there are three quarter-wave sections connected between a 100-ohm load and a 50-ohm line. Its re¯ection coef®cient characteristics can be found from (6.3.4), as illustrated in Figure 6.4. There are three zeros in it, one at b`  p=2 and the other two symmetrically located around this point. In other words, zeros occur at b`  p=4, p=2, and 3p=4. When the number of quarter-wave sections is increased from 3 to 6, the rb` plot changes as illustrated in Figure 6.5. UNIFORMLY DISTRIBUTED SECTION REFLECTION COEFFICIENTS 195 For a six-section transformer, Figure 6.5 shows ®ve minor lobes between two main peaks of rb`. One of these minor lobes has its maximum value (peak) at b`  p=2. Six zeros of this plot are symmetrically located, b`  np=7, n  1; 2; .; 6. Thus, characteristics of rb` can be summarized as follows:  Pattern of rb` repeats with an interval of p.  There are N nulls and (N À 1) minor peaks in an interval. Figure 6.4 Re¯ection coef®cient versus b` of a three-section transformer with equal section re¯ection coef®cients for R L  100 O and Z o  50 O. Figure 6.5 Re¯ection coef®cient versus b` for a six-section transformer with equal section re¯ection coef®cients (R L  100 O and Z o  50 O). 196 IMPEDANCE TRANSFORMERS  When N is odd, one of the nulls occurs at b`  p=2 (i.e., `  l=4).  If r M is speci®ed as an upper bound on r to de®ne the frequency band then points P 1 and P 2 bound the acceptable range of b`. This range becomes larger as N increases. Since w N1 À 1  Y N n0 w À w n 6:3:5 where w n  e j 2pn N1 ; n  1; 2; .; N 6:3:6 (6.3.1) may be written as follows: G G N  Y N n1 w À w n  Y N n1 w À e j 2pn N1  6:3:7 This equation is of the form of (6.2.9). It indicates that when section re¯ection coef®cients are the same, roots are equispaced around the unit circle on the complex w-plane with the root at w  1 deleted. This is illustrated in Figure 6.6 for N  3. It follows that r r N  Y N n1 w À e j 2pn N1             Y N n1 w À e j 2pn N1        6:3:8 Figure 6.6 Location of zeros on a unit circle on the complex w-plane for N  3. UNIFORMLY DISTRIBUTED SECTION REFLECTION COEFFICIENTS 197 Since w  e jj is constrained to lie on the unit circle, distance between w and w n is given by d n j e jj À e j 2pn N1       6:3:9 For the case of N  3, d n j  0) is illustrated in Figure 6.7. From (6.3.3), (6.3.8), and (6.3.9), we get rj 1 N  1 R L À Z o R L  Z o         Y N n1 d n j6:3:10 Thus, as j À2b` varies from 0 to 2 p, w  e jj makes one complete traverse of the unit circle, and distances d 1 ; d 2 ; .; d N vary with j.Ifw coincides with the root w n , then the distance d n is zero. Consequently, the product of the distances is zero. Since the product of these distances is proportional to the re¯ection coef®cient, rj n  goes to zero. It attains a local maximum whenever w is approximately halfway between successive roots. Example 6.2: Design a four-section quarter-wavelength impedance transformer with uniform distribution of re¯ection coef®cient to match a 100-O load to a 50-O air-®lled coaxial line at 900 MHz. Determine the range of frequencies over which the re¯ection coef®cient remains below 0.1. Compare this bandwidth with that obtained for a single-section impedance transformer. Figure 6.7 Graphical representation of (6.3.9) for N  3 and j  0. 198 IMPEDANCE TRANSFORMERS [...]... end, we recast the above equation as follows: G ˆ G0 ‡ G1 w ‡ G2 w2 ‡ Á Á Á ‡ GN wN ˆ GN N Q nˆ1 …w À wn † …6:2:9† where j ˆ À2b` …6:2:10† w ˆ e jj …6:2:11† and, Note that for b` ˆ 0 (i.e., l 3 I ), individual transformer sections in effect have no electrical length and load RL appears to be directly connected to the main line Therefore, Gˆ N P nˆ0 Gn ˆ RL À Zo ; RL ‡ Zo …, w ˆ 1† …6:2:12† and, only N . The chapter begins with the single-section impedance transformer that provides perfect matching at a single frequency. Matching bandwidth can be increased. increased number of quarter-wave sections. Chebyshev transfor- mers can provide even wider bandwidth with the same number of sections but the re¯ection