Tài liệu RF và mạch lạc lò vi sóng P6 ppt

6.2MULTI-SECTION QUARTER-WAVE TRANSFORMERS Consider an N-section impedance transformer connected between a transmission line of characteristic impedance of Zoand load RL, as shown in Fig

IMPEDANCE TRANSFORMERS

In the preceding chapter, several techniques were considered to match a given loadimpedance at a ®xed frequency These techniques included transmission line stubs aswell as lumped elements Note that lumped-element circuits may not be practical athigher frequencies Further, it may be necessary in certain cases to keep there¯ection coef®cient below a speci®ed value over a given frequency band Thischapter presents transmission line impedance transformers that can meet suchrequirements The chapter begins with the single-section impedance transformerthat provides perfect matching at a single frequency Matching bandwidth can beincreased at the cost of a higher re¯ection coef®cient This concept is used to designmultisection 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 aboutthe center frequency and increase gradually on either side A wider bandwidth isachieved with an increased number of quarter-wave sections Chebyshev transfor-mers can provide even wider bandwidth with the same number of sections but there¯ection coef®cient exhibits ripples in its pass-band This chapter includes aprocedure to design these multisection transformers as well as transmission linetapers The chapter concludes with a brief discussion on the Bode-Fano constraints,which provide an insight into the trade-off between the bandwidth and allowedre¯ection coef®cient

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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 problemearlier in Example 3.5 This section presents a detailed analysis of such circuits.Consider the load resistance RL that is to be matched with a transmission line ofcharacteristic impedance Zo Assume that a transmission line of length ` andcharacteristic impedance Z1 is connected between the two, as shown in Figure6.1 Its input impedance Zin is found as follows

Zinˆ Z1RL‡ jZ1tan…b`†

For b` ˆ 90 (i.e., ` ˆ l=4 ) and Z1ˆpZoRL; Zin is equal to Zo and, hence,there is no re¯ected wave beyond this point toward the generator However, itreappears at other frequencies when b` 6ˆ 90 The corresponding re¯ectioncoef®cient Gin can be determined as follows

RL Zosec…b`†

!28

…6:1:2†

Figure 6.1 A single-section quarter-wave transformer

Variation in rinwith frequency is illustrated in Figure 6.2 For b` near 90, it can

…RL Zo† 1 r2

Mp

In the case of a TEM wave propagating on the transmission line, b` ˆp2ff

o,where fo is the frequency at which b` ˆp2 In this case, the bandwidth( f2 f1† ˆ Df is given by

Df ˆ … f2 f1† ˆ 2… fo f1† ˆ 2 fo 2fo

p y1

…6:1:5†and the fractional bandwidth is

…6:3:2†and, from (6.2.12),

PN nˆ0Gnˆ …N ‡ 1†rN ˆRL Zo

Consider that there are three quarter-wave sections connected between a 100-ohmload 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` ˆ p=2 andthe other two symmetrically located around this point In other words, zeros occur atb` ˆ p=4, p=2, and 3p=4 When the number of quarter-wave sections is increasedfrom 3 to 6, the r…b`† plot changes as illustrated in Figure 6.5

For a six-section transformer, Figure 6.5 shows ®ve minor lobes between twomain peaks of r…b`† One of these minor lobes has its maximum value (peak) atb` ˆ p=2 Six zeros of this plot are symmetrically located, b` ˆ np=7,

n ˆ 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 1) minor peaks in an interval

Figure 6.4 Re¯ection coef®cient versus b` of a three-section transformer with equal sectionre¯ection coef®cients for RLˆ 100 O and Zoˆ 50 O

Figure 6.5 Re¯ection coef®cient versus b` for a six-section transformer with equal sectionre¯ection coef®cients (RLˆ 100 O and Zoˆ 50 O)

 When N is odd, one of the nulls occurs at b` ˆ p=2 (i.e., ` ˆ l=4 ).

 If rM is speci®ed as an upper bound on r to de®ne the frequency band thenpoints P1and P2 bound the acceptable range of b` This range becomes larger

G

GN ˆ

YN nˆ1

r

rN ˆ

YN nˆ1

w ej 2pn N‡1

... 1†b`‡ GNe j2Nb` …6:2:6†or,

Note that for b` ˆ (i.e., l ! ), individual transformer sections in effect have

no electrical length and load RL appears