AP21-6

EVALUATION OF INPUT IMPEDANCE OF CYLINDRICAL DOUBLET ANTENNAS

Xu Xiaowen and Ruan Yinqzheng (Chengdu Institute of Radio .Engineering, Chengdu, Sichuan, China)

ABSTRACT

By means of electrostatics and transmission line theory,the equivalent transmission line method has been complemented and corrected so that to be used to evaluate the input im- pedance of cylindrical doublet antennas. A more accurate calculation formula has been developed,in which the l o s s along the equivalent transmission line and the end-effects have been taken into account. Numerical and experimental re- sults obtained show that this corrected formula is adaptable to thick cylindrical antennas and gives more accurate result with less mathematical complication than both Hallen's in- tegral equation method and Schelkunoff's biconical antenna theory.

INTRODUCTION

In equivalent transmission line method,a doublet antenna is treated as a homogeneous opened-out transmission line with radiation loss,and then the input impedance of antenna is obtained from the transmission line the0ry.l~~ The advantage of this approach is that engineers can use the familiar transmission line theories with simple mathematical opera- tions. The obtained results,however,are not accurate,espe- cially for thick antennas.

To solve this problem rigorously,the following three methods are usually employed.1-4 The first method starts from Max- well's field equations and treats the antenna as a boundary value problem. This exact way,unfortunately,can be used only for a few antennas not including the commonly atopted cylin- drical doublet antennas. The second one is Hallen's integral method which is universal and accurate,but it is quite dif- ficult to yet the final results because of the mathematical complication. The third one is Schelkunoff's biconical an- tenna theory which requires a good understanding of spheri-

CH2435-6/87/0000-0844$01.00 @ 1987 IEEE

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cal wave mode theory and a rich experience of using spheri- cal function as well as the necessary tables of spherical function.

In order to avoid the mathematical difficulties from these methods above and obtain satisfactory accuracy,the tradi- tional equivalent transmission line method is complemented and amended in this study by use of the static electricity and l o s s y transmission line theory. The numerical results from this analysis agree very well with the experimental da- ta obtained resently,even for thick cylindrical antennas.

CORRECTED TRANSMISSION LINE REPRESENTATION OF DOUBLET ANTENNAS

In consideration of the energy radiation loss along the line and the end-effects,the traditional input impedance expres- sion' for a dipole antenna can be modified as follows5

zin= Zc'coth [ ( c% '+ j j3 ' )Lq)

where d ' , p ' and Zce -represent attenuation constant,phase shift constant and characteristic impedance for the equiva- lent lossy transmission line,respectively; L' is equivalent line length taking the end-effects into account.

d' , p ' ,Zc' and L' are given, respectively, by5

where pis the wave phase shift constant in lossless 1ine;Rr is radiation resistance of the antenna referred to the loop current;Zco is the average characteristic impedance defined by

with L denoting the half-length and r denoting the radius of the dipole.

where the second term on the right-hand side comes from the influence of end-surface electrostatic capacity ce; the

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t h i r d term r e p r e s e n t s t h e i n f i u e n c e o f t h e d i s t r i b u t i o n ca- p a c i t y Cd i n t h e r e g i o n n e a r t h e e n d . 5

S u b s t i t u t i n g ( 2 ) , ( 4 ) , ( 5 ) a n d ( 6 ) i n t o (1) , o n e h a s t h e f i n a l e x p r e s s i o n f o r i n p u t i m p e d a n c e o f d o u b l e t a n t e n n a

NUMERICAL A N D EXPERIMENTAL RESULTS

F i g . 1 s h o w s t h e c o m p u t a t i o n r e s u l t s o f i n p u t r e s i s t a n c e a n d r e a c t a n c e o f d i p o l e a n t e n n a a c c o r d i n g t o t h e c o r r e c t e d e q u i - v a l e n t t r a n s m i s s i o n l i n e f o r m u l a ( 7 ) .

F i g . 1 I n p u t i m p e d a n c e o f d i p o l e a n t e n n a s

F i g . 2 compares t h e r e s u l t s from f o r m u l a ( 7 ) w i t h t h o s e from S c h e l k u n o f f 3 a n d H a l l e n l f o r t h e case of Zc0=600n. I t i s shown t h a t i n t h i s case o f t h i n d i p o l e , t h e t h r e e c u r v e s agree w e l l w i t h e a c h o t h e r .

Fig.2 Comparison of three calculation methods Legend : from formula(7) in this paper

------- from Schelkunoff -.-.- from Hallen

Fig.3 is about the resonance length for a half-wave dipole with which length the imaginary part of input impedance goes t o zero. In Fig.3, curve(1) comes from Schelkunoff without end-surface capacity Ce;4 curve (2) includes the capacity;4 Curve (3) is from Hallen1;curve(4) is the results from for- mula(7) in this paper: curve ( 5 ) denotes some experimental results obtained resently.6 It is clearly shown that the corrected transmission line representation (7) is in good accordance with experimental results for thick antennas. Be- cause of neglecting the end-surface capacity, curves(1) and (3) give larger resonance length than experimental results. On the other hand,curve(2) overstates the end-surface capa- city(to keep the same average characteristic impedance), and,therefore,gives smaller resonance length.

Fig.4 takes the feed-end surface capacity Ci into account which gives some additional influence on the input impedance (admittance) as well as resonance length.

CONCLUSION

Corrected equivalent transmission line method provides an engineering approach to calculate input ihedance of thick doublet antennas with more satisfactory accuracy and less mathematical complication. Also, this method can be used to monopole as well as the cylindrical antennas with arbitrary cross-section. The analysis of end-effects here can be adap- ted to some other kinds of transmission line, such as coaxi- al cable and microstrip transmission lines.

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