The voltage is proportional to Eo and the current is proportional to H on the surface of the sphere,so Z can be write as a continued fraction: the normalized impedance Z,is: r 1 元网 jp2m-1,1 jp 2h-3 ，=h1-+ h, je ph]=[ph +h ]ph-nh 乙，= 3 ph 10
' ( ) ( ) n n n E h Z j H h ' 1 ' ' 1 1 1 [ ] [ ] n n n n n n n n n n n n n h h h h h h h nh Z j h h nh The voltage is proportional to and the current is proportional to on the surface of the sphere , so the normalized impedance is: n E H Z can be write as a continued fraction: Z n
Z can be interpteted as a cascade of series capacitances and shunt inductances terminated with a unit resistance c品 20-31c In -()(" j(pkn) L -(月(") Zn=j(pha)'/phn. Equivalent circuit of TM spherical wave. RADIUS OF SPHERE c:VELOCITY OF LIGHT c. Equivalent circuit of electric dipole for n=1
can be interpteted as a cascade of series capacitances and shunt inductances terminated with a unit resistance Z n for n=1
Near the operating frequency,the equivalent circuit can be approximate by a simple series RLC circuit Z.=B.+JX.-R.+J(ol-oC Using h in-inn 1 J and n are the spherical Bessel functions nntiIn Intinn= of the first and second kind Xn=[pin(gin)'+pnn(pnn)'phn2, Rn=phn-2, -+
Near the operating frequency, the equivalent circuit can be approximate by a simple series RLC circuit 1 ( ) n n n n n n Z R jX R j L C 1 1 2 1 n n n n n n n h j jn n j j n Using and are the spherical Bessel functions of the first and second kind n n j n
For a certain mode n,the average power dissipation in Z is: the average electric energy stored in Z is -(目'%t 2(2n十1) so,the Q for TMo wave is 20Wn 1, =p and the total Q is: (n+1) when ka <1,for lowest TM mode ∑'An2 Qn(p) 2%+1 1+2k2a2 (n+1) ka(1+ka2) ∑'An2 2n+1 CHU's limit
For a certain mode n, the average power dissipation in Zn is: the average electric energy stored in Zn is so, the Qn for TMn0 wave is and the total Q is: when 1, for lowest TM mode 2 2 Chu 3 3 2 2 ka 1+2k a Q = k a (1+k a ) CHU’s limit
Criterion I:Maximum Directivity c()- 0-.0/2n 2n+1 G.- [P(0)]2 9=N (+1) 27 23 Ifka>N,Q≈1or<l 9 If ka<N,O>1 d 3 If kaN,1 N 9 The tansition occurs when ka~N, 5 10 Aa corresponding to D≈ 2w台 10 20 25 for 0>>1 2常0/入 FIG.6.Q of omni-directional antenna.Criterion: Max.gain with fixed number of terms. (2=-3dB bandwidth)
Criterion I: Maximum Directivity If , 1 or <1 If , 1 If , 1 The tansition occurs when , corresponding to 4 D ka N Q ka N Q ka N Q ka N a 0 2 for 1 f f Q Q 2 1 (2 | | 3dB bandwidth) f f f