Proceedings of the 38th European Microwave Conference

Analytical Equations for the Analysis of Folded

Dipole Array Antennas
Hubregt J. Visser#1
#
Holst Centre
P.O. Box 8550, 5605 KN Eindhoven, The Netherlands
1
visser@ieee.org

Abstract— An accurate analytical model has been derived for equations for these radiators combined into series arrays of
a linear array of wire folded dipole antennas. The model folded dipoles.
combines closed form analytical equations for the folded dipole
antenna, the re-entrant folded dipole antenna, the two-wire II. ANALYSIS
transmission line, the mutual coupling between two folded dipole In the equations to be derived, we will use analytical
antennas and the mutual coupling between two thin dipole
antennas.
expressions for the input impedance of an ordinary dipole
antenna. To include mutual coupling effects in the array
I. INTRODUCTION antenna analysis, we will separate the radiating elements from
Wire antennas may still be found in numerous applications, the feeding structure and analyse them separately. For the wire
ranging from large broadcasting antennas, [1] to in-clothing- folded dipole antenna coupling, use will be made of analytical
integrated antennas for use in the ISM frequency bands, [2]. equations for the coupling between thin ordinary dipole
Single wire radiators may be employed for small-band antennas.
applications. Broadband wire antennas may be realised as log A. Single Folded Dipole
periodic arrays of monopole or dipole elements, [1], [3] and
A single folded dipole radiator is shown in Figure 1.
may be used, for example, for broadcast applications, [1]. To
enhance the range in UWB communication systems, higher
D
gain antennas are needed. Log periodic dipole array antennas
may then be used if OFDM-like modulation schemes are
IA IA
being employed, [4]. When, for UWB communication, L IT IT
different length dipole elements are being used in an array, a I
2 2
better correlation will be achieved as compared to one single
+ -
dipole, [5]. For low-power applications, not only in UWB, but V+ V V+ +V
also in the context of RFID and rectenna systems, an accurate V 2 2 2 2
- - + - -
estimation of the antenna input impedance must be made or a
desired impedance value must be synthesised [6]. So, a
regained interest in the analysis of wire array antennas has d d=2a
been created. Although these antennas may be analysed by
(commercial) full-wave methods, e.g. the Method of Moments Folded dipole Transmission line Antenna
(MoM), it is still opportune to develop analytical analysis antenna mode mode
methods, especially if we want to employ these methods in Fig. 1 Decomposition of an equal radius wire folded dipole antenna (left) into
multi-analysis-iteration optimisation schemes for a transmission line mode (middle) and an antenna mode (right).
automatically designing wire array antennas.
In [9] it is shown that the current on the folded dipole
For the above mentioned log periodic array antennas, the
radiator may be considered as composed of a transmission line
feeding at successive junctions of the dipoles with the feeding
mode and a dipole mode, see Figure 1. The input impedance is
line should be reversed, [7], to provide a 1800 phaseshift
then calculated as
between adjacent dipoles. By employing folded dipoles to
create a log-periodic antenna, the cumbersome twist of the
feeder line between two adjacent dipoles is avoided by 4 ZT Z D
Z in = , (1)
providing the required 1800 phase shift through the folded ZT + 2 Z D
dipole configuration, [8].
where ZD is the impedance of a cylindrical, ordinary, dipole
In the following we will review analytical equations and
antenna with effective radius ae, where
improvements for single folded dipole radiators and derive

978-2-87487-006-4 © 2008 EuMA 706 October 2008, Amsterdam, The Netherlands

 1 ⎛ D⎞ The array antenna may be analysed by constructing ABCD
ln (ae ) = ln (a ) + ln ⎜ ⎟ . (2) matrices for the re-entrant folded dipoles [12] and for the
2 ⎝a⎠ interconnecting transmission lines [13], after which a chain
matrix analysis may be applied to the array antenna. This
Herein, a is the radius of the wire and D is the separation of analysis however, does not include mutual coupling effects,
the wires, see Figure 1. In our analyses we use the analytic [2], [12].
equation for the dipole impedance as given in [7], that is based
on the work of C.T. Tai

⎡ L ⎛ L⎞ ⎤
2
I1 I2
Z D = ⎢122.65 − 204.1k0 + 110⎜ k0 ⎟ ⎥
⎣⎢ 2 ⎝ 2 ⎠ ⎥⎦ 1 2
V1 V2
⎡ ⎛ ⎛ L⎞ ⎞ ⎛ L⎞ ⎤ 1’
⎢120⎜⎜ ln ⎜ ⎟ − 1⎟⎟ cot⎜ k0 ⎟ − 162.5 + ⎥ , (3) 2’
⎝ ⎝a⎠ ⎠ ⎝ 2⎠
− j⎢ ⎥
⎢ 2 ⎥
⎢140k0 L − 40⎛⎜ k0 L ⎞⎟ ⎥
⎢⎣ 2 ⎝ 2 ⎠ ⎥⎦
Fig. 2 Series array of re-entrant folded dipole antennas (left) and
where k0=2π/λ0 is the free space wave number and L is the modification of a folded dipole antenna into a re-entrant folded dipole antenna.
length of the radiator.
ZT is the impedance of a short-circuited two-wire For the inclusion of mutual coupling effects, the radiating
transmission line of length L/2 elements and the feeding structure are separated as described
in [14], [15].
The admittance matrix of an N-elements folded dipole array
⎛k L⎞
ZT = − jZ 0 tan⎜ 0 ⎟ , (4) may be written as a 2Nx2N array [Y]
⎝ 2 ⎠
[Y ] = [YF ] + [YA ] , (7)
where Z0 is the characteristic impedance of the two-wire
transmission line, that is given by where [YF] is the admittance matrix of the feed network,
that has the form
⎡ D + D 2 + 2a 2 ⎤
Z 0 = 120 ln ⎢ ⎥. (5) ⎡0 0 0 L 0 0⎤
⎢⎣ 2a ⎥⎦ ⎢0
⎢ [YF1 ] 0 L 0 0 ⎥⎥
To broaden the range of wire separations, the folded dipole ⎢0 0 [YF 2 ] L 0 0⎥
length is replaced by an equivalent length according to [10], [YF ] = ⎢ ⎥. (8)
so that wire separations up to λ0/6 instead of λ0/100 are ⎢0 0 0 O 0 0⎥
allowable. The equivalent length is given by, [11]1 ⎢M
⎢
M M [YF ( N −1) ] 0⎥
⎥
⎢⎣0 0 0 L 0 YL ⎥⎦
Leq = L + 0.39 D . (6)

In Equation (8), YL is the load admittance of the array. For

B. Series Array of Folded Dipoles the array shown in Figure 2, YL=∞. The 2x2 submatrices [YFi],
If the folded dipole antennas are arranged into a series array, i=1,2,…,N-1, are defined by, [13]
see Figure 2, the folded dipole antenna needs to be modified
into a so-called re-entrant folded dipole [12], see also Figure 2. YFi11 = YFi 22 = − jY0i cot(k0li )
, (9)
YFi12 = YFi 21 = jY0i csc(k0li )

where Y0i and li are, respectively, the characteristic

1 admittance and length of two-wire transmission line i.
This length extension allows for larger wire separations in the folded dipole.
For two-wire transmission lines connecting folded dipoles in a series array,
the wire separation should still be limited to λ0/100. The reason that the wires [YA] is the admittance matrix of the network of re-entrant
may be separated further in the folded dipole antenna is due to the dominant folded dipoles and has the form
character of the dipole mode, see equation (1).

707
 ⎡ [YA11 ] [C A12 ] L [C A1N ]⎤ u0 = k0 ⎛⎜ d 2 + (l1 + l2 ) − (l1 + l2 )⎞⎟
2
⎢[C ] [YA 22 ] L [CA 2 N ]⎥⎥ ⎝ ⎠
[YA ] = ⎢ A 21 , (10) v0 = k0 ⎛⎜ d + (l1 + l2 ) + (l1 + l2 )⎞⎟
2 2
⎢ M M O ⎥ ⎝ ⎠
⎢ ⎥
⎣[C AN 1 ] [CAN 2 ] L [YANN ]⎦ u0 ' = k0 ⎜ d + (l1 − l2 ) − (l1 − l2 )⎞⎟
⎛ 2 2

⎝ ⎠
v0 ' = k0 ⎛⎜ d + (l1 − l2 ) + (l1 − l2 )⎞⎟ .
2 2
where the 2x2 submatrices [YAii], i=1,2,…,N, are defined by (16)
⎝ ⎠
( )
[11]
u1 = k0 d + l1 − l1
2 2

v =k ( d +l )
YAii11 = YAii 22 = 12 YTi + 14 YDi
, (11)
2
+l2

w =k ( d +l )
1 0 1 1
YAii12 = YAii 21 = − 12 YTi + 14 YDi 2
+l 2

y =k ( d −l )
1 0 2 2

with YTi and YDi, respectively, the transmission line 1 0

2
+l 2
2 2
admittance and the dipole admittance of re-entrant folded
dipole i. They are calculated using Equations (3) and (4). The mutual admittance follows from the mutual impedance
The 2x2 submatrices [CAij], i,j=1,2,…,N, i≠j, contain the from
mutual admittances between the folded dipole elements.
Following [16], the mutual admittance between two folded
Z12
dipole antennas may be calculated from the mutual admittance Y12 = , (17)
between two equivalent dipole antennas Z D1Z D 2 − Z12

Y21 folded − dipole− to − folded − dipole = 14 Y21dipole− to − dipole . (12) where ZD1 and ZD2 are the dipole impedances of the two
coupled dipoles.
In our effort to solve the folded dipole array with analytical III. VERIFICATION
equations, we take for the equivalent dipole antennas,
infinitely thin dipole antennas having the same lengths as the The above analysis has been implemented in software and
folded dipole antennas they represent. The mutual impedance several wire folded dipole array antennas have been analysed.
between two non-staggered dipoles of half-lengths l1 and l2, Comparisons have been made with a Method of Moment
separated by a distance d, is then given by, [17] analysis. Typical analysis results are shown for a four element
wire folded dipole array antenna, see Figure 3.
Z12 = R12 + jX12 , (13)
2a
where
L1 L2 L3 L4
⎧ ⎡Ci (u 0 ) + Ci (v 0 ) ⎤ ⎡Ci (u 0 ') + Ci (v 0 ') ⎤ ⎫
⎪ ⎢ − Ci (u ) − Ci (v ) ⎥ ⎢ ⎥ ⎪, (14)
⎪cos k (l + l )⎢ 1 1 ⎥ + cos k (l − l )⎢ − Ci (u1 ) − Ci (v1 ) ⎥ ⎪
⎪ 0 1 2
⎢ − Ci (w1 ) − Ci ( y1 )⎥ 0 1 2
⎢ − Ci (w1 ) − Ci ( y1 )⎥ ⎪
⎪⎪ ⎢ ⎥ ⎢ ⎥ ⎪⎪
R12 = 30⎨ ⎣ + 2Ci (k 0 d ) ⎦ ⎣ + 2Ci (k 0 d ) ⎦⎬
⎪ ⎪
⎪ ⎡− Si(u 0 ) + Si(v0 )⎤ ⎡− Si(u 0 ') + Si(v 0 ')⎤ ⎪
⎢ ⎥ ⎢
⎪+ sin k 0 (l1 + l 2 ) + Si(u1 ) − Si(v1 ) + sin k 0 (l1 − l 2 ) + Si(u1 ) − Si(v1 ) ⎪ ⎥
⎪ ⎢ ⎥ ⎢ ⎥⎪
⎪⎩ ⎢⎣− Si(w1 ) + Si( y1 )⎥⎦ ⎢⎣+ Si(w1 ) − Si ( y1 ) ⎥⎦ ⎪⎭

and

⎧ ⎡ − Si(u 0 ) − Si(v 0 ) ⎤ ⎡− Si(u0 ') − Si(v0 ')⎤ ⎫ D TL D TL D T D

⎪ ⎢ ⎥ ⎢ ⎥ ⎪ .(15)
⎪cos k (l + l )⎢ + Si(u1 ) + Si(v1 ) ⎥ + cos k (l − l )⎢+ Si(u1 ) + Si(v1 ) ⎥ ⎪
⎪ 0 1 2
⎢ + Si(w1 ) + Si( y1 )⎥ 0 1 2
⎢+ Si(w1 ) + Si( y1 ) ⎥ ⎪
⎪⎪ ⎢ ⎥ ⎢ ⎥ ⎪⎪
X 12 = 30⎨ ⎣ − 2 Si(k 0 d ) ⎦ ⎣− 2Si(k 0 d ) ⎦ ⎬ Fig. 3 Four elements series array of re-entrant folded dipole antennas.
⎪ ⎪
⎪ ⎡ − Ci (u 0 ) + Ci (v 0 ) ⎤ ⎡− Ci (u 0 ') + Ci (v 0 ')⎤ ⎪
⎪ + sin k 0 (l1 + l 2 )⎢ + Ci (u1 ) − Ci (v1 ) ⎥ + sin k 0 (l1 − l 2 )⎢+ Ci (u1 ) − Ci (v1 ) ⎥ ⎪ The real part of the input impedance as a function of
⎪ ⎢ ⎥ ⎢ ⎥⎪
⎩⎪ ⎣⎢ − Ci (w1 ) + Ci ( y1 )⎦⎥ ⎣⎢+ Ci (w1 ) − Ci ( y1 ) ⎦⎥ ⎭⎪ frequency is shown Figure 4, together with Method of
Moments (MoM) analysis results. The imaginary part of the
In Equations (14) and (15), Si(x) and Ci(x), are the sine and input impedance is shown in Figure 5, again with MoM
cosine integral of argument x, respectively and analysis results.

708
 The array dimensions are: L1=11mm, L2=15mm, L3=19mm, antennas and the feeding network is separated is successfully
L4=23mm, D=0.15mm, TL=10mm, a=3μm. The analysis has applied to combine all the aforementioned closed-form
been performed for frequencies ranging from 9GHz to 12GHz. analysis equations. The analysis may be employed in an
The Figures clearly show a very good agreement between optimisation scheme to synthesise desired input impedance
the analysis results of our model and those obtained with a characteristics.
Method of Moments for the first resonance. Although MoM is
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