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Polarization Properties of Sprial Antennas: A TutorialUlrich R. Kraft aa Deutsche Aerospace AG, Space Division , D-81663 Munich, GermanyPublished online: 29 Oct 2007.
To cite this article: Ulrich R. Kraft (1994) Polarization Properties of Sprial Antennas: A Tutorial, Electromagnetics, 14:3-4,259-284, DOI: 10.1080/02726349408908386
To link to this article: http://dx.doi.org/10.1080/02726349408908386
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POLARIZATION PROPERTIES OF SPRIAL ANTENNAS: A TUTORIAL
Ulrich R. Krafl Deufsche Aerospace AG, Space Division 081663 Munich, Germany
ABSTRACT
The basic polarisation properties of spiral antennas are considered by means of circularly polarised spherical waves (CPSW). At first, specific properties of the CPSW and their impact on the antenna polarisation are discussed and conditions for the radiation of pure circular pohrisation are identaied. Then. svmmetrical multi-arm structures are intmduced as a oeneral class of . , ~- ~ - ~
antennas which permils an optimisalion of the polarisation perlormance by a suitable control of the CPSW spectra. Finally, the specific CPSW spectra of phnar spirals -one member of this class - are discked. An explanation of the polari&tion i f planar spirals and some design guidelines are derived from this discussion.
1. INTRODUCTION
Since the early sixties, spiral antennas have served as both transma and receive antennas for a variety of applications in fields like radar, direction finding, navigation, electronic countermeasures and satellite communications.
The distinct advantage of this antenna type is its simplicity. Broadband operation ard circular polarisation can be obtained withcut the need for a very sophisticated feed arrangement or a hiohlv oolimised antenna confiouration. In addition. the s~iral antenna can ooerale in dierent - . . radiation modes. This offers th;opportunity to obtain diffe;ent types of radiatidn panern with the same antenna structure which has applications in areas like direction finding, homing etc.
Descriptions of the design and the key properties of spiral antennas have been published since 1959 (Dyson. 1959). Whereas the papers of the early inventors were mainly based on experimental work (Dyson. 1959. Kaiser, 1960. Donnelhn. 1960. Wheeler. 1961. Dyson and Mayes. 1961, Swan-Sussman. 1963 and Dyson. 1965). numerical analysis techniques were used as swn as wfficientlv wwertul comouters became available (Yeh and Mei. 1967. 1968 and Atia and Me, 1971) ~owad'ays, method &nnnen t (MOM) basea comp~ter programs permlt a relaole pred~ct~on ol the propen es of splral anlennas as shown by Nakam (Nakano el al . 1982. 1963) and others.
Thouoh the cornouter simulation of sdral antennas can be reoarded as a solved oroblem. it still remais essent~al to unaerstand lhe bass propcn.cs as well &the lhmlts of thls antenna lype, In order lo perform a successful deslgn or even to decide, whether a spwal 1s su~lable for a certam application or not. Those questions can not be answered quickly through computer simuhtions
An explanation of some aspects of spiral antennas - the polarisation properties - are the subject of this paper, which focusses on two and four-arm planar spirals. Scattered over different publications from the past 30 years, many ideas have been published already in this field. Here. the concept of circularly polarised spherical waves (CPSW) is added to the work of earlier authors in an attempt to close some gaps and to improve the completeness of the picture.
Elec+rornagnebcs. 14'259-284.1994 CoWnght 0 1994 Taylor 6 Franm 02726Y394 St0 00 + 00
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The paper is composed of three major sections :
U. R. KRAFT
In the first section, circularly polarised spherical waves (CPSW) are introduced as a means to represent the radiation fields of circularlv wlarised antennas. The orooerties ol these , . . . elementary waves and the~r Impact on the antenna polansatlon are d~scussed and condn~ons for the radlatlon of pure clrcular polansat~on are derweo In addltlon. Ihe ~nlluenco of antenna size and planarity on the CPSW mode spectrum is considered.
In the second section, the specific CPSW s~ectra of svmmetrical multi-an antennas like spirals are invesligated. Fundamental differences in the polarisation properties of two-arm and four-arm structures are presented. The impact of feed voltages and antenna geometly on the CPSW spectrum is discussed
The s w i a l properties associated wilh the ammetrv of planar soiral antennas are the subiect , . of the'third s'ection. Here, both two-arm andfour-an spirals are discussed. An explanatio;l of the basic polarisatbn properties and some guidelines for the design of spiral antennas are derived from these considerations.
CIRCULARLY POLARISED SPHERICAL WAVES
Introduction to Circularly Polarised Spherical Waves
The representation of antenna fields as a superposition of elementary spherical waves is a well- established technique, which is routinely used for the determination of antenna farfields from measurements of the corresponding near4ields (near-field far-field transformation. Ludwig. 1971).
Whereas in most cases. the ex~ansion of the antenna field into a sum of soheriial waves is used as a pure mathematical means to formulate a certain problem conveniently, this approach can also provide some insight into more general antenna features, if existing symmetries and other basic properties of the elementary waves are considered in more detail
Usually, a set of transversal electric (TE) and transversal magnetic (TM) elementary waves or modes is used for the described expansion. These modes arise from the solution of the vectorial wave.equation by means of an electrical and a magnetical vedor potential with radial orientation. With respect to the far-field each TM or TE mode represents one linearly polarised elementary wave.
For the description of circularly polarised antennas, it is more convenient to express their radiation fields as a superposition of circularly polarised spherical waves (CPSW). This approach has some distinct advantages :
The two sets of elementaly waves i.e. left-handed (LH) and right-handed (RH) waves. describe directlv the co-wlarised and the cross-wlarised comwnent of the antenna field. The sum ot the m.poar~sed waves dolormlnes the co-polarsad antenna pattem, whereas tne sum of the cross-polar~sed waves descr bes the cross-polansed antenna patlem . For a given direction, the ellipticity or axial ratio (AR) of the antenna field can be derived directly from the ampliudes of the sums of LH and RH waves. An additional phase information is not required.
It is simple to identify those directions for which pure circular polarisation is radiated by the antenna because the sum of either the LH waves or the RH waves vanishes for such directions.
An elegant formulation of the CPSW expansion has been given by von Winterfeld (von Winterfeld, 1986). where the electrical farfield of en antenna is wriien as
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 261
Here Cnnp denotes the expansion coefficient of each individual mode and O and q are the conventional spherical coordinates. The unit vector ep describes LH and RH circular polarisatmn for p=+l and .I. respectively.
Any set of the indices n, m and p defines an idivaual CPSW mode which is represented by the corresponding spherkl function Snnp. This function can be separated further inlo a sub-function of the azimuth (q) and a sub-function of the OH-axis angle (6)
with
Lnmp(6) = mp [dnml(cos 6 ) I sin 131 - sin 6 [d ~ y ( c o s 6) 1 d cos 6 ) l (2b)
wherein the functions Pare the associated Legendre-po~noms
Whereas the properties of the exponentkl funclion for the azimuth are welkknown. the functions Lnmp(B) require some more consideration. Normalised absolute values 01 these functions are shown h F~gure 1 for n U and /m%3 These aagrams represent the mdulaual rad~atnon patterns 01 the adferent olernenlaly waves n an arbltrary az~mutnal plane
FIGURE 1. Normalised amplitude of the individual CPSW d e s as a function of the OH-axis angle 6 (pattern for constant azimuth).
It is evident from the figure that several symmetries with respect to the indices m and p exist and that only tew modes contribute to the radiation for the poles of the coordinate system (O=OO. 180'). These properties and their impact on the antenna polarisation are discussed in the next two sections.
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U. R. KRAFT
2.2 Specific Properlies of Circularly Polarised Spherical Waves
Before we discuss the impact of the CPSW mode spedrum on the antenna polarisation. the specific properlies of these waves will be considered.
The first interesling class of properties are specxic symmetries of the CPSWs and directions lor general zeros in their patterns. These properties are clearly shown in Figure 1. Mathematically they can be formulated as follows :
. a simultaneous change in the signs of the indices m and p does not change the pattern
a change in the sign for the index m causes a rotation of the pattern by 180' and a phase shin by 18090r 360"
Ln.rnp (180•‹-6) = (-l)n'lm"l L nmp (O) (4)
only lunctions with m d l contribute to !he on-axis radiation (8.0'. 160')
L.mp (i)=OO) = 0 for m p +l (5a)
Lnmp (i9=18O0) = 0 for mp* -1 (Sb)
Whereas eq. (3) and (4) imply that four elementary waves form a specific, symmetrical group of modes for any given set of indices n and I d . eqs. (5a) and (5b) offer the opportunity to formulate simple conditions lor the radiation of pure circular polarisation in on-axis direction. Proofs of all 3 equations are given in (Krafl. Diss.1988).
The second area of interest is the relation between the size of the antenna and the existence of dominant elementary waves in their radiation field. It will be shown in section 4 ol this paper, that fundamental dinerences in the pobrisation properties of small and large spiral antennas exist because the antenna size implies a restriction of the number of available dominant modes. The inpact of the antenna size on the CPSW mode spectrum is derived by the following consideration.
According to Ludwig (Ludwig. 1971). the free space can be regarded as a waveguide for spherical waves, which exhibis a parlicular cut-off radius Rc lor each individual spherical wave mode. This cut-off radius is determined by the index n of the corresponding spherical wave and describes the minimum radius of an antenna structure, which is neccessary in order to excite this spherical wave wilhout an excessive storage of reactive energy in the near-field. Rc is defined by the equation
R c = n l ( 2 n : / A ) (6)
wherein A denotes the free space wavelength.
For a given radius R of the antenna, eq (6) implies, that the number of dominant modes in the CPSW spectrum is restricted to waves with n < N = (2n I A) R. As stwwn in (Ludwig. 1971). the contribution of modes with nzN is less than 0.1 %of the total radiated power i.e.
%rnp <<I for n > N = ( Z n / A ) R 5 Iml (7)
For this reason, the CPSW spectrum of small, low-gain antennas like spirals is dominated by a small number of modes. Since no CPSW for Imbn exist, if was possible to focus the discussion of the specific propellies of spiral antennas within section 4 of this paper on CPSW modes with nW i.e imld3.
The third area of interest is the impact of planarity of the antenna structure on the corresponding CPSW mode spectrum.
Within this paper we consider planar spiral antennas i.e wire structures, which have no extension in on-axis (B=0". 180") direction. Such a geometry implies, that the electrical current. which
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 263
'generates' the antenna radiation. has also no component in on-axis direction. Consequently, the radiated fields exhibi some symmetries with respect to the plane of the antenna structure (B=90•‹).
A formulation of this symmetry in terms of the circularly polarised field components ERH and ELH
is given in equation (8). A proof of this equation can be found in (Krafl. Diss. 1988)
With respect to the CPSW mode spectrum of a planar spiral or any other planar wire antenna, eq. (8) imlies that the corresponding LH and the RH modes lor any given set of indices n.m have to be e x k d with equal anipliiude in order to satisfy the symnieiy condition. The mathematical formulation of this relation is
- (-l)"-InJ Crimp for planar wire antennas located in the plane B=9OS (9) Cnm-p - Eq (9) can be aeducw. 14 eq (8) 1s forrnulatw in terms of CPSW and then the syrnmetnes, glven .n eqs (3) and (4) are used Agaln a prool of Inns equallon can be found in (Krah Das. 1988)
It should be noted that a practical antenna structure is always non-planar in a stricter sense, since it has a finite thickness and needs some feed and sup~orl structures. Nevertheless, the ~~ ~- ~ . . assumption of perlect planarity is a good approximation for a description of the antenna properties - at least for directions around the antena axis. The quality of the approximation degrades somewhat lor directions cbse to the antenna plane (B=9O0) depending on the actual configuration . . to be considered.
2.3 Circularly Polarised Spherical Waves and Antenna Polarisation
AHer we have discussed the specific properties of the CPSWs, the impact of these properties on the antenna polarisation will be considered in this section.
Together, the properties expressed in the eqs. (3) - (9) imply serbus restrictions for the degree of freedom in the mode spectrum of a low gain (i.e. small), planar wire structure like the spiral antenna. Impacts of these restrictions are demonstrated on the basis of two examples :
As a first example, let us consider a planar wire antenna without azimuthal dependence of the current distribution.
It is evident, that an antenna without any azimuthal dependence of the current radiates a field, which has also no azimuthal dependence. Such a field consists completely of CPSWs with the azimuthal index m=O. Because the antenna is planar, the LH waves (p=l) and the RH waves (p=- I ) are excited with equal amplitudes (eq. (9)) and because of the CPSW symmetries (eq. (3).(4)) these waves have the same panems, if the field does not depend on the azimuth (m=O). For these reasons. tne LH and RH components of lhe rad~aled flela nave the same amplltdae f& any glven dlrectlon I e the radtaled flela 1s pertectly hear polarlsed for all d.rect~ons
A practical example for a planar wire antenna without azimuthal dependence is a small loop antenna, located in the plane B=90•‹. The radiated field of this antenna is - in fact - linearly polarised for all directions, if the current distribution is suff'iently homogenous.
The first example shows that CPSWs with md) are required to obtain an elliptically or circularly polariied antenna field because these modes have different patterns for the LH and RH component. In this case. the balance of the two circularly polarised components of the radiated field can change with the direction. Pure circular polarisation is radiated for those directions where one of the two components vanishes.
This behavior is illustrated by the second example, an antenna with an azimuthal dependence of the current but a sufficiently small extension to restrict the number 01 dominant modes to the group n=l, m=fl, p d l . Eq. (6) shows. that the eflective radius of such an antenna has to be smaller tnan about (tin) wavelengths in order to avod domlnant modes wllh n.2 For reasons of s~mpllc~ly 11 is assdmed that modes n.1, m=O p-ll are not excned by the antenna Practical examples for such antennas are small crosseddipole antennas and small spirals. In practice, the excit&ion of
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264 U. R. KRAFT
the rnodes m=O can be avoded by a suitable feeding arrangement.
Because of the planarity of the antenna, the LH and RH rnodes ( w 1 ) for each set n=l, m=?l must have equal amplitudes. Consequently, the balance of the 4 possible dominant CPSWs is determined by one single parameter k=C1.l+l/Cll+l=C1.l.l/C1,.l which describes the relative
ampliaude of the LH and RH waves for linearly decreasing phase in +p (m=-1) and -q (m=+l) direction (c.1. eq. pa)). The impact of this balance on the total LH and RH radiation pattern is visualised in Figure 2 for different values of the parameter k. The figure shows the LH and RH patterns for the blane Q = 0".
h = 0.1
-= LHCP - - - : RHCP
FIGURE 2. ldealised circularty polarised antenna patterns for a small, planar wire antenna. The parameter k describes the balance of CPSWs for m=l and m=-1.
It the CPSW lor 1;nearly decreasing phase In +v (m=-I) and .Q (m=+l) directan are excited with equal amplnude (k-1). the total radiation patterns for the LH and RH component of the field are identical. In this case, the entire antenna field is linearly polarised, because the planarity of the antenna enforces that for each set n,m RH and LH wave6 have equal amplitudes
A different radiation panern for the LH and RH component of the field is obtained, if the CPSWs for linearly decreasing phase in +p (in=-1) and -9 (m+l) direction are not excited with equal ampllude ( k l ) . In this case the polarisetion depends on the coordinates 19 and Q . The arrows shown in Figure 2 indicafe directions, where the total LH component of the field vanishes i.e Dure IRH) circular wlarisation is radiated. We call those directins 'wints of circular polarisaton' ~ P C P ~ ar;d n $6 lnteresImg to note, that we have found such PCPs f i r all lnvehtlgated circularly polarised antennas I e cms~ed dipoles, spuals and hellces (Kraft, 1988. 1990. Kraft and ~dnich..iss?. 1990) even in cases wherethe polirisation In on axi; direction is far from being circular.
It is ewdent from the figure, that the PCPs move towards the poles of the coordmate system, one amup of CPSWs (m-1 or m-1) becomes superior They reach the axis. I one of the groups is ~om&etelv suoore&ed. This means that the radiation of Derfect circular wlarisation for the on-axis
7 , .. d i i i o n can be obtained if either all modes with m=+; or all modes'with m=-1 are completely suppressed. In the first case pure RHCP and LHCP are radiated for *=OD and B=1SOQ. respectively. In the second case ll is vice vem.
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 265
Since all modes with m e 1 do nct contribute to the on-axis radiation of the antenna, the behavior shown in Figure 2 reflects a general rule. A mathematical fonuktion of this 'mode condilion' lor the excitation ol pure LH (p=+l) or RH (p=-1) circular polarisation in the direction &=OD is
- 0 for m=-p and n=1.2,3 ...., /rn/ Cnrnp - (10)
So far, we have focussed our anention on the polarisation for the on-axis direction. For the cases k.0.5. 0.1 and 0.0. the total polarisation state for the complete upper halfqhere is visualised in Figure 3 by means of the 'polarisation map' (Kratl and M6nich. 1987). Within this representation. the axial ratio (AR) and the mayor half-axis angle (r) of the polarisation ellipse are shown as a lunction of the coordinates 0 and 9. PCPs can be identlied by an intersection of the contour lines lor r . This is explained by that fact. that the mayor hall-axis is not defined for perfect circular polarisation, where the polarisation ellipse becomes a circle.
(a) k = 0.5 (b) k = 0.1
FIGURE 3. Polarisation map lor an idealised small, planar wire antenna. The parameter k dewribes the balance of CPSWs for m=l and m=-1. (3) k=0.5. (3b) k=0.1. (X) k=O.
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266 U. R. KRAFT
Again, it is evidenl that an optimisation of the polarisation purity for the directions around B =Om requires the suppression of one group of CPSWs (m=+l for perfect RHCP, m=-1 for perfect LHCP) as formulated in eq. (10). However, even if one of these groups is totally suppressed (Fig. 3c). the AR still deteriorates for off-axis directions. This is explained by the fact, that the suppression of one CPSW group enforces an on-axis zero in the cross.polarised antenna pattern bul does no1 supress the entire cross-pobrisation lor all directions. Such a complete suppression is no! possible with planar structures because of the symmetry conditions described in eqs. (8) and (9). Figure 3c shows the best achievable case. Though a slower deterioration of the AR with lhe angle B can be obtained in rn azimuthal plane if the parameter k is selected acmrdingly, the AR will then degrade faster in the orthogonal plane. Such an example is shown in Figure 3b.
In conclusion it can be summansed that an optimisation of the polarisation performance for the upper half sphere and especially the generation of pure circular polarisation for the on-axis direction 8=0•‹ or 180" requires antenna structures, which offer the opportunity to -EMldh
of the radiated fields i.e. the index m of the CPSWs.
One general class of such antennas is discussed in the next section of the paper. Spiral antennas are a member on this class.
3. MODE SPECTRA OF SYMMETRICAL MULTI-ARM ANTENNAS
3.1 Definition and General Properties of Symmetrical Multi-Arm Antennas
In the previous sections we have shown that the optimisation of the polarisation performance for directions close to the antenna axis (B=O0) requires anlenna strunures which permit a control of the azimuthal dependence of the radiated fields. One general class of such structures are symmetrical multi-am antennas (SMA). These antenms mnsist of N separately fed arms and show a N-fold symmetry with respect to the z-axis (B=OO). A rotation of a SMA by an angle 0=2n/N about its axis leaves the antenna structure invariant.
Figure 4 shows an example for a SMA with lour arms ( N 4 ) and an abitrary shape of the individual anlenna arms. The parameters v denote the actual feed voltages applied to the arms 1 lo 4 with reference to a center reference conductor.
FIGURE 4. Symmel i i l muni-am antenna (SMA) wlh N=4, an a!biirary basic geometry and feed voltages v1 to v.,.
The special properties of SMAs can be subdivided into two groups : Because of the symmetry of the structures, general properties exisl which depend only on the number of the arms and the actual f ew volliges ~&s&qdentl~. these properties applytor such dltferent antenna types as lhe dipole and the two-am spnal for the case N=2 or the crossed dipole and the lour-arm spiral for lhe case N.4. Such general properties are discussed in this section. Apart from them, specific
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 267
properties exist - of course - which are related to shape of the antenna arms. Such properlies are considered in section 4 of this paper for various types of planar spiral antennas.
Basic symmetry properlies of SMAs and their impact on the mode spectrum of the radialed antenna fields have been considered already since the 60s and 70s (Dyson and Mayes. 1961. Sivan-Sussman. 1963. Deschamps and Dyson. 1971. Atia and Mei. 1971). The basis for the following mathematical formulations has been established at this time.
Any possible excitation state of a SMA can be described by a feed vector V = (vl. vp... vN) which
contains the N actual feed voltages applied to the antenna. Since the sum 01 all possible feed vectors V forms a N-dimensional vector space. any feed vector V can be expressed by a linear combination of the basic vectors of this vector space - the so-called basic excitation modes VM
(M=O,l ..... N-1) - as described by eq. (11)
N-1
V = a, VM with aM = [V VM'] I [V] (11)
M=O
Here, the parameler aM denotes the expansion coefficient of the actual excilation vector V in terms
of the basic excitation mode Vw whereas [V] denotes the normalisation factor of this veclor space.
In order to obtain a symmetry for both the antenna structure d the excitation voltages, it is convenient to select the basic excitation modes to
VM = [ I . expu M 0) , expu 2M 0). .... expu (N-1) @)I wilh 0=2dN (12)
An excnalan of a SMA w~th one of these basc modes lmplles a conl~gurat.on wnh perfect N-fold symmelry of the clectrcal antenna currents wnlch generate the radlallon helds Smce a rolaton of such a confiwration bv an ande 0=2dN about its axis leaves the antenna qwmetw invaknt but causes a phase shin of the-antenna currents by j M 0. the change of the radiated fields. implied by such a rotation, is also limited to a phase shin by j M 0. Consequently. the CPSW spectrum of a SMA wilh N arms and an excitation wilh V=VM is restricted to modes with
m = M + N k with k=O, f 1. C!.... (13)
wherein the factor N k accounts for the ambiguity of the exponential function.
It will be shown below that it is possible to excite CPSWs with m=l and to avoid, simullaneously. modes for m=.l or vce versa 01 the n~mber of arms (N) IS suffc~ently large In these cases perfect circular polarlsatlon 1s radtatw for the on axls d~rect~on as d~scussed in the prev.ous sectlon
Since lwo-arm and four-arm antennas are of mayor interest for practical applications, we will focus the following considerations on these special cases (N=2 and N.4).
3.2 Specific Properties of Two-Arm and Four-Arm SMA
At first, let us consider symmetrical two-arm antennas :
For the two-arm case (N.2) two possible basic excitalion modes exist which can be written as
V o = [ l . l ] . V1 =[l,-11 (14)
Whereas Vo describes an unbalanced in-phase excitation of the two arms. which requires the
physical existence of an additional reference conductor. V1 is the usual, balanced excitation mode
of the antenna, whereby the two arms are led in anti-phase.
The possible CPSW spectra whih correspond to these excitation modes are reslricted to
m = ..., -4, -2, 0, +2, +4, ... for V = Vo (15)
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268 U. R. KRAFT
and m = ..., -5, -3, -1, +1, +3, +5, ... for V = V, (16)
Since the spectrum for V = Vo does not contain any mode with m+l (eq. (15)), the corresponding
radiation field of the antenna vanmhes for the z-axis (see eq. (5a) and (5b)). For spiral antennas, tnis operating mode has been demonstrated amady in 1960 by Oonnelan (Donnellan. 1960). who measured the described 'beacon-type' radiation of a I&-arm archimedean spiral antenna with a small ground plane and an equi-phase excitation of the arms.
For a perfect excitation with V, the CPSW spectrum of a twoarm SMA can comprise modes with
m=l and m=-1 (eq (16)) If these modes dom~nate, whlch applles for almost all lnterestlng cases, the maxlmum of the rad~aton panern occurs for the z-axa
The polarisation for the on-axis direction depends on the balance of the CPSWs with m d l as Shown in section 2.3. This balance, however, is not determined by the wmmetw and the excitation (eq (16)) but depends only on tne shape of the antenna arms Two represeniitwe exanples for thls behavor are a two-arm splral and a s~nple d~pole If the same excnatlon V, 1s applled to these
two antennas, the radiation of the spiral antenna is elliplbal or even circular whereas the dipole radiates pure linear polarisation.
A practical feed netvvork will not perfectly realise a balanced excitation. In this case, the actual excitation is dominated by V, but contains also Vo .The resun of such a mixture is a radiation
pattern with a maximum close to the z-axis but a slight unsymmatry referred to this axis. The polarisation lor the on-axis direction is not affected by such imperfections.
In conclusion it can be stated that the on-axis polarisation of a two-an SMA does not depend on a specific excitation i.e. an optimisation of the polarisation properties has to focus on the specific properties associated with the geometry of the antenna arms. Such aspects of the specificspiral geometry are discussed in section 4 of this paper.
The situation is different for the case of a four-arm SMA. For a four-arm SMA (N=4) four possible excitation modes exist which can be wrinen as
Again V,, describes an unbalanced in-phase excitation of the antenna arms, which requires the
physical existence of an additional reference conduclor. V2 is a symmetrical anti-phase excitalion,
whereas Vl and V3 are excitation modes with a linearly decreasing phase in +q (V3) and -q (V,)
direction.
The possible CPSW spectra for these excitation modes are restricted to
m = ..., -8, -4, 0, +4, +8, ... for V = VO
m = ..., -7, -3, +I . +5, ... for V = V l
m = ..., -6, -2. +2, +6, ... for V = V2
and m = ..., -5, -1. +3, +7, ... for V = V3
Since the spectra for V = Vo and V = V2 do not contain any mode for m d l , the corresponding
fields vanishes for the z-axis (see eq. (54 and (5b)). Usually, these d e s are not desired for most applications but they may be present with minor amplhude due to imperfections of the applied feed network structure. The polarisation for the on-axis direction is not affected by these excitation states.
For a perfect excitation with V, the CPSW spectrum contains modes with m=+l but does not contain &es for m-1. Such a field is perfectty circular polarised for the z-axis without a dependence on a specifk shape of ihe antenna arms. Whereas pure LH &rimtion is radiated for *=OD pure RH polarisation is achieved for &;180D.
A similar result can be obtained through a perfect excitation with V3. In this case, the CPSW
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 269
spectrum contains modes with m - 1 but it does not contain modes for m=+l i.e. the field is perfectly RH circular polarised for 6.O' and perfectly LH circular pokrised for 6=180•‹.
From th~s consideration II can be concluded that - in general - SMAs w~th 4 arms and a perfect excital~on with V, or V3 provide an oplimsed clrcular polarisallon performance for direct~ons close
to the antenna axis and permil the radtalon of essentially pure circular polarisalion for the on.axis d~reclion ilseil Smce this prqxny does not depend on a specik geometry of the antenna arms. lhere slwuld be no s~bstantial d.lference in the polansdon pmpenies of a spiral, a crossed dpole . . antenna or even an antenna like the one shownin figure 4.
In Dracfice. however. there are - of course - substantial differences which are associated with the specific ge&netry oflhe antenna arms. Though all SMAs with four arms have the general ability to generate pure circular polarisation for the on-axis direction, this abiliy depends on two additional conditions which have been assumed a prioriin the previous discussion
At first, it has been assumed that the antenna radiates any field for the on-axis direction, which presupposes that the CPSW with m d l are present in as SpectNm. This is lrue lor most practical cases and especially for low gain antenna structures - but it can not be assumed a priori An example lor an exception from the rule is a SMA which is formed by two crossed dipoles with an individual arm length of 1 A. Each of the two dipoles has an extension of 2 A and radiates no energy in the direction of the SMA axis. Consequently, the total radiation of the SMA vanishes also for this direction -even lor perfect excitations with Vl or V3.
The second a prioriassumption is the realisation of a perfect excitation with V1 or V3 withoffl any
amplitude and phase error. Though it may be possible to reduce the imperfections of the feed network by a very careful design, a perfed excilation wilhout anyerror is not possible in practice. A deviation from the perfect case. however. causes a deterioration of the polarisation purity and this deterbration depends on the specific geometry of the antenna arms.
For these reasons, it is essential lo find a description of the polarisation properties of four-arm SMAs which includes the discussed general symmetry properties as well as the specific pmpenies of the wire geometry lor all possible excitations. Such a description is developed below. In order to obtain simple and measurable description parameters we will focus the investigation on the desired main-beam direction, which is of most practical interest.
For a general excilation V, the LH and RH components of the radiated field can be expressed as
N-1
ERH,LH (WP. V) = C ERH,LH (fir% V=VM) (22)
M=O
wherein, the parameter aM denotes again the expansion coelficient of the actual excitation vector
V in terms of the basic excitation modes VM (eq.(ll)).
Because of the CPSW properties described by eqs (5a) and (Sb), the series for the LH and the RH components of the radiated field (eq. 22) degrades to one single term for the LH component and one single term for the RH componenl for the desired main-beam direction 6=0•‹.
Consequently, the cmss polarisalion discrimination XPD for this direction can be written as
XPD = 20 lg IERH/ELHI = XPDf + XPD* [dB] (24)
with XPD, = 20 lg la3 1 a,l (25)
and XPD, = 20 1g IERH (+0•‹, V=V3) / ELH (+On, V=V,)I (26)
Whereas the first term in eq. (24) describes the influence of the actual feed vector V on the
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polarisation, the second term represents the influence of the antenna arm geometry or structure. The total purity of the polarization is determined by the sum of both factors which permits the possibility to improve a poor polarisation performance caused by a basically unsuited antenna shape through a proper feeding or to improve a poor feed network quality by a suitable antenna shape.
An example for the first case is the crossed dipole antenna. Though the basic antenna geometry - a dipole - is a linearly polarised antenna structure, it is posible to generate pure circular polarisation, if the feed network of the antenm realises a perfect excitation with V=V3 or V=V1
An example for the second case is the spiral antenna which is able to generate circular polarisation - even for cases where the feed network is far from being perfect. This important advantage of the spiral antenna permits a polarisation purity which is sufficient lor many applications without the need for a sophisticated feed arrangement or a highly optimised antenna structure.
Defining a as the quotient of the difference voltages Ux=vl-v3 and Uy=v2-v4. XPDf can be written
as
XPDf = 20 lg la3 I a, l = 20 lg l(1 +j a) 1 (1 -j u)l with a = (v2-v4) I (v,-v3) (27)
This equation is visualised in Figure 5 which shows contour lines for the factor XPDf drawn in the
complex a-plane. Whereas the upper haw plane represents the radiation of LH elliptical polarisation for 6-0" (XPD c We). the lower hall plane corresponds to RH polarisation. For a+j i.e. Uy=iiU, the actual leed vector V is equal to the basic excitation modes V, or V3 and.
conseauentlv. oedect LH or RH wlarisation is generated for the on-axis direction. If the actual ampliltke al;d bhase errors of th; applied leed network are known or can be estimated. Figure 5 provides a graphical means to determine !he corresponding feed network factor XPD1.
FIGURE 5. Representation of the feed network factor XPDf of a four arm SMA
as a function of the parameter a = (v2-v4)/(vl.v3)
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 271
If the antenna itself does not contribute further lo the polarisation purity i.e if it does not exhibit additional relevant mode selection ~rooerties. the total XPD is complelelv given bv the feed . network factor If aodltlonal mode seleclmn pmpertles are provded by the antenna geometry. the totally ach~eved cross-polansatlon suppression exceeds the flgures predcted by tne lactor XPD,
This additional suppression of the unwanted polarisation component can be described by the second term in eq. (24) - the factor XPDSt.
According lo eq. (26). XPDSI will be 0 dB for a given antenna structure, il this antenna reacts on
the excitations V, and V3 with exactly the same radiation pattern but a change in the sense of the
polarisation. In this case the sense of the dominant pobrisation component is completely determined by the sxcitalion and does not depend on the geometry of the antenna arms. If the antenna reacts with dinerent radiation patterns for the two excitation states, it exhibits additional mode selective properlies and. consequently, the structure has an impact on the on.axis pobrisation i.e. XPD,, will not be equal to 0 dB.
Examples for two extreme cases ere shown in Figure 6 which respresents the relevant radiation panerns for a small crossed-dipole antenna and a moderately sized. RH archimedean spiral antenna. Since both antennas are planar structures, the radiation patterns for the corresponding oflhogonal polarisation components, which are not shown in the figure, are given by a 180" shift of the co-polarised panern (eq. (8)).
(a) half-wave crossed-dipole
- (b) moderately sized archimedean spiral
FIGURE 6. Calculated ratintion pattern of s p e d case antennas for excilahns with V, and V3. Figure 68 represents a crossed dipole with a tinmeter of 0.5 A. figure 6b represents a planar RH lour-arm archimedean spiral with a diameter of 1.45 A.
Whereas the two pattern for the excilations V, and V3 are identical for the crossed dipole, the
properly designed spiral antenna generates two considerably dinerent panerns. This implies that XPDSt is 0 dB for crossed dipole antennas whereas it can be considerably large for spirals.
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As shown by this consideration. the structural factor XPDSt provides a simple means to describe
such mode selective propellies which are relevant for the main-beam polarisation purity and which are related to the geometry of the antenna arms. Since lhis parameter does not depend on the actual excitation state V of ihe antenna (eq.(26)), it can be measured easily in practice:
According to eq. (24). the total cross-pobrisation suppression XPD (in dB) of a lour-arm SMA is given by the of the feed network factor and the structural tactor. A measurement of the total XPD lor an actual excitation slate which enforces a vanishing leed network factor (0 dB) provides. therelore. a direct measurement of lhe structural factor of this antenna.
XPDst [dB] = XPD [dB] for XPDf = 0 dB (28)
Such an exciletion can be realised by a balanced feeding of two oppositely located antenna arms (8.g. 1 and 3) and a shorling 01 the remaining two arms to the central relerence conductor. The corresponding leed vector is V = (1.0:l .O) which implies la3/all =1 i.e. XPDl = 0 dB.
Unbalanced comwnents of the excitation which mav be present -in practice- be-cause of the . . ~mperfecl.ons of any practical feea network do not alfect this approach because such unbalanced components would ~mp y an adaltonal exctatlon ot tho bas c modes V, and V, whicn 00 no1 have " - an impact on the maimbeam polarisation (eq. (18). (20)).
Successful measurements of the structural factor XPDS1 have been reported by the author (Krafi.
Diss.1988. 1990) for archimedean four-arm spiral antennas.
4. TWO AND FOUR-ARM SPIRAL ANTENNAS
4.1 CPSW Mode Selection Mechanism of Spiral Antennas
Apart from the CPSW mode selection properlies which are common to all SMA types. spiral antennas exhibii an additional selection mechanism which ensures that a orooerlv desianed spiral . . , radiates predominantly circular polarisation for directions close to the antenna axis (B=&) wheieby the sense of the polarisation is determined by the winding sense of the spiral arms.
The basic effects which determine this behavior can be explained by the radiation mechanism of a spiral antenna :
A1 the teed poinl of each antenna arm, a current wave is generated which travels along the wire conductor wifh a phase velocity close to that of light in freaspace. If this forward travelling wave can couple into radiation modes of the antenna, energy is radiated and, consequently. the amplilude ot the antenna current decreases since less energy is then concentrated in the bounded current wave. If the forward travellina current wave is not sulficientiv attenuated bv the radiation of - energy, a the remamng fractton reaches the opon ond of the anlenna w.ro ond is totslfy reflected oack tnlo the dlrectton 01 the antenna center This rellected Nave travels in lhe opposlte dlrectlon and can elso radiate, the corresponding radiation modes are possible in the spectrum of the considered entenna.
The question whether the forward travelling current wave FTCWI and the reflected. backward ~ - . - ~ - -. travelling current wave (BTCW) are able lo couple Into radialan d e s of lhe antenna I e. whether they are able to generate rad~ation is determined by four factors
the winding sense of the spiral wire the size of the antenna the number of antenna arms the actual excitation
The specific shape of the antenna arms has an impact on the strength of the attenuation i.e. the strength of the coupling between the bounded current wave and the radiation modes of the antenna but it does not alfed the basic question whether the current waves generate radiation or not. Therefore these more general mode selection propallies apply for both the archimedean and the logarithmic spiral type.
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POLARIZATION PROPERTIES OF SPIRAL AWENNAS 273
The factor which detenines the sign of the index m of those CPSWs which can be excited by the forward and backward travelling current waves is the winding sense of the spiral.
If a rlght-hand splral 1s wnadered the forward travelling wave produces an antenna current wdh lrnearly decreasng phase In crp dnecton, whereas the backward travellmg wave 1s associated w~th a linearly decreashg phase in -v direction. This dependency from the azimuth can be written as
I&) = IFo (9) exp (-kg) forthe FTCWon a RH-Spiral (29)
Ig(q) = leo(cp) exp (+kq) for the BTCW on a RH-Spiral (30)
wherein k denotes the effective phase velocity of the current waves with respect to the azimuthal direction whereas IFO and IBo denote the corresponding arnpldude of the forward travelling wave
and the backward travelling wave, respectively.
As discussed in section 2, the CPSW modes are characterised by the functions
Snmp(%cp) = Lnmp(@ exx (im9)
for anv given set of the indices n,m and D. If one compares the azimuthal dependence of the CPSW A h the eqs. (29) and (30). 11 becomes obvlous that the forwam travelling wave is able to codple Into CPSW modes with m<O, whereas the backward traveljng wave wll couple into CPSW with m>O. For the left-hand spiral it is vice versa.
The sfrendh of the couolina and the determination of the dominant CPSW modes demnd on the aclual a z ~ k l h a l phase ;el<y of the current waves at different regons of the antenn; and on the actual exc~tatan as well If the bcal az~muthal phase vebclry 1s close to the veloclty of l ~ h t in free sDace. an azimuthal phase deoendence rim. f2im. f3im urn occur for those sections ofihe s~iral ,.. ,.. antenna, where the circumferince of the antenna reaches 1,2 and 3 A.
This situalion is visualised in Figure 7 on the next page which shows the azimuthal phase dependence of the CPSW modes m-1 . -2 and .3 toaether with the azimuthal ohase demndance of tne forward travelling wave on both two-arm and 6ur-arm RH sprrals wdh ddlarent eicitations An azimuthal phase velocly equal to the velochy of hght in tree space (cO) has been assumw
It is evident from the figure that the azimuthal phase dependence of the different CPSW modes and the corresponding phase dependence of the forward travelling wave can be identical for cenain circumferences of the spiral, if a wecific excitation is applied to the antenna. In these cases, the coupling between the bounded current wave and tnecorrespond~n~ radiation modes reaches a maxlmum 1.e. a slgnificant redraton takes place. The figure indicates that such regions of the antenna occur at circumferences C=lml A for a cowling of the current wave into the CPSW . - wlth the index m. These reglono have been identnf~ed previously as the so-called 'active regions' of a spiral antenna (Kaiser. 1960. Dyson. 1965) and it is interesting to mte that the correspond~ng antenna radll are dent~cal to the cu-off radd for the lowest order CPSW modes Imkn as described by eq (6) For the modes m?st, r 2 and r3 these actlve reglons occur lor R=O 16 A. R=O 32A and R;; 028 b, respectively.
If the antenna size, number of arms, winding sense and excitation permit the existence of cenain CPSW modes characterised by the index m, the main coupling of the current waves into these modes will occur within the 'active region' for these modes i.a. for an antenna radius which is approximately equal to the cut-off radiusfor these modes.
For the hvo-an RH spiral a coupling of the forward travelling wave into CPSWs with the index m -1 and m=-3 occurs, if the antenna is excited in its usual balanced mode V1 and the antenna size
is sufficient)y large to permit the existence of active regions at a circumference of 1 A and 3A, respectively. The backward travelling wave can couple into CPSWs with the index m+l and m=+3 whereby the active regions are the same as for the forward travelling wave. For a LH spiral. the sign for the index m of the CPSWs has to be changed.
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qm. .IC""."l W.Y. Opn End Feed Perd @ F e d V d t s p .
FIGURE 7. Azimuthal phase dependence of CPSWs and lhe forward travelling currenl waves on two- and four-arm spirals for dinerent circumferences and excitations of the antennas. Azimuthal phase velocity of co has been assumed.
In case of the RH four-arm spiral, the coupling ot lhe current waves depends also on the excitation state of the antenna. The forward travelling wave can couple into the CPSWs with m=-1, m=-2 and m=-3, if the excitations V3. V1 and V, are applied to the antenna. A condition for this coupling is.
again, an antenna size which is sulficiently large lo permit lhe existence of the relevant active regions. The backward travelling wave can couple into the CPSWs with the indices m=+t, m=+2 ond m=+3. 11 the requared exc~tat~on and antenna slze are grven Agaln. the same argumentallon holds tor o LH splral 11 the slgn tor the Index m of tho CPSWs is changed
Due lo this specific behavior of spirals which is associated with the travelling wave nature of the antenna currents and the azimuthal orientation of the wires. a spiral antenna exhibits CPSW mode selection properlies whiih are not common lo all SMAs. Depending on antenna size, winding sense, number of arms end the actual excitation only some of the CPSW modes which are possible for a general SMA (eq. (15)-(16) and (18)-(21)) will be dominantly present in the mode spectrum of a spiral antenna.
The impact of these mode selection properlies on the polarisation of two-arm and four-arm spiral antennas is discussed in the following two sections.
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 275
4.2 Two-Arm Planar Spiral Antennas
For the discussion of two.arm planar spiral antennas let us assume a right-hand spiral which is excited with the basic excitation mode V1.
AS discussed in section 3.2 the general CPSW mode spectrum of such a two-arm SMA is restricted lo the rnodes with m = .... -5. -3. -1. +t , +3. +5. ... (eq. (16)). In addiiion it has been shown in the previous section that the forward travelling current wave (FTCW) couples into the rnodes m=-1, -3. -5, etc. whereas a possible reflected backward travelling current wave (BTCW) couples into the modes m=l. 3, 5, etc. if the size of the antenna permits the formation of the relevant 'active zones'. The existence of minor contributions from !he basic excitation vector Vo in
the actual excitation V due to imperfections of the feed netwol* does MI affect the on-axis polarisation as discused in section 3.2.
In order to optimise the polarisation performance of the antenna for directions close to the antenna axis and to obtain pure RHCP lor the axis itself, it is required to suppress the CPSW modes for m=+l mmpletely. Since the excitation of these modes is not affected by the anlenna feed voltages. the only possibility for the desired suppression is to avoid - as far as possible - the generation of a reflected wave which would couple into the unwanted CPSWs.
This eftect can be obtained - in principle - by three different means :
If the antenna geometry can be designed in such a way that the coupling of the forward travelling current wave into the rnodes m=-1 yields a complete radiation of this wave, no energy reaches the end of the antenna arms and no backward travelling wave is formed. . If the antenna size is sufficient lo ~ermi l the existence of additional dominant modes i.e. the modes md3 , the forward travelling wave is additionally attenuated by the coupling into the modes m=-3, whereas the reflected wave is attenuated by a coupling into the modes m=+3 before it can reach the active reqion for the modes m=+l. Since the modes m d 3 do not affecl the radiation in on-axis direElion the polarisation purity is improved.
11 the open end ol the antenna wre 1s meddled in order to Increase lhe radiatan or to absorn the energy bounded in the forward travellmg wave before the wave reaches the opon end. the reflected wave is reduced and, mnsequently, the polarisation purity is improved.
Whereas the first approach is very diiicun to realise in practice, the remaining two can be used to obtain a two.arm spiral antenna with wnsiderable polarisation purity for directions close to the antenna axis.
An example is shown in Figure 8 which presents the on-axis axial ratio (AR) for different archimedean spiral (ASP) and logar'ihmic spiral (LSP) antennas as a function of the maximum antenna size. The geometry of the individual antenna arms is given by
r(v) = ro + a9 for the ASP (31)
and r(v) = ro exp(a9) for the LSP (32)
Whereas the LSP data is based on unpublished simulations made by the author. the figures for both the conventional and the modified ASP have been compiled from the literature (Nakano and Yamauchi. 1982).
At first, let us wnsider the conventional ASP and LSP case.
It is evident from the figure that the polarisation purity for both spiral types is improved, when the antenna size reaches the cut-aft region for the modes m t 3 . In this case an active zone for the radiation of these modes is formed which improves lhe polarisation properties through two eftects. Firstly, the forward travelling wave is additionally attenuated by the generation of the CPSW modes with m - 3 . Secondly the reflected, backwards travelling wave is attenuated by the generation of CPSW modes with m=+3 before it reaches the 'active zone' for the m=+l mode. Both radiation lypes do not affect the on-axis polarisation but decrease the generation of the unwanted m=+l modes. Consequentb, a conventional two-arm spiral antenna should have a
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0.2 0.3 0.4 0.5 0.6
Maximum Antenna Radius (in Wavelengths)
FIGURE 8. On-axis axial ratio lor dinerent types of planar two-arm spiral antennas as a function of the maximum antenna radius.
minimum radius of 0.5 - 0.6 A in order to permit a significant suppression of the unwanted CPSW modes. A significant panern degradation due to the radiation of the mentioned higher modes does not occur as long as the wanted excitation of the CPSW modes with m=-1 remains the dominant radiation effect.
An example lor the other previously mentioned approach to reduce the backward travelling current wave is also included in the figure. Nakano and Yemauchi (Nakam and Yamauchi. 1982) have shown that the polarisation purity of a smaller two-arm archimedean spiral anlenna can be improved, if the outerwsl part 01 the antenna arms is replaced by a zig-zag section which enforces the radiation 01 the lomard travelling wave before it reaches the end of the antenna arms. The AR of this modilied ASP is also shown in Figure 8. It is evident from the figure, that such an increased radiation permits a better axial ratio lor spirals which are not sufficiently large to radiate the modes m=i3.
A final basic possibilny for the improvement of the AR is a resistive termination of the antenna arms. I1 the remaining forward travelling wave can be attenuated by resistive absorption, the reflected wave is decreased and the polarisation purity is improved. The problem asxriated with this appmach is the construction of a proper termination lor the forward travelling current wave which is not trivial since the current wave is only loosly bound to the wire.
In conclusion it can be stated that optimum polarisation purity is obtained tor two-arm spiral antennas il the generation 01 a reflected current wave is avoided - as far as possible. This can be achieved by
en antenna geometry which ensures a very good coupling of the forward travelling current wave into the modes m=-1 . an antenna which is sufficiently large to permit an exckation of the modes m d 3 i.e a maximum antenna radius 01 at least 0.5 - 0.6 A
I1 the antenna size has to be limited because of the actual system requirements, a modification of the antenna arms with a zig-zag section or a resislive termination are possible options. In both cases the design of the mcddied antenna section is not trivial and will require experimental optimisation.
Errors in the actual excitation of the two-arm s~iral does not affect the ~olarisation for directions close to the antenna axls Therefore an optmusaton of the feed network can mprove the pattern symmetry (by a suppression of V,,) but n mll not Improve the polansatan properties
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS
4.3 Four-Arm Planar Spiral Anlennas
The case of a four.arrn planar spiral antenna is different from the already discussed twoarm case because the permitted CPSW spectrum of a four-arm spiral can be controlled by the actual feed voltages applied to the antenna.
In contrast lo a twoarm SMA, a feeding of any lour-arm SMPI wilh either Vl or V3 enforces a
CPSW spectrum which contains only the modes m=+l pl the modes m=-I. If the feed network realises these basic excitation vectors without amplitude and phase errors. Ihe polarisation for directions close to the antenna axis is optimised and perfect circular polarisation is radiated for the on-axis direction itself.
Any practical realisation, however, can not be perfect. Since in this case the achieved axial ratio (AR) or the achieved cross-polarisation discriminatkn (XPD) depends on both the actual excitation and the geometry of the antenna arms, it is not useful to discuss the specific mode selection properties of four-arm spirals on the basis of the AR or the total XPD. A better basis is the geometry factor XPDSt as discussed in section 3.2. This parameter does not depend on the actual
excitation of the spiral antenna and it provides a direct measure for the specific, polarisation relevant mode selection properties of the structure. Optimum conditions for an excellent polarisation performance are achieved when XPD* reaches large positive figures. In this case lhe
XPD-contribution from the feed-network is improved by the mode selection properties of the spiral i.e. even non-optimised feed networks can be sufficient to obtain very pure polarisation for directions close lo the antenna axis
The basic mechanism which leads to a large geometry lactor is -again - the coupling between the forward and the backward travelling current waves and the different CPSW modes.
For reasons of simplicity let us assume a right-hand planar four-an spiral. If this antenna is excited with the basic excitation modes Vl or V3. only the modes
m = .... -5, -1. +3. +7. ... lor V = V3 and m = .... -7. -3. +I. +5. ... for V = V1
are possible as shown in eq. (19) and (21).
For a right-hand spiral, the forward travelling cunent wave (FTCW) can couple into the CPSW modes m=-1. -5 etc. for the excitation with V3 and into the CPSW modes 171-3. -7 etc. for the
excitation with V1. The reflected, backward travelling current wave (BTCW) can couple into the
CPSW modes m=+3, +7 etc. for the excitation with V3 and into the CPSW modes m=+l. +5 etc.
for the excitation with V,.
The size of the antenna delenines whether the mentioned CPSW modes are radiated or not i.e whether the relevant 'active zones' can be formed. This behaviour is visualised in Figure 9 which shows the calculated cunent distributions on one of the arms of an archimedean four-arm spiral antenna for perfect excitations with V1 and V3. The parameters of the spiral are a=0.02A and
r0=0.025A for all displayed cases. The maximum radius of the antenna is 0.275 A. 0.525A and
0.72% for the Figures 9 a, b and c, respectively. For reasons of simplicity, only the ampliiudes of the calculated currents are displayed in the figure. The corresponding phase diagrams can be found in the lterature (Krall. 1990) for the cases (a) and (c).
If the maximum antenna radius is smaller than the cut-aff radius for the CPSW modes m e 3 (Fig. 9a), the radiation of the modes m+l is possible, whereas the formation of active zones for the modes m*3. *5 does not take place.
In case of an excitation with V3, the forward travelling current wave couples into the mdes m=.l
and is attenuated through the radiation of energy. The remaining current wave which reaches the open end of the antenna arm is reflected. Since the modes m=+l are not possible because of the symmetry and the excitation wilh V3 and the higher modes m=+3. +7 etc. can not be excited with
signaicant ampliiudes because of the limited antenna size. the reflected backward travelling cunent wave does not produce any relevant radiation. Consequently, it is not significantly
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0.0 Normallsed Length 01 One Antenna Arm 1 .o
FIGURE 9. Calculated current distributions for an archimedean four-arm spiral antenna with a.0.02L r0=0.025A and perfect excitations wilh V, and V3. The maximum radius of the antenna is
(a) 0.275A. (b) 0.5251 and (c) 0.725A.
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 279
attenuated by the radiation of energy and the standing wave ratio of the total current distribution remains almost constant over the entire antenna arm.
In case of an excitation with V1. the forward travelling current wave can not radiate significantly
since the modes m=-1 are not wssible for this excitation and the hiqher modes m=-3. -7 etc can not be excited with relevant amplitudes because of the limited size oi the antenna. Therefore, the forward travelling current wave is not significantly attenuated and almost the complete energy is reflected at the open end of the antenna arm. The backward travelling current wave cou~lesinto the modes m=+t and rad~ates mosl of ns energy Snce only the backward travelhng wave is Slgn~lanlly attenuated through the rad.at~on of energy. the stanalng wave ra1.o of the lotal currenl distribution increases towards the end ol the antenna arm where the amplitude of the reflected wave reaches its maximum.
The described behavior of the spiral antenna changes when the maximum antenna radius reaches the cut-off radius for the modes m=i3 i.e. an adive zone for these modes becomes possible. Fig. 9 b shows a corresponding case where the spiral radius is slightly larger than the cut-off radius for . . . these modes.
In case of an excitation with V,, the forward travelling current wave couples into the modes m=-1 - and is attenuated through the radiation of energy. The reflected backward travelling cunent wave couples into the modes m=+3 and is also attenuated. As a consequence the standing wave ratio of the current distribution is smaller than the one presented for the small spiral.
In case of an excitation with V1, the forward travelling current wave couples into the modes m - 3
and is attenuated when it reaches the relevant reaions of the antenna whereas the backward " travelling current wave couples into the modes m+l within the corresponding active zone. Since the antenna size is just sufficient to permit an active zone for the modes m=f3 buf the extension of this zone is limited. the attenuation of the forward travellins cunent wave is not verv hish and, as a - . - consequence, the standing wave ratio in the current distribution is still significant.
It is interesting to note that this standing wave remains almost constant for a larger part of the antenna wire whereas is decreases rapidly in the region close to the feed-point, where the active zone for the modes m= +l is located. This indicates that - in fact -almost all radiation takes places in the 'active zones' of the antenna whereas the remaining part of the arms acts mostly as a pure transmission medium for the current waves.
Finally. Fig. 9 c shows a case where the antenna size is considerably larger than the cut-ofl radius for the modes m 3 3 .
For an excitation with V3 the current distribution does not differ very much from the previous case
since the dominant part of the radiation takes place in the active region for the modes m=-1 which is located close to the feed-point. Therefore, an increase of the antenna radius from 0.525.I. (Rg 9b) to 0.725A. (Fig 9c) has almost no effect.
For an excitation with V1 the standing wave in the current distribution is signilicantly reduced when
compared to the previous case. This indicates that only a small reflected wave is present. The effect can be explained by the larger size of the spiral which permits the formation of a sufficiently extended active zone for the modes m=-3. Due to the increased radiation of these modes the forward travelling current wave Is considerably attenuated before it reaches the open end of the antenna arms and the amplitude of the backward travelling current wave becomes very small.
The impact of the discussed mode selection properties on the polarisation of the antenna can be explained from a view on the circulariy polarised radiation patterns. Figure 10 shows the relevant patterns lor the three previously discussed spiral antenna sizes and the excitations V1 and V3.
Since the anlenna is planar, the orthogonal polarised patterns, which are not shown in figure, can be obtained by a 180" shii of the presented diagrams (eq. (8)).
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FIGURE 10. Radiation patterns for an archimedean lour-arm spiral antenna with a=0.02A, r0=0.025A and excitations with V, and Vg. The maximum radius of the antenna is (a) 0.275?., (b)
0.525Aand (c) 0.725L
For the case of the small spiral (Fig. 10a) the patterns for the excitations V, and V3 are similar
since the radiation is dominated by the modes m=fl in both cases. The right-hand spiral radiates ' RHCP for the on-axis direction (B=O0) in case of an excitation with V3 but it radiates also LHCP for
an excitation wilh V, in contrast to its winding sense. This behavior is explained by the fact that in
the former case the fomard travelling current wave radiates, whereas the backward travelling current wave generates the radiation in the latter case. As a consequence, the polarisation for directions close to the antenna axis is mainly determined by the actual excilation and does not depend significantb on the winding sense of the spiral. Therefore, the factor XPDSt is small i.e.
both RHCP and LHCP can be obtained from the same antenna but on-axis polarisation purity requires a high quality of the feed network.
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 281
If the maximum radius of the antenna reaches the cut-off radius for the modes md3, the pattern for the excitation V1 begins to change due to the increased radiation of the modes m=-3. This
effect can be seen clearly in the Figures 10 b and 10 c. Whereas the corresponding panern In Fig 10 b IS a mlxtLre of the fyplcal rad~ation patterns lor the modes m=+l and rn=-3 (c f Faure I). the larger antenna presented in Fig. 10 c radiates with the typical panern of the highe; modes
For the exckatlon V3, the change of the radiation patterns wiih the antenna size is not vely
signficant, since in all cases the maximum radiation takes place in the same active region for the modes m= -1
As a conseauence of these effects. the wlarisation ~uritv for directions close to the antenna axis 1s not ser1o"siy affected by the amplnude and phase err& of the feed network prowded that the antenna 1s sutflctently large If the actual excrtatbx conslsts of a m a r e of the modes V, and V, ~. " because of the non-perlect nature of the applied excitation network, the cross-polarised components generated by the unwanted exchation V, will not contribute to the radiation in on-axis
direction, if the modes m=+l are not present in the corresponding CPSW spectrum. Whereas the contribution of these modes to the antenna pattern is already reduced in Figure 10 b it is negligible in Figure 10 c.
Here, it should bs noted that the described general effects have been recorded firstly in 1965. whereas measurements of radiation patterns similar to those presented in Figure 10 have been published by Kim and Dyson in 1971 (Kim and Dyson. 1971). MOM calculations of the antenna currents on archimedean four-arm spirals were reported by Nakano 81. al. (Nakano el, al., 1983) in 1983.
Quanttatively. the dscussed mode selection propartles of four-arm splrals can be described by the structural factor XPD, as shown before. Calculated flgures for th16 parameter are owen In -. - Figure 11 for both archimedean (ASP) end logarithmic (LSP) spiral antennas. XPDa is presented
as a function of the maximum antenna radius. The cut-off radius for the modes m e 3 in a free- space medium is indicated in the figure.
3 5
30- Cut-Off Radlus
2 5-
E 20- u E 15- - -
(A
g lo- I X
5-
-5-1 I 0 dl 0.2 013 014 d5 016 0'7 0.b ab 1.0
Maximum Antenna Radius (in Wavelengths)
FIGURE 11. Cakubted structural factor XPDd for different archimedean (ASP) and logarithmic
(LSP) four-arm spiral antennas as a funt i in of the maximum antenna radius. The nominal cut-off radius for the modes m i 3 in a free-space environment is indicated.
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It is obvious from the figure lhal a minimum antenna radius of aboul 0.5 - 0.6 A is required in order to obtain the desired effect of the specific mode selection properties on the polarisalion punty. This limit is associated with the formation of an active zone for the modes m=i3 as described ~reviouslv. Whereas this minimum antenna size for an optimised four-arm spiral is identical to the bne ideiified for the two-arm case, the polarisation of the four-am spiral can be improved additionally by an optimised excitation as expressed in eq. (24) which is not possible for the twc- arm spiral:
Figure 11 indicates that geometry factors as high as 25 dB should be possible through a proper design of the spiral structure. Measured figures of the geometry factor XPDSl have been published
by the aufhor in 1990 (Krall. 1990) for dinerent ASP. Whereas - in practice -the curves shown in Figure 11 are slightly shined toward smaller radii due lo the ellective dielectric constant of the practical spiral carrier medium, maximum figures of 25 dB have been verified.
In conclus~on 11 canoe staled that an optmum Impact of lhe spec.1~ mode selection properties 01 four-arm spirals on the polarsalon puriy for d~rectons close lo lhe antenna ax6 is obtained. d the forward travelling current wave can radiate for the excitation V, and V3 and a backward
travelling current wave is avoided
Similar to the case of a two-am spiral, this behavior requires
an antenna geometry which ensures a good coupling ol the forward travelling current wave into the modes m=.l and m=-3
an antenna s m wncn is sJff,c:enlty large lo permil an excilal~on of the modes m z 3 I e. a rnaxirn~m anlenna radlJs of at east 0 5 - 0 6 h (belle, 0 7 - 0 8 A )
In conlrasl lo the lwo-arm case. tne polarlwl.on punly prowded by tho anlenna arm geometry of a four-arm splral can be improved through a sullable excitallon nelwork whch ensures a suffctenlly high feed nelwork lanor XPD, Ths property ollers the opportunity for t radeds belvreen anlenna
size and geometry on the one hand and the feed network quality on the other hand which are not possible for the two-arm case.
The selection of a suitable spiral design for a specific application depends - of course - on the requirements implied by the actual scenario :
If the polariition purity requirements are moderate, ehher a larger conventional two-arm spiral, a smaller, modified two-arm spiral or a smaller four-arm spiral with a suiliciently optimised feed- nelwork are the basic choices.
For more severe polarisation requirements a two-arm spiral becomes more dinicult to use whereas a lour-arm spiral can be adapted to the requirements by either an increase of the anlenna size or a further reduction of the leed network amplitude and phase erron.
If the radiation of both LHCP and RHCP is required in order lo obtain polarisation diversity, only small spiral antennas can be used. Here, two-arm spirals are unsuitable because their polarisation can not be controlled through !he feed network. In contrast, small four-arm spirals can radiate either lell-hand or right-hand polarised waves depending on the applied feed-voltages. In this case, however. the polarisalion purity of the radiation depends solely on the quality of the feed network. Consequently, the simultawus realisation of polarisation diversity very pure circular polarisation requires the implementation of highly optimised excitation arrangements.
5. CONCLUSIONS
The basic polarisalion properties of spiral antennas have been mnsidered by means of circularty polarised spherical waves. It has been shown that optimum conditions for the radiation of RHCP (LHCP) are obtained for directions close lo the antenna axis, if the CPSW spectrum of the anlenna contains the modes m=-1 (m=+l) and, simulatenously, the modes m=+l (m=-1) are avoided.
These condiiions can be fulfilled by symmetrical four-arm antennas, provided that the feed network permits a perfect control of the amlied excitation voltages. For practical cases, the total XPD of ihe antennas is delemined by the sum of a leed network term and a strudural term which
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POLARIZATION PROPERTIES OF SPIRAL ANTENNAS 283
accounts lor the specifi properties asscciated w%h the geometry of the antenna arms. For two arm antennas, the applied excitation does not support the suppression of one of the polarisation components. Therefore, the polarisation of a twoarm antenna depends exclusively on the specilii mcde selection properties determined by the wire geometry.
Apart from the general properlies of symmetrical multi-arm antennas, spirals exhibit specific mode selection properties which are associated with the travelling wave nature ot their antenna currents. ~eoendina on the windina sense of the soiral arms and on the size and excitation of the antenna sorile of l i e CPSW maies are not radiated. Because of this effect the 'preferred' polarisation sense of a larger spiral is always determinded by the winding sense of the antenna arms, whereas a small spiral can radiate both RHCP and LHCP.
Based on these considerations, general guidelines for the design of both two-arm and lour-arm spirals have been derived for different antenna requirements. Whereas a two-arm spiral has to be sufliciently large or requires specific modifications of the wire geometry in order to radiate very pure circular polarisation, a four-arm spiral offers the opportunity for trade-offs between an optimisation of the spiral geometry and an optimisation of the feed network. If the effect of the geometry has to be optimised, larger four-arm spiral antennas are required.
6. ACKNOWLEDGEMENT
The author wishes to thank the editors for the opportunity to present the basic results of his earlier work on circularly polarised antennas within one comprehensive paper. He also wishes to acknowledge the contributions of Prof. G. M6nich from the Technical University of Berlin. Germany, who stimulated the authors investigations, co-invented the 'polarisation map' and supervised the corresponding activities from 1985 to 1989. Finally. the author whishes to thank Alice Kran for her patience and encouragement during the busy time needed to prepare this contribution.
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