2006 INTERNATIONAL RF AND MICROWAVE CONFERENCE PROCEEDINGS, SEPTEMBER 12 - 14, 2006, PUTRAJAYA, MALAYSIA
The Bode-Fano Integrals as an Objective Measure of Antenna Bandwidth
Reflection Coefficient Product Limit
A. Ghorbani 1, M. A. Ansarizadeh 2, N. J. McEwan 3, Raed A. Abd-alhameed 4, D. Zhou
12Amirkabir University of Technology, Tehran, Iran.,
3'4 Bradford university, Bradford BD7 lDP, UK
ghorbani@aut.ac.ir, ansarizade@hotmail.com
nj mcewangbradford. ac. uk,
r.a.a.abd@bradford.ac.uk
Abstract -The Bode-Fano integral is proposed as an
objective tool for assessing the bandwidth of antennas
and special schemes for antenna bandwidth
improvement. The limiting gain-bandwidth product
is a measure of optimum power transfer from source to
loads. For loads represented in Darlington canonical
form, optimum tolerance can be calculated using the
Bode-Fano theory. In this paper for the first time the
Fano limit has been calculated for loads with up to
four reactive elements. It was note that even though
increasing the substrate height of coaxially feed
microstrip patch antennas will increase the antenna
bandwidth; however this is only true up to some
specific heights and afterwards the antenna potential
bandwidth decreases for future increase in substrate
height. Also for double resonance circuits it was
concluded that the maximum potential bandwidth is
obtained if the two resonant frequencies are the same.
Keywords: bandwidth-reflection coefficient product; BodeFano ;double resonant antennas
1. Introduction
The increasing scale of radio communications,
especially digital mobile systems using advanced
spread spectrum modulation techniques, now makes it
very demanding to design antennas with sufficient
bandwidth. Problems are particularly compounded in
mobile or portable consumer applications where
antennas are increasingly required to be
multifunctional and multiband, and yet there is
enormous pressure to reduce their physical size. Many
researchers are now addressing these issues and few
techniques for achieving more bandwidth have been
made [1-3].
The Bode-Fano integral is an objective tool for
assessing the bandwidth of antennas and specially
schemes for bandwidth improvement [4], As a result,
the maximum achievable bandwidth- reflection
coefficient product for antennas can be estimated using
0-7803-9745-2/06/$20.00 (©)2006 IEEE.
11 ohm
E
Fig. 1: load model in the gain-bandwidth analysis.
I ohm
1 ohm
L
matching
III
InetworkI
I
reactive
network
Fig. 2: load and matching network.
the Bode-Fano theory if an equivalent circuits for
antennas are provided. Bode first treated this problem
[5], his work includes load structures with single
reactive elements, Fano [6] extended the work of Bode
to cover loads that can be represented in Darlington
canonical form. Later, Youla [7] continued the work of
Fano for loads in its full generality. The above authors
modeled the generator and load by a voltage source in
series with a pure resistance and a lossless network
terminated in a unit resistance respectively, as shown
in Figi.
For loads with single reactive element Bode and
Fano obtained analytical formulas which is related to
the time constant of the network, but in general, this
limit has to be calculated numerically if the number of
reactive elements is more than one. As an example,
Fano obtained optimum tolerance numerically for a
second order low-pass structure.
The equivalent circuit of antennas such as dipole
microstrip, consists of more than one reactive
element. Recently some authors have considered the
Bode-Fano integral for single resonance antennas [8,
4]; however wideband antennas usually manifest
multiple resonance behaviors so it is interesting to
or
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apply the Bode-Fano theory in order to find the
maximum potential bandwidth versus the reflection
coefficient for antennas with dual resonances, such as
U and E-shaped antennas. In section 1 the basics of the
Fano theory is presented, in section 2 this theory is
used to obtain the limits for antennas with two adjacent
resonant frequencies and in section 3 conclusions are
made.
3. Application of the Bode-Fano theory to
antennas
(a)
(b)
2. Fano Gain-bandwidth Theory
With refer to Fig. 2 and applying the Fano theory,
the best possible match over desired frequency band is
if
obtained
we
the
function
expand
F(s)=ln(l/pl(s)) where s =T+ j
is the
complex frequency variable, at the zeros of
transmission (ZOT) of N', which are usually at zero
or infinity for antennas. The Taylor series expansion of
F (s ) in 1/s and s respectively, leads to:
AI
s +A s +...
ln (1, p (s))= r jA +
jA0° +A Os +A 0s33 +...
(c)
(d)
Fig. 3: (a) 2nd order low-pass, (b), (c) single resonant with
inductive and capacitive feed respectively, (d) double
resonant circuit.
(1)
3.1 First order lowpass network
If the antenna equivalent can be represented as a
simple first order network, such as a simple shunt RC
/LR load, then the first order Bode-Fano integral is:
+00
fInll/pl(jw) dc = r/(RC) -TyA, (4)
The Taylor series coefficients are obtained from
the equivalent circuit (in our case from the antenna
model). Usually Af and A 0 are equal to 0 or zT
0
jo2
depending on the sign of arg (p1 (j)) . An0 and
Af must be real since p1 (1w) is an even and
arg(p (jc)) is an odd function of c . Fano
showed the a ZOT of N' of order k is also a ZOT
of N with the same order, therefore the first 2k
Taylor series coefficients of p1 (s ) at the ZOT, are
determined exclusively by the load N' independently
of the matching network N", except in the
degenerative case (i.e. the right hand element of N' is
the of the same type as the left hand element of N" ),
where only first 2k -1 coefficients are determined
exclusively by N', same is true for ln(I/p1(s))
and its derivatives. With this assumption, the first 2k
Taylor series coefficients are:
A n0 = (II W)d nln(Ilp(s))Id (11S)n
A
00
n
=
(II W)dnIn Ilp(s) Ids
(2)
nlp/p1(jwo) dc
rTL -;Ty ";.
i
R
(5)
i
Where p1 is the reflection coefficient of the over-all
network and Ari are the zeros of p1 (s ) which occur
in the right-half plane (RHP). These equations state
that the area under the curve lnl/ P' max can not
exceed a value fixed by the normalized capacitor or
inductor. If there are zeros of p1 (s ) in RHP, the area
will decrease accordingly. It is therefore evident that
the optimum tolerance is achieved when
lpI Pconst within the frequency band c1 < c < )2
and |P1 = 1I otherwise. Thus (4), (5) leads to:
OK A=AlO
(6)
Klct =A10
(7)
where K is defined by:
n
(3)
K =(2/7T)Iln(l/p1)
max
(8)
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And cc is the upper frequency of the passband.
Utilizing equations (2) and (3) yields:
A °° =2/C
(9)
A10 2L
(10)
(t) /Ao') K- -3A3 / Al °)
+2 (Tr IA o
4
0.06
0 09.
V
3.2 Second order lowpass network
Fano extended the analysis to cover a second
order network and showed that as the number of the
ZOT of N'increases, additional restrictions in the
form of simultaneous nonlinear equations are imposed.
Writing Fano equations for the network shown in Fig.
(3.a), we obtain:
2y Ari
K = -3A °° + 2y Ai
COK = A,' '
3
2/ C1
Now
0.2
.- - - - - - - - - - -
0.6
0.4
Wc/A1 Inf
After some mathematical manipulation, (15) and
(16) leads to the following third order equation:
K3 -3( Al /w)c K2 +(4 +3(A
Al
L2-2(A3/(Al ))+1J/(Al lc)
ifA3 < 0 andWc /AX
<
)
) )K
0 (17)
Having solved (17) the optimum value of K for
the network in Fig. (3.a) versus normalized bandwidth
is obtained, Results are plotted in Fig. 4 for the
(14)
parameter-AA) / (A )3.
3A /(Aj)
then K is still given by equation (6), which indicates
that presence of L2 does not limit the bandwidth of
the first order lowpass circuit. Otherwise, it is
necessary to introduce RHP zeros. Generally speaking,
the problem is optimized when minimum number of
(RHP) zeros of p1 are introduced [5]. To solve (11)
and (12) two degrees of freedom are necessary with
minimum RHP zeros. If we choose c / AJ as the
independent parameter then we have only control
over K, so another degree of freedom is created by
introducing a single real RHP zero rr this will
optimize the value of K. If we chose a pair of
complex conjugate zeros, it would create two degrees
of freedom resulting in an underdetermined system,
therefore (1 1) and (12) becomes:
(c /XAj)K = ]-2 (Ur /AO)
1
(12)
(13)
(Cl /L2 -1/3)/4
0.8
Fig. 4: optimum matching exponent for a second order
lowpass load.
And
A3 / (A1 )
0
- - - - - - - - - - -
(11)
i
Applying equation (3) to Fig. (3.a) yields:
A
(16)
(15)
With refer to Fig. 4 one concludes that as far as
L2 is less than Cl 44(oc /AX ) + I /3J then the
optimum bandwidth is controlled only by Cl,
otherwise, the bandwidth of the antenna is dictated by
both Cl and L2 .
3.3 Parallel tuned network
In the case of a parallel tuned circuit (a single
parallel tuned network can model most simple
antennas in the vicinity of their resonant frequency),
again writing the Fano gain-bandwidth equations, we
have:
(w2 -c1) K Alj _ 2 Y2ri
(18)
(01-I -c -1 )K :A10 -2 A
(19)
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A1 is still given by (13), applying (2) to Fig. (3.b)
we have:
A10= 2L2
(20)
If no RHP zeros of p1 exists, the maximum value of
K is achieved and is given by:
K=l/bw
(21)
bw =(C)2 - )1)I/AX
(22)
1=/(2C1 ) = A
=
/A1
(23)
Thus, the normalized bandwidth becomes:
W)2
B =
(w)2
-
1/
W1
+c,1)/2
+(KQ)
(24)
Where Q is the quality factor of the parallel resonant
circuit. For KQ >> then (24) reduces to:
bw =2 -
=
coo
I / (KQ)
(25)
Equation (21) reveals that in the absence of RHP
zeros, presence of L2 does not change the normalized
bandwidth; it only shifts the position of the passband
along the real frequency axis. The presence of the RHP
zeros reduces the potential bandwidth and RHP zeros
may be introduced if the load resistor is larger than the
source resistor.
3.4 Parallel tuned circuit in series with an
inductor
The antenna equivalent circuit can be modified by
the series inductorL3 to account for the antenna feed
effect as shown in Fig. (3.b). In this case the Fano
gain-bandwidth equations are:
After some mathematical manipulation, we observe if:
bw <(1/2) 3C1/L3 1
(30)
a single tuned network (antenna), Otherwise, L3 is too
large and limits the optimum bandwidth of a single
It is frequently mentioned in the literature that
increasing the substrate height of single patch
microstrip antennas will increases the antenna
bandwidth, our results supports this idea, however it
should be emphasized that there is limitation on the
amount of increase in bandwidth due to substrate
height increase. Since for large heights, the value of
would
become
more
than
L3
Ci
/L4(cc lAo
+
If (30) does not hold then (26)-(28) are optimized
by introducing a pair of complex conjugate RHP zeros
in p1 thus above equations become:
,
-~o
C6200TCtlC)260
09' ( 0T ' ) + 3 Ct) Ct2
2ar3 cos 30
) Al
(602 - 't /
= Al0Al(
(27)
where
(28)
Arl = Ar2 = Aj"Oa.e'j
0
1
w1w2
K
3
A.°°
+
(32)
(c2 -c1t ) K =-3Aj + 2Z Ai23
c2-1 K :A1 -2
(31)
((co2 -)/AC)K =1 -2acos0
(26)
_
1/3J, as a result the potential
bandwidth is reduced. This is an important conclusion
obtained from the Fano theory in antenna design and
manufacturing. Also having got the maximum limit on
L3the maximum height of the substrate can be
calculated for optimal bandwidth operation.
(co2 -c1,) K = Al -2y ri
(co-1
(29)
tune network.
Also (18) and (19) implies that:
B
3A /(A1)=
[c1 (L2+L3)/(L2L3)-1/3]/4
holds, L3 neither limits nor increases the bandwidth of
Where
w2.w,
Applying (3) to Fig. (3.b) yields:
-
(21/a ) cos 0 (33)
/(A1 )
(34)
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and denotes complex conjugate. Equations (31) and
(33) will become identical if we choose:
4
*
A,-A
a =1/
01
0.03
(35)
Substitution of (35) into (31) and (32) yields:
K3 -3/bwK2 +(1+3/bw2)K+
1
7
3KAljOA1
-
IJ/bw3
(36)
0
0
D
02
04
06
Wc/A1 Inf
08
1
(a)
4A
_
.
Having solved (36), the optimum value of K for
.-A -
0.06
desired values of- A /(AJ versus bw are plotted
in Fig. (5).
Form this figure one can easily calculate the potential
limit of antenna by knowledge of the substrate height.
-X
-
-_
3.5 Parallel tuned circuit in series with a
capacitor
Most antennas at low frequency behave as open
circuit. Therefore for better representation and
modeling a series capacitor should be introduced
which results in the equivalent circuit similar to Fig.
(3.c). Following the same procedure as in section 2.4
the Fano gain-bandwidth equations are:
(CO21-,) K = A1 -2 Ari
3
4
002
06
04
Wc/A1 Inf
(-I
2-1
(b)
(37)
0O12~
i
(0)1-3_-0) -3 )K = -3A3° + 2y, Ari-3
(39)
0.2
where
0.6
0.4
Wc/A1 Inf
0.8
(c)
-A3O / (A1o) = [(C1 + C3)IL2 -1/3]/4
(40)
V>3C3 /L2-1 C3 does
limit the bandwidth of the tuned section and the
again if bw < (1 / 2)
o.1
(38)
Al-+2y yri-1
1
0.03 \ oo '009
-
)K =
08
,
not
matching network may start with a degenerate
capacitor. Otherwise, C3 is too small and to achieve
optimum tolerance a pair of complex conjugate RHP
zeros must be introduced. The result are plotted in
Fig. 5 by replacing -A3 /(Al )3 with -A3) /(Al)3
.
Fig. 5: (a, b, c) optimum tolerance versus the normalized
bandwidth for various values of
-A,-/(Aj)
0, 0.06, 0.l5,andparameter
I / Aj°Al°°
3.6 Inductor and capacitor in series with
Parallel tuned circuit.
In general case, antennas equivalent circuits either
dipole or microstrip can be presented in the form of
Fig (3.d). The Fano gain-bandwidth limitation for the
above load can be presented as:
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-2
(wO2 -c)KAKA4)3
(41)
=
_')
A1K0
o
i 1
(42)
2 ri3
(2o3 1c) K =-3A3j + 2y
(43)
(co -3 j)c-3 K =-3A30+ 2
(44)
co
=
yri-3
Solving (41) - (44) numerically shows the optimum
tolerance is obtained when the series and parallel
resonant frequencies are the same. (i.e. L1C2 = L3C4
), which is equivalent to:
A3 /(A1
A3 /(A10)
(45)
whereAf /(AJ) and -A3 /(A1 )are given by
(29) and (40) respectively. Therefore, if (45) is
satisfied then the optimum gain-bandwidth is given by
Fig. (5).
On the other hand, if (45) is not met, the presence of
L3 and C4will generally reduce the potential
bandwidth. Even though there are some exceptions,
such as when C4 is large, then the degenerative
element can reduce its value, therefore its possible to
make resonant frequencies the same so the optimum
tolerance will be reached. But if C4 is too small since
we have half control over the element (i.e. can be
reduced) therefore optimum tolerance would not be
reached. Also in the cases that:
If
A ° /(A A°
o/(A °)
)) >
L3 L1C2 /C4
3. Conclusions
The Bode-Fano theory can be usefully applied to
calculate the limiting gain-bandwidth product for
antennas. It can also be used as an objective measure
for antenna bandwidth definition.
In this paper, it was shown that antennas
bandwidth is controlled mainly by A
which is
related to its time constant. For single resonant circuit
the bandwidth is determined by the capacitor Cl and
maximum tolerance can be easily reached unless the
antenna is over coupled/under coupled. In which the
degree of coupling controls the potential bandwidth. If
antenna equivalent circuit in addition to a parallel
resonant circuit have a series inductor or capacitor
within a specific range, then again the optimum
tolerance can be reached otherwise the presence of
these elements can reduce the potential bandwidth and
finally for cases which the antenna equivalent circuit
consists inductor as well as capacitor in series with the
parallel resonant circuit the optimum bandwidth can be
reached if the parallel and series resonant frequencies
are the same but if they are not equal the existence of
L3 and C4 will reduce the potential band width.
References
[1] H. A. Wheeler, "Small antennas", IEEE Trans.
Antennas Propagat. , VOL. Ap-23, pp 462-469,
JULY 1975.
[2] A. K. Shackelford, et al, "Design of small-size
wide-bandwidth microstrip patch antennas", IEEE
trans. Antenna and Propagat., VOL, 45, NO. 1,
February 2003.
[3] F. Yang, "Wideband E-shaped Antennas for
(46)
else
C4 L1C2 IL3
Therefore, in these cases with the new values L3
and C4 then optimum tolerance can be reached. As
shown in Fig. 5. Finally our calculation shows that as
long as
L1C2 = L3C4 and
ifL3 <3C1/ 1+4 ((w2 -
Cnuori
I)/A )J, or
C4 >L2 (4/K2 +1)/3
Hold then the presence of L3 and C4 do not limit
maximum potential bandwidth of the parallel tuned
network; which is uniquely determined by Cl .
wireless communications", IEEE Trans. Antennas
Propagat., VOL. 49, pp 1094-1100, JULY 2001.
[4] A. Ghorbani, "An approach for calculating the
limiting bandwidth-reflection coefficient product
for microstrip patch antennas", IEEE Trans.
Antennas and Propagat., Vol. 54, NO.4, APRIL
2006
[5] H. W. "Bode, Network Analysis and Feedback
Amplifier Design", Van Nostrand, New York,
1945, sec. 16.3.
[6] R. M. Fano, "Theoretical limitations on the
broadband matching of arbitrary impedances",
MIT technical report 41, January, 1950.
[7] D. C. Youla "A new theory of broadband
matching" IEEE Trans. Circuit theory, VOL. CT11, pp. 30-50, Mar. 1964.
[8] A. Hujanen, et al, "Bandwidth limitations on
impedance matched ideal dipoles ", IEEE Trans.
Antennas and Propagat., VOL. 53, NO. 10,
October 2005.
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