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7.2 Design and practice of ESA	97

Centerline
21 mm	1
	1		Matching			A: Feeding point
	Proposed		Matching			A: Feeding point
	Proposed		section 2	A	B	B: Shorting point
15	meandered loop			t = 14		0.8
	antenna			t = 14		0.8
	antenna		4			7
27			4			7
				w = 4.5		1
				w = 4.5
6	13.5
7	System ground			Matching		Bending line
	System ground			section 1
	(45	100 mm2)		section 1
				1 mm
	50	microstripline				1
	on back side		2.5	2.5		1
						1

(a)				2	(b)
(a)					(b)

1 mm thick plastic housing (εr = 3.5, σ = 0.02 S/m)

Ground plane length L = 100

Proposed antenna

h =7 mm

0.8 mm thick FR4 substrate

3 mm

(c)

Figure 7.8 A MLA having a loop structure (from [8], copyright C 2006 IEICE).

7.2.1.1.1.1.3 Folded-type meander line antenna

As can be seen in Figure 7.5, Q of a meander line antenna becomes higher as the size becomes smaller, meaning that the smaller the antenna size becomes, the narrower the bandwidth will be. In order to increase the bandwidth, a folded type can be useful [10], since a folded structure has substantially two modes, balanced and unbalanced modes, and by arranging the susceptances of these two modes appropriately, bandwidth can be increased. Figure 7.9 is an antenna model, constituting a folded structure with two meander lines, which are connected at the top. The folded structure is decomposed into two parts; balanced and unbalanced modes as shown in Figure 7.10 [11], in which

(a)shows the folded structure, on which currents I1 and I2 ﬂow on each element, and

(b)illustrates decomposed modes; the unbalanced mode (the current Iu) plus balanced mode (the current Ib). The unbalanced mode current Iu equals (I1 + I)2/(1 + γ ), and the balanced mode current Ib equals (γ I1 – I2)/(1 + γ ), where γ stands for the ratio of the unbalanced current on the element 1 and that on the element 2 [10]. This folded model is

equivalently expressed by a circuit shown in Figure 7.11, in which the circuit parameters such as the source voltage V, currents I1, the decomposed mode currents Iu and Ib, and the impedances for both modes, Zu (unbalanced) and Zb (balanced), are provided.

98	Design and practice of small antennas I

	d
Element 1	w
Element 1
Element 2	t
	t
h	p
Ground	Ground
Ground
plane	y

	x	Feed
	x	point
		point

Figure 7.9 A folded antenna model with two meander lines (from [10], copyright C 1999 IEICE).

Element

1	2		1	2		1	2
					γ Iu
I1		I2	Iu			Ib		Ib
I1		I2	Iu			Ib		Ib

	(a)					(b)

Figure 7.10 Currents on a folded structure and on the decomposed balanced and unbalanced modes ([10], copyright C 1999 IEICE).

			Impedance
			step up ratio
			2 : 1
I	Ib	Iu
	1
V		2Zb	Zu

Figure 7.11 Equivalent circuit corresponding to a folded structure ([10], copyright C 1999 IEICE).

7.2 Design and practice of ESA	99

By using these circuit parameters, Zu and Zb respectively, are given by
n2 Zu = V/Iu	(7.6)
and
Zb = V/(2Ib)	(7.7)

where Iu = (I1 + I2)/(1 + γ ), Ib = (γ I1 – I2)/(1 + γ ), and γ denotes a ratio of unbalanced currents on the element 1 and 2, and n = 1 + γ .

From (7.6), the unbalanced impedance n2 Zu = V/In2, and is given by using radiation resistance R, antenna Q, and resonance frequency f0 as

n2 Zu = R + j Ru						(7.8)
where u = Q( f / f0 − f0/ f ) = Q(2 f )/ f0 ( f = f − f0).						(7.9)
Admittance (unbalanced) Yu is expressed from (7.8) as
Yu = 1/(n2 Zu ) = Gu + j Bu						(7.10)
and Gu = 1/{R(1 + u2)}
Bu = −u/{R(1 + u2)}						(7.11)
In the same way, the balanced mode admittance Yb is
Yb = 1/Zb = jBb						(7.12)
and Bb = −B0u/Q.						(7.13)
Here B0 is deﬁned by using a parameter K as
	B0 = K Q.					(7.14)
Then Bb = K u.						(7.15)
Now the total susceptance Bt = Bu + Bb will be
1			u		+ K u.	(7.16)
Bt = −
	R	(1	+	u2)

From (7.11) and (7.14), when u is very small, Bu approaches 1/R, and by selecting an appropriate value for K, Bt can be made to be zero. The reﬂection coefﬁcient of the line is given by

(u) = {Y0 + Yi (u)}/{Y0 − Yi (u)}.

(7.17)

(u) is a function of u, and Y0 is the characteristic admittance of the line. The voltage standing wave ratio (VSWR) S as function of (u) is

S(u) = {1 + (u)}/{1 − (u)}.

(7.18)

100	Design and practice of small antennas I

	3						R = 25
							R = 25
							R = 40
							R = 50
	2.5						R = 75
	2.5						R = 100
							R = 100
VSWR	2
VSWR
	1.5
						um
		K = 0.01
	1−3	−2	−1	0	1	2	3
				u

Figure 7.12 VSWR characteristics in relation to parameter u with variation of input resistance R ([10], copyright C 1999 IEICE).

VSWR

5
4			K = 0.05
					0.001
3
					0.005
2					0.01
2

	R = 25 Ω
1
1	−2 −1	0		1	2	3
−3	−2 −1	0		1	2	3

Figure 7.13 VSWR characteristics in relation to parameter u with variation of K (from [10], copyright C 1999 IEICE).

Now desired relative bandwidth RBW is expressed by using bandwidth	fm speciﬁed at
a required value of Sm and um that correspond to Sm as
RBW = 2 fm / f0	(7.19a)
= um /Q.	(7.19b)

Then, (7.19b) shows that increase of RBW is possible by setting um at its maximum value.

S with respect to u is illustrated in Figure 7.12, where R is taken as a parameter, and in Figure 7.13, where K is used as a parameter, respectively. Design parameters can be known from Figure 7.14 and Figure 7.15, in which relationships between Sm and K, and

7.2 Design and practice of ESA	101

0.015

0.01

0.005	2	2.5	3	3.5
1.5

VSWR

	3.5
	3
	3
max	2.5
	2.5

u	2


	1.5
	1.5
	1
	1	2	2.5	3	3.5
	1.5	2	2.5	3	3.5

VSWR

Figure 7.15 Relationship between umax and Sm (from [10], copyright C 1999 IEICE).

Sm and um, respectively, are provided. RBW in relation to Sm is given in Figure 7.16, where Q is used as the parameter.

In practical designs, one of these antennas may be decomposed into two modal parts: balanced and unbalanced modes. The balanced mode can be treated equivalently as a two-wire transmission line of length L with characteristic impedance Z0. Here the length L is that of the meander line from the feed point to its end. The input susceptance Bb of this line is written as

Bb = −(1/Z0) cot(2π L/λ).

(7.20)

By using the wavelength λ0 at the resonance,

Bb = −(1/Z0) cot{(2π L/λ0)(1 + f / f0)}.

(7.21)

102	Design and practice of small antennas I

	30
	25
[%]	20		Q = 10
	20
				20
max	15			20
	15

RBW	10			30
RBW	10			50
				50
	5			100
				100
	0	2	2.5	3	3.5
	1.5	2	2.5	3	3.5
			VSWR Sm

Figure 7.16 Relationship between Qmax and Sm (from [10], copyright C 1999 IEICE).

]	40
[Ω			L/λ0 = 0.75
c			L/λ0 = 0.75
Z
impedance	30
	30		Q = 10
Characteristic	20
	20
	10		20
	10		20
			50
	0
	0.005	0.01	0.015

Figure 7.17 Relationship between Zc and K with variation of Q (from [10], copyright C 1999 IEICE).

L can be written by using number n of the resonance mode as

L = (2n − 1)λ0/4(n = 1, 2, 3, . . .).	(7.22)
Then, Bb = (1/Zt ) tan{(2π L/λ0)( f / f0)}	(7.23)
= K u.	(7.24)
Here K = (1/Z0)(L/λ0)/Q	(7.25)

where the approximate expression taking only the ﬁrst-order variable for tangent is used. From this, K can be determined by selecting Z0 and L/λ. Z0 in relation to K is expressed by Figure 7.17, where Q is taken as the parameter. By using the meander line parameters w (width) and a (wire radius) in terms of t (separation of two lines), which are given in

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