Lumped Elements for RF and Microwave Circuits phần 6 pot

Interdigital Capacitors 239

Figure 7.7 Interdigitated capacitor’s |S11 | and |S21 | responses.

7.2 Design Considerations

In this section we discuss several design considerations such as compact size,

high-voltage operation, multilayer structure, and voltage tunable capacitor.

7.2.1 Compact Size

The capacitor size can be reduced by reducing the dimensions of the structure

or by using a high dielectric constant value substrate. The achievable Q-value

and fabrication photoetching limit on the minimum line width and separation

dictate the size of the capacitor. For ceramic and GaAs substrates, these limits

are about 12 and 6 mm, respectively. It is well known that the wavelength of

a signal is inversely proportional to the square root of the dielectric constant

of the medium in which the signal propagates. Hence, increasing the dielectric

constant of the medium a hundred-fold will reduce the component dimensions

240 Lumped Elements for RF and Microwave Circuits

Figure 7.8 Interdigitated capacitor’s ∠S11 and ∠S21 responses.

Table 7.2

Physical Dimensions and Equivalent Model Values for Interdigital Capacitors

Physical Dimensions INDIG80 INDIG180 INDIG300 INDIG400 UNITS

Finger length, , 80 180 300 400 mm

Finger width, W 12 12 12 12 mm

Finger spacing, side, S 8888 mm

Finger spacing, end, S ′ 12 12 12 12 mm

Finger thickness, t 5555 mm

Number of fingers, N 20 20 20 20 mm

Substrate thickness, h 125 125 125 125 mm

Capacitance, C 0.126 0.252 0.405 0.527 pF

Inductance, L 0.001 0.025 0.064 0.101 nH

Resistance, Rdc 1.89 0.850 0.500 0.441 V

Shunt capacitance, Cs 0.028 0.052 0.080 0.104 pF

Interdigital Capacitors 241

Figure 7.9 The measured performance of an interdigital capacitor compared with the present

model and Touchstone model: (a) reflection coefficient and (b) transmission coeffi￾cient.

by a factor of 10. This simple concept is being exploited extensively as distributed

circuit technology is being adopted at RF and lower microwave frequencies.

7.2.2 Multilayer Capacitor

Gevorgian et al. [22] have reported closed-form expressions for interdigital

capacitors, on two- and three-layered substrates, using conformal mapping

242 Lumped Elements for RF and Microwave Circuits

technique. Figure 7.10 shows the interdigital capacitor configuration, and the

total capacitance is given by

C = C3 + Cn + Cend (7.12)

where C3 , Cn , and Cend represent the three-finger capacitance, capacitance of

the periodical (n − 3) structure, and a correction term for the fringing fields of

the ends of the strips, respectively. Closed-form expressions for these capacitance

components are given next.

C3 capacitance:

C3 = 4e 0 e e3

K (k ′

03 )

K (k 03 )

, (7.13)

Figure 7.10 (a) Physical layouts and (b) cross-sectional view of the interdigital capacitor.

(From: [22].  1996 IEEE. Reprinted with permission.)

Interdigital Capacitors 243

where , is the length of strip fingers and

e e3 = 1 + q 13

e 1 − 1

2 + q 23

e 2 − e 1

2 + q 33

e 3 − 1

2 (7.14a)

qi3 = K (ki3 )

K (k ′

i3 )

K (k ′

03 )

K (k 03 )

, for i = 1, 2, 3 (7.14b)

k 03 = S

S + 2g

√

1 − S (S + 2g)

(S + 2S1 + 2g)D

2

1 − S S

(S + 2S1 + 2g)D

2 (7.14c)

ki3 =

sinh S

pS

2hi

D

sinh S

p(S + 2g)

2hi D

?

√

1 − sinh2

F

p(S + 2g)

2hi G ⁄ sinh2

F

p(S + 2S1 + 2g)

2hi G

1 − sinh2

F

pS

2hi

G ⁄ sinh2

F

p(S + 2S1 + 2g)

2hi G

(7.14d)

and k ′

i3 = √1 − k 2

i3 , i = 1, 2, 3. In the preceding formulas, S1 = S should be

used where the widths of the external and middle fingers are the same.

Cn capacitance:

Cn = (n − 3)e 0 e en

K (k 0 )

K (k ′

0 )

, (7.15)

where

e en = 1 + q 1n

e 1 − 1

2 + q 2n

e 2 − e 1

2 + q 3n

e 3 − 1

2 (7.16a)

qin = K (kin )

K (k ′

in )

K (k ′

0 )

K (k 0 )

, for i = 1, 2, 3 (7.16b)