Lumped Elements for RF and Microwave Circuits phần 2 pot

Inductors 29

Figure 2.6 Spiral inductors and their coupled-line EC models: (a) circular 2 turns, (b) rectangu￾lar 2 turns, and (c) rectangular 1.75 turns.

3

I1

I2

I3

I4

4 =

3

Y11 Y12 Y13 Y14

Y21 Y22 Y23 Y24

Y31 Y32 Y33 Y34

Y41 Y42 Y43 Y44 43V1

V2

V3

V4

4 (2.20)

30 Lumped Elements for RF and Microwave Circuits

Figure 2.7 Rectangular 1.75-turn spiral inductor: (a) physical layout and (b) coupled-line EC

model.

Inductors 31

Figure 2.8 The network model for calculating the inductance of a planar rectangular spiral

inductor.

Figure 2.9 A four-port representation of the coupled-line section of an inductor.

This matrix can be reduced to two ports by applying the boundary condi￾tion that ports 2 and 4 are connected together:

V2 = V4 (2.21a)

I2 = −I4 (2.21b)

By rearranging the matrix elements, the two-port matrix can be written

as follows:

F

I1

I3

G = F

Y ′11 Y ′13

Y ′31 Y ′33

GFV1

V3

G (2.22)

where

32 Lumped Elements for RF and Microwave Circuits

Y ′11 = Y11 − (Y12 + Y14 )(Y21 + Y41 )

Y22 + Y24 + Y42 + Y44

(2.23)

Y ′13 = Y13 − (Y12 + Y14 )(Y23 + Y43 )

Y22 + Y24 + Y42 + Y44

(2.24)

and

Y ′33 = Y ′11 (2.25)

Y ′31 = Y ′13 (2.26)

due to symmetry.

The admittance parameters for a coupled microstrip line are given by [45]

Y11 = Y22 = Y33 = Y44 = −j [Y0e cot ue + Y0o cot uo ]/2 (2.27a)

Y12 = Y21 = Y34 = Y43 = −j [Y0e cot ue − Y0o cot uo ]/2 (2.27b)

Y13 = Y31 = Y24 = Y42 = j [Y0e csc ue − Y0o csc uo ]/2 (2.27c)

Y14 = Y41 = Y23 = Y32 = j [Y0e csc ue + Y0o csc uo ]/2 (2.27d)

where e and o designate the even mode and the odd mode, respectively.

An equivalent ‘‘pi’’ representation of a two-port network is shown in

Figure 2.10 where

YA = −Y ′13 (2.28)

YB = Y ′11 + Y ′13 (2.29)

and

Figure 2.10 Pi EC representation of the inductor.

Inductors 33

YA = −j

1

2 5

Y0e cot ue + Y0o cot uo (2.30)

+

FY0e S

1 − cos ue

sin ue D + Y0o S

1 + cos uo

sin uo DG

2 FY0e S

1 − cos ue

sin ue D − Y0o S

1 + cos uo

sin uo DG6

YB = 2jY0e Y0o (1 − cos ue )(1 + cos uo )

[Y0o sin ue (1 + cos uo ) − Y0e sin uo (1 − cos ue )] (2.31)

Because the physical length of the inductor is much less than l/4, sin

ue, o ≅ ue, o and cos ue, o ≅ 1 − u2

e, o /2. Also Y0o > Y0e ; therefore, (2.30) and

(2.31) are approximated as follows:

YA ≅ −j

Y0e

2ue

(2.32)

YB ≅ jY0eue (2.33)

which are independent of the odd mode. Thus the ‘‘pi’’ EC consists of shunt

capacitance C and series inductance L as shown in Figure 2.11. The expressions

for L and C can be written as follows:

YA = 1

jvL = −j

Y0e

2ue

(2.34)

or

L = 2ue

vY0e

(2.35)

Figure 2.11 Equivalent LC circuit representation of the inductor.