R. Ludwig and G. BogdanovRF Circuit Design: Theory and Applications

2nd edition

Figures for Chapter 2

Figure 2-1 Voltage distribution as a function of time (z = 0) and as a function of space (t = 0).

20

10

0

10

200 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0 20 40 60 80 100 120 140 160 180 200

V(z,

t)

20

10

0

10

20

V(z,

t)

z = 0

t = 0

t, s

z, m

Figure 2-2 Amplitude measurements of 10 GHz voltage signal at the beginning (lo-cation A) and somewhere along a wire connecting load to source.

Channel 1 Channel 2

A B

l

V

zz = l

VA

VB

VG

RG

R L

VA

0

V

0 lz

Figure 2-3 Partitioning an electric line into small elements over which Kirchhoffs laws of constant voltage and current can be applied.

z z + z

z + z

R1 L1

R2 L2G C

z

I(z) I(z + z)

V(z + z)V(z)+

+

z

Figure 2-4 Geometry and field distribution in two-wire parallel conductor transmission line.

2aD

Electric Field (solid lines)

Magnetic Field (dashed lines)

r

Figure 2-5 Coaxial cable transmission line.

2a

2b2c

r

Figure 2-6 Microstrip transmission line representation.

t

h w

(a) Printed circuit board section (b) Microstrip liner

Figure 2-7 Electric flux density field leakage as a function of dielectric constants.

(a) Teflon epoxy ( = 2.55) (b) Alumina ( = 10.0)r r

Figure 2-8 Triple-layer transmission line configuration.

(a) Sandwich structure (b) Cross-sectional field distribution

r r

Figure 2-9 Parallel-plate transmission line.

z

x

0(b) Field distribution(a) Geometric representation

Figure 2-10 Segmentation of two-wire transmission line into z-long sections suit-able for lumped parameter analysis.

G C

z

z

z + z

z + z

I(z) I(z + z)

V(z + z)V(z)+

+

G C

R1 L1

R2 L2

R1 L1

R2 L2

R1 L1

R2 L2G C

z

1

2

1

2

Figure 2-11 Segmentation of a coaxial cable into z length elements suitable for lumped parameter analysis.

+

+

z z + z

z z + z

I(z) I(z + z)

G C G CV(z + z)V(z)C

R1 R1 R1L1 L1 L1

R2 R2 R2L2 L2 L2G

Figure 2-12 Generic electric equivalent circuit representation.

+

_

+

_

z z + z

I(z) I(z + z)

V(z + z)V(z) C

R L

G

Figure 1-1 Ampres law linking the current flow to the magnetic field.

I

I

Line path lH(r)

H(r)

I H dl=

Figure 2-13 Magnetic field distribution inside and outside of an infinitely long wire of radius a = 5 mm carrying a current of 5 A.

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

r, mm

H, A

/m

r a=

=HIr

2pia2

=HI

2pir

Figure 2-14 The time rate of change of the magnetic flux density induces a voltage.

BE

Path l+

_

V

B

S

Figure 2-15 Parallel-plate transmission line geometry. The plate width w is large compared with the separation d.

x

yzd

dp

w

Table 2-1 Transmission line parameters for three line types

Parameter Two-Wire Line Coaxial Line Parallel-Plate Line

R/m

1piacond-----------------------

1piacond-----------------------

12picond-----------------------

1a---

1b---+

2

wcond--------------------

LH/m

pi---

cosh 1 D2a------

2pi------

ba---

ln d

w----

GS/m

pidiel

cosh 1 D 2a( )( )------------------------------------------

2pidielb a( )ln-------------------- diel

w

d----

CF/m

pi

cosh 1 D 2a( )( )------------------------------------------

2pib a( )ln--------------------

w

d----

Figure 2-16 Segment of a transmission line with voltage loop and current node.

R L

G C

z z + z

I(z) I(z + z)

V(z) V(z + z)+

_

+

_

a

Figure 2-17 Integration surface element for Faradays law application.

x

yz

J

Hy

I

z

Iw

d

z + z

ith cell plate 2

plate 1

Figure 2-18 Surface element used to apply Ampres law.

x

yz

I

z

Iw

d

z + z

ith cell plate 2

plate 1

Figure 2-19 Microstrip characteristic impedance as a function of w/h.

1

10

100

1k

Char

acte

ristic

impe

danc

e Z 0

,

0.1 100.3Line width to dielectric thickness ratio, w/h

= 1

= 12

= 4.6= 7= 10

= 2= 3

31

r

r

r

rr

r

r

Figure 2-20 Effective dielectric constant of the microstrip line as a function of w/h for different dielectric constants.

0.1 1 100.3 3Line width to dielectric thickness ratio, w/h

0

2

4

6

8

10

12

Effe

ctiv

e di

elec

tric

cons

tant

, ef

f

= 1

= 12

= 10

= 7

= 4= 3= 2

r

r

r

r

r

r

r

Figure 2-21 Effect of conductor thickness on the characteristic impedance of a mi-crostrip line placed on a 25 mil thick FR4 printed circuit board.

0.1 1 100.3 3Line width to dielectric thickness ratio, w/h

0

50

100

150

Char

acte

ristic

line

impe

danc

e Z 0

,

t = 0

t = 1.5 mil

h = 25 mil= 4.6

FR4

r

Figure 2-22 Terminated transmission line at location z = 0.

Zin

Z0 ZL

0z = lz

0

Figure 2-23 Short-circuited transmission line and new coordinate system d.

Zin

Z0 ZL = 0

0d = l

d

Figure 2-24 Standing wave pattern for various instances of time.

0 0.5pi1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

pi 1.5pi 2pi 2.5pi 3pi 3.5pid

t = 1/4pi + 2pin

t = 2pin

t = 1/2pi + 2pint = 3/8pi + 2pin

t = 1/8pi + 2pinV(

d)/(2j

V+)

Figure 2-25 SWR as a function of load reflection coefficient

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

|0|

SWR

0 .

Figure 2-26 Voltage, current, and impedance as a function of line length for a short-circuit termination.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

1.5

1

0.5

0

0.5

1

1.5

2

Short circuit

Open circuit

Short circuit

Open circuit

Short circuit

Zin(d)jZ0

V(d)2jV+

2V+/Z0I(d)

d/

Figure 2-27 Magnitude of the input impedance for a 10 cm long, short-circuited transmission line as a function of frequency.

1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

300

350

400

450

500

f, GHz

|Z in|,

Figure 2-28 Voltage, current, and impedance as a function of line length for an open-circuit termination.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

1.5

1

0.5

0

0.5

1

1.5

2

d/

Zin(d)jZ0

V(d)2V +

2jV+/Z0I(d)

Short circuit

Open circuit

Open circuit

Open circuit

Short circuit

Figure 2-29 Impedance magnitude for a 10 cm long, open-circuited transmission line as a function of frequency.

1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

300

350

400

450

500

f, GHz

|Z in|,

Figure 2-30 Input impedance matched to a load impedance through a line seg-ment.

/4

ZLZ0 = ZLZin

Zin = desired ZL = given

4

Figure 2-31 Input impedance of quarter-wave transformer.

ZlineZ0 = 50

l = /4

w

ZLZin

Figure 2-32 Magnitude of Zin for frequency range of 0 to 2 GHz and fixed length d.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40

45

50

f, GHz

|Z in|,

Figure 2-33 Generic transmission line circuit involving source and load terminations.

in

Z0 ZL

ZG

VG

S

out

L = 0

Figure 2-34 Equivalent lumped input network for a transmission line configuration.

VG

ZG

ZinV

Figure 2-35 Impedance of a coaxial cable terminated by a resistor: (a) network analyzer measurement, (b) theoretical prediction.

200 400 600 800 100020

30

40

50

60

70

80

90

100

f, MHz|Z in

|, (a) (b)

100

Figure 2-36 Network analyzer with the resistive 100 test load attached.