ERF L1 Introduction to Distributed Circuit Design...o Double-stub matching o Quarter-wave impedance transformer Radiofrequency Engineering 1 C. Collado, J.M. González-Arbesú EETAC-UPC

1. INTRODUCTION TO DISTRIBUTED CIRCUIT DESIGN

Microwave circuit elements and analysis• Frequency bands [COLLIN 1.1]• RF circuit analysis [COLLIN 1.3]

Transmission lines• Transmission line types• Propagation equations [COLLIN 3.1] [POZAR 3.1]

o lossless lineso lossy lineso low-loss lines

• Reflection coefficient [POZAR 3.3]• Power and losses: return loss [POZAR 3.3]• Voltage standing wave ratio [POZAR 3.3]• Impedance [POZAR 3.3]• Generator mismatch [POZAR 3.6]• Smith Chart [COLLIN 5.1]• Impedance matching [COLLIN 5.2-5.5]

o lumped elementso Single-stub matchingo Double-stub matchingo Quarter-wave impedance transformer

1Radiofrequency EngineeringC. Collado, J.M. González-Arbesú

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forward and reverse propagating waves

impedance of transmission lines

attenuation

1. INTRODUCTION TO DISTRIBUTED CIRCUIT DESIGN

Transmission line design• Balanced and unbalanced lines• Homogeneous and non-homogeneous lines• Coupled lines [POZAR 8.6]• Line design

Application notes• Coaxial cables• Connectors

2Radiofrequency EngineeringC. Collado, J.M. González-Arbesú

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[COLLIN] R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York)[POZAR] D.M. Pozar, Microwave Engineering, Addison-Wesley Publishing Company, 2nd Edition, 1993 (Reading, Massachusetts)

• : attenuation constant [m-1]• : phase constant [rad·m-1]• Cd : distributed capacitance per unit length [F/m]• 0 : electric permittivity of vacuum [8.85·10-12 F/m]• f0 : frequency [Hz]• : propagation constant [m-1]• G : distributed conductance per unit length [S/m]• i(z,t) : current in time domain [V]• I0

+ : current amplitude of progressive wave at z=0 [A]• l : transmission line length [m]• Ld : distributed inductance per unit length [H/m]• : wavelength [m]• 0 : magnetic permeability of vacuum [4·10-7 H/m]• : angular frequency [rad/s]• R : distributed resistance per unit length [/m]• T : period [s]

3

GLOSSARY

Radiofrequency EngineeringC. Collado, J.M. González-Arbesú

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• RS : surface resistivity [/square]• S : skin depth [m]• : conductivity [Sm]• d : dielectric conductivity [Sm]• tan : loss tangent [adim]• RL : return loss [dB]• : (voltage) reflection coefficient [adim]• G : (voltage) generator reflection coefficient [adim]• IN : (voltage) reflection coefficient at input port [adim]• L : (voltage) load reflection coefficient [adim]• v(z,t) : voltage in time domain [V]• VG : voltage at generator [V]• V0

+ : voltage amplitude of progressive wave at z=0 [V]• vp : phase velocity [m/s]• VSWR : Voltage Standing Wave Ratio [adim]• ZG : generator impedance []• ZIN : impedance at the input port of the transmission line []• ZL : load impedance []• Z0 : transmission line characteristic impedance []

4

GLOSSARY


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5

MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS Frequency bands

• International classification of the frequency bands.

• Radar classification of frequency bands. The old one (WW II) is still widely used.

[Tables taken from: R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York)]


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MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis

• At frequencies where is several orders of magnitude larger than the greatest dimension of the circuit or system: To transmit, receive, and/or process data the basic building blocks are

capacitors, inductors, resistors, and transistors. Loop currents and node voltages are enough to analyse the circuits. To analyse the circuits no propagation effects have to be considered: the delay

in the propagation of signals at different points in the circuit is negligible compared with the period of the applied signal. Lumped circuit models are valid.

• At microwave frequencies is compared with the circuit dimensions and: Propagation effects can not be ignored: there is a delay in the propagation of

signals among different points in a circuit. There are distributed capacitances and inductances in the circuit. There is an increase in the impedance of terminals and connectors. Unshielded circuits with dimensions compared with become effective radiators. Distributed circuit models are used.


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• Wave propagation along a line considering the propagation delays:

• Some remarks: Wave propagation along a line considering the propagation delays. Each point in the line has a different voltage/current at the same time t. Periodicity in time or period T. Spatial periodicity or wavelength . Dimensions use to be defined with respect to .

0

2

T00

2fv

v

tAtvG 0cos Source voltage:

vztAtzv 0cos ,

Voltage:

t = 0 L: line length

v: propagation speedvG(t)

z = 0 z

propagation delay


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Example: Propagation at Low Frequencies. Consider a circuit having a transmission line length of 0.003 (3 lines, 0.001-length each) fed with a sinusoidal wave of 1 GHz.


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2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V1,

mV

2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V2,

mV

2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V3,

mV

2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V4,

mV

V4

TranTran1

MaxTimeStep=1.0 psecStopTime=10.0 nsec

TRANSIENTRR1R=50 Ohm

RR2R=50 Ohm

VtSineSRC1

Delay=0 nsecFreq=1 GHzAmplitude=1 VVdc=0 V

TLINTL2

F=1 GHzE=0.36Z=50.0 Ohm

TLINTL3

F=1 GHzE=0.36Z=50.0 Ohm

TLINTL1

F=1 GHzE=0.36Z=50.0 Ohm

V1 V2 V3

9


Example: Propagation at High Frequencies. Consider a circuit having a transmission line length of 15 (3 lines, 5-length each) fed with a sinusoidal wave of 1 GHz.


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2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V1,

mV

2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V2,

mV

2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V3,

mV

2 4 6 80 10

-400

-200

0

200

400

-600

600

time, nsec

V4, m

V

V4

TLINTL1

F=1 GHzE=1800Z=50.0 Ohm

TLINTL3

F=1 GHzE=1800Z=50.0 Ohm

TLINTL2

F=1 GHzE=1800Z=50.0 Ohm

TranTran1

MaxTimeStep=1.0 psecStopTime=10.0 nsec

TRANSIENTRR1R=50 Ohm

RR2R=50 Ohm

VtSineSRC1

Delay=0 nsecFreq=1 GHzAmplitude=1 VVdc=0 V

V1 V2 V3

10

TRANSMISSION LINES Transmission line types

• Transmission lines are physical devices whose purpose is to guide electromagnetic waves (carry RF power) from one place to another.

• They are capable of guiding TEM waves (TEM waves can only exist in structures containing two or more separated conductors).

• Two-wire transmission lines are inefficient for transfering electromagnetic energy at high frequencies due to the lack of confinement in all directions.

• Coaxials are more efficient than two-wire lines in those cases.

shielded pair

two-wire ribbon line(twin lead)

twisted pair

flexible coaxial line

air coaxial line

[Images from: http://www.techlearner.com/Apps/TransandGuides.pdf]


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• There are lots of planar structures used as transmission lines. Metallic parts are supported by dielectrics (fiberglass, ceramics, foams,...).

microstrip

stripline

coplanar transmission line


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• Waveguides are the most efficient. They are fabricated with just one conductor. Waveguides do not support TEM waves.

• Two-wire lines are less bulky and less expensive than waveguides.

waveguides

[Image from: http://www.techlearner.com/Apps/TransandGuides.pdf]


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TRANSMISSION LINES Propagation equations

• Field modes: electromagnetic fields configurations supported by a structure.• A coaxial transmission line supports a TEM mode (electric field orientation, magnetic

field orientation, and energy propagation direction for a triad).

electromagnetic field distribution(TEM mode)

E

H

b

a

L

physical structure

z

abstract model(ideal transmission line)

L

z

+

-

tzi , tzv ,

ldtzHtzi

ldtzEtzv

,,

,,


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TRANSMISSION LINES Propagation equations: lossless lines

• The knowledge of voltage and current waves propagating along the transmission line allows the use of a distributed circuit model to analyze its performance.

• The model represents an infinitesimally short segment of the transmission line• This model is convenient to explore properties of lines without knowing the fields in

detail. However, the structures should be analyzed in detail if accurate performances have to be known.

+

-

tzi ,

tzv ,

+

-

tzzv ,

tzzi ,

z 0 that casein or zdz

• Ld and Cd are the distributed inductance [H/m] and capacitance [F/m] associated to the coaxial structure and materials. No losses are assumed in this example (meaning that there is no distributed resistance).

CL

zCCzLL

d

d

+

-

tzi ,

tzv ,+

- tzzv ,

tzzi ,


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• Applying Kirchhoff’s voltage and current laws:


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CL

zCCzLL

d

d

+

-

tzi ,

tzv ,+

- tzzv ,

tzzi ,

tzzit

tzzvzCtzi

tzzvt

tzizLtzv

d

d

,,,

,,,

• Dividing by z and taking the limit z 0:

t

tzvCz

tzit

tziLz

tzv

d

d

,,

,,

• Considering sinusoidal steady-state condition (cosine based phasors) (TRANSIENTS NOT CONSIDERED):

VCjzI

ILjzV

d

d

16


• The wave equations can be solved simultaneously:


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IzI

VzV

-

-

22

2

22

2

being: ddCL

phase constant

• Solutions are: zjzj

zjzj

eIeIzI

eVeVzV

00

00being:

0

00

0

00

ZVI

ZVI

and: d

d

CLZ 0

characteristic impedance

• Meaning that there are waves propagating in opposite directions along the transmission line (positive and negative waves).

V V

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• Wave propagation solutions in time domain are:

• Wavelength and phase velocity on the line are:

ddCLfv 1 2

p

• Do not forget… that the electrical model parameters depend on line geometry.

H

E

physical structure

d

d

CLdistributed

model parameters 0Z

electrical model

parameters

0000

0000

argcosargcos,

argcosargcos,

IztIIztItzi

VztVVztVtzv

the wave propagation on the line means a delay (in fact a phase delay)

lengths l are given in terms of or in degrees (l)

vp does not change with frequency: NO DISTORTION

(each frequency component of a signal travels at the same vp

along the line)

18

Take your time…Characteristic impedance. Given the solutions of the voltage and current wave

equations derived for a two-wire lossless transmission line differential equations:

Find the characteristic impedance of the line and the relation between the voltage and current waves.


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VCjzI

ILjzV

d

d

zjzj

zjzj

eIeIzI

eVeVzV

00

00

being: ddCL

19

Take your time…Solution: Characteristic impedance.


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ILjzV

d

zjzjd

zjzj eIeILjeVjeVj 0000

Sustituting the solution for the current into the differential equation for the voltage:

zjzj

zjzj

eIeIzI

eVeVzV

00

00

0

00

0

0000

ZVI

ZV

CL

VL

VI

d

dd

characteristic impedance

20

Take your time…Lossy two-wire transmission line. In previous slides we presented the model of a two-

wire lossless transmission line. Suggest a model for a lossy two-wire transmission line (e.g. lossy coaxial cable).


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dielectric insulator (lossy) metallic shield and core (lossy)

Losses in metals are characterized by their electrical conductivity (units: -1m=Sm) or by their surface resistivity RS (units: /square). RS represents the mean power absorbed by a unit area (1 m2) and can be used on planar surfaces or surfaces with curvature radius smaller than the skin depth S. Currents on metals flow mainly in their outer “skin” an a level called skin depth.

Losses in dielectrics are characterized by their loss tangent tan . It represents the tangent of the angle in the complex plane between the electric field resistive losses and its lossless reactive component.

SSR 1

fS 1

'''tan

d

Hints:

21

Take your time…Solution. Lossy two-wire transmission line.


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dL

dCG

R

z

depends on and/or RS

depends on tan S

22

Take your time…Telegrapher’s equations. Derive the propagation equations for a lossy two-wire

transmission line having the lumped-circuit model of the figure. The resulting set of differential equations are called the telegrapher’s equations and are due to Oliver Heaviside in 1880.

Ld distributed inductance per unit length [H/m]Cd distributed capacitance per unit length [F/m] R distibuted resistance per unit length, for both conductors [/m]G distibuted conductance per unit length, for both conductors [S/m]


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[Imag

e fro

m:

http

://es

.wik

iped

ia.o

rg/w

iki/O

liver

_Hea

visi

de]

dL

dCG

R

z

23

Take your time…Solution: Telegrapher’s equations.


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• According to Kirchhoff’s laws:

tzzit

tzzvzCtzzvzGtzi

tzzvtzizRt

tzizLtzv

d

d

,,, ,

,, ,,

• Dividing by z and taking the limit z 0:

t

tzvCtzvGz

tzit

tziLtziRz

tzv

d

d

,, ,

,, ,

Telegrapher’s equations

tzi ,

tzv ,+

-

tzzv ,

tzzi ,

+

-

dL

dCG

R

z

24

TRANSMISSION LINES Propagation equations: lossy lines

• From the lumped-circuit model of a lossy two-wire transmission line we get the telegrapher’s equations.


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t

tzvCtzvGz

tzit

tziLtziRz

tzv

d

d

,, ,

,, ,

• Considering sinusoidal steady-state condition (cosine based phasors):

VCjGzI

ILjRzV

d

d

tzi ,

tzv ,+

-

tzzv ,

tzzi ,

+

-

dL

dCG

R

z

25


• Wave equations are now:


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IzI

VzV

22

2

22

2

zzzz

zz

eVeVZ

eIeIzI

eVeVzV

000

00

00

1 being: d

d

CjGLjRZ

0

• The solutions are:

• And in time domain:

0000 argcosargcos, VzteVVzteVtzv zz

being: dd CjGLjRj

complex propagation

constant

attenuation phase constant

vp depends with frequency: DISTORTION (each frequency component of a signal

travels at different vp along the line)

26


• Lumped-circuit parameters required to model some common lines as a function of their dimensions, surface resistivity (RS), and materials filling the space between the conductors (permitivitty = ’-j’’ and permeability = 0r).

• Here follows a table showing some classical transmission line:


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[Table from: [POZAR]]

parameters for other transmission lines are found in the literature or computed by using specific software

27

TRANSMISSION LINES Propagation equations: low-loss lines

• In practical lines losses are small. The equations for attenuation and propagation factor can be simplified.


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The complex propagation factor can be re-arranged:

dddd

dddd CLRG

CG

LRjCLjCjGLjRj 21

For a low-loss line we can assume: and dLR dCG

dddd

dddd C

GLRjCLj

CG

LRjCLj

21 1

Finally:

d

d

d

d

CLG

LCR

21 ddCL

d

d

CLZ 0

d

d

d

d

CL

CjGLjRZ

0 same values than lossless lines

DISTORTIONLESS

28

TRANSMISSION LINES Reflection coefficient

• Wave reflection on a transmission line can be illustrated by considering a lossless transmission line loaded with an arbitrary impedance ZL. Z0 is the characteristic impedance of the transmission line.


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• A voltage reflection coefficient can be defined for any point in the line as the amplitude of the reflected voltage wave normalized to the amplitude of the incident voltage wave.

zVzVz

• Because at the load (z=0) the impedance of the line is ZL: 0

00

00

00 Z

VVVV

IVZL

0

00ZZZZ

L

LL

voltage reflection coefficient at load

zjeVV 0zjeVV 0

VG

ZG=Z0

ZL

z=-l z=0

Z0

29


• One has to be careful with the coordinate axis chosen to define the reflection coeficient along the line.


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zVzVz

zjeVV 0

zjeVV 0VG

ZG=Z0

ZL

z=-l z=0

Z0

z=0 z=l

by definition:

0

0

0

00ZZZZ

VV

L

LL

coefficient at load: 0

02

0

0

ZZZZe

VVl

L

LljL

ljIN e

VVl 2

0

0

coefficient at input port:

0

00VV

IN

ljLIN e 2 in both cases:

however: zjLez 2 zlj

Lez 2

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• Because at the input port of the line (z=-l) the impedance is:

lj

Llj

lj

lj

IN eeeVeV

lVlVl

22

0

0 0

ljLIN e 2

voltage reflection coefficient at input port

• At any point in the line: zjL ez 2

• Reflection coefficient is a complex number.• For a passive load the magnitude of the reflection coefficient is always lower than 1.

10

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ljlLl

l

IN eeeVeV

lVlVl

22

0

0

The reflection coefficient is attenuated when the line increases its length.

Example: Reflection coefficient of a loaded lossy transmission line. Taking into consideration the equations of the voltage and current waves flowing in a lossy transmission line find the equation for the input impedance on the line.

32

Take your time…Standard loads. Which is the reflection coefficient corresponding to, respectively, an

open circuit, short circuit, and reference impedance (Z0)?


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Line transition. Is the voltage reflection coefficient the same at both sides of the transitions between the transmission lines?

LZL 0LZL0ZZL

L

1 2

'0Z 0Z

33

Take your time…Mismatched lossless transmission line. By appliying boundary conditions to the ports

of the lossless transmission line of the figure, find the voltage reflection coefficients INand L, and the magnitude of the progressive wave V0

+.


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zjeVV 0zjeVV 0VG

ZGZL

z=0 z=l

Z0

34

Take your time…Solution. Mismatched lossless transmission line.


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zjeVV 0zjeVV 0

VG

ZGZL

z=0 z=l

Z0

ljljLL

ljlj eVeVZZZIeVeVlVlV 00

000 0

voltage:

ljL e

VV

lVlV 2

0

0

defining the load reflection coefficient L…

0

0

ZZZZ

L

LL

… we get:

• at the output port of the line:

• and at the input port:

000

000

00

IZ

VZ

V

ZIVVV GG

voltage and current:

defining IN…

ljLIN e

VV 2

00

…we get:

GINGG ZZ

ZVV

11

0

00

and G…

0

0

ZZZZ

G

GG

35

TRANSMISSION LINES Power and losses: return loss


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• To avoid the existence of reflected waves (=0) on a loaded transmission line the load impedance ZL should be equal to the characteristic impedance Z0. Such a load is said to be a matched load.

• The time-average power flow along the line at point z is:

PPZ

VZ

VzIzVPav

0

2

0

0

2

0*

22Re

21

• Incident power to the line:0

2

0

2ZV

Pinc

• Reflected power at the load: 2

0

2

0

2 Lref ZV

P

2

0

2

0 12

zZ

VPav

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• When the load is mismatched not all the available power is the delivered to the load. This loss is called return loss (RL):

dB log20 LRL

• Some considerations derive from the previous equations:

,00

L

L

ZZZ

0

0

L

L

PPP

PPP

• The power carried by the positive flowing wave can be greater that the average power flowing on the line

power on the load

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• What happens in case of a lossy transmission line?

For a forward propagating wave in a transmission line with length l:

z

z

eVZ

zI

eVzV

00

0

1 lePIVlP 2* 0Re

21

incoming power at input port

attenuation

Wave attenuation (L) between planes separated l:

in decibels [dB]:

in Nepers [Neper]:

lellP

PL 686.8log20 0log10dB

llP

PL

0ln21Nepers

1 Neper = 8.686 dB

38

Take your time…Power losses: dB and Nepers. A RG59/U coaxial cable has an attenuation of 39.3

dB/100m (this means 39.3 dB each 100 m of cable) at 1 GHz. Find the attenuation in Nepers when having 200 m of cable, and the value of the attenuation coefficient (in Np/m).


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Take your time…Solution. Power losses: dB and Nepers.


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Attenuation in dB for 200 m of cable:

dB 6.78m 200 mdB 393.0dB

L

Np 05.9

NpdB .6868

dB 6.78Np

L

Attenuation in Nepers for 200 m of cable:

mNp 045.0

m 002Np .059

Attenuation coefficient:

40

Take your time…Feeding and antenna. A 50- antenna is fed with a signal of 100 W by means of a 75

cable. Find the power finally arriving to the antenna.


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a) 4 W b) 20 W c) 80 W d) 96 W

Phasors. Find the phasor of the current flowing along the open transmission line of the figure.

VG

ZG

z=0 z=l

Z0

a) lzejVzI lj sin2 0

lzeZVjzI lj

sin20

0c)

zZVzI cos2

0

0

b)

lzZVzI

cos20

0d)

When the transmission line is matched, the magnitude of the voltage on the line is constant:

When the transmission line is not matched the overlap of an incoming and a reflected wave leads to a standing wave whose magnitude oscillates with the position on the line:

41

TRANSMISSION LINES Voltage standing wave ratio


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• The voltage on each point of a line depends on the load attached at its end:

zjLeVzV 2

0 1

0VzV

LVV 1 0max

LVV 1 0min

• A measure of the line mismatch is the Voltage Standing Wave Ratio (VSWR):

L

L

VVVSWR

11

min

max

• SWR is a real positive number: VSWR1

10

L

L

VSWRVSWR 1matched load

total reflection

42

TRANSMISSION LINES Voltage standing wave ratio


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Example: Voltage on a transmission line. Plot the normalized voltage and current on a transmission line 2 in length as a function of position on the line (z). Consider the following loads: L=0,0.2,0.4,0.6,0.8,1 and L=0.8, 0.8 ej90, 0.8 ej180, 0.8 ej270 .

zjLe

VzV 2

0

1

zjLe

IzI 2

0

1

0

90

180270

180

270

090

decreasing L

decreasing L

43

TRANSMISSION LINES Impedance


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• The impedance in every position (z) of the line is:

zj

L

zjL

zjzj

zjzj

eeZ

eVeVeVeVZ

zIzVzZ

2

2

000

000 1

1

• The input impedance of a line loaded with ZL is:

ljZZ

ljZZZljZlZljZlZZ

lIlVZ

L

L

L

LIN

tantan

sincossincos

0

00

0

00

INZ ZL

z=-l z=0

Z0

L zZ

z

INZ

z=-l z=0

Z0

zZ

z

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ljZZIN tan0

ljZZIN tan

0

short-circuited stub:

open-circuited stub:

inductive impedance

capacitive impedance

• The case of highly reflective loads:

• Quarter-wave long transmission lines perform as quarter-wave (impedance) transformers or impedance inverters, being the input impedance inversely proportional to the load impedance.

LIN Z

ZZ20,...2,1,0

24 nnl

When:

• Otherwise, any line whose length is any multiple of /2 does not transform the load impedance, regardless of the characteristic impedance.

LIN ZZ ,...3,2,1 2

nnl When:

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Example: Emulating lumped elements with electrically-short transmission lines.Suggest what lumped elements can be simulated by means of short transmission lines when these line end, respectively, with an open circuit and with a short circuit.

short-circuited short-stub:

CjljZZIN

10

open-circuited short-stub:

being:0Zv

lCp

INZ Z0

l

INZC

LjljZZIN 0 being:pvlZL 0 INZ Z0

l

INZL

46



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Example: Input impedance of a loaded lossy transmission line. Taking into consideration the equations of the voltage and current waves flowing in a lossy transmission line, find the equation for the input impedance on the line.

zzzz

zz

eVeVZ

eIeIzI

eVeVzV

000

00

00

1

lZlZ

lZlZZZL

LIN

sinhcoshsinhcosh

0

00

ZIN lZZ tanh0

0

1tanhZZL

Z

INZ ZL

z=-l z=0

Z0

L zZ

zat z=-l

that is, also

47

TRANSMISSION LINES Generator mismatch


EETAC-UPC

• The voltage on the input port of the line can be calculated from the source voltage and depends on the load impedance.

IN

ININ Z

IVZ

11

00

0and being:

INGIN

ING VVV

ZZZVV

10 000at z=0:

GINGG ZZ

ZVV

11

0

00

0

0

ZZZZ

G

GG

being:

VG

ZG

ZL

z=0 z=l

Z0

LIN

ZIN

G

48

TRANSMISSION LINES Generator mismatch


EETAC-UPC

• Consequently:

the power delivered from the source to the line (and if it has no losses, delivered to the load): IN

GIN

Gav Z

ZZVIVP Re

2100Re

21

2*

22

2

2 GINGIN

INGav XXRR

RVP

being: INININ jXRZ and: GGG jXRZ

when: 0ZZG

0ZZIN

20GVV

GG ZZ

ZVV

0

00

Power delivered to the load. Consider a lossless transmission line connected to a fixed source impedance ZG=RG+jXG. Find the power delivered to the load (or the power delivered to the transmission line) in the following two cases: when the load is matched to the line (ZL=Z0); and when the generator is matched to the loaded line (ZG=ZIN).

49

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EETAC-UPC

Solution: Power delivered to the load.

50

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EETAC-UPC

• Load matched to the line (ZL=Z0).

220

02

2 GG

Gav XRZ

ZVP

• Generator matched to the loaded line (ZG=ZIN).

22

2

8 GG

GGav XR

RVP

Impedance for maximum power transfer or available power. Assuming that the generator series impedance is fixed, find the input impedance ZIN to achieve the maximum power transfer to the load (lossless transmission line). In that case, find the power delivered to the load.

51

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EETAC-UPC

Solution: Impedance for maximum power transfer.

52

Take your time…


EETAC-UPC

0

0

IN

av

IN

av

XPRP

00222

INGIN

INGING

XXRXXRR

GIN

GIN

XXRR *

GIN ZZ

• Maximizing the power transfer…

… we get that the input impedance of the line should be the complex conjugate of the source impedance. This condition is called conjugate matching.

• In this case, the power transferred to the load is (lossless transmission line):

G

Gav R

VP

8

2

53

TRANSMISSION LINES Smith Chart


EETAC-UPC

• A graphical tool very helpful when dealing with impedance transformation and matching network design is the Smith Chart (invented by Phillip H. Smith in 1939 while working for Bell Telephone Laboratories). Because working with (almost) infinite values for resistances and reactances is usual in Microwaves, a

graphical plot of these impedances is not practical in a rectangular coordinate system. However, operating with the reflection coefficient associated to a given normalized passive impedance (impedance Z with possitive resistance normalized to the characteristic impedance of the line Z0) leads to a graphical representation of the loads inside a circle of unity radius.

• The transformation between impedances and reflection coefficients leads to a Z-chart:

• The transformation between admitances and reflection coefficients leads to a Y-chart.

11

11 z

zz

0ZZz being:

11

11 y

yy

00

YZYYy being:

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EETAC-UPC

• In the Z-(Smith) Chart, lines of constant reactance map into circumferences.

r

x

Z-Chart xjrz

11

zz

11z

0x

0x

impedances corresponding to

active loads

f g h i

f g h i

abcde

a

b

c

d e

Re

Im

55



EETAC-UPC

• In the Y-(Smith) Chart, lines of constant susceptance map into circumferences

Y-Chart g

b

bjgy

0b

0b

yy

11

11y

abcde a b

c

d

e

f g h i

fg

hi Re

Im

56



EETAC-UPC

[Images from: http://upload.wikimedia.org/wikipedia/commons/d/df/Smith_chart_explanation.svg]

• Z-Smith chart: main loads and their correspondence with the reflection coefficient.

57



EETAC-UPC

transmission line length in wavelengths

(periodicity /2)

phase of reflection coefficient in degrees

(periodicity 180)

shift towards LOAD

shift towards GENERATOR

• Z-Smith chart.

ljLIN e 2

angle

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EETAC-UPC

50 5050

0ZjZ

Example: Impedance on Smith Chart. Plot the load ZL=50+j50 on the Smith chart (reference impedance 50 ). Indicate the value of the reflection coefficient, the return loss, and the VSWR.

jz 1

63.5º

Graphically from the Smith Chart:

dB 7RLdB 8.5VSWR 2.6 VSWR

63.5º)angle( 46.0

Numerically from equations:

dB 99.6log20RL

º4.63 45.011 je

zz

dB 36.862.211

VSWR

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EETAC-UPC

Example: Loaded coaxial. A 2 cm-length coaxial operating at 3 GHz is loaded with ZL=50+j50 . Use the Smith chart to find the input impedance seen from the coaxial transmission line. Consider the dielectric constant of the line is 2.56.

Graphically from the Smith Chart:

32.0·103

56.2·103 02.0 8

9

c

flfv

ll r

p

jz 1Normalized load impedance (point A):Transmission line electrical length:

0.32 from load to generator

A

B

Normalized input impedance (point B): 10.041.0 jz Unnormalized input impedance: 0.55.20 jZin

Numerically from equations: º20.115360 ll

j

ljZZljZZZZ

L

LIN 9.43.19

tantan

0

00

60

TRANSMISSION LINES Impedance matching


EETAC-UPC

• Though the Smith chart is useful to avoid tedious computations involving complex numbers, nowadays this is not a problem thanks to the widespread use of calculators and computers.

• The main advantage of using Smith chart is that it allows drawing conclusions without requiring complex calculations.

• Smith chart is very useful when designing matching networks. Several alternatives are possible when trying to match a load (or a generator) to the reference impedance of a transmission line. Some of them are: Impedance matching with lumped elements. Single-stub matching. Double-stub matching. Triple-stub tuner. Quarter-wave impedance transformer.

• In a practical design of a matching network the technology to be used and the frequency bandwidth of the solution should be considered.

• Conjugate matching can also be treated by using the Smith Chart.

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EETAC-UPC

• When designing matching networks use to be helpful employing CAD tools.

• An unlicensed tool (with limited functionalities) called Smith (designed by Prof. Fritz Dellsperger and Michael Baud from Bern University) is an example.

[Image from: http://www.fritz.dellsperger.net/]

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EETAC-UPC

• Impedance matching using reactive elements is desirable due to the absence of losses.

• The matching solution depends on the technology to be used for its implementation.

A

BL

moving CW or CCW a load along a constant-R circle is equivalent to add

a L or a C in series, respectively

-1/(C)

DZ-Smith chart

moving CCW or CW a load along a constant-G circle is equivalent to add a

L or a C in shunt, respectively

A

D

B -1/(L)

C

Y-Smith chart

Z-chart Y-chart

L

C

63

TRANSMISSION LINES Impedance matching: lumped elements


EETAC-UPC

• Impedance matching using two lumped elements. ZL inside the circle can be matched using shunt-series elements.xj1

CL

0Z LZCL0Z LZ

YL inside the circle can be matched using series-shunt elements.bj1

C

L0Z LZ

CL0Z LZ

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EETAC-UPC

1C0Z LZ

2C1C

0Z LZ2C

1C0Z LZL C

L0Z LZ

ZL outside the circle and YL outside the circle can be matched using series-shunt and shunt-series elements.

xj1 bj1

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EETAC-UPC

• Impedance matching using three lumped elements networks (only some possible solutions are shown).

2CL0Z LZ1C

3C0Z LZ1C

2C

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EETAC-UPC

• Impedance matching using four lumped elements networks (only some possible solutions are shown).

• When designing matching networks, remember that shorter paths in the Smith chart provide a wider operational bandwidth.

2C

2L0Z LZ

1C

1L

4C0Z LZ2C

3C 1C

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EETAC-UPC

• Lumped elements (should be smaller than /10) used to design the matching networks can be:

Capacitors: chip capacitors, MIM capacitors, interdigital capacitors, open-circuited stubs.

Inductors: chip inductors, loop inductors, spiral inductors, short-circuited stubs.

Resistor: chip resistors, planar resistors.

• Lumped elements have parasitics in the microwave range.• The standard units used when describing the size of a lumped element is the mil:

1 mil=0.001 in=25 m=1/40 mm

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[Images from [POZAR]]

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EETAC-UPC

Example: Matching a dipole. Consider a dipole with input impedance 82+j45 and operating at 2.45 GHz. Consider all the possibilities of matching the dipole to the line using a two-lumped elements network when fed with a 50 transmission line. Solve the problem analytically and check the results using the application Smith.exe.

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EETAC-UPC

Example: Matching a monopole. Consider a monopole with input impedance 30+j20 operating at 2.45 GHz. Consider all the possibilities of matching the monopole to the line using a two-lumped elements network when fed with a 50 transmission line. Solve the problem using the application Smith.exe. Make a frequency sweep from 2 to 3 GHz and decide which solution has a better bandwidth (assume that the monopole impedance does not change in the proposed frequency bandwidth).

worst RL is around 20 dB

worst RL is around 14 dB

70

TRANSMISSION LINES Impedance matching: single-stub matching


EETAC-UPC

• Single-stub matching. A stub is a short-circuited or open-circuited section of a transmission line. There are two alternatives: single-shunt-stub and single-series-stub. There are two parameters to adjust: distance l from load to stub and the shunt impedance.

ZL

l

Z0Z0 Z0

ZL

l

Z0Z0

Z0

ZL

l

Z0Z0

Z0

ZL

l

Z0Z0 Z0

ZL

d

Z0Z0 Z0 Z0

71

TRANSMISSION LINES Impedance matching: double-stub matching


EETAC-UPC

• Double-stub matching.

Forbidden region where loads can’t be matched.

d

d

d

ZLZ0Z0 Z0 Z0

d

d

Single-stub tuning can be a problem for a variable matching network due to the variable line length l between the load and the stub. A double-stub with fixed separation d between stubs and variable stub lengths is used to

solve this problem. Unfortunately, there is a region where impedances can’t be matched (itcan be tuned out by adding a certain line length). Although double-shunt-stub tuning is shown, double-series-stub tuning is also possible.

72

TRANSMISSION LINES Impedance matching: triple-stub tuner


EETAC-UPC

• Triple-stub tuner. A triple-stub tuning network does not have regions where loads can not can be

matched. This networks has more degrees of freedom (three variable stub lengths) in order

to increase the bandwith of the tuner.

ZLZ0 Z0

d

Z0 Z0Z0 Z0

d

ZLZ0 Z0

d

Z0 Z0Z0 Z0

d

73

TRANSMISSION LINES Impedance matching: quarter-wave transformer


EETAC-UPC

A resistive load RL can be matched to a transmission line with reference impedance Z0 by means of a /4 section of a transmission load having a reference impedance of Z0’: 0

'0 ZRZ L

• Quarter-wave impedance transformer.

If the load is not purely resistive a series or paraler reactive element (lumped element or transmission line section) should be added to make it purely resistive before including the quarter-wave transformer.

ZL

l

Z0

/4

Z0’

ZL=RL +jXL

/4

Z0’ -jXL

YL=GL +jBL

/4

Z0’ -jBL

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EETAC-UPC

Example: Matching a monopole with a microstrip single-stub network. Consider a monopole with input impedance 30+j20 operating at 2.45 GHz. Consider the possibility of matching the monopole using a single-stub network made with microstrip technology (avoid using vias) and a Rogers Duroid 4003C substrate (r=3.55). Solve the problem using the application Smith.exe.

High impedance matching. Which of the following options are possible to match an impedance of 188 to a 50 transmission line?

75

Take your time…


EETAC-UPC

Z0=50 188 Matching Network

Z0’=97

/4

C

L CL

LL

a) b) c) d)

76

TRANSMISSION LINE DESIGN Balanced and unbalanced lines


EETAC-UPC

• A balanced line is a transmission line having two conductors with the same voltage magnitude but a phase shift of 180º with respect to ground. Impedance of both conductors is equal with respect to ground.

• The balanced and unbalanced character of transmission lines has to be accounted for proper connection to circuits or devices.

• An example of a balanced line is a twin-lead line, whereas an unbalanced line is a coaxial cable.

• The transition from a balanced to an unbalanced structure, or viceversa, requires a BALUN transformer.

• BALUNS are used to connect balanced to unbalanced lines or structures. They are required irrespective of transmission line technology.

• Baluns are usually narrowband devices. It is difficult to design wide bandwidth baluns.

V/2 -V/2V

0 Volts

UNbalancedBALanced

0 Volts

77

TRANSMISSION LINE DESIGN Homogeneous and non-homogeneous lines


EETAC-UPC

• Homogeneous dielectric media are uniform in all points and its physical properties are unchanged. A transmission line in a homogeneous medium has a propagation velocity that

depends only on material properties (dielectric permittivity r and magnetic permeability r). The principal wave existing in these kind of transmission lines could be a TEM

(transversal electromagnetic) wave.

• Non-homogeneous media contain multiple materials with different dilectric constants. Wave propagation velocity in non-homogeneous transmission lines depends

on material properties and structure dimensions An effective r,eff dielectric constant is often used to represent an average

dielectric constant. These line do not propagate pure TEM modes.

78

TRANSMISSION LINE DESIGN Homogeneous and non-homogeneous lines


EETAC-UPC

• Cross sections of some common transmission lines:

microstrip

embedded microstrip

coupled microstrip

coplanar

coplanar strip

non-

hom

ogen

eous

centered stripline dual stripline

coaxial shielded two-conductor

two-conductor

circular WGrectangular WG

hom

ogen

eous

79

TRANSMISSION LINE DESIGN Coupled lines


EETAC-UPC

• Coupled lines are balanced-type lines.

• These structures are analyzed by means of an odd and an even excitation mode that superpose. In the even mode the currents in the strip conductors are equal in amplitude

and flow in the same direction. In the odd mode the currents in the strip conductors are equal in amplitude

and flow in opposite directions. Each strip conductor is characterized by its

characteristic impedance (relative to ground) and its propagation constant. Both parameters are different for the excited

modes.

eZ0

oZ0

e0

o0 odd mode parameters

even mode parameters

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e fro

m: [

PO

ZAR

]]

80

TRANSMISSION LINE DESIGN Line design


EETAC-UPC

• Homogeneous lines propagate purely TEM modes and have analytical equations helping to match the characteristic impedance and propagation constants of the lines to given specific requirement.

• Non-homogeneous lines do not propagate purely TEM modes (but under certain circumstances they propagate quasi-TEM modes). The characteristic parameters of the line have to be numerically computed or other approximate techniques have to be used.

http://web.awrcorp.com/Usa/Products/Optional-Products/TX-Line/

• Nowadays, CAD software helps the designer... (e.g. TX-Line Calculator of AWR, LineCalc part of ADS of Agilent Technologies).

81

TRANSMISSION LINE DESIGN Line design


EETAC-UPC

• TX-Line Calculator of AWR is a license free software for transmission line design.

82

APPLICATION NOTES Coaxial cables


EETAC-UPC

• The coaxial cable was invented by Oliver Heaviside at the end of the 19th century for transmitting telegraphic signals.

• The first propagating mode in a coaxial cable is TEM. It has a cut-off frequency of zero.• Next mode is TE11. Its cut-off frequency depends on the mean radius between the

conductors and the material filling this gap.• The characteristic impedance of the cable also depends on the geometry of the cable

section and the materials.

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electric field distribution inside a coaxial

TEM mode TE11

rrinout

TEc rr

f

22

111

dDZ

r

r ln21

0

83

APPLICATION NOTES Coaxial cables


EETAC-UPC

• Characteristic impedance, frequency bandwidth, attenuation, wave speed of propagation, and maximum power handling capability should be carefully considered when adquiring a cable for a given application.

[Table from SSi Cable Corp.]

TNC

84

APPLICATION NOTES Connectors


EETAC-UPC

• Some 50 impedance connectors typically used in RF and microwave equipment. Others exist.

Warning! RP connectors(RP: reverse polarity)

11 (18) GHz 11 GHz 18 GHz2 GHz

BNC N

18 (26.5) GHz

SMA

APC-7

6 GHz

MCX

6 GHz

MMCX

40 (45) GHz

K BNC: Bayonet Neill-ConcelmanMCX: Micro CoaXialMMCX: Micro-Miniature CoaXialN: Neill connectorTNC: Threaded Neill-ConcelmanAPC-7: Amphenol Precission Connector

with 7 mm diameterSMA: SubMiniature version AAPC-3.5: Amphenol Precission

Connector with 3.5 mm diameterSMK: SubMiniature version K (also

called 2.92 mm)

26.5 (34) GHz

3.5

Maximum power transfer. Two complex impedances, ZS=(25-j15) and ZL=(100-j25) ,have to be matched by means of a LC network in order to have maximum power transfer between them. Calculate the values of the inductance and capacitance at 100 MHz using a reference impedance of 25 .

85

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EETAC-UPC

ZLZS

Impedance matching. Use the Smith chart and an operating frequency of 1 GHz to find the capacitance and the transmission line length l that matches an impedance of ZL=(100-j200) to 50 by means of the network of the figure.

86

Take your time…


EETAC-UPC

Smith chart. A 50 lossless bifilar transmission line is loaded with an impedance of (100-j200) . Line length is 2.25 m and the frequency of the propagated harmonic wave is 100 MHz.

a) Find the input impedance of the line using the Smith chart.b) Find the input impedance using the analytical equation that provides the input

impedance of a transmission line as a function of its length, load and its electrical model parameters.

c) Calculate the power dissipated at the load when the available power is 100 W.

87

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EETAC-UPC

Documents

ERF L1 Introduction to Distributed Circuit Design...o Double-stub matching o Quarter-wave impedance transformer Radiofrequency Engineering 1 C. Collado, J.M. González-Arbesú EETAC-UPC