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RF Communication Circuits. Ali Fotowat-Ahmady. Office hours: Sat Mon 1:30pm to 2pm Sat Mon 6:00pm 6:30pm Telephone contact: 0912-111-7364 (= RFMi ) Calls OK in reasonable hours! Technical Q’s on SMS OK 24 hours! Please DO NOT KNOCK ON MY DOOR BEFORE MY CLASSES!

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RF Communication Circuits

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Rf communication circuits

RF Communication Circuits

Ali Fotowat-Ahmady

Office hours: Sat Mon 1:30pm to 2pm

Sat Mon 6:00pm 6:30pm

Telephone contact: 0912-111-7364 (=RFMi)

Calls OK in reasonable hours!

Technical Q’s on SMS OK 24 hours!

Please DO NOT KNOCK ON MY DOOR BEFORE MY CLASSES!

Do the HW’s every week and U will do well.


Rf communication circuits1

RF Communication Circuits

Lecture 2: Transmission Lines

Dr. Fotowat-Ahmady

Sharif University of Technology

Fall-1391

Prepared by: Siavash Kananian & AlirezaZabetian


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Waveguiding Structures

A wave guiding structure is one that carries a signal (or power) from one point to another.

There are three common types:

  • Transmission lines

  • Fiber-optic guides

  • Waveguides


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Transmission Line

Properties

  • Has two conductors running parallel

  • Can propagate a signal at any frequency (in theory)

  • Becomes lossy at high frequency

  • Can handle low or moderate amounts of power

  • Does not have signal distortion, unless there is loss

  • May or may not be immune to interference

  • Does not have Ez or Hz components of the fields (TEMz)

Twin lead

(shown connected to a 4:1 impedance-transforming balun)

Coaxial cable (coax)


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Transmission Line (cont.)

CAT 5 cable

(twisted pair)

The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities.


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Transmission Line (cont.)

Transmission lines commonly met on printed-circuit boards

w

er

h

er

h

w

Microstrip

Stripline

w

w

w

er

h

er

h

Coplanar waveguide (CPW)

Coplanar strips


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Transmission Line (cont.)

Transmission lines are commonly met on printed-circuit boards.

Microstrip line

A microwave integrated circuit


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Fiber-Optic Guide

Properties

  • Uses a dielectric rod

  • Can propagate a signal at any frequency (in theory)

  • Can be made very low loss

  • Has minimal signal distortion

  • Very immune to interference

  • Not suitable for high power

  • Has both Ez and Hz components of the fields


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Fiber-Optic Guide (cont.)

Two types of fiber-optic guides:

1) Single-mode fiber

Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength.

2) Multi-mode fiber

Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).


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Higher index core region

Fiber-Optic Guide (cont.)

http://en.wikipedia.org/wiki/Optical_fiber


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Waveguides

Properties

  • Has a single hollow metal pipe

  • Can propagate a signal only at high frequency: >c

  • The width must be at least one-half of a wavelength

  • Has signal distortion, even in the lossless case

  • Immune to interference

  • Can handle large amounts of power

  • Has low loss (compared with a transmission line)

  • Has either Ez or Hz component of the fields (TMzorTEz)

http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)


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Transmission-Line Theory

  • Lumped circuits: resistors, capacitors,inductors

neglect time delays (phase)

  • Distributed circuit elements: transmission lines

account for propagation and time delays (phase change)

We need transmission-line theory whenever the length of a line is significant compared with a wavelength.


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Dz

Transmission Line

2 conductors

4 per-unit-length parameters:

C= capacitance/length [F/m]

L= inductance/length [H/m]

R= resistance/length [/m]

G= conductance/length [ /m or S/m]


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+ + + + + + +

B

- - - - - - - - - -

x

x

x

Transmission Line (cont.)

RDz

LDz

i(z,t)

i(z+z,t)

+

+

v(z+z,t)

CDz

v(z,t)

GDz

-

-

z


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Transmission Line (cont.)

RDz

LDz

i(z,t)

i(z+z,t)

+

+

v(z+z,t)

CDz

v(z,t)

GDz

-

-

z


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TEM Transmission Line (cont.)

Hence

Now let Dz 0:

“Telegrapher’sEquations”


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TEM Transmission Line (cont.)

To combine these, take the derivative of the first one with respect to z:

Switch the order of the derivatives.


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TEM Transmission Line (cont.)

Hence, we have:

The same equation also holds for i.


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TEM Transmission Line (cont.)

Time-Harmonic Waves:


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TEM Transmission Line (cont.)

Note that

= series impedance/length

= parallel admittance/length

Then we can write:


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TEM Transmission Line (cont.)

Let

Then

Solution:

 is called the "propagation constant."

Convention:


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TEM Transmission Line (cont.)

Forward travelling wave (a wave traveling in the positive z direction):

The wave “repeats” when:

Hence:


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Phase Velocity

Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest.

vp

(phase velocity)


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Phase Velocity (cont.)

Set

In expanded form:

Hence


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I+(z)

+

V+(z)

-

z

Characteristic Impedance Z0

A wave is traveling in the positive z direction.

so

(Z0is a number, not a function of z.)


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Characteristic Impedance Z0 (cont.)

Use Telegrapher’s Equation:

so

Hence


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Characteristic Impedance Z0 (cont.)

From this we have:

Using

We have

Note: The principal branch of the square root is chosen, so that Re (Z0) > 0.


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General Case (Waves in Both Directions)

wave in +z direction

wave in -z direction

Note:


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I -(z)

+

V -(z)

-

z

Backward-Traveling Wave

A wave is traveling in the negative z direction.

so

Note: The reference directions for voltage and current are the same as for the forward wave.


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I(z)

+

V(z)

-

z

General Case

Most general case:

A general superposition of forward and backward traveling waves:

Note: The reference directions for voltage and current are the same for forward and backward waves.


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Summary of Basic TL formulas

guided wavelength  g

phase velocity  vp


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Lossless Case

so

(real and indep. of freq.)

(indep. of freq.)


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Lossless Case (cont.)

In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that

(proof given later)

The speed of light in a dielectric medium is

Hence, we have that

The phase velocity does not depend on the frequency, and it is always the speed of light (in the material).


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Terminated Transmission Line

Terminating impedance (load)

Ampl. of voltage wave propagating in positive z direction at z = 0.

Ampl. of voltage wave propagating in negative z direction at z = 0.

Where do we assign z = 0?

The usual choice is at the load.

Note: The length l measures distance from the load:


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Terminated Transmission Line (cont.)

Terminating impedance (load)

What if we know

Can we use z = - las a reference plane?

Hence


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Terminated Transmission Line (cont.)

Terminating impedance (load)

Compare:

Note: This is simply a change of reference plane, from z = 0 to z = -l.


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Terminated Transmission Line (cont.)

Terminating impedance (load)

What is V(-l )?

propagating backwards

propagating forwards

The current at z = - lis then

l distance away from load


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Terminated Transmission Line (cont.)

Total volt. at distance l from the load

Ampl. of volt. wave prop. towards load, at the load position (z = 0).

Ampl. of volt. wave prop. away from load, at the load position (z = 0).

L Load reflection coefficient

l Reflection coefficient at z = - l

Similarly,


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Terminated Transmission Line (cont.)

Input impedance seen “looking” towards load at z = -l .


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Terminated Transmission Line (cont.)

At the load (l= 0):

Recall

Thus,


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Terminated Transmission Line (cont.)

Simplifying, we have

Hence, we have


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Terminated Lossless Transmission Line

Impedance is periodic with period g/2

tan repeats when

Note:


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Terminated Lossless Transmission Line

For the remainder of our transmission line discussion we will assume that the transmission line is lossless.


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Matched Load

A

Matched load: (ZL=Z0)

No reflection from the load

For any l


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Short-Circuit Load

Short circuit load: (ZL = 0)

B

Note:

Always imaginary!

S.C. can become an O.C. with a g/4 trans. line


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Using Transmission Lines to Synthesize Loads

This is very useful is microwave engineering.

A microwave filter constructed from microstrip.


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Example

Find the voltage at any point on the line.


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Example (cont.)

Note:

Atl= d:

Hence


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Example (cont.)

Some algebra:


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Example (cont.)

Hence, we have

where

Therefore, we have the following alternative form for the result:


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Example (cont.)

Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load).


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Example (cont.)

Wave-bounce method (illustrated for l = d):


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Example (cont.)

Geometric series:


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Example (cont.)

Hence

or

This agrees with the previous result (setting l = d).

Note: This is a very tedious method – not recommended.


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Time- Average Power Flow

At a distance l from the load:

Note:

If Z0 real (low-loss transmission line)


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Time- Average Power Flow

Low-loss line

Lossless line (= 0)


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Quarter-Wave Transformer

Z0T

ZL

Z0

Zin

This requires ZL to be real.

so

Hence


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Voltage Standing Wave Ratio


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Coaxial Cable

Here we present a “case study” of one particular transmission line, the coaxial cable.

a

b

Find C, L, G, R

For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics.

We will assume no variation in the z direction, and take a length of one meter in the z direction in order top calculate the per-unit-length parameters.


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h = 1[m]

Coaxial cable

a

l0

-l0

b

Coaxial Cable (cont.)

Find C (capacitance / length)

From Gauss’s law:


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h = 1[m]

Coaxial cable

a

l0

-l0

b

Coaxial Cable (cont.)

Hence

We then have


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Coaxial Cable (cont.)

Find L(inductance / length)

h = 1[m]

From Ampere’s law:

I

Coaxial cable

Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency.

I

z

I

Magnetic flux:

center conductor

S

h


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Coaxial Cable (cont.)

h = 1[m]

I

Coaxial cable

Hence


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Coaxial Cable (cont.)

Observation:

This result actually holds for any transmission line.


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Coaxial Cable (cont.)

For a lossless cable:


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h = 1[m]

Coaxial cable

a

l0

-l0

b

Coaxial Cable (cont.)

Find G (conductance / length)

From Gauss’s law:


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a

l0

-l0

b

Coaxial Cable (cont.)

We then have

or


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Coaxial Cable (cont.)

Observation:

This result actually holds for any transmission line.


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Coaxial Cable (cont.)

As just derived,

To be more general:

This is the loss tangent that would arise from conductivity effects.

The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral.

Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies.


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Coaxial Cable (cont.)

General expression for loss tangent:

Effective permittivity that accounts for conductivity

Loss due to molecular friction

Loss due to conductivity


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h = 1[m]

Coaxial cable

a

b

Coaxial Cable (cont.)

Find R (resistance / length)

Rs = surface resistance of metal


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General Transmission Line Formulas

(1)

(2)

(4)

(3)

Equations (1) and (2) can be used to find L and C if we know the material properties and the characteristic impedance of the lossless line.

Equation (3) can be used to find G if we know the material loss tangent.

Equation (4) can be used to find R (discussed later).


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General Transmission Line Formulas (cont.)

Al four per-unit-length parameters can be found from


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Common Transmission Lines

Coax

a

b

Twin-lead

h

a

a


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w

t

er

h

Common Transmission Lines (cont.)

Microstrip


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w

t

er

h

Common Transmission Lines (cont.)

Microstrip


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ZTH

+

Z0

ZL

-

Limitations of Transmission-Line Theory

At high frequency, discontinuity effects can become important.

transmitted

incident

Bend

reflected

The simple TL model does not account for the bend.


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Limitations of Transmission-Line Theory (cont.)

At high frequency, radiation effects can become important.

We want energy to travel from the generator to the load, without radiating.

When will radiation occur?

ZTH

+

Z0

ZL

-


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a

b

z

Limitations of Transmission-Line Theory (cont.)

The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, or under any circumstances.

The fields are confined to the region between the two conductors.


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Limitations of Transmission-Line Theory (cont.)

The twin lead is an open type of transmission line – the fields extend out to infinity.

The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair.”)

+

-

Having fields that extend to infinity is not the same thing as having radiation, however.


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Limitations of Transmission-Line Theory (cont.)

The infinite twin lead will not radiate by itself, regardless of how far apart the lines are.

S

reflected

incident

h

No attenuation on an infinite lossless line

The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency.


Rf communication circuits

Limitations of Transmission-Line Theory (cont.)

A discontinuity on the twin lead will cause radiation to occur.

Incident wave

pipe

obstacle

h

Reflected wave

Note: Radiation effects increase as the frequency increases.

Incident wave

h

bend

bend

Reflected wave


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Limitations of Transmission-Line Theory (cont.)

To reduce radiation effects of the twin lead at discontinuities:

  • Reduce the separation distance h (keeph << ).

  • Twist the lines (twisted pair).

h

CAT 5 cable

(twisted pair)


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