A LOW LOSS TRANSMISSION LINE IS ONE FOR WHICH

R<<шl

AND                      G<<шC

Thus                     Z=R+jшL- jшL

And                     Y=G+jшC – jшC

(a)    Characteristic impedance,

Z₀=R+jшL/G+jшC

= jшL/jшC

Z₀= L/C

Z₀ is pure resistance for low-loss line.

(b)   Propagation constant

Ỳ= (R+jшL)(G+jшC)

= (jшL)(jшC)

Ỳ=jш LC

As                          Ỳ=α+jβ

By comparing real and imaginary parts of above two equations, we get

Attenuation constant, α=0

And phase shift constant, β=ш LC

Let us also discuss another concept called BALUN

BALUN consists of two words bal(short form of balanced) and up (short form of unbalanced).It is a form of a quarter wave transformer matching balancedline  to an unbalanced transmission circuit.this problem arises when an ungrounded antenna is being fed by a coaxial cable.The cable from the dipole antenna offers reasonably high output impedance which does not match with the input impedance of the TV receiver.A transformer is thus required that matches the output impedance a parallel wire cable with input impedance of the TV set.

This transformer is known as balun and has got few turns of winding on a ferrite core. Balun is also called as Bazooka.

The post RADIO FREQUENCY LINES OR LOW LOSS LINES first appeared on Winner Science.

The outermost scale around the perimeter of Smith chart is called the Wavelengths towards generator(WTG) scale.It has been constructed to denote movement on the transmission line toward the generator,in units of the wavelength,λ.Thus ,l is measured in wavelengths and one complete rotation corresponds to l=λ/2.

In some transmission line problems,it may be necessary to move from some point on the transmission line toward another point closer to the load,in which case the phase is increased and which corresponds to counter clockwise direction.So Smith chart contains one more scale around its perimeter ,in between the θ_p and WTG scale, for accommodating such a need. This scale is called the wavelengths toward load (WTL) scale.

Why circumference of Smith chart is calibrated for λ/2?

By rotation of λ/4 or 0.25 λ on the Smith chart transform Z_n in to Y_nor vice versa. The points representing Y_n and Z_n will be diametrically opposite to each other on the SWR circle. This is the reason that the circumference of Smith chart is calibrated for λ/2.

The post USE OF “WAVELENGTHS TOWARDS GENERATOR”(WTG) SCALE AND “WAVELENGTHS TOWARDS LOAD”(WTL) SCALE ON SMITH CHART first appeared on Winner Science.

REFLECTION COEFFICIENT

Definition:- Reflection co-efficient is defined as the ratio of the reflected voltage to the incident voltage.

p=V_r/V_i

where                            V_r=Reflection voltage

V_i= Incident Voltage

The reflection co-efficient can also be defined in terms of the ratio of reflected current and incident current. However , it is observed that p as defined from current ratio is negative with respect to that defined from voltage ratio. The reason being that the reflected current suffers a 180⁰ phase shift at the receiving end while the reflected voltage does not.

Thus                                         p=- I_r/I_i

Where                                       I_r=Reflected Current

I_i= Incident Current

It is a vector quantity having magnitude and direction both.

Relation Between Voltage Standing Wave Ratio and Reflection Coefficient

The points of voltage maximum are those ,where the incident and reflected voltage are in phase and add up,

That is                              |V_max|=|V_i|+|V_r|                                  (1)

Where V_i is the r.m.s value of incident voltage and

V_r is the r.m.s value of the reflected voltage.

Also                           |V_min|=|V_i|-|V_r|                                           (2)

Because the points of voltage minimum are those ,where the incident and reflected voltage are out of phase and thus ,will have opposite sign

From definition of VSWR ,                             VSWR=|V_max|/|V_min|

Substituting the value of V_max and V_min from equation (1) and (2) in above equation,we get

VSWR=|V_i|+|V_R|/|v_I|-|v_R|

Dividing numerator and denominator by V_i,we have

VSWR = (1+V_r/V_i)/(1-V_r/V_i)                                          (3)

As reflection co-efficient,

P=V_r/V_i

By substituting value of p in equation (3),we get

VSWR=1+|p|/1-|p|                   (4)

Or            VSWR (1-|p|)=1+|p|

Or            VSWR-1=|p|(VSWR+1)

Thus           |p|=VSWR-1/VSWR+1              (5)

Expressions (4) and (5) represents the relation between voltage standing wave ratio and reflection coefficient.

For current standing wave ratio

CSWR= 1-|p|/1+|p|

The post Relation Voltage Standing Wave Ratio and Reflection Coefficient first appeared on Winner Science.

STANDING WAVES

When a transmission line is not correctly terminated ,the travelling wave at the receiving end is reflected completely or partially at the termination. The combination of incident and reflected waves give rise to standing waves of current and voltage along the line, with definite maxima and minima of current and voltage along transmission line.

At some points , the incident and reflected signals are in phase and both the components add together and at these points voltage and current will be maximum . on the other hand,at some other points ,the two components may oppose each other and at these points voltage and current will be minimum.

The resultant graphical profile of both these incidents and reflected waves is called standing waves.

The points where the resultant signal (voltage or current) is maximum are known as voltage or current maxima.On the other hand, the points where the resultant signal (voltage or current) is minimum are called voltage or current minima. Thus

|V_max|=|V_i|+|V_r|

|V_min|=|V_i|-|V_r|

|I_max|=|I_i|-|I_r|

The magnitude of standing waves provides an idea of the amount of reflection.

Standing Wave Ratio

The ratio of the maximum and minimum magnitudes of current or voltage on a line having standing waves is called the standing wave ratio.

SWR=|V_max|/|V_min|

SWR=|I_max|/|I_min|

Voltage standing wave ratio:VSWR

The ratio of maximum and minimum magnitude of voltage on a line having on a line having standing waves is called voltage standing wave ratio.

Therefore                         VSWR=|V_max|/|V_min|

Generally , VSWR is always greater than 1 and when it is equal to 1,the line is correctly terminated and there is no reflection.

Current Standing Wave Ratio: CSWR

The ratio of maximum and minimum magnitude of current on a line having standing waves is called current standing wave ratio.

Therefore                                         CSWR=|I_max|/|I_min|

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IMPEDANCE MATCHING

A transmission line is matched when the load impedance ,Z_ris equal to the characteristic impedance ,Z₀ of the line that is

Z_R=Z₀

As                                     p= Z_R-Z₀/ Z_R+Z₀

If                                                Z_R=Z₀

Then,reflection coefficient,

P=0

Thus, there is no reflected wave so the incident power is fully absorbed by the load. Therefore ,the maximum power transfer is possible.

So in communication networks, the elements of network should be designed such that maximum power transfer takes place between the source (or transmitter)and load (or antenna) .This means that if maximum power transfer has to take place between the source and the load,the resistance of the load should be equal to that of the source and the reactance of the load should be equal to that of the source but opposite in sign. That is if the source is inductive ,the load should be capacitance and vice-versa. When this condition is achieved,it is referred to as impedance matching , and the methods employed to achieve this are termed as impedance matching devices.

Stub

A stub is a piece of transmission line.

It is possible to connect sections of open or short circuited line called stub in shunt with the main line at some point or ponts to effect impedance matching. This is called stub matching.

It has two advantages:

(a)    The length and characteristic impedance of the line remains unchanged.

(b)   Adjustable susceptance can be added in shunt with the transmission line.

Stub matching is of two types:

(i)                 Single stub matching

(ii)               Double stub matching

(i) Single stub matching:-

(a) Connection of stub in parallel with transmission line

It consists of an open or short circuited section of transmission line of length l_t,connected in parallel with the main line at distance l_s from the load Z_R.Stub has the same characteristic impedance as the main line.

Generally short circuited stubs are preferred comparable to open circuited stub as open circuited stub radiates some energy at high frequencies.

Use of a single stub to provide impedance matching. It is shown that a transmission line having characteristic impedance of ‘Z₀’ is terminated in a complex load admittance of (g_R+jb_R)

First step:-

Locate a point nearest to the load on the transmission line where the normalized admittance is (1+jb_R)

Second step:

A stub (short or open circuited transmission line) is added in parallel across the transmission line at a point so as to offer a suscetpance of –jb_R. Thus the transmission line with a characteristic impedance of ‘Z₀’ gets matched to a complex load upto that point.

We connect the stub in parallel with the main line as it is easier to deal with the admittance as they can be added up.

(b) Connection of stub in series with transmission line

Here also the first step is to locate a point on the transmission line where the normalized impedance looking towards the load end is(1+j X) .At that point,a stub is added with the stub offering a normalised reactance of (-j X). The feed line needs to be cut for insertion of series stub. This technique is therefore not commonly used as it is difficult to fabricate in co-axial and striplines.

(ii) Double stub matching

In single stub matching, the stub is placed on the line at a specified point.Its location varies with Z_R and frequency. This creates some difficulties as the specified point may occur at an undesirable location. In such cases, double stubs are used. Here the distance between the two stubs is fixed such as λ/16,λ/8,3λ/16,3λ/8 or even closer and the lengths of the two stubs are adjusted to match the load.

The post IMPEDANCE MATCHING USING STUB first appeared on Winner Science.

A transmission line is said to be quarter wave transmission line when its length equals quarter wavelength.

That is                                             l=λ/4

Or                                                   l=(2n-1)λ/4 (odd multiple of λ/4)

Or                                                   βl=2π/λ * λ/4                                              [ β=2π/λ]

=π/2

β   l=(2n-1)λ/2

therefore                                  tan βl= tan [(2n-1) π/2] +- ∞

as                                                           Z_i=Z₀[ Z_r+j Z₀ tan βl/ Z₀+j tan βl]

divide numerator and denominator by j tan βl

Z_i=Z₀[ Z_r/j tan βL+ Z₀/Z₀/j tan βl+Z_r]

As                             βl= π/2

Thus                                  Z_i=Z₀[0+Z₀]/(0+Z_r)

Thus                                  Z_i=Z_u² / Z_R

Thus quarter waves loss-less line transform the load impedance (Z_t) to input terminals as its inverse multiplied by the square of Z₀ . It is also called as quarter wave transformer. An open circuit quarter wave line appears as short circuit at the input terminals and short circuit appears as open circuit.

2. Input Impedance Z_i for half-wave transmission line:-

A ransmission line is said to be half- wave transmission line when its length equal half wavelength.

That is                              l=λ/2

Βl= 2π/λ*λ/2

=π

When   l=nλ/2 (integral multiple of λ/2)

Then                                  βl=nπ

Thus                           tan  βl= tan(nπ)

= 0

As                                   Z_i=Z₀[Z_r+jZ₀tanβl/Z₀+jZ_rtanβl]

Substituting value of tan βl=tan nπ=0 in above expression,we get

Z_i=Z_R

Thus,half wave lossless line transforms the Z_Rto Z_i without any change.

The post Input impedance for quarter wave and half wave transmission line first appeared on Winner Science.

Ultra High Frequency lines commonly abbreviated as U.H.F lines are one of the types of the transmission lines. Ultra high frequency lines have operational frequency range from 300 to 3000 MHz or wavelength from 100 cm to 10 cm.

Under normal frequencies the transmission lines are used as wave guides for transferring power and information from one point to another.

At Ultra High Frequencies, the transmission lines can be used as circuit elements like capacitor or an inductor . It means they can be used in circuits like a capacitor or an inductor.

Assuming line to be lossless ,that is α=0.

As Input impedance of a lossless line

Z_i=z₀[Z_r+j Z₀tab β l/Z₀+jZ_Rtan β l]

Cases

1. Input Impedance Z_i for open circuited line

A voltage difference will exist between two wires but no current can flow in open circuit.Thus at the open end termination of this line there exist a maximum voltage and zero current. Therefore impedance at the open end will be infinite.

That is                                        Z_R=∞

I_r=0                                                                      [z_r=V_r/I_R]

Now dividing numerator and denominator of R.H.S in expression of Z_j,we get

Z_i=Z₀[1+jZ₀/Z_rttan βL/Z₀/Z_R+j tan βL]

= Z₀[1+0/0+jtan βL]                              (Z_R=∞]

=Z₀/j tan βL

Or                                                           (Z_i)OC= -j Z₀cot βl

2. Input Impedance Z_i for short circuited line

For the short –circuited end between the two transmission lines wires there will be no voltage difference ,but there will be a maximum current flow. Therefore ,at the short –circuited termination ,the current is maximum ,the voltage is zero and impedance will also be zero.

That is             Z_R=0

This implies                              V_­r=0                                            [z_r=v_R/i_R]

THUS EQUATION 31(B) BECOMES

Z_i=Z₀[0+j Z_otan βl/Z₀+0]

Thus                                                 (Z_i)SC=jZ₀tan βl

As                                       Z_i=R₀+j X₀

Thus by comparing above two equations, we get

X₀=Z₀tan βl

Thus Z_iof a short circuited line is pure reactive.

For a short length line,

Tan βl= βl

Therefore Z_i=j Z₀ βl

= j L/C ш LC l       [Z₀=L/C and β=ш LC]

Z_i= jшLl               (inductive Reactance]

The post ULTRA HIGH FREQUENCY (U.H.F) TRANSMISSION LINES-1 first appeared on Winner Science.

A transmission line is said to be distortionless when attenuation constant ‘α’ is frequency independent and the phase shift constant ‘β’ is linearly dependent on the frequency.

Condition for line to be distortionless

R/L=G/C

(a)    Propagation constant

Ỳ= (R+jωL)(G+jωC)

Or                     Ỳ= RG(1+jωL/R) (1+jωC/G)

If                       R/L=G/C

Put value of R/L in equation of Ỳ

Thus                 Ỳ= RG(1+jωC/G)(1+jωC/G)

Or                     Ỳ= RG(1+jωC/G)

Also                   Ỳ=α+jβ

Comparing Real and Imaginary parts, we get

α=RG

and                       jβ= RGjωC/G

=jω RC/G

Or                                 β=ω L/C C                                      (R/G=L/C)

Thus                         β=ω LC

The above results show that α is frequency independent and β is frequency dependent

(b)   Characteristic impedance

Z₀= √R+jωL/G+jωC

Z₀ =R/G=L/C

(c)    Phase velocity:-

V_p=ω/β

Substituting value of β in above expression,we get

V_p=ω/ω LC

Thus                              v_p=1/ LC

Note: If you do not know about the basics of transmission line then please read the article transmission line and its types.

Last time I have also discussed the lossless transmission line and its condition.

The post DISTORTIONLESS LINE first appeared on Winner Science.

DEFINITION I:- Z₀ is defined as the ratio of the square root of series impedance per unit length ,Z to the square root of shut admittance per unit length,Ỳ

Z₀=Z/Y =R+jωL/G+jωc

DEFINITION 2:- The characteristic impedance , Z₀ of a line is defined as the ratio of the forward voltage wave ,V_fto the forward current wave ,I_f at any point on the line.

Z₀=V_f/I_f

DEFINITION 3:- z₀ IS defined as the minus of the ratio of the reflected voltage wave . V_r to the reflected current wave , I_r at any point on the line,

Z₀= -V_r/I_r

Characteristic impedance ,Z₀ is also called Surge imperdance.

LOSSLESS TRANSMISSION LINES

A transmission line is said to be lossless if the conductors of line are perfect that is cnductivity σ_c=∞ and the dielectric medium between the lines is lossless that is conductivity σ_d=0

Condition for a line to be lossless

R=0=G

For loss less line,

(a)    Attenuation Constant       α=0

(b)   Propagation constant

Ỳ=α+jβ=jβ                                               (α=0)

Also as                      Ỳ=(R+jωL)(G+jωC)

As                              R=0, G=0

Thus propagation constant     Ỳ=j ω LC

(c) Phase shift constant

By comparing imaginary parts of Ỳ, we get

Phase shift constant β=ω LC

(d) Characteristic impedance,

Z₀=R+jωL/G+jωc

As                                          R=0=G

Z₀=L/C

Thus Z₀ is pure reactance (that is in the form of inductance or capacitance).

(e) Phase velocity or the velocity of propagation in lossless line,

V_p=ω/β

By putting value of β,we get

Thus                          v_p=ω/ω LC

Or                                      v_p=1/LC

The post LOSSLESS TRANSMISSION LINE AND ITS CONDITION first appeared on Winner Science.

Transmission lines are used to carry electromagnetic energy from one point to another . It means it transfer from one point to another. Generally it consists of two conductors. It is used to connect a source to a load. The source may be a transmitter or an electric generator and the load may be an antenna. Transmission lines are used for operational frequencies equal to or less than about 3GHz. Above 3GHz they will be replaced by waveguides.

Transmission lines are commonly used in power systems for power transmission. These lines can be used as circuit elements like capacitors, inductors at ultra high frequencies(300-3000MHz)

Types of Transmission Lines

(a)    Two wire transmission line:- This transmission line consists of a pair of parallel conducting wires separated by a uniform distance .These are used in power systems or telephones lines.

(b)   Coaxial transmission line:- This consists of an inner and a coaxial outer conducting sheath separated by a dielectric medium . They are used as TV cables, telephones cables and power cables.

(c)    Parallel plate transmission line or planar line:- It has two parallel conducting plates separated by a dielectric slab of uniform thickness

(d)   Optical fibres transmission line:- It consists of core and cladding . Information passes through the core in the form of totally internal reflected TEM waves.

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