# Transmission lines-part 2

Jun 17 • General • 2116 Views • 1 Comment on Transmission lines-part 2

TRANSMISSION LINES-PART 2

SINGLE AND DOUBLE STUB MATCHING:

In this case, we use a length of transmission line to transform the admittance of the load such that its conductance has a very normalized value of 1. (ie. for about 50 ohm transmission line, the conductance would be jst 0.02 ohms. We then have to cancel the susceptance at that point by adding a susceptance to ground. Let us consider the following example:
We wish to match a 100 + j50 ohm load to a 50 ohm transmission line. We do the following steps:

2 – Draw a VSWR circle

3 – Draw a line from the center of the chart, through the load (green dot), to the outside of the chart. Note the lambda on the “wavelengths toward the generator” scale.

4 – Draw a line from the center of the chart, through the point where VSWR circle intersects the g = 1 circle (magenta dot), to the outside of the chart. Note the lambda on the “wavelengths toward the generator” scale.

5 – The distance we have traveled is the difference between the lambda from step 4 and that from step 3. i.e the electrical length of the cable we will place in series with the load.

6 – Get the susceptance of the reflected load admittance (magenta dot). You will cancel it with its complement. (In the example, Y = 1+j1. The susceptance B = j1 Siemens. We will now cancel it with an inductor whose susceptance is -j1 S. Single stub matching figure

Double Stub Matching

The major problem with single stub matching is that it is very difficult to adjust once built. The length of the series transmission line cannot be changed easily. The answer to this problem is double stub match. This kind of a match is typical for microstrip amplifiers. Double stub matching figure

You will notice two stubs and a length of transmission line between them. The stubs now could be lengths of transmission line or they could be an inductor or a capacitor. It is a normal practice to place the first stub as close as possible to the load. Practical problems might be prevented as we have shown a short length of line between the load and the first stub. There comes a new circle on the chart; the shifted to G=1.0 circle. If you get onto that circle (by adding the correct first stub), then you will be on original G=1.0 circle when you move to the stub 2 position. Then you get to go to the center of the chart by adding the second stub.

Here are the few steps for doing a double stub match:

2 – Move along the VSWR circle by the length required to get to the stub 1 position. In this case we move from 0.087 to 0.137 for a distance of 0.5 .

3 – Move along a line of constant admittance ’til you get to the shifted G=1.0 circle. In this case we can move from a normalized admittance of -j0.65 mho to j0.1 mho. This means the capacitive admittance we have to add is j0.75 mho. (The reactance would be -j67 ohms not normalized.)

4 – Construct a VSWR circle for the point you reached in step 3. Move along with it until you reach the original G=1.0 circle. In doing this, we have travelled just 0.125 .

5 – Now move to the center of the chart by adding the admittance of stub 2. We start at just j0.55 and move to zero so we have moved -j0.55 mho. That results in an inductive reactance of j91 ohms not normalized.

PARAMETERS OF LINE AND  COAXIAL CABLE:

Coaxial cable, or coax, is an electrical cable with an inner conductor surrounded by a much flexible, tubular insulating layer, surrounded by a tubular conducting shield. The term coaxial arises from the inner conductor and the outer shield sharing the same geometric axis. Coaxial cable was first invented by a English engineer and mathematician THE Oliver Heaviside, who first patented the design in 1880.

Coaxial cable is used as a transmission line for radio frequency signals, in applications such as connecting the radio transmitters and receivers with their antennas, computer network (Internet) connections, and distributing the cable television signals. One other advantage of coaxial over other types of radio transmission line is that in an ideal coaxial cable the electromagnetic field carrying the signal exists only in the space between the inner and outer conductors. This allows the coaxial cable runs to be installed next to metal objects such as gutters without the power losses that occur in other types of transmission lines, and provides the protection of the signal from external electromagnetic interference.

Coaxial cable differs from the other shielded cable used for carrying lower frequency signals such as audio signals, in that the dimensions of the cable are controlled to give a precise, constant conductor spacing, which is needed for it to function efficiently as a radio frequency transmission line.

DISSIPATION LESS LINE:

It is defined as that line which has no dissippation.

INPUT IMPEDANCE OF OPEN AND SHORT CIRCUIT:

The open circuit test, or “no-load test”, is one of the methods used in electrical engineering to determine the no-load impedance in the excitation branch of a transformer.

A short circuit (sometimes abbreviated to short or s/c)which is an electrical circuit that allows a current to travel along an unintended path, often where essentially no (or a very low) electrical impedance is encountered. The electrical which is opposite of a short circuit is an “open circuit”, which is an infinite resistance between the two nodes. It is common to misuse the”short circuit” to describe any electrical malfunction, regardless of the actual problem.

EIGTH WAVE , QUARTER WAVE AND HALF LINE:

In either of the case, the line has voltage anti nodes at both the ends, and has current nodes at both the ends. That means to say, there is a maximum voltage and a  minimum current at either end of the line, which corresponds to the condition of an open circuit. The fact that this condition exists for both the ends of the line which tells us that the line faithfully reproduces its terminating impedance at the source end, so that the source “sees” an open circuit where it connects to the transmission line, j as if it were directly open-circuited.

Positive real function, LC, RL, RC, and RLC network synthesis:

Any passive driving the point impedance, such as the impedance of a violin bridge, is a positive real.
A Positive real functions have been studied extensively in the continuous-time case in the context
of the network synthesis [1, 9]. Very little,we can say that however,it seems to be available in the discrete time case.
The purpose of this note (an excerpt from ) is that to collect together some facts about the positive real
transfer functions for discrete-time linear systems.
Deﬁnition of A complex valued function of a complex variable f(z) is said to be positive real
(PR) if the conditions are
1. z real =⇒ f(z) real
2. |z| ≥ 1 =⇒ re{f(z)} ≥ 0
We now specialize to the subset of functions f(z) is representable as a ratio of ﬁnite-order polynomials in z. This class of “rational” functions is the set of all transfer functions of the ﬁnite-order
time-invariant linear systems, and we write H(z) to denote a member of this class. We can use the
convention that stable, minimum phase systems are analytic and nonzero in the strict outer disk.1
Condition (1) implies that for H(z) to be the PR, the polynomial coeﬃcients must be the real, and therefore complex poles and zeros must exist in conjugate pairs.

FOSTER AND CAUER  METHOD:

For the calculation of hot spot temperatures or temperature
fields in electronic systems with rapidly varying chip heat
source strength usually reduced models are used. Numerous
compact static [1-5] and transient [6-10] thermal models have
been established for a rapid calculation of temperatures. The
notion of “compact” thermal model usually implies boundary
condition independence (BCI) [1, 2], i.e. the model is valid for
all (or nearly all) the reasonable temperatures,such as heat flows and also
heat transfer coefficients which are applied to the thermal contact areas.
An advantage of the model presented in [11, 12] is its the ease of
the parameter determination by the simple linear least square fit to
measured or simulated heating curves (thermal impedances).
The model was extended in  to include the effects of varying surface or the ambient temperature and varying the heat flows at the thermal contact areas. However, it is not possible to use the arbitrary heat transfer coefficients” α” as external bound.c. parameters independently of the model parameters. Thus the new
model parameters have to be determined, when the” α” changes. On
the other hand the compact models of [1, 2, 4, 5] deal with the small packages with the small thermal contact areas in comparison to the multi chip module of Fig. 1, which is mounted on the cooling radiator. Such modules are always have a large temperature variation along the bottom side of the module base plate that depending upon the different heating cases. Thus the module can hardly be approximated thermally by a compact model which is independent of” α”.Most of the reduced models,are also [11, 12], apply to the systems with negligible nonlinearities.

MINIMUM POSITIVE REAL FUNCTION:

Positive-real functions, often abbreviated to PR function, are usually a kind of mathematical function that first arose in electrical network analysis. They are some complex functions, Z(s), of a complex variable, s. Rational function is defined to have the PR property if it has a positive real part and is analytic in the right half plane of the complex plane and takes on real values on the real axis.

Bott-Duffin method, Synthesis-Coefficient.:

The ‘constrained inverse’ was introduced by Bott and Duffin in 1953. Since then the various generalizations have been risen. In this paper, we propose a generalization of the Bott–Duffin inverse. Applications of the projection methods we are using for  solving the sparse linear systems, the truncated methods we are using for solving the least squares problems, and generalized saddle point problems are also presented. The relationships with the other generalized Bott–Duffin inverses are explored. The results indicates that the generalized Bott–Duffin inverse may be a useful for the  matrix computations.

GATE Syllabus-

IES Syllabus-

### One Response to Transmission lines-part 2

1. Dinesh says:

Transmission details which you have provided is very useful…

and these notes are given a brief detailed information about the lines