Abaque de Smith – Download as PDF File .pdf), Text File .txt) or read online. EXERCICE ABAQUE DE – Download as PDF File .pdf), Text File .txt) or read online. fr. abaque de Smith, m diagramme de Smith, m diagramme polaire d’impédance, m. représentation graphique en coordonnées polaires du facteur de réflexion.

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The Smith chartinvented by Phillip H. Smith —[1] [2] is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency RF engineering to assist in solving problems with transmission lines and matching circuits. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

While the use of paper Smith charts for solving the complex mathematics involved in matching problems has been largely replaced by software based methods, the Smith chart display is still the preferred method of displaying how RF parameters behave at one or more frequencies, an alternative to using tabular information.

Thus most RF circuit analysis software includes a Smith chart option for the display of results and all but the simplest impedance measuring instruments can display measured results on a Smith chart display. The Smith chart is plotted on the complex reflection aaque plane in two dimensions and is scaled in normalised impedance the most commonnormalised admittance or both, using different colours to distinguish between them. The most commonly used normalization impedance is 50 ohms. Once an answer is obtained through the graphical constructions described below, it is straightforward to convert between normalised impedance or absque admittance and the corresponding unnormalized value by multiplying by the characteristic impedance admittance.

Reflection coefficients can be read directly from the chart as they are unitless parameters. The Smith chart has circumferential scaling in wavelengths and degrees. The wavelengths scale is used in distributed component problems and represents the distance measured along the transmission line connected between the generator or source and the abwque to the point under consideration.

The degrees scale represents the angle of the voltage reflection coefficient at that point. The Smith chart may also be used for lumped element matching and analysis problems.

Use of the Smith chart and the interpretation of the results obtained using it requires a good understanding of AC circuit theory and transmission line theory, both of which are pre-requisites for RF engineers. As impedances and admittances change with frequency, problems using the Smith chart can only be solved manually using one frequency at a time, the result being represented by a point. Provided the frequencies are sufficiently close, the resulting Smith chart points may be joined by straight lines to create a locus.

A locus of points on a Smith chart covering a range of frequencies can be used to visually represent:. The accuracy of the Smith chart is reduced for problems involving a large locus of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these.

Normalised impedance and normalised admittance are dimensionless. Actual impedances and admittances must be normalised before using them on a Smith chart. Once the result is obtained it may be de-normalised to obtain the actual result.

Using complex exponential notation:. All terms are actually multiplied by this to obtain the instantaneous phasebut it is conventional and understood to omit it.

For the loss free case therefore, the expression for complex reflection coefficient becomes. This equation shows that, for a standing wave, the complex reflection coefficient and impedance repeats every half wavelength along the transmission line.

### Smith chart – Wikipedia

The complex reflection coefficient is generally simply referred to as reflection coefficient. The outer circumferential scale of the Smith chart represents the distance from the generator to the load scaled in wavelengths and is therefore scaled from zero to 0.

These are the equations which are used to construct the Z Smith chart. They both change with frequency so for any particular measurement, the frequency at which it was performed must be stated together with the characteristic impedance.

Any actual reflection coefficient must have a magnitude of less than or equal to unity so, at the test frequency, this may be expressed by a point inside a circle of unity radius. The Smith chart is actually constructed on such a polar diagram.

The Smith chart scaling is designed in such a way that abaaque coefficient can be converted to normalised impedance or vice versa. Using the Smith chart, the normalised impedance may be obtained with appreciable accuracy by plotting the point representing the reflection coefficient treating the Smith chart as a polar diagram and then reading its value directly using the characteristic Smith chart scaling.

This technique is a graphical alternative to substituting the values in the simth.

### File:Smith chart – Wikimedia Commons

S,ith substituting the expression for how reflection coefficient changes along an unmatched loss free transmission line. Versions of the transmission line equation may be similarly derived for the admittance loss free case and for the impedance and admittance lossy cases.

Sjith path along the arc of the circle represents how the impedance changes whilst moving along the transmission line. In this case the circumferential wavelength scaling must be used, remembering that this is the wavelength within the transmission line and may differ from the free space wavelength. If a polar diagram is mapped on to a cartesian coordinate system it is conventional to measure angles relative to the positive x -axis using a counterclockwise direction for positive angles.

The magnitude of a abaqus number is the length of a straight line drawn from the origin to the point representing it.

The region above the x-axis represents inductive impedances positive imaginary parts and the region below the x -axis represents capacitive impedances negative imaginary parts. If the termination is perfectly matched, the reflection coefficient will be zero, represented effectively by a circle of zero radius or in fact a point at the centre of the Smith chart.

If the termination was a perfect open circuit or short circuit the magnitude of the reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle. The normalised impedance Smith chart is composed of two families of circles: In the complex reflection coefficient plane the Smith chart occupies a circle of unity radius centred at the origin. Substituting these into the equation relating normalised impedance and complex reflection coefficient:.

This is the equation which describes how the complex reflection coefficient changes with the normalised impedance and may be used to construct both families of circles. The Y Smith chart is constructed in a similar way to the Z Smith chart case but by expressing values of voltage reflection coefficient in terms of normalised admittance instead of normalised impedance. The normalised admittance y T is the reciprocal of the normalised impedance z Tso. The region above the x -axis represents capacitive admittances and the region below the x -axis represents inductive admittances.

Capacitive admittances have positive imaginary parts and inductive admittances have negative imaginary parts. Again, if the termination is perfectly matched the reflection coefficient will be zero, represented by a ‘circle’ of zero radius or in fact a point at the centre of the Smith chart.

If the termination was a perfect open or short circuit the magnitude of the voltage reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle of the Smith chart.

A point with a reflection coefficient magnitude 0. The length of the line would then be scaled to P 1 assuming the Smith chart radius to be unity. The following table gives some similar examples of points which are plotted on the Z Smith chart. For each, the reflection coefficient is given in polar form together with the corresponding normalised impedance in rectangular form.

The conversion may be read directly from the Smith chart or by substitution into the equation. In RF circuit and matching problems sometimes it is more convenient to work with admittances representing conductances and susceptances and sometimes it is more convenient to work with skith representing resistances and reactances. Solving a typical matching problem will often require several changes between both types of Smith chart, using normalised impedance for series elements and normalised admittances for parallel elements.

For these a dual normalised impedance and admittance Smith chart may be used. Alternatively, one type may be used and the scaling converted to the other when required. In order to change from normalised impedance to normalised admittance or vice versa, the point representing the value of reflection coefficient under consideration is moved through exactly degrees at the same radius.

## International

For example, the point P1 in the example representing a reflection coefficient of 0. To graphically change this to the equivalent normalised admittance point, say Q1, a line is drawn with a ruler from P1 through the Smith chart centre to Q1, an equal radius in the opposite direction. This is equivalent to moving the point through a circular path of exactly degrees. Once a transformation from impedance to admittance has been performed, the scaling changes to normalised admittance until a later transformation back to normalised impedance is performed.

Again, these may be obtained either by calculation or using a Smith chart as shown, converting between the normalised impedance and normalised admittances planes.

The choice of whether to use the Z Smith chart or the Y Smith chart for any particular calculation depends on which is more convenient. Impedances in series and admittances in parallel add while impedances in parallel and admittances in series are related by a reciprocal equation. Dealing with the reciprocalsespecially in complex numbers, is more time consuming and error-prone than using linear addition.

In general therefore, most RF engineers work in the plane where the circuit topography supports linear addition. The following table gives the complex expressions for impedance real and normalised and admittance real and normalised for each of the three basic passive circuit elements: Using just the characteristic impedance or characteristic admittance and test frequency an equivalent circuit can be found and vice versa. Here the electrical behaviour of many lumped components becomes rather unpredictable.

This occurs in microwave circuits and when high power requires large components in shortwave, FM and TV Broadcasting. For distributed components the effects on reflection coefficient and impedance of moving along the transmission line must be allowed for using the outer circumferential scale of the Smith chart which is calibrated in wavelengths. The smitth example shows how a transmission line, terminated with an arbitrary load, may be matched at one frequency either with a series or parallel reactive component in each case connected at precise qbaque.

How may the line be matched? This is plotted on the Abaqud Smith chart at point P As the transmission line is loss free, a circle centred at the centre of the Smith chart is drawn through the point P 20 to represent the path of xe constant magnitude reflection coefficient due to the termination.

At point P 21 the circle intersects with the unity circle of constant normalised resistance at. An alternative shunt match could be calculated after performing a Smith chart transformation from normalised impedance to normalised admittance. Point Q 20 is the equivalent of P 20 but expressed as a normalised admittance. Reading from the Smith chart scaling, remembering that this is now a normalised admittance gives.

In fact this value is not actually used. The earliest point at which a shunt conjugate match could be introduced, moving towards the generator, would be at Q 21the same position as the previous P 21but this time anaque a normalised admittance given by.

From the table it can be seen that a negative admittance would require an inductor, connected in parallel with the transmission line. A suitable inductive shunt matching would therefore be a 6. The analysis of lumped element components assumes that the wavelength at the frequency of operation is much greater than the dimensions of the components themselves. The Smith chart may be used to analyze such circuits in which case the movements around the chart are generated by the normalized impedances and admittances of the components at the frequency of operation.

In this case the wavelength scaling on the Smith chart circumference is not used. At this frequency the free space wavelength is 3 m.

The component dimensions themselves will be in the order of millimetres so the assumption of lumped components will be valid.

If there were very different values of resistance present a value closer to aabaque might be a better choice. The analysis starts with a Z Smith chart looking into R 1 only with no other components present.

## Interactive Smith chart

The first transformation is OP 1 along the line of constant normalized resistance in this case the addition of a normalized reactance of – j 0. Points with suffix P are in the Z plane and points with suffix Q are in the Y plane.

The following table shows the steps taken to work through the remaining components and transformations, returning eventually back to the centre of the Smith chart and a perfect 50 ohm match.