An impedance Smith chart (with no data plotted)

The Smith Chart, invented by Phillip H. Smith (1905-1987),^[1][2] is a graphical aid or nomogram designed for electrical and electronics engineers specialising in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits.^[3] Use of the Smith Chart utility has grown steadily over the years and it is still widely used today, not only as a problem solving aid, but as a graphical demonstrator of how many RF parameters behave at one or more frequencies, an alternative to using tabular information. The Smith Chart can be used to represent many parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability.^[4][5] The Smith Chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.^[6] Image File history File links Download high-resolution version (2416x3030, 257 KB) A smith chart. ... Image File history File links Download high-resolution version (2416x3030, 257 KB) A smith chart. ... Phillip Hagar Smith (April 29, 1905–August 29, 1987) was an electrical engineer, who became famous for his invention of the Smith chart. ... Smith chart which shows how the complex impedance of a transmission line varies along its length This article is about the graphical devices called nomograms. ... Electrical Engineers design power systems… … and complex electronic circuits. ... It has been suggested that this article or section be merged with Radio waves. ... A transmission line is the material medium or structure that forms all or part of a path from one place to another for directing the transmission of energy, such as electromagnetic waves or acoustic waves, as well as electric power transmission. ... Impedance matching is the practice of attempting to make the output impedance of a source equal to the input impedance of the load to which it is ultimately connected, usually in order to maximize the power transfer and minimize reflections from the load. ... A table is a mode of visual communication that maps the logical structure of a set of data into a hierarchical matrix. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... In electrical engineering, the admittance (Y) is the inverse of the impedance (Z). ... The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. ... Scattering parameters or S-parameters are terminology used in electrical engineering, electronic engineering, and communications systems engineering describe the electrical behavoir of linear electrical networks when under various steady state stimulii by small signals. ... In telecommunication, noise figure (NF) is the ratio of the output noise power of a device to the portion thereof attributable to thermal noise in the input termination at standard noise temperature (usually 290 K). ... In mathematics, stability theory deals with the stability of the solutions of differential equations and dynamical systems. ... Look up one in Wiktionary, the free dictionary. ... Circle illustration In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its boundary. ... Cross coupled LC oscillator with output on top An electronic oscillator is an electronic circuit that produces a repetitive electronic signal, often a sine wave or a square wave. ... In mathematics, stability theory deals with the stability of the solutions of differential equations and dynamical systems. ...

Overview

The Smith Chart is plotted on the complex reflection coefficient plane in two dimensions and is scaled in normalised impedance (the most common), normalised admittance or both, using different colours to distinguish between them. These are often known as the Z, Y and YZ Smith Charts respectively.^[7] Normalised scaling allows the Smith Chart to be used for problems involving any characteristic impedance or system impedance, although by far the most commonly used is 50 Ohms. With relatively simple graphical construction it is straighforward to convert between normalised impedance (or normalised admittance) and the corresponding complex voltage reflection coefficient. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. ... 2-dimensional renderings (ie. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... In electrical engineering, the admittance (Y) is the inverse of the impedance (Z). ... The characteristic impedance of a uniform transmission line is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. ... A multimeter can be used to measure resistance in ohms. ...

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 load 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. The circumference is the distance around a closed curve. ... The wavelength is the distance between repeating units of a wave pattern. ... A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized °, is a measurement of plane angle, representing 1／360 of a full rotation. ... A signal generator, also known variously as a test signal generator, tone generator (in audio only), waveform generator, or frequency generator is an electronic instrument that generates repeating electronic signals (in either the analog or digital domains). ... The lumped element model of electronic circuits makes the simplifying assumption that each element is an infinitesimal point in space, and that the wires connecting elements are perfect conductors. ...

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. City lights viewed in a motion blurred exposure. ...

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. This is often adequate for narrow band applications (typically up to about 5% to 10% bandwidth) but for wider bandwidths it is usually necessary to apply Smith Chart techniques at more than one frequency across the operating frequency band. Provided the frequencies are sufficiently close, the resulting Smith Chart points may be joined by straight lines to create a locus. FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... The frequency range is defined as the range of frequencies in which the device is allowed to operate. ... This article does not cite any references or sources. ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ...

A locus of points on a Smith Chart covering a range of frequencies can be used to visually represent:

• how capacitive or how inductive a load is across the frequency range
• how difficult matching is likely to be at various frequencies
• how well matched a particular component is.

The accuracy of the Smith Chart is reduced for problems involving a large spread of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these. Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. ...

Mathematical Basis

Actual and Normalised Impedance and Admittance

A transmission line with a characteristic impedance of may be universally considered to have a characteristic admittance of where

Any impedance, expressed in Ohms, may be normalised by dividing it by the characteristic impedance, so the normalised impedance using the lower case z, suffix T is given by

Similarly, for normalised admittance

The SI unit of impedance is the Ohm with the symbol of the upper case Greek letter Omega () and the SI unit for admittance is the Siemens with the symbol of an upper case letter S. 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. Cover of brochure The International System of Units. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... A multimeter can be used to measure resistance in ohms. ... The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ... Look up Ω, ω in Wiktionary, the free dictionary. ... In electrical engineering, the admittance (Y) is the inverse of the impedance (Z). ... The siemens (symbol: S) is the SI derived unit of electric conductance. ... In the physical sciences, a dimensionless number (or more precisely, a number with the dimensions of 1) is a quantity which describes a certain physical system and which is a pure number without any physical units; it does not change if one alters ones system of units of measurement...

The Normalised Impedance Smith Chart

Using transmission line theory, if a transmission line is terminated in an impedance () which differs from its characteristic impedance (), a standing wave will be formed on the line comprising the resultant of both the forward () and the reflected () waves. Using complex exponential notation: Termination as a technical term has different meanings. ... A standing wave, also known as a stationary wave, is a wave that remains in a constant position. ... For the technique in organ building, see Resultant (organ). ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The exponential function is one of the most important functions in mathematics. ...

and

where

is the temporal part of the wave and

where

is the angular frequency in radians per second (rad/s)

is the frequency in Hertz (Hz)

is the time in seconds (s)

and are constants

is the distance measured along the transmission line from the generator in metres (m)

Also A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... It has been suggested that this article or section be merged into Angular velocity. ... Some common angles, measured in radians. ... Look up second in Wiktionary, the free dictionary. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... The hertz (symbol: Hz) is the SI unit of frequency. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... The metre or meter is a measure of length. ...

is the propagation constant which has units 1/m

where For an electromagnetic field mode varying sinusoidally with time at a given frequency, the propagation constant is the logarithmic rate of change, with respect to distance in a given direction, of the complex amplitude of any field component. ...

is the attenuation constant in Nepers per metre (Np/m)

is the phase constant in radians per metre (rad/m)

The Smith Chart is used with one frequency at a time so the temporal part of the phase () is fixed. All terms are actually multiplied by this to obtain the instantaneous phase, but it is conventional and understood to omit it. Therefore In telecommunication, the term attenuation constant has the following meanings: 1. ... A neper (Symbol: Np) is a unit of ratio. ... The metre or meter is a measure of length. ... In electromagnetic theory, the phase constant is one component of the propagation constant for a plane wave. ... Some common angles, measured in radians. ... In signal processing, a general sinusoidal signal with constant amplitude can be defined as: where is the amplitude, and is the instantaneous phase (or local phase or simply phase) . The simplest useful form is: which is effectively the same as the cyclical form: , where mod is the Modulo_operation. ...

and

The Variation of Complex Reflection Coefficient with Position Along the Line

The complex voltage reflection coefficient is defined as the ratio of the reflected wave to the incident (or forward) wave. Therefore

where C is also a constant. For a uniform transmission line (in which is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is lossy ( is finite) this is represented on the Smith Chart by a spiral path. In most Smith Chart problems however, losses can be assumed negligible () and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes This article or section does not cite its references or sources. ... This article does not cite its references or sources. ...

The phase constant may also be written as

where is the wavelength within the transmission line at the test frequency. Therefore

This equation shows that, for a standing wave, the complex reflection coefficient and impedance repeats every half wavelength along the transmission line. 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 0.50.

The Variation of Normalised Impedance with Position Along the Line

If and are the voltage across and the current entering the termination at the end of the transmission line respectively, then

and

.

By dividing these equations and substituting for both the voltage reflection coefficient

and the normalised impedance of the termination represented by the lower case Z, subscript T

gives the result:

.

Alternatively, in terms of the reflection coefficient

These are the equations which are used to construct the Z Smith Chart. Mathematically speaking and are related via a Möbius Transformation. In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...

Both and are expressed in complex numbers without any units. 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. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

may be expressed in magnitude and angle on a polar diagram. 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 reflection 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 equations. The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Look up Unity in Wiktionary, the free dictionary. ...

By substituting the expression for how reflection coefficient changes along an unmatched loss free transmission line

for the loss free case, into the equation for normalised impedance in terms of reflection coefficient

.

and using Euler's identity Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... In mathematics, the term identity has several important uses: identity can refer to an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ...

yields the impedance version transmission line equation for the loss free case:^[8]

where is the impedance 'seen' at the input of a loss free transmission line of length l, terminated with an impedance

Versions of the transmission line equation may be similarly derived for the admittance loss free case and for the impedance and admittance lossy cases.

The Smith Chart graphical equivalent of using the transmission line equation is to normalise , to plot the resulting point on a Z Smith Chart and to draw a circle through that point centred at the Smith Chart centre. The 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.

Regions of the Z Smith Chart

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 counter-clockwise direction for positive angles. The magnitude of a complex number is the length of a straight line drawn from the origin to the point representing it. The Smith Chart uses the same convention, noting that, in the normalised impedance plane, the positive x-axis extends from the center of the Smith Chart at to the point . The region above the x-axis represents inductive impedances and the region below the x-axis represents capacitive impedances. Inductive impedances have positive imaginary parts and capacitive impedances have negative imaginary parts. Fig. ... A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

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. Given that Smith's chart isn't accurate enough, Rafay's chart can be used to find the inverse impedance. Open circuit can mean:- In electronics, where there is nothing connected to a load and no current can flow. ... For alternate meanings see Short circuit (disambiguation) A short circuit (sometimes known as simply a short) is a fault whereby electricity moves through a circuit in an unintended path, usually due to a connection forming where none was expected. ...

Circles of Constant Normalised Resistance and Constant Normalised Reactance

The normalised impedance Smith Chart is composed of two families of circles: circles of constant normalised resistance and circles of constant normalised reactance. In the complex reflection coefficient plane the Smith Chart occupies a circle of unity radius centred at the origin. In cartesian coordinates therefore the circle would pass through the points (1,0) and (-1,0) on the x-axis and the points (0,1) and (0,-1) on the y-axis.

Since both ρ and are complex numbers, in general they may be expressed by the following generic rectangular complex numbers:

Substituting these into the equation relating normalised impedance and complex reflection coefficient:

gives the following result:

.

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.^[9]

The Y Smith Chart

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_T, so

Therefore:

and

The Y Smith Chart appears like the normalised impedance type but with the graphic scaling rotated through , the numeric scaling remaining unchanged.

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. Disambiguation page Complex number Concept in Social Theory ...

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.

Practical Examples

Example points plotted on the normalised impedance Smith Chart

A point with a reflection coefficient magnitude 0.63 and angle , represented in polar form as , is shown as point P₁ on the Smith Chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the graduation and a ruler to draw a line passing through this and the centre of the Smith Chart. The length of the line would then be scaled to P₁ assuming the Smith Chart radius to be unity. For example if the actual radius measured from the paper was 100 mm, the length OP₁ would be 63 mm. Image File history File links Download high-resolution version (2459x2453, 210 KB) Smith Chart Example 2 I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (2459x2453, 210 KB) Smith Chart Example 2 I, the creator of this work, hereby release it into the public domain. ...

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.

Working with Both the Z Smith Chart and the Y Smith Charts

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 impedances (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 180 degrees at the same radius. For example the point P1 in the example representing a reflection coefficient of has a normalised impedance of . 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 180 degrees. Reading the value from the Smith Chart for Q1, remembering that the scaling is now in normalised admittance, gives . Performing the calculation Electrical conductance is the reciprocal of electrical resistance. ... In electrical engineering, the susceptance (B) is the imaginary part of the admittance. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... It has been suggested that Electric reactance be merged into this article or section. ... This article or section does not adequately cite its references or sources. ... This article or section does not adequately cite its references or sources. ...

manually will confirm this.

Once a transformation from impedance to admittance has been performed the scaling changes to normalised admittance until such time that a later transformation back to normalised impedance is performed. In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ...

The table below shows examples of normalised impedances and their equivalent normalised admittances obtained by rotation of the point through . Again these may either be obtained by calculation or using a Smith Chart as shown, converting between the normalised impedance and normalised admittances planes.

Values of complex reflection coefficient plotted on the normalised impedance Smith Chart and their equivalents on the normalised admittance Smith Chart

Image File history File links Download high-resolution version (2459x2453, 209 KB) Example impedance points on a normalised impedance Smith Chart with their equivalent normalised admittance points I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (2459x2453, 209 KB) Example impedance points on a normalised impedance Smith Chart with their equivalent normalised admittance points I, the creator of this work, hereby release it into the public domain. ...

Choice of Smith Chart Type and Component Type

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 whilst impedances in parallel and admittances in series are related by a reciprocal equation. If Z_TS is the equivalent impedance of series impedances and Z_TP is the equivalent impedance of parallel impedances, then

For admittances the reverse is true, that is

Dealing with the reciprocals, especially 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: resistance, inductance and capacitance. Knowing just the characteristic impedance (or characteristic admittance) and test frequency can be used to find the equivalent circuit from any impedance or admittance, or vice versa. The reciprocal function: y = 1/x. ... In the study of electricity, it is often necessary to reduce a complex circuit into a simpler form. ...

Using the Smith Chart to Solve Conjugate Matching Problems With Distributed Components

Usually distributed matching is only feasible at microwave frequencies since, for most components operating at these frequencies, appreciable transmission line dimensions are available in terms of wavelengths. Also the electrical behavior of many lumped components becomes rather unpredictable at these frequencies. Microwaves are electromagnetic waves with wavelengths longer than those of terahertz (THz) frequencies, but relatively short for radio waves. ...

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 following 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 positions.

Smith Chart construction for some distributed transmission line matching

Supposing a loss free air-spaced transmission line of characteristic impedance Z₀ = 50Ω, operating at a frequency of 800 MHz, is terminated with a circuit comprising a 17.5 Ω resistor in series with a 6.5 nanohenry (6.5 nH) inductor. How may the line be matched? Image File history File links Download high-resolution version (2459x2453, 214 KB) Smith Chart distributed line conjugate matching examples I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (2459x2453, 214 KB) Smith Chart distributed line conjugate matching examples I, the creator of this work, hereby release it into the public domain. ...

From the table above, the reactance of the inductor forming part of the termination at 800 MHz is

so the impedance of the combination (Z_T) is given by

and the normalised impedance (z_T)is

This is plotted on the Z Smith Chart at point P₂₀. The line OP₂₀ is extended through to the wavelength scale where it intersects at the point . As the transmission line is loss free, a circle centred at the centre of the Smith Chart is drawn through the point P₂₀ to represent the path of the constant magnitude reflection coefficient due to the termination. At point P₂₁ the circle intersects with the unity circle of constant normalised resistance at

.

The extension of the line OP₂₁ intersects the wavelength scale at , therefore the distance from the termination to this point on the line is given by

Since the transmission line is air-spaced, the wavelength at 800 MHz in the line is the same as that in free space and is given by

where is the velocity of electromagnetic radiation in free space and is the frequency in hertz. The result gives , making the position of the matching component 29.6 mm from the load. A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic...

The conjugate match for the impedance at P₂₁ () is

z_match = − j(1.52),

As the Smith Chart is still in the normalised impedance plane, from the table above a series capacitor is required where

Rearranging, we obtain

.

Substitution of known values gives

To match the termination at 800 MHz, a series capacitor of 2.6 pF must be placed in series with the transmission line at a distance of 29.6 mm from the termination.

An alternative shunt match could be calculated after performing a Smith Chart transformation from normalised impedance to normalised admittance. Point Q₂₀ is the equivalent of P₂₀ 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). However, the extension of the line OQ₂₀ through to the wavelength scale gives . The earliest point at which a shunt conjugate match could be introduced,moving towards the generator, would be at Q₂₁, the same position as the previous P₂₁, but this time representing a normalised admittance given by

.

The distance along the transmission line is in this case

which converts to 123 mm.

The conjugate matching component is required to have a normalised admittance (y_match) of

.

From the table it can be seen that a negative admittance would require to be an inductor, connected in parallel with the transmission line. If its value is , then

This gives the result

A suitable inductive shunt matching would therefore be a 6.5 nH inductor in parallel with the line positioned at 123 mm from the load.

Using the Smith Chart to Analyse Lumped Element Circuits

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 analyse such circuits in which case the movements around the chart are generated by the (normalised) 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. The following circuit will be analysed using a Smith Chart at an operating frequency of 100 MHz. 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. Despite there being no transmission line as such, a system impedance must still be defined to enable normalisation and de-normalisation calculations and is a good choice here as . If there were very different values of resistance present a value closer to these might be a better choice.

A lumped element circuit which may be analysed using a Smith Chart

Smith Chart with graphical construction for analysis of a lumped circuit

The analysis starts with a Z Smith Chart looking into R₁ only with no other components present. As is the same as the system impedance, this is represented by a point at the centre of the Smith Chart. The first transformation is OP₁ along the line of constant normalised resistance in this case the addition of a normalised reactance of -j0.80, corresponding to a series capacitor of 40 pF. Points with suffix P are in the Z plane and points with suffix Q are in the Y plane. Therefore transformations P₁ to Q₁ and P₃ to Q₃ are from the Z Smith Chart to the Y Smith Chart and transformation Q₂ to P₂ is from the Y Smith Chart to the Z Smith Chart. 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. Image File history File links Download high-resolution version (1635x651, 3 KB) Lumped element circuit for Smith Chart analysis I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (1635x651, 3 KB) Lumped element circuit for Smith Chart analysis I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (2459x2453, 213 KB) Smith Chart construction example I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (2459x2453, 213 KB) Smith Chart construction example I, the creator of this work, hereby release it into the public domain. ...

References

1. ^ Smith, P. H.; Transmission Line Calculator; Electronics, Vol. 12, No. 1, pp 29-31, January 1931
2. ^ Smith, P. H.; An Improved Transmission Line Calculator; Electronics, Vol. 17, No. 1, p 130, January 1944
3. ^ Ramo, Whinnery and Van Duzer (1965); "Fields and Waves in Communications Electronics"; John Wiley & Sons; pp 35-39. ISBN
4. ^ Pozar, David M. (2005); Microwave Engineering, Third Edition (Intl. Ed.); John Wiley & Sons, Inc.; pp 64-71. ISBN 0-471-44878-8.
5. ^ Gonzalez, Guillermo (1997); Microwave Transistor Amplifiers Analysis and Design, Second Edition; Prentice Hall NJ; pp 93-103. ISBN 0-13-254335-4.
6. ^ Gonzalez, Guillermo (1997) (op. cit);pp 98-101
7. ^ Gonzalez, Guillermo (1997) (op. cit);p 97
8. ^ Hayt, William H Jr.; "Engineering Electromagnetics" Fourth Ed;McGraw-Hill International Book Company; pp 428 433. ISBN 0-07-027395-2.
9. ^ Davidson, C. W.;"Transmission Lines for Communications with CAD Programs";Macmillan; pp 80-85. ISBN 0-333-47398-1

P.H.Smith 1969 Electronic Applications of the Smith Chart. Kay Electric Company

External links

• A Collection of Smith Chart Resources Tutorials, graphics and other info on Smith Chart
• linSmith Smith charting program for Linux.
• Smith Chart Print free Smith Charts from your computer.
• Black Magic Smith Chart - Vector-graphic (infinitely scalable) Smith Chart for practical use.
• The Java Smith-Chart-Tool - A free Java-Tool to paint s-parameters in a Smith-Chart.
• Smith Chart Applet - a Java applet that simulates operations on a Smith chart
• Smith Excel Graph plots reflection coefficient data in real and imaginary formats on a customizable Smith Chart (Microsoft Excel Spreadsheet 53K)
• PostScript functions Functions to plot dots, lines, gamma circle, constant real and imaginary path in PostScript format to make vectorial images.
• An online educational interactive Smith chart. A choice of impedance and admittance charts with a chart marker.
• A Matlab m file to generate a standard color smith chart.