Smith chart
Encyclopedia
The Smith chart, invented by Phillip H. Smith (1905-1987), is a graphical aid or nomogram
Nomogram
A nomogram, nomograph, or abac is a graphical calculating device developed by P.E. Elyasberg, a two-dimensional diagram designed to allow the approximate graphical computation of a function: it uses a coordinate system other than Cartesian coordinates...

 designed for electrical and electronics engineers
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

 specializing in radio frequency
Radio frequency
Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...

 (RF) engineering to assist in solving problems with transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...

s and matching
Impedance matching
In electronics, impedance matching is the practice of designing the input impedance of an electrical load to maximize the power transfer and/or minimize reflections from the load....

 circuits. 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
Table (information)
A table is a means of arranging data in rows and columns.Production % of goalNorth 4087102%South 4093110% The use of tables is pervasive throughout all communication, research and data analysis. Tables appear in print media, handwritten notes, computer software, architectural...

 information. The Smith chart can be used to represent many parameters including impedances
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

, admittance
Admittance
In electrical engineering, the admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance . The SI unit of admittance is the siemens...

s, reflection coefficient
Reflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...

s, scattering parameters
Scattering parameters
Scattering parameters or S-parameters describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals....

, noise figure
Noise figure
Noise figure is a measure of degradation of the signal-to-noise ratio , caused by components in a radio frequency signal chain. The noise figure is defined as the ratio of the output noise power of a device to the portion thereof attributable to thermal noise in the input termination at standard...

 circles, constant gain contours and regions for unconditional stability
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

. The Smith chart is most frequently used at or within the unity radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 region. However, the remainder is still mathematically relevant, being used, for example, in oscillator
Electronic oscillator
An electronic oscillator is an electronic circuit that produces a repetitive electronic signal, often a sine wave or a square wave. They are widely used in innumerable electronic devices...

 design and stability
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

 analysis.

Overview

The Smith chart is plotted on the complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 reflection coefficient
Reflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...

 plane in two dimensions
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

 and is scaled in normalised impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

 (the most common), normalised admittance
Admittance
In electrical engineering, the admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance . The SI unit of admittance is the siemens...

 or both, using different colours to distinguish between them. These are often known as the Z, Y and YZ Smith charts respectively. Normalised scaling allows the Smith chart to be used for problems involving any characteristic
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...

 or system impedance which is represented by the center point of the chart. The most commonly used normalization impedance is 50 ohm
Ohm
The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...

s. Once an answer is obtained through the graphical constructions described below, it is straightforward to convert between normalised impedance (or normalised 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
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

 scaling in wavelengths and degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

s. The wavelengths scale is used in distributed component
Distributed element model
In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...

 problems and represents the distance measured along the transmission line connected between the generator
Signal generator
Signal generators, also known variously as function generators, RF and microwave signal generators, pitch generators, arbitrary waveform generators, digital pattern generators or frequency generators are electronic devices that generate repeating or non-repeating electronic signals...

 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
Lumped element model
The lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions...

 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
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....

 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
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

 at a time, the result being represented by a point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

. 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
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

.

A locus of points on a Smith chart covering a range of frequencies can be used to visually represent:
  • how capacitive
    Capacitance
    In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

     or how inductive
    Inductance
    In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

     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 locus of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these.

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
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

 of impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

 is the ohm
Ohm
The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...

 with the symbol of the upper case Greek letter
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

 Omega
Omega
Omega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to omicron, which means "little O"...

 (Ω) and the SI unit for admittance
Admittance
In electrical engineering, the admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance . The SI unit of admittance is the siemens...

 is the siemens
Siemens (unit)
The siemens is the SI derived unit of electric conductance and electric admittance. Conductance and admittance are the reciprocals of resistance and impedance respectively, hence one siemens is equal to the reciprocal of one ohm, and is sometimes referred to as the mho. In English, the term...

 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.

The normalised impedance Smith chart

Using transmission line theory, if a transmission line is terminated
Electrical termination
Electrical termination of a signal involves providing a terminator at the end of a wire or cable to prevent an RF signal from being reflected back from the end, causing interference...

 in an impedance () which differs from its characteristic impedance (), a standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...

 will be formed on the line comprising the resultant of both the forward () and the reflected () waves. Using complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

 notation: and
where is the temporal
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

 part of the wave is the spatial part of the wave and where is the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

 in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s per second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

 (rad/s) is the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

 in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

 (Hz) is the time in seconds (s) and are constant
Constant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...

s is the distance measured along the transmission line from the generator in metres (m)
Also is the propagation constant
Propagation constant
The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...

 which has units 1/m
where is the attenuation constant in neper
Neper
The neper is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. It has the unit symbol Np. The unit's name is derived from the name of John Napier, the inventor of logarithms...

s per metre (Np/m) is the phase constant in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s 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
Instantaneous phase
The notions of Instantaneous Phase and Instantaneous Frequency are important concepts in Signal Processing that occur in the context of the representation and analysis of time-varying signals....

, but it is conventional and understood to omit it. Therefore 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
Spiral
In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

 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
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.

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.

may be expressed in magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

 and angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 on a polar diagram
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

. 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.

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
yields the impedance version transmission line equation for the loss free case:
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
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

 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
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

 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 (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
Open circuit
The term Open circuit may refer to:*Open-circuit scuba, a type of SCUBA-diving equipment where the user breathes from the set and then exhales to the surroundings without recycling the exhaled air...

 or short circuit
Short circuit
A short circuit in an electrical circuit that allows a current to travel along an unintended path, often where essentially no electrical impedance is encountered....

 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.

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 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.

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 yT is the reciprocal of the normalised impedance zT, so
Therefore:
and
The Y Smith chart appears like the normalised impedance type but with the graphic scaling rotated through 180°, 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
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

 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.

Practical examples

A point with a reflection coefficient magnitude 0.63 and angle 60° 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.

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.
Some examples of points plotted on the normalised impedance Smith chart
Point Identity Reflection Coefficient (Polar Form) Normalised Impedance (Rectangular Form)
P1 (Inductive)
P2 (Inductive)
P3 (Capacitive)

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 susceptance
Susceptance
In electrical engineering, susceptance is the imaginary part of admittance. The inverse of admittance is impedance and the real part of admittance is conductance. In SI units, susceptance is measured in siemens...

s) and sometimes it is more convenient to work with impedances (representing resistances
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

 and reactances). Solving a typical matching problem will often require several changes between both types of Smith chart, using normalised impedance for series
Series and parallel circuits
Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently. Components connected in series are connected along a single path, so the same current flows through all of the...

 elements and normalised admittances for parallel
Series and parallel circuits
Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently. Components connected in series are connected along a single path, so the same current flows through all of the...

 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
manually will confirm this.

Once a transformation
Transformation (mathematics)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...

 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.

The table below shows examples of normalised impedances and their equivalent normalised admittances obtained by rotation of the point through 180°. 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 reflection coefficient as normalised impedances and the equivalent normalised admittances
Normalised Impedance Plane Normalised Admittance Plane
P1 () Q1 ()
P10 () Q10 ()


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 is the equivalent impedance of series impedances and is the equivalent impedance of parallel impedances, then
For admittances the reverse is true, that is
Dealing with the reciprocals
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

, 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. Using just the characteristic impedance (or characteristic admittance) and test frequency an equivalent circuit
Equivalent circuit
In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that is the simplest form of a more complex circuit in order to aid analysis. In its most common...

 can be found and vice versa.
Expressions for Real and Normalised Impedance and Admittance with Characteristic Impedance Z0 or Characteristic Admittance Y0
Element Type Impedance (Z or z) or Reactance (X or x) Admittance (Y or y) or Susceptance (B or b)
Real () Normalised (No Unit) Real (S) Normalised (No Unit)
Resistance (R)
Inductance (L)
Capacitance (C)

Using the Smith chart to solve conjugate matching problems with distributed components

Usually distributed matching is only feasible at microwave
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...

 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.

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.
Supposing a loss-free air-spaced transmission line of characteristic impedance , operating at a frequency of 800 MHz, is terminated with a circuit comprising a 17.5 resistor in series with a 6.5 nanohenry (6.5 nH) inductor. How may the line be matched?

From the table above, the reactance of the inductor forming part of the termination at 800 MHz is
so the impedance of the combination () is given by
and the normalised impedance () 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
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

and is the frequency in hertz. The result gives , making the position of the matching component 29.6 mm from the load.

The conjugate match for the impedance at P21 () is
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 () of.
From the table it can be seen that a negative admittance would require 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 analyze 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 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. The following circuit will be analyzed 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 normalization and de-normalization 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.
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 normalized resistance in this case the addition of a normalized 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.
Smith chart steps for analysing a lumped element circuit
Transformation Plane x or y Normalized Value Capacitance/Inductance Formula to Solve Result
Capacitance (Series)
Inductance (Shunt)
Z Capacitance (Series)
Y Capacitance (Shunt)

3D Smith Chart

A generalized 3D Smith chart based on the extended complex plane, Riemann sphere and inversive geometry was recently proposed. The chart unifies the passive and active circuit design on little and big circles on the surface of a unit sphere using the stereographic conformal mapping of the reflection coefficient's generalized plane. Considering the point at infinity, the space of the new chart includes all possible loads. The north pole is the perfect matching point, while the south pole is the perfect mismatch point.

External links

  • Excel Smith Chart Non-commercial, interactive Smith Chart that looks best in Excel 2007+.
The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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