Smith chart

The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator 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. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, formula_1 scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. 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.
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 is still a very useful method of showing 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 plot measured results on a Smith chart display.
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. Normalised scaling allows the Smith chart to be used for problems involving any characteristic or system impedance which is represented by the center point of the chart. 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 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 a scale around its circumference or periphery which is graduated 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.
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 prerequisites 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. 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.
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.
Mathematical basis.
Actual and normalised impedance and admittance.
A transmission line with a characteristic impedance of formula_2 may be universally considered to have a characteristic admittance of formula_3 where
Any impedance, formula_5 expressed in ohms, may be normalised by dividing it by the characteristic impedance, so the normalised impedance using the lower case "z"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 (Ω) 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.
The normalised impedance Smith chart.
Using transmission-line theory, if a transmission line is terminated in an impedance (formula_5) which differs from its characteristic impedance (formula_2), a standing wave will be formed on the line comprising the resultant of both the incident or forward (formula_10) and the reflected or reversed (formula_11) waves. Using complex exponential notation:
where
Also
where
The Smith chart is used with one frequency (formula_26) at a time, and only for one moment (formula_27) at a time, so the temporal part of the phase (formula_14) is fixed. All terms are actually multiplied by this to obtain the instantaneous phase, but it is conventional and understood to omit it. Therefore,
where formula_20 and formula_21 are respectively the forward and reverse voltage amplitudes at the load.
The variation of complex reflection coefficient with position along the line.
The complex voltage reflection coefficient formula_33 is defined as the ratio of the reflected wave to the incident (or forward) wave. Therefore,
where is also a constant.
For a uniform transmission line (in which formula_35 is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is lossy (formula_24 is non-zero) this is represented on the Smith chart by a spiral path. In most Smith chart problems however, losses can be assumed negligible (formula_37) and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes 
where formula_39 is the reflection coefficient at the load, and formula_22 is the line length from the load to the location where the reflection coefficient is measured. The phase constant formula_25 may also be written as 
where formula_43 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 formula_45 and formula_46 are the voltage across and the current entering the termination at the end of the transmission line respectively, then
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 formula_53 and formula_54 are related via a Möbius transformation.
Both formula_33 and formula_56 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.
formula_33 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.
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 formula
yields the impedance-version transmission-line equation for the loss free case: 
where
formula_62 is the impedance 'seen' at the input of a loss free transmission line of length formula_63, terminated with an impedance formula_64
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 formula_64, 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 counterclockwise 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 formula_66 to the point formula_67. The region above the x-axis represents inductive impedances (positive imaginary parts) and the region below the -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.
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 formula_68 and formula_69 are complex numbers, in general they may be written as:
with "a", "b", "c" and "d" real 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 "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 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 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 formula_77, is shown as point P1 on the Smith chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the formula_78 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.
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 formula_77 has a normalised impedance of formula_80. 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 formula_81. Performing the calculation
manually will confirm this.
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.
The table below shows examples of normalised impedances and their equivalent normalised admittances obtained by rotation of the point through 180°. Again, these may be obtained either by calculation or using a Smith chart as shown, converting between the normalised impedance and normalised admittances planes.
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 while impedances in parallel and admittances in series are related by a reciprocal equation. If formula_83 is the equivalent impedance of series impedances and formula_84 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. Using just the characteristic impedance (or characteristic admittance) and test frequency an equivalent circuit can be found and vice versa.
Using the Smith chart to solve conjugate matching problems with distributed components.
Distributed matching becomes feasible and is sometimes required when the physical size of the matching components is more than about 5% of a wavelength at the operating frequency. 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 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 formula_89, operating at a frequency of 800 MHz, is terminated with a circuit comprising a 17.5 formula_90 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 (formula_92) is given by
and the normalised impedance (formula_94) 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 formula_96. 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 formula_98, 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 formula_101 is the velocity of electromagnetic radiation in free space and formula_18 is the frequency in hertz. The result gives formula_103, making the position of the matching component 29.6 mm from the load.
The conjugate match for the impedance at P21 (formula_104) is 
As the Smith chart is still in the normalised impedance plane, from the table above a series capacitor formula_106 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 formula_111. 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 (formula_114) 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 formula_116, 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 formula_119 is a good choice here as formula_120. 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 formula_120 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 -"j"0.80, 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 "P"1 to "Q"1 and "P"3 to "Q"3 are from the Z Smith chart to the Y Smith chart and transformation "Q"2 to "P"2 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. 
3D Smith chart.
A generalized 3D Smith chart based on the extended complex plane (Riemann sphere) and inversive geometry was proposed in 2011. 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 map 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.