Smith chart

Overview

A network analyzer (HP 8720A) showing a Smith chart.

The Smith chart is plotted on the complex reflection coefficient plane in two dimensions and is scaled in normalised impedance (the most common), normalisedadmittance 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 anycharacteristic 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 circumferential scaling in wavelengthsand 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 pre-requisites for RF engineers.

As impedances and admittances change with frequency, problems using the Smith chart can only be solved manually using onefrequency 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:

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 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 $Z_0\,$ may be universally considered to have a characteristic admittance of $Y_0\,$ where

$Y_0 = \frac{1}{Z_0}\,$

Any impedance, $Z_T\,$ 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

$z_T = \frac{Z_T}{Z_0}\,$

Similarly, for normalised admittance

$y_T = \frac{Y_T}{Y_0}\,$

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 thesiemens 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 ( $Z_T\,$ ) which differs from its characteristic impedance ( $Z_0\,$ ), a standing wave will be formed on the line comprising the resultant of both the forward ( $V_F\,$ ) and the reflected ( $V_R\,$ ) waves. Using complex exponential notation:

$V_F = A \exp(j \omega t)\exp(-\gamma l)\,$ and

$V_R = B \exp(j \omega t)\exp(\gamma l)\,$

where

$\exp(j \omega t)\,$ is the temporal part of the wave

$\exp(-\gamma l)\,$ is the spatial part of the wave and

$\omega = 2 \pi f\,$ where

$\omega\,$ is the angular frequency in radians per second (rad/s)

$f\,$ is the frequency in hertz (Hz)

$t\,$ is the time in seconds (s)

$A\,$ and $B\,$ are constants

$l\,$ is the distance measured along the transmission line from the generator in metres (m)

Also

$\gamma = \alpha + j\beta\,$ is the propagation constant which has units 1/m

where

$\alpha\,$ is the attenuation constant in nepers per metre (Np/m)

$\beta\,$ 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 ( $\exp(\omega t)\,$ ) is fixed. All terms are actually multiplied by this to obtain the instantaneous phase, but it is conventional and understood to omit it. Therefore

$V_F = A \exp(-\gamma l)\,$ and

$V_R = B \exp(\gamma l)\,$

The variation of complex reflection coefficient with position along the line

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

$\rho = \frac{V_R}{V_F} = \frac{B \exp(\gamma l)}{A \exp(-\gamma l)} =C \exp(2 \gamma l)\,$

where C is also a constant.

For a uniform transmission line (in which $\gamma\,$ is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is lossy ( $\alpha\,$ is finite) this is represented on the Smith chart by a spiral path. In most Smith chart problems however, losses can be assumed negligible ( $\alpha = 0\,$ ) and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes

$\rho = \rho_0 \exp(2j \beta l)\,$

The phase constant $\beta\,$ may also be written as

$\beta = \frac{2\pi}{\lambda}\,$

where $\lambda\,$ is the wavelength within the transmission line at the test frequency.

Therefore

$\rho = \rho_0 \exp\left(\frac{4 j \pi}{\lambda}l\right)\,$

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 $V\,$ and $I\,$ are the voltage across and the current entering the termination at the end of the transmission line respectively, then

$V_F + V_R = V\,$ and

$V_F - V_R = Z_0I\,$ .

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

$\rho=\frac{V_R}{V_F}\,$

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

$z_T=\frac{V}{Z_0I}\,$

gives the result:

$z_T=\frac{1+\rho}{1-\rho}\,$ .

Alternatively, in terms of the reflection coefficient

$\rho=\frac{z_T-1}{z_T+1}\,$

These are the equations which are used to construct the Z Smith chart. Mathematically speaking $\rho\,$ and $z_T\,$ are related via a Möbius transformation.

Both $\rho\,$ and $z_T\,$ 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.

$\rho\,$ 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

$\rho = \frac{B \exp(\gamma l)}{A \exp(-\gamma l)} =\frac{B \exp(j \beta l)}{A \exp(-j \beta l)}\,$

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

$z_T=\frac{1+\rho}{1-\rho}\,$ .

and using Euler's identity

$\exp(j\theta) = \cos \theta + j \sin \theta\,$

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

$Z_{IN} = Z_0 \frac{Z_L + j Z_0 \tan (\beta l)}{Z_0 + j Z_L \tan (\beta l)}\,$

where $Z_{IN}\,$ is the impedance 'seen' at the input of a loss free transmission line of length l, terminated with an impedance $Z_L\,$

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 $Z_L\,$ , 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 $z_T = 1 \pm j0\,$ to the point $z_T = \infty \pm j\infty\,$ . The region above the x-axis represents inductive impedances (positive imaginary parts) and the region below the x-axis represents capacitive impedances (negative imaginary parts).

If the termination is perfectly matched, the reflection coefficient will be zero, represented effectively by a circle of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open circuit or short circuit the magnitude of the reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle.

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 $z\,$ are complex numbers, in general they may be expressed by the following generic rectangular complex numbers:

$z = a + jb\,$

$\rho = c + jd\,$

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

$\rho=\frac{z-1}{z+1}\,$

gives the following result:

$\rho = c + jd = \frac{a^2+b^2-1}{(a+1)^2+b^2} + j \left(\frac{2b}{(a+1)^2+b^2}\right)\,$ .

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

$y_T=\frac{1}{z_T}\,$

Therefore:

$y_T = \frac{1-\rho}{1+\rho}\,$

and

$\rho = \frac{1-y_T}{1+y_T}\,$

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

Example points plotted on the normalised impedance Smith chart

A point with a reflection coefficient magnitude 0.63 and angle 60° represented in polar form as $0.63\angle60^\circ\,$ , is shown as point P₁ on the Smith chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the $\angle60^\circ\,$ 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)	$0.63\angle60^\circ\,$	$0.80 + j1.40\,$
P₂ (Inductive)	$0.73\angle125^\circ\,$	$0.20 + j0.50\,$
P₃ (Capacitive)	$0.44\angle-116^\circ\,$	$0.50 - j0.50\,$

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 andsusceptances) 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 serieselements 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 $0.63\angle60^\circ\,$ has a normalised impedance of $z_P = 0.80 + j1.40\,$ . 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 $y_P = 0.30 - j0.54\,$ . Performing the calculation

$y_T=\frac{1}{z_T}\,$

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.

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 ( $z = 0.80 + j1.40\,$ )	Q₁ ( $y = 0.30 - j0.54\,$ )
P₁₀ ( $z = 0.10 + j0.22\,$ )	Q₁₀ ( $y = 1.80 - j3.90\,$ )

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

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 $Z T S$ is the equivalent impedance of series impedances and $Z T P$ is the equivalent impedance of parallel impedances, then

$Z_{TS} = Z_1 + Z_2 + Z_3 + ... \,$

$\frac{1}{Z_{TP}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \frac{1}{Z_3} + ... \,$

For admittances the reverse is true, that is

$Y_{TP} = Y_1 + Y_2 + Y_3 + ... \,$

$\frac{1}{Y_{TS}} = \frac{1}{Y_1} + \frac{1}{Y_2} + \frac{1}{Y_3} + ... \,$

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 topographysupports 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.

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)

Sent from my iPad

NoBs - أخبار التجارة و الأعمال

Wednesday, August 10, 2011