Overview
The Smith chart, invented by Phillip H. Smith (1905–1987) is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. 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 simultaneously display multiple parameters including impedances, admittances, reflection coefficients, 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
The Smith chart, invented by Phillip H. Smith (1905–1987) is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. 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 simultaneously display multiple parameters including impedances, admittances, reflection coefficients, 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
Mathematical basis
Actual and normalised impedance and admittanceA transmission line with a characteristic impedance of may be universally considered to have acharacteristic 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 (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.
[edit]The normalised impedance Smith chartUsing 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 exponentialnotation:
andwhere
is the temporal part of the wave is the spatial 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
is the propagation constant which has units 1/mwhere
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
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 non-zero) 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
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.
[edit]The variation of normalised impedance with position along the lineIf 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 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 identity
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.
[edit]Regions of the Z Smith chartIf 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 (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.
[edit]Circles of constant normalised resistance and constant normalised reactanceThe 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 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.[9]
[edit]The Y Smith chartThe 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 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.
Actual and normalised impedance and admittanceA transmission line with a characteristic impedance of may be universally considered to have acharacteristic 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 (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.
[edit]The normalised impedance Smith chartUsing 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 exponentialnotation:
andwhere
is the temporal part of the wave is the spatial 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
is the propagation constant which has units 1/mwhere
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
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 non-zero) 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
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.
[edit]The variation of normalised impedance with position along the lineIf 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 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 identity
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.
[edit]Regions of the Z Smith chartIf 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 (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.
[edit]Circles of constant normalised resistance and constant normalised reactanceThe 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 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.[9]
[edit]The Y Smith chartThe 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 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.
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