Smith chart

A transmission line with a characteristic impedance of ${\displaystyle Z_{0}\,}$ may be universally considered to have a characteristic admittance of ${\displaystyle Y_{0}\,}$ where

${\displaystyle Y_{0}={\frac {1}{Z_{0}}}\,}$

Any impedance, ${\displaystyle Z_{\text{T}}\,}$ 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

${\displaystyle z_{\text{T}}={\frac {Z_{\text{T}}}{Z_{0}}}\,}$

Similarly, for normalised admittance

${\displaystyle y_{\text{T}}={\frac {Y_{\text{T}}}{Y_{0}}}\,}$

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 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.

Transmission lines terminated by an open circuit (top) and a short circuit (bottom). A pulse reflects perfectly off both these terminations, but the sign of the reflected voltage is opposite in the two cases. Black dots represent electrons, and arrows show the electric field.

Using transmission-line theory, if a transmission line is terminated in an impedance (${\displaystyle Z_{\text{T}}\,}$) which differs from its characteristic impedance (${\displaystyle Z_{0}\,}$), a standing wave will be formed on the line comprising the resultant of both the incident or forward (${\displaystyle V_{\text{F}}\,}$) and the reflected or reversed (${\displaystyle V_{\text{R}}\,}$) waves. Using complex exponential notation:

${\displaystyle V_{\text{F}}=A\exp(j\omega t)\exp(+\gamma \ell )~\,}$ and

${\displaystyle V_{\text{R}}=B\exp(j\omega t)\exp(-\gamma \ell )\,}$

where

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

${\displaystyle \exp(\pm \gamma \ell )\,}$ is the spatial part of the wave and

${\displaystyle \omega =2\pi f\,}$ where

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

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

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

${\displaystyle A\,}$ and ${\displaystyle B\,}$ are constants

${\displaystyle \ell \,}$ is the distance measured along the transmission line from the load toward the generator in metres (m)

Also

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

where

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

${\displaystyle \beta \,}$ is the phase constant in radians per metre (rad/m)

The Smith chart is used with one frequency (${\displaystyle \omega }$) at a time, and only for one moment (${\displaystyle t}$) at a time, so the temporal part of the phase (${\displaystyle \exp(j\omega t)\,}$) is fixed. All terms are actually multiplied by this to obtain the instantaneous phase, but it is conventional and understood to omit it. Therefore,

${\displaystyle V_{\text{F}}=A\exp(+\gamma \ell )\,}$ and

${\displaystyle V_{\text{R}}=B\exp(-\gamma \ell )\,}$

where ${\displaystyle A\,}$ and ${\displaystyle B\,}$ are respectively the forward and reverse voltage amplitudes at the load.

Looking towards a load through a length ${\displaystyle \ell \,}$ of lossless transmission line, the impedance changes as ${\displaystyle \ell \,}$ increases, following the blue circle. (This impedance is characterized by its reflection coefficient ${\displaystyle V_{\text{reflected}}/V_{\text{incident}}}$.) The blue circle, centered within the impedance Smith chart, is sometimes called an SWR circle (short for constant standing wave ratio).

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

${\displaystyle \Gamma ={\frac {V_{\text{R}}}{V_{\text{F}}}}={\frac {B\exp(-\gamma \ell )}{A\exp(+\gamma \ell )}}=C\exp(-2\gamma \ell )\,}$

where C is also a constant.

For a uniform transmission line (in which ${\displaystyle \gamma \,}$ is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is lossy (${\displaystyle \alpha \,}$ 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 (${\displaystyle \alpha =0\,}$) and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes

${\displaystyle \Gamma =\Gamma _{\text{L}}\exp(-2j\beta \ell )\,}$

where ${\displaystyle \Gamma _{\text{L}}\,}$ is the reflection coefficient at the load, and ${\displaystyle \ell \,}$ is the line length from the load to the location where the reflection coefficient is measured. The phase constant ${\displaystyle \beta \,}$ may also be written as

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

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

Therefore,

${\displaystyle \Gamma =\Gamma _{\text{L}}\exp \left({\frac {-4j\pi }{\lambda }}\ell \right)\,}$

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 .

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

${\displaystyle V_{\text{F}}+V_{\text{R}}=V\,}$ and

${\displaystyle V_{\text{F}}-V_{\text{R}}=Z_{0}I\,}$.

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

${\displaystyle \Gamma ={\frac {V_{\text{R}}}{V_{\text{F}}}}\,}$

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

${\displaystyle z_{\text{T}}={\frac {V}{Z_{0}I}}\,}$

gives the result:

${\displaystyle z_{\text{T}}={\frac {1+\Gamma }{1-\Gamma }}\,}$.

Alternatively, in terms of the reflection coefficient

${\displaystyle \Gamma ={\frac {z_{\text{T}}-1}{z_{\text{T}}+1}}\,}$

These are the equations which are used to construct the Z Smith chart. Mathematically speaking ${\displaystyle \Gamma \,}$ and ${\displaystyle z_{\text{T}}\,}$ are related via a Möbius transformation.

Both ${\displaystyle \Gamma \,}$ and ${\displaystyle z_{\text{T}}\,}$ 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.

${\displaystyle \Gamma \,}$ 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

${\displaystyle \Gamma ={\frac {B\exp(-\gamma \ell )}{A\exp(\gamma \ell )}}={\frac {B\exp(-j\beta \ell )}{A\exp(j\beta \ell )}}\,}$

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

${\displaystyle z_{\text{T}}={\frac {1+\Gamma }{1-\Gamma }}\,}$.

and using Euler's formula

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

yields the impedance-version transmission-line equation for the loss free case:[10]

${\displaystyle Z_{\text{in}}=Z_{0}{\frac {Z_{\text{L}}+jZ_{0}\tan(\beta \ell )}{Z_{0}+jZ_{\text{L}}\tan(\beta \ell )}}\,}$

where ${\displaystyle Z_{\text{in}}\,}$ is the impedance 'seen' at the input of a loss free transmission line of length ${\displaystyle \ell }$, terminated with an impedance ${\displaystyle Z_{\text{L}}\,}$

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 ${\displaystyle Z_{\text{L}}\,}$, 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.

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 ${\displaystyle z_{\text{T}}=1\pm j0\,}$ to the point ${\displaystyle z_{\text{T}}=\infty \pm j\infty \,}$. 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.

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 ${\displaystyle \Gamma }$ and ${\displaystyle z\,}$ are complex numbers, in general they may be written as:

${\displaystyle z=a+jb\,}$

${\displaystyle \Gamma =c+jd\,}$

with a, b, c and d real numbers.

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

${\displaystyle \Gamma ={\frac {z-1}{z+1}}\,}$

gives the following result:

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

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.[11]

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

${\displaystyle y_{\text{T}}={\frac {1}{z_{\text{T}}}}\,}$

Therefore:

${\displaystyle y_{\text{T}}={\frac {1-\Gamma }{1+\Gamma }}\,}$

and

${\displaystyle \Gamma ={\frac {1-y_{\text{T}}}{1+y_{\text{T}}}}\,}$

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.

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 ${\displaystyle 0.63\angle 60^{\circ }\,}$, is shown as point P₁ on the Smith chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the ${\displaystyle \angle 60^{\circ }\,}$ 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)	${\displaystyle 0.63\angle 60^{\circ }\,}$	${\displaystyle 0.80+j1.40\,}$
P2 (Inductive)	${\displaystyle 0.73\angle 125^{\circ }\,}$	${\displaystyle 0.20+j0.50\,}$
P3 (Capacitive)	${\displaystyle 0.44\angle -116^{\circ }\,}$	${\displaystyle 0.50-j0.50\,}$

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 ${\displaystyle 0.63\angle 60^{\circ }\,}$ has a normalised impedance of ${\displaystyle z_{P}=0.80+j1.40\,}$. 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 ${\displaystyle y_{P}=0.30-j0.54\,}$. Performing the calculation

${\displaystyle y_{\text{T}}={\frac {1}{z_{\text{T}}}}\,}$

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.

Values of reflection coefficient as normalised impedances and the equivalent normalised admittances

Normalised Impedance Plane	Normalised Admittance Plane
P1 (${\displaystyle z=0.80+j1.40\,}$)	Q1 (${\displaystyle y=0.30-j0.54\,}$)
P10 (${\displaystyle z=0.10+j0.22\,}$)	Q10 (${\displaystyle y=1.80-j3.90\,}$)

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

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 ${\displaystyle Z_{\text{TS}}}$ is the equivalent impedance of series impedances and ${\displaystyle Z_{\text{TP}}}$ is the equivalent impedance of parallel impedances, then

${\displaystyle Z_{\text{TS}}=Z_{1}+Z_{2}+Z_{3}+...\,}$

${\displaystyle {\frac {1}{Z_{\text{TP}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+{\frac {1}{Z_{3}}}+...\,}$

For admittances the reverse is true, that is

${\displaystyle Y_{\text{TP}}=Y_{1}+Y_{2}+Y_{3}+...\,}$

${\displaystyle {\frac {1}{Y_{\text{TS}}}}={\frac {1}{Y_{1}}}+{\frac {1}{Y_{2}}}+{\frac {1}{Y_{3}}}+...\,}$

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 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.

Expressions for Impedance and Admittance
Normalised by Impedance Z₀ or Admittance Y₀

Element Type	Impedance (Z or z) or Reactance (X or x)		Admittance (Y or y) or Susceptance (B or b)
	Real (${\displaystyle \Omega \,}$)	Normalised (No Unit)	Real (S)	Normalised (No Unit)
Resistance (R)	${\displaystyle Z=R\,}$	${\displaystyle z={\frac {R}{Z_{0}}}=RY_{0}\,}$	${\displaystyle Y=G={\frac {1}{R}}\,}$	${\displaystyle y=g={\frac {1}{RY_{0}}}={\frac {Z_{0}}{R}}\,}$
Inductance (L)	${\displaystyle Z=jX_{\text{L}}=j\omega L\,}$	${\displaystyle z=jx_{\text{L}}=j{\frac {\omega L}{Z_{0}}}=j\omega LY_{0}\,}$	${\displaystyle Y=-jB_{\text{L}}={\frac {-j}{\omega L}}}$	${\displaystyle y=-jb_{\text{L}}={\frac {-j}{\omega LY_{0}}}={\frac {-jZ_{0}}{\omega L}}\,}$
Capacitance (C)	${\displaystyle Z=-jX_{\text{C}}={\frac {-j}{\omega C}}\,}$	${\displaystyle z=-jx_{\text{C}}={\frac {-j}{\omega CZ_{0}}}={\frac {-jY_{0}}{\omega C}}\,}$	${\displaystyle Y=jB_{\text{C}}=j\omega C\,}$	${\displaystyle y=jb_{\text{C}}=j{\frac {\omega C}{Y_{0}}}=j\omega CZ_{0}\,}$