Smith chart

The Smith chart (english Smith Chart) is a tool of the complex AC circuit analysis, with the calculations of complex resistances ( impedances ) can be attributed to a geometric construction. It was first introduced in 1939 by Phillip Smith.

The Smith chart is also used in the line impedance matching theory. The Smith chart used there differs only in the interpretation of the axes and the axis label from the one shown here.

Construction

The diagram is circular and provided with a complex coordinate system. It is based on the conformal mapping

The complex plane onto itself It results from the definition of the reflection coefficient.

In this figure, the right half-plane is mapped to the interior of the unit circle. The left half-plane is of no importance, since it corresponds to negative ohmic resistors, which do not occur in passive components.

In mathematics, this transformation of a plane is known to another as a Möbius transformation. She obeys the general form

The figure has the special property that the image of a number z (for example ) and their reciprocal point symmetrical to the center of the circle (for example ). ( In electrical engineering, is the symbol for the imaginary unit j used )

The Smith chart can thus be used both as an impedance and admittance chart.

The calculation of a parallel circuit, the inverse value of the total impedance is the sum of the reciprocal values ​​of the partial impedances. These reciprocal of education is thus replaced geometrically in the Smith chart created by mirroring at the center. In the Smith chart is always used standardized variables. This has the advantage that it is independent of variables such as the actual frequency, wavelength, or impedance.

In the transmission line theory, eg impedance matching problems can be reflection coefficient and standing wave ratio ( SWR ) is simply determined from the Smith chart without complex calculation. For this purpose, we measure the length of the connecting line between the origin and the point of intersection of the two circles of the normalized impedance. The phase of the reflection coefficient can be read from the extension of the line on the outer scale of the Smith chart. The SWR can be determined indirectly through the reflection factor, but it can also be read directly from the Smith chart - the intersection of the real axis to the right of center of the circle with the circle, which is given by the magnitude of the reflection coefficient.

One would now like to the reflection factor at an arbitrary position calculated on a line, this corresponds to a rotation of the reflection factor to the normalized line length at the end of the line on the reflection factor circuit either towards the generator, that is clockwise or towards the load, ie in the counter-clockwise direction.

The Smith chart is in the upper half starting from inductive and capacitive in the lower half of the impedance values ​​.

3D Smith chart

There are also three-dimensional generalized Smith charts project the active and passive networks together at the Riemann sphere.

Using the Smith Chart

• Normalization: All elements are normalized, ie impedances are divided by their characteristic impedance, admittances registered with multiplied and then in the Smith chart.
• Series-connected impedances can be added directly.
• Parallel impedances must first be brought to admittance form, i.e. mirrored about the center. Alternatively two Smith charts may be used one above the other, wherein a Smith chart is rotated by 180 °. Order reflections can be achieved around the center of a transition of a chart to the other.
• Stub: The stub has to be converted into an equivalent impedance and add depending on the arrangement, such as a serial or parallel impedance.
• Movement on the line: impedance or admittance chart diagram to rotate the appropriate cable length to the generator (clockwise) or to the load ( counterclockwise).
• SWR: The SWR is obtained by clockwise around the center rotates the point at this place on the real axis and read off the corresponding value.
• Short circuit: the leftmost point in the impedance diagram, or far right of the admittance diagram
• Idle: the rightmost point in the impedance diagram, or left in the admittance diagram

Smith charts on paper for the graphical determination are used primarily in the field of education and teaching and for documentation. In the practical application of Smith charts are usually present in the context of relevant programs. Even complex instruments such as network analyzers can display directly measured data usually in the form of Smith charts.

The original advantage and the intention to simplify the complex calculations in the form of graphical determination of the numerical values ​​has been eliminated by the wide availability of powerful calculators and computers with appropriate software packages. What remains as the primary application is a graphical representation of impedance curves in technical documents and data sheets.

Example

An ohmic resistor R = 150 Ω and a capacitor C = 10 uF are connected in series, in parallel, is a coil L = 0.5 H. The circuit is connected to a generator whose frequency is f = 79.6 Hz.

The angular frequency is ω = 2? F = 500 then s -1.

For the complex resistance ( impedance ) of the capacitor follows

For the impedance of the coil is calculated one

For the series connection of a resistor and capacitor values ​​are simply added together to give

To enter the values ​​into the Smith chart in which can no longer be large numbers represent one normalized with a suitable reference resistance, eg Z0 = 100 Ω, by dividing all values ​​by him. Then

And

These two impedances in parallel. For the total impedance X is therefore

This reciprocal values ​​are obtained in the Smith chart by mirroring the circle center.

They amount to

The addition of the two reciprocals is done by calculation or in the Smith chart by " counting " on the coordinate grid.

Obtained

To determine the total impedance of X, the inverse of which is again to be formed. So you just received reflects the point at the center of the circle.

As a result, one finds

Since it has been divided by 100 Ω, one has to multiply so again. Final impedance of the entire circuit is thus

Therefore, it can alternatively be represented by a series circuit of a resistor of 375 Ω and a coil of 125j Ω (with ω = 500 s-1 corresponds to an inductance of 0.25 H).