Smith Chart Free Download

<div>The Smith chart is a mathematical transformation of the two-dimensional Cartesian complex plane. Complex numbers with positive real parts map inside the circle. Those with negative real parts map outside the circle.If we are dealing only with impedances with non-negative resistive components, our interest is focused on the area inside the circle.The transformation, for an impedance Smith chart, is:</div><div></div><div></div><div>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.</div><div></div><div></div><div></div><div></div><div></div><div>smith chart free download</div><div></div><div>Download Zip: https://t.co/Gf3YP7kne9 </div><div></div><div></div><div>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.</div><div></div><div></div><div>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.</div><div></div><div></div><div>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.</div><div></div><div></div><div>For a uniform transmission line (in which γ \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 ( α \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 ( α = 0 \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</div><div></div><div></div><div>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.</div><div></div><div></div><div>Γ \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.</div><div></div><div></div><div>The Smith chart graphical equivalent of using the transmission-line equation is to normalise Z L , \displaystyle \,Z_\mathsf 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.</div><div></div><div></div><div>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.</div><div></div><div></div><div></div><div></div><div></div><div></div><div>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 yT is the reciprocal of the normalised impedance zT, so</div><div></div><div></div><div>The Y Smith chart appears like the normalised impedance, type but with the graphic nested circles rotated through 180, but the numeric scale remaining in its same position (not rotated) as the Z chart.</div><div></div><div></div><div>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.</div><div></div><div></div><div>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.</div><div></div><div></div><div>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.</div><div></div><div></div><div>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 Z TS \displaystyle Z_\textTS is the equivalent impedance of series impedances and Z TP \displaystyle Z_\textTP is the equivalent impedance of parallel impedances, then</div><div></div><div></div><div>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.</div><div></div><div></div><div>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 L 1 = 0.098 λ \displaystyle L_1=0.098\lambda \, . 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</div><div></div><div></div><div>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</div><div></div><div></div><div>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 Z 0 = 50 Ω \displaystyle Z_0=50\ \Omega \, is a good choice here as R 1 = 50 Ω \displaystyle R_1=50\ \Omega \, . If there were very different values of resistance present a value closer to these might be a better choice.</div><div></div><div></div><div>The analysis starts with a Z Smith chart looking into R1 only with no other components present. As R 1 = 50 Ω \displaystyle R_1=50\ \Omega \, 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 -j0.80, 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 P1 to Q1 and P3 to Q3 are from the Z Smith chart to the Y Smith chart and transformation Q2 to P2 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.</div><div></div><div> df19127ead</div>