![]() Draw the g=1 circle this is where yB should be located. lA, lB used to tune the network ECE357 / Prof.Be very careful of: – If you need to work on an impedance or an admittance chart – Where the open/short circuit locations are on each chart ECE357 / Prof.Shorter line sections and stubs give better performance in terms of bandwidth – Shorter transmissions lines have less variation of electrical parameters with frequency.HumĮxample: Match a load impedance ZL = 100 + j80 Ω to a 50 Ω line using a single series open-circuit stub.Įxample: For a load impedance ZL = 15 + j10 Ω, design two tuning networks based on shunt short-circuited stubs to match this load to 50 Ω. Determine stub length ℓ between the open / short circuit point and the points representing ±jx’ (±jb’) ECE357 / Prof. Determine load-section length d from angles between point representing zL (yL)and the point on the rL=1 (gL=1) circle 4. Draw the |Γ| circle and translate the impedance along the line to the rL=1 (gL=1) circle (2 solutions) => r’ = 1±jx’ (y’ = 1±jb’) 3. Plot the normalized load impedance on the Smith Chart (and convert to admittance for shunt stub tuning) 2. Single-Stub Matching: Steps (shunt version in parentheses) 1. The parameters of the line are as follows: Z0 = 75 Ω, α = 0.029 Np/m, and β = 0.2π rad/m. Input Impedance / Input Reflection Coefficient from a Lossless LineĮxample: What is the input impedance seen into a 0.2λ line terminated in ZL?Įxample: Determine the input impedance of a 2 m long line terminated in a load impedance ZL = 67.5 – j45 Ω. Intersection of rLcircle and xLcircle defines a normalized load impedance.rL- and xL- circles are orthogonal to each other.The process of plotting admittance is essentially reversed - where adding an inductor to a series circuit would move the impedance value clockwise along a constant resistance circle, a shunt inductor would move it counter-clockwise along a constant admittance circle shunt capacitors similarly move your values clockwise on an admittance chart, where a series capacitor would be counter-clockwise. This is an important step too, as by flipping it over, you now have a chart that will assist you in dealing with shunt components rather than those in series. It is actually surprisingly easy to plot the equivalent chart for admittance - all you have to do is mirror the Chart horizontally. The terms corresponding to resistance and reactance are called conductance and susceptance, respectively. Let's get started by writing the equation for the reflection coefficient of a load impedance, given a source impedance: Once we get past the derivation, there will be a few simplified images showing how those equations can be used and combined to get the final product. That's all the Smith Chart really is: a collection of circles, each one centered in a different place in (or outside) the plot, and each one representing either constant resistance or constant r eactance. By taking the standard reflection coefficient formula and manipulating it so that it provides us with the equations for circles of various radii, we'll be able to construct the basic Smith Chart. ![]() That said, even if you don't fully understand the derivation below, you can still use the chart to help you with your own design. In order to understand the construction of the chart, you'll need to understand high school algebra and the basics of complex numbers, as well as have a basic understanding of impedance in electronic circuits. ![]() The Smith Chart has been in use since the 1930s as a method to solve various RF design problems - notably impedance matching with series and shunt components - and it provides a convenient way to find these solutions without the use of a calculator. This article covers the mathematics behind creating the chart and its physical interpretation. Smith Charts are an extremely useful tool for engineers and designers concerned with RF circuits.
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