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{{projectinfo|Tutorial| Network Analysis of a Simple Transmission Line Circuit & the Smith Chart |RF27RFTUT3_16A.png|In this project, you will perform network analysis of simple transmission line circuits using the Smith chart.|

*Scattering [[Parameters]]*Impedance [[Parameters]]

|All versions|{{download|http://www.emagtech.com/contentdownloads/project-file-download-repository|ProjectRepo/RFLesson3.zip RF Tutorial Lesson 3|[[RF.Spice A/D]] R15}} }}

=== What You Will Learn === In this tutorial you will run a network analysis test of the simple transmission line circuit you built in the previous tutorial lessons. You will examine the Z- and S-parameters of your circuit and study the Smith chart.

[[File:RFTUT3_1.png|thumb|500pxleft|550px|The quarter-wave impedance transformer circuit tuned for f₀ = 2GHz.]]

As a first step, you will run a "Network Analysis" test of your RF circuit. Network analysis calculates the S/Z/Y [[parameters]] of your circuit based on the port(s) you define for your RF circuit. In this case, you will define a one-port network with an input port established at the input of the T-Line between Node 2 and ground. In the Toolbox, select the Test Panel and check the "Network Analysis" checkbox. Open the Test's Settings dialog. It has three tabs at the top: Connections, Sweep and Output. In the first tab, '''Connections''', you define the port(s) of your circuit. In this case, Port 1 is defined between Node 2 and the ground. Accept the default value of 50Ω for the "Reference Impedance". In the second tab, '''Sweep''', set the start and stop frequencies to 1GHz and 5GHz, respectively. Select a linear scale interval and set the step size to 10MHz. This will provide a smooth graph of the port characteristics. In the third tab of the dialog, '''Output''', go to the "Parameter Set" section and choose the '''S''' radio button to compute the scattering [[parameters]]. Since your circuit is a one-port, you will have the S11-parameter only. From the top "Graph Type" options, choose '''Cartesian (Mag/Phase)'''. Check the checkboxes labeled '''Decibels''' and '''Degrees''' for magnitude and phase, respectively.

[[File:RFTUT3_2.png|thumb|left|230px|The "Connections" tab of Network Analysis Test Panel.]]

[[File:RFTUT3_3.png|thumb|left|230px|The "Sweep" tab of Network Analysis Test Panel.]]

[[File:RFTUT3_4.png|thumb|left|230px|The "Output" tab of Network Analysis Test Panel.]]

[[File:RFTUT3_5.png|thumb|left|720px|Cartesian graph of the magnitude and phase of the S11-parameter.]]

Run a network analysisNext, go back to the '''Output''' tab of the Network Analysis Test Panel and choose the '''Z''' radio button in the "Parameter Set" section. An output graph like Make sure you remove the one shown below is generatedcheck mark from the "Decibels" check box. This graph looks like a resonant circuitFrom the top "Graph Type" options, where choose '''Cartesian (Real(z11/Imag) peaks around f = 2GHz''', and Imag(z11) crosses the zero at because you are more interested in the same frequency. This can easily be explained by response of the fact that at f = 2GHz, λ_g = λ₀ = 150mm real and len = 0.5λ_g. In other wordsimaginary parts of the Z11-parameter, your T-Line part acts as a half-wave transmission line resonator at 2GHz. The which are indeed the input impedance of the line segment becomes pure real at this frequency and equals the load resistance: Z_in = R2 = 100Ωyour circuit.

[[File:RFTUT3_8RFTUT3_6.png|thumb|720pxleft|Cartesian graph of the real and imaginary parts 230px|The "Connections" tab of the Z11-parameterNetwork Analysis Test Panel.]]

==Plotting Run another network analysis and view the S-Parameter on Smith Chart==output graph as shown below. Note that at 2GHz, the real part of the impedance is 50Ω and its imaginary part vanishes as you would expect for perfect impedance match.

The Smith chart is a very useful graphical tool for RF engineers. <table><tr><td>[[RFFile:RFTUT3_8.Spice]] allows you to plot the S-[[parameters]] png|thumb|left|720px|Cartesian graph of your circuit on the Smith Chart. For this purpose, open the Network Analysis test panel of the Toolbox once again real and this time choose the S radio button in the "Parameter Set" section imaginary parts of the Output tab. For the "Graph Type", check the "Smith" checkbox. In order to better view the data points on the Smith chart, change the step size to 100MHz in the Sweep tab of the test panel. Run a network analysis of your RF circuit and view the resulting Smith chart as shown belowZ11-parameter. ]]</td></tr></table>

{{Note| The Smith chart is available a very useful graphical tool for RF engineers. [[RF.Spice A/D]] allows you to plot the S-parameter set onlyparameters of your circuit on the Smith Chart.}} For this purpose, open the Network Analysis Test Panel of the Toolbox once again, and this time choose the '''S''' radio button in the "Parameter Set" section of the Output tab. For the "Graph Type", check the "Smith" checkbox. In order to better view the data points on the Smith chart, change the step size to 500MHz in the Sweep tab of the test panel. Run a network analysis of your RF circuit with the following parameters:

| valign="bottomtop"|[[File:RF18.png|thumb|180px|Choosing Smith chart as the graph type in Test Panel.]]| valign="bottom"|[[File:RF20.png|thumb|600px|The s11-parameter plotted on a Smith Chart.]]

[[File:RF21At the end of the simulation, view the resulting Smith chart.png|thumb|400px|Reading Sometime, the s11 data on program tries to fit the graph to the size of the graph window. If your graph window is elongated, the Smith chart will look elliptical. To make it circular, click on the graph window's title tab to make it active. Then open the '''Edit Graph''' tab of the Toolbox and press the {{key|Circle}} button/tab at the top of the panel. Then check box labeled '''Plot Circular Chart(Not Elliptical)'''.]]

Let's analyze the s11-parameter data plotted on the {{Note| The Smith chart in a little bit more detail. The s11-parameter data points over the frequency range 1-3GHz form a perfect circle around the center of the circular chart. The distance from the center is available for the magnitude of the reflection coefficient s11 (also known as the return loss). Since your TS-Line segment is lossless (alpha = 0), its delivers the input signal to the load without attenuationparameter set only. However, it gives rise to a phase shift in the signal, which can also be interpreted as a time delay. You can read the values of the data points on the Smith chart using the "Tracking Crosshairs" feature. For this purpose, click the }} <table><tr><td>"Track Selected Plot" [[File:b2TrackCH_ToolRFTUT3 12.png]] button of the '''[[Toolbars#Graph_Toolbar |Graph Toolbar]]''' and then move the mouse onto the surface of the thumb|left|720px|The S11-parameter plotted on a Smith Chart. A bar appears on the graph the connect the current cursor position to the center of the Smith chart. The tracking bar jumps over shows the data points. On the Status Bar, you can view the frequency associated with each data point as well as the real and imaginary parts of the reflection coefficient at that point. The 2GHz point is located on the horizontal axis of the Smith chart to the right of the origin where im(s11) = 0. The start and stop frequencies, i.e. the 1GHz and 3GHz point coincide on the horizontal axis of the Smith chart to the left of the origin. The opposite figure also shows the data point corresponding to 12GHz.5GHz and 2.5GHz, where the reflection coefficient are pure imaginary. ]]</td></tr></table>

==Effect You can read the values of the data points on the Smith chart using the "Tracking Crosshairs" feature. For this purpose, click the '''Track Selected Plot''' [[File:b2TrackCH_Tool.png]] button of the '''Graph Toolbar''' and then move the mouse onto the surface of the Smith Chart. A tracking bar appears on the graph that connects the current cursor position to the data points. On the Status Bar, you can view the frequency associated with each data point as well as the real and imaginary parts of the reflection coefficient at that point. The 2GHz point is located at the center of Varying the Line Segment Length==Smith chart.

[[File:RF22.png|thumb|320px|Associating == Exploring the s11 data on the Smith Chart with line segment lengths at the fixed frequency Effect of 2GHz.]] From the transmission line theory, Varying the input impedance and input reflection coefficient can be expressed as: Line Segment Length ==

<math>\Gamma_{in} = \frac{1 - \zeta Gamma_L e^{-2j \beta L}}= \frac{\zeta_L - 1 + } {\zeta zeta_L + 1} e^{-2j \beta L}}</math>

where &zeta; = Z_L/Z₀, L is the length of the line segment (same as the len parameter), and &beta; = 2&pi;/&lambda;_g = 2&pi;&radic;(&epsilon;_eff)/&lambda;₀. Note that both input impedance and input reflection coefficient are functions of the product &beta;L or the product 2&pi;&radic;(&epsilon;_eff)(f.L)/c, where c is the speed of light. For a given fixed value of &epsilon;_eff (same as the eeff parameter), both Z_in and &Gamma;_in are functions of the product (fL) rather then functions of f or L individually. This property has a practical application. If you fix the length of the transmission line segment and vary the frequency, you will get a circular plot on the Smith chart that is equivalent to changing the line segment length at a fixed frequency. For example, in the Smith chart of the previous step, L = 75mm was fixed. s11 at 2GHz is where the circle intercepts the horizontal axis to the right of the origin. The intercept points of the circle with the positive and negative vertical axes correspond to 1.5GHz and 2.5GHz, respectively.   Now let's interpret this the same circular curve at a fixed frequency of f = 2GHz, assuming that L is now varies instead varied. The intercept point origin of the circle with the positive horizontal axis Smith chart obviously corresponds to L = 75mm37.5mm. The intercept points point that corresponded to a frequency of 1GHz in the circle with the positive vertical axis above Smith chart now corresponds to L = 75*would represent a line length of (11GHz &times; 37.5GHz/2GHz5mm) = 56.25mm, while the intercept points with the negative vertical axes corresponds to L = 75*(2.5GHz/2GHz) = 9318.75mm. To verify these resultsSimilarly, you can change the length of the T-Line segment first point that used to 56.25mm and then correspond to 93.75GHz, and run a network analysis over the 1-3GHz frequency rangeof 4GHz now would represent a line length of (4GHz &times; 37. The Smith chart results for the two cases are shown in the figures below. The data points corresponding to f 5mm)/2GHz = 2GHz has been marked in both figures, which are identical to the previous results75mm.

| valign="bottomtop"|[[File:RF27.png|thumb|300px|Computed s11 data for line segment length L = 56.25mm over the frequency range 1-3GHz.]]| valign="bottom"|[[File:RF28.png|thumb|290px|Computed s11 data for line segment length L = 93.75mm over the frequency range 1-3GHz.]]

|} ==Effect of an Inductive Load== As a next step, you will add an inductive component to your resistive load. Make sure that the length of the T-Line segment is the original 75mm value. Add a new inductor of value L = 6.37nH between the resistive load and the output of the T-Line segment. At 2GHz, your termination load now has an impedance Z_L = 100 + j80 Ohms. Run a new AC sweep test of this circuit over the same frequency range 1-3GHz with a linear step size of 100MHz. The figures below show both the modified circuit and the resulting s11-parameter data plotted on the Smith chart. In the next tutorial lesson you will design a single-stub matching network for this circuit.  {| borderclass="0wikitable"

| valign! scope="bottomcol"|[[File:RF23Point No.png]]| valign! scope="bottomcol"|Frequency @ L = 37.5mm[[File:RF24.png! scope="col"|thumb|500px|Computed s11 data for line segment length L Length @ f = 75mm and Z_L = 100 + j80 &Omega; over the frequency range 1-3GHz.]]2GHz

==Investigating the Effect of a Capacitive an Inductive Load==

As the final a next step of his tutorial lesson, you will add a capacitive an inductive component to your resistive load. Change Make sure that the resistive load's length of the T-Line segment is the original 75mm value to 60&Omega; and add . Add a new capacitor inductor of value C L = 06.995pF 37nH between the resistive load and the output of the T-Line segment.  <table><tr><td>[[File:RFTUT3_15.png|thumb|left|550px|The quarter-wave impedance transformer circuit with an inductive load.]]</td></tr></table> At 2GHz, your termination load now has an impedance Z_L = 60 - 100 + j80 Ohms. Run a new AC sweep test network analysis of this your circuit over the same frequency range 1-3GHz with a linear step size of 100MHz. The figures below show both the modified circuit and the resulting s11-parameter data plotted on the Smith chart. In the next tutorial lesson you will design single-stub and dual-stub matching networks for this circuit.parameters specified below:

| valign="bottomtop"|[[File:RF25.png]]|-{| valignclass="bottomwikitable"|-[[File:RF26.png! scope="row"|thumbStart Frequency|500px1G|Computed s11 data for line segment length L -! scope= 75mm and Z<sub>L<"row"| Stop Frequency| 5G|-! scope="row"| Steps/sub> Interval| 250Meg|-! scope= 60 "row"| Interval Type| Linear|- j80 &Omega; over the frequency range 1-3GHz.]]! scope="row"| Parameter Set| S

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