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{{projectinfo|Tutorial| Transient Analysis of a Simple Transmission Line Circuit |RF176RFTUT2 6.png|In this project, the basic concepts you will perform transient simulation and Fourier analysis of RF.Spice A/D are demonstrated, and a simple voltage divider is modeled and examinedtransmission line circuits.|

*[[CubeCAD]]Transmission Line*VisualizationGeneric T-Line*[[EM.Tempo#Lumped Sources | Lumped Sources]]Transient Test*[[EM.Tempo#Scattering Parameters and Port Characteristics | S-Parameters]] Sinusoidal Waveform*[[EM.Tempo#Far Field Calculations in FDTD | Far Fields]] Pulse Waveform*[[Advanced Meshing in EM.Tempo]] Characteristic Impedance *Reflection Coefficient*Fourier Analysis*Fundamental Frequency*Harmonic|All versions|{{download|http://www.emagtech.com/contentdownloads/project-file-download-repository|EMProjectRepo/RFLesson2.Tempo zip RF Lesson 1|[[EM.Cube]] 14.82}} }}

=== What You Will Learn ===

[[File:RF160.png|thumb|500pxleft|550px|The basic transmission line circuit with three voltage probe markers.]]

Run a Transient Test of your circuit with the specified [[parameters]] below. Note that your Plot Edit List must already contain v(SOURCE), v(IN) and v(OUT). The node voltages of all voltage probe markers are automatically added to the Plot Edit List.

<table><tr><td>[[File:RF162RF161A.png|thumb|400pxleft|720px|The property dialog graph of the voltage source, input and output voltages for a sinusoidal waveform with f₀ = 2GHz when RL = 50&Omega;.]]</td></tr></table>

[[File:RF161ARFTUT2 5.png|thumb|left|900px640px|The graph Changing the waveform in the property dialog of the voltage source, input and output voltages for a sinusoidal waveform with f₀ = 2GHz when RL = 50&Omega;.]]

Next, you will try out a rectangular pulse waveform as your voltage source. Open the property dialog of VS and change the waveform type to "Pulse". Set the period waveform parameters as specified below:  {| border="0"|-| valign="top"||-{| class="wikitable"|-! scope="row"| Initial Voltage| 0|-! scope="row"| Peak Voltage| 1|-! scope="row"| Delay Time| 0|-! scope="row"| Rise Time| 1p|-! scope="row"| Fall Time| 1p|-! scope="row"| Pulse Width| 50p|-! scope="row"| Pulse Period| 500p|} The duty cycle of the pulse train to 500ps and set the pulse width to 100ps. This means a duty cycle of 20waveform is therefore 10%. Set the rise time and fall time of the pulse both to 1ps. Set the initial and peak voltages of the pulse to 0 and 1V, respectively. Run a new transient test of your circuit with the same test parameters as before and compare the results to the previous case of a sinusoidal waveform. Here, too, there is a 200ps delay between the input ad and output voltages. Due to the perfect impedance match at both the input and output, v(in) and v(out) have equal amplitudes of 0.5V. Moreover, the v(in) is simply is half-scaled replica of the source voltage.

[[File:RF163RFTUT2_4.png|thumb|left|900px720px|The graph of the source, input and output voltages for a pulse waveform with a period of T = 500ps when RL = 50&Omega;.]]

== Investigating the Effect of Load Mismatch==

Now change the load resistor to RL = 100&Omega;. Run the Transient Test with the same settings as before for both cases of sinusoidal and pulse waveforms. : {| border="0"|-| valign="top"||-{| class="wikitable"|-! scope="row"| Start Time| 0|-! scope="row"| Stop Time| 3n|-! scope="row"| Linearize Step| 1p|-! scope="row"| Step Ceiling| 1p|-! scope="row"| Preset Graph Plots| v(source), v(in), v(out)|} The results of the two case cases are shown in the figures below:

[[File:RF166RFTUT2 7.png|thumb|left|900px720px|The graph of the source, input and output voltages for a sinusoidal waveform with f₀ = 2GHz when RL = 100&Omega;.]]

[[File:RF164RFTUT2 6.png|thumb|left|900px720px|The graph of the source, input and output voltages for a pulse waveform with a period of T = 500ps when RL = 100&Omega;.]]

In the top graph, the "Show Maxima" feature of the graph window has been enabled. This can be done from the first tab of the "Edit Plots" Panel. As you can see from these figures, the effect of the load mismatch in the sinusoidal signal case is the difference in the amplitudes of the input voltage and load voltage. This is due to the nonzero load reflection coefficient &Gamma;: <sub>L</submath> \Gamma_L = (Z_L \frac{ Z_L - Z₀) / (Z_LZ_0 }{ Z_L + Z<sub>0Z_0 } = \frac{ 100 - 50 }{ 100 + 50 } = \frac{1}{3} </submath>) = 1/3.  In the case of the pulse waveform, the impedance mismatch introduces a dispersion of the waveform as can be clearly seen in the shape of v(in). Besides the additional reflected pulse in each period, you can also see a sizable overshoot in the following close-up of the last graph. Note that the second smaller pulse has reaches node "IN" after a delay of 400ps, which is the round-trip time from the input point to the load and back. Moreover, the The amplitude of the smaller reflected pulse (166.7mv) is 1/3 the amplitude of the larger incident pulse (500mV) as you would expect from the value of the reflection coefficient.  ==Analyzing a Quarter-Wave Impedance Transformer == In RF Tutorial lesson 1, you designed a quarter-wave impedance transform to match an arbitrary resistive load to a 50&Omega; source. Set the length of the T-Line segment to L = &lambda;₀/4 = 37.5mm for an operating frequency of 2GHz. Also, set the characteristic impedance of the T-line to Z₀ = &radic;(100.50) = 70.71&Omega;. Run a new transient test with the same settings as before for both cases of sinusoidal and pulse waveforms: {| border="0"|-| valign="top"||-{| class="wikitable"|-! scope="row"| Start Time| 0|-! scope="row"| Stop Time| 3n|-! scope="row"| Linearize Step| 1p|-! scope="row"| Step Ceiling| 1p|-! scope="row"| Preset Graph Plots| v(source), v(in), v(out)|}

[[File:RF165RF167.png|thumb|left|900px550px|A closeThe quarter-up of the graph of the input voltage wave impedance transformer circuit designed for a pulse waveform with a period of T = 500ps when RL = 100&Omega;2GHz operation.]]

[[File:RF167.png|thumb|450px|The Quarter-Wave Impedance Transformer circuit.]]==Analyzing a Quarter-Wave Impedance Transformer and Effect of Source Mismatch == In Tutorial lesson 1 you designed a quarter-wave impedance transform to match an arbitrary resistive load to a 50&Omega; source or a 50&Omega; transmission line. Set the length of the T-Line segment to L = &lambda;₀/2 = 37.5mm. Also, set the characteristic impedance of the T-line to Z₀ = &radic;(100.50) = 0.71&Omega;. Run a new Transient Test with the same settings as before for both cases of sinusoidal and pulse waveforms. The results are shown and compared in the figure figures below.:

[[File:RF168RFTUT2 8.png|thumb|left|900px720px|The graph of the source, input and output voltages for a sinusoidal waveform with f₀ = 2GHz when the T-Line segment acts as a quarter-wave transformer.]]</td>

[[File:RF169RFTUT2 9.png|thumb|left|900px720px|The graph of the source, input and output voltages for a pulse waveform with a period of T = 500ps when the T-Line segment acts as a quarter-wave transformer.]]

[[File:RF170.png|thumb|450px|The Fourier Transform Settings dialog.]]As you can see from the above figure for , in the case of the "pure harmonic" sinusoidal excitation, the source voltage is equally split between the source resistor RS and the input port of the T-Line. However, the situation is slightly different in the case of pulse waveform. The dispersive effects of the transmission line are in full display in this case. In other words, the matching condition is satisfied only at 2GHz and not at its harmonics present in the pulse waveform. Note that both load and source reflection coefficients are nonzero for this circuit:

<table><tr><td>[[File:RF170.png|thumb|left|420px|The Fourier Transform Settings dialog.]]</td></tr></table> To better understand this point, you can use RF.Spice's Fourier analysis feature, which is part of the transient test. You will run the "Transient Test " for the pulse waveform two more times, once with the output plot set to v(source) and next with the output plot set to v(in). However, this time you will also enable the "Fourier Analysis" feature of Transient Test. This can be done from enabled in the Transient Test Panel. Check the "Apply Fourier" checkbox and click the "Fourier Setup" button to open the Fourier Transform Settings dialog. Set the Fundamental Frequency to 2GHz and set the reference output node to "0" for the ground. Then, set the positive output node to "1" for the source voltage in the first run and then to "2" for the input voltage in the second run. At the end of the simulation, additional bar chart graphs are added to the Data Manager window. The spectral contents of the source and input voltages are shown in the figures below. You can clearly see that both signal signals have significant DC contents as well as sizable higher harmonics.

[[File:RF171RFTUT2 10.png|thumb|left|900px720px|The spectral contents of the source voltage v(source) with a pulse waveform.]]

[[File:RF172RFTUT2 11.png|thumb|left|900px720px|The spectral contents of the input voltage v(in) with a pulse waveform.]]

[[File:RF173.png|thumb|450px|The Quarter-Wave Impedance Transformer circuit with a series short stub before the load.]][[File:RF175.png|thumb|450px|The Quarter-Wave Impedance Transformer circuit with a shunt open stub before the load.]]== Investigating the Effect of Inductive and a Capacitive Loads Load ==

In Tutorial Lesson 3 you used open and short stubs for tuning and matching the last part of loads to [[Transmission Lines|transmission lines]] and sources. You saw that a given frequencythis tutorial lesson, an open or short stub may behave as an inductive or capacitive you add a 0.1pF shunt capacitor called "CL" to the load depending on and see its lengtheffect in the case of pulse train signal. The modified circuit is shown in the opposite figure. The impedance of an open or short stub the shunt capacitor at the <I>n</I>th harmonic of the source's 2GHz fundamental frequency is given by:

<math> Z_{openCap} = -jZ_0 cot\frac{j}{\omega C_L} = -\frac{j}{2\pi n (2\beta Ltimes 10^9) (0.1\times 10^{-12})} \approx -j(796/n) \ \Omega </math>

In this part of the tutorial lesson you will add very small open and short stubs with len {| border= 5mm and Z_"0 "|-| valign= 70.71&Omega; to the 100&Omega; load of the previous circuit. At f_{"top"||-{| class="wikitable"|-! scope="row"| Start Time| 0} |-! scope= 2GHz, you have tan"row"| Stop Time| 3n|-! scope="row"| Linearize Step| 1p|-! scope="row"| Step Ceiling| 1p|-! scope="row"| Preset Graph Plots| v(&beta;Lsource) = tan, v(2&pi; L/&lambda;_gin) = 0.213 and cot, v(&beta;Lout) = 4.70.|}

First, you will add a series short stub between Enable the output "Fourier Analysis" for Node 2 with a fundamental frequency of 2GHz. The figures below show the T-Line segment and simulation results. You can see from the resistive load RL as shown in top figure that the above figures. The small short stub introduces output voltage has been slightly deformed with a series reactance of -j15rounded rising edge.03&Omega; to be added to Also, the 100&Omega; resistive loadreflected pulse now has significant undershoot and overshoot at its rising and falling edges, respectively. Next, you will add a shunt open stub between compare the output spectral contents of v(in) for the T-Line segment two cases without and with the resistive load RL also shown in the above figuresshunt capacitor. The small open stub introduces a shunt susceptance of j0.003S to be added to the 0.01S conductance of the resistive load. Run new Transient [[Tests]] with these additional series short contents at DC, fundamental frequency and shunt open stubs over first few harmonics are almost the time interval [0 - 3ns]. The time-domain voltages have been plotted in same, but the figures below. The dispersive effects gravely affect the input and output voltagesresults start to differ at higher harmonics as we justified earlier.

[[File:RF174RFTUT2_13.png|thumb|left|900px720px|The graph of the source, input and output voltages in the quarter-wave transformer circuit with a series short stub before the load.]]

[[File:RF176RFTUT2_14.png|thumb|left|900px720px|The graph of the source, input and output voltages in the quarter-wave transformer circuit with a shunt open stub before the load.]]

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