*Fundamental Frequency
*Harmonic
|All versions|{{download|http://www.emagtech.com/contentdownloads/project-file-download-repository|ProjectRepo/RFLesson2.zip RF Tutorial Lesson 2|[[RF.Spice A/D]] R15}} }}
=== What You Will Learn ===
In this tutorial you will explore the transient response of transmission line circuits with various load configurations. You will also perform a Fourier analysis of non-sinusoidal signals.
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[[File:RF160.png|thumb|500pxleft|550px|The basic transmission line circuit with three voltage probe markers.]]
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== Transient Simulation of a Simple Transmission Line Circuit ==
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.
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<table><tr><td>[[File:RFTUT2 5.png|thumb|500pxleft|640px|Changing the waveform in the property dialog of the voltage source.]]</td></tr></table> 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 waveform [[parameters]] as specified below:
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! scope="row"| initial Initial Voltage
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The duty cycle of the pulse train waveform is therefore 10%. 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 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, v(in) is simply is half-scaled replica of the source voltage.
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In the case of the pulse waveform, the impedance mismatch introduces a reflected pulse that reaches node "IN" after a delay of 400ps, which is the round-trip time from the input point to the load and back. 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 and Effect of Source Mismatch == [[File:RF167.png|thumb|500px|The quarter-wave impedance transformer circuit designed for 2GHz operation.]]
In RF Tutorial lesson 1, you designed a quarter-wave impedance transform to match an arbitrary resistive load to a 50Ω source. Set the length of the T-Line segment to L = λ<sub>0</sub>/4 = 37.5mm for an operating frequency of 2GHz. Also, set the characteristic impedance of the T-line to Z<sub>0</sub> = √(100.50) = 70.71Ω. Run a new transient test with the same settings as before for both cases of sinusoidal and pulse waveforms:
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[[File:RF167.png|thumb|left|550px|The quarter-wave impedance transformer circuit designed for 2GHz operation.]]
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The results are shown and compared in the figures below:
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[[File:RFTUT2 8.png|thumb|left|720px|The graph of the source, input and output voltages for a sinusoidal waveform with f<sub>0</sub> = 2GHz when the T-Line segment acts as a quarter-wave transformer.]]</td>
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[[File:RFTUT2 9.png|thumb|left|720px|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.]]
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<math> Z_{in} = \frac{Z_{0c}^2}{Z_L} = \frac{70.71^2}{100} = 50\Omega </math>
[[File:RF170.png|thumb|380px|The Fourier Transform Settings dialog.]]
As you can see from the above figure, 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:
<math> \Gamma_S = \frac{ Z_S - Z_{0c} }{ Z_S + Z_{0c} } = \frac{ 50 - 70.71 }{ 50 + 70.71 } = -0.172 </math>
<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 with the "Fourier Analysis" 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 signals have significant DC contents as well as sizable higher harmonics.
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[[File:RFTUT2 10.png|thumb|left|720px|The spectral contents of the source voltage v(source) with a pulse waveform.]]
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[[File:RFTUT2 11.png|thumb|left|720px|The spectral contents of the input voltage v(in) with a pulse waveform.]]
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== Investigating the Effect of Inductive and a Capacitive Loads Load == [[File:RFTUT2_12.png|thumb|500px|The quarter-wave impedance transformer circuit with a parallel capacitor at the load.]]
In the last part of this tutorial lesson, you add a 0.1pF shunt capacitor called "CL" to the load and see its effect in the case of pulse train signal. The modified circuit is shown in the opposite figure. The impedance of the shunt capacitor at the <I>n</I>th harmonic of the source's 2GHz fundamental frequency is given by:
It can be seen that at the fundamental frequency, the capacitor has very negligible effect, but at higher harmonics, its impedance becomes comparable to the 100Ω resistive loads.
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[[File:RFTUT2_12.png|thumb|left|550px|The quarter-wave impedance transformer circuit with a parallel capacitor at the load.]]
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Run a transient test of your modified circuit with the same settings as before only for the case of pulse waveform:
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[[File:RFTUT2_13.png|thumb|left|720px|The graph of the source, input and output voltages in the quarter-wave transformer circuit with a series short stub before the load.]]
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[[File:RFTUT2_14.png|thumb|left|720px|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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