{{projectinfo|Tutorial| Examining Coupled Transmission Lines |RF90.png|In this project, you are will be exposed to the concepts concept of coupled transmission lines and will examine their properties.|
*Coupled [[Transmission Lines]]
*Even Mode
*Odd Mode
*System Characteristic Impedance
*AC Frequency Sweep
|All versions|{{download|http://www.emagtech.com/contentdownloads/project-file-download-repository|ProjectRepo/RFLesson6.zip RF Tutorial Lesson 6|[[RF.Spice A/D]] R15}} }}
=== What You Will Learn ===
In this tutorial you will learn about [[RF.Spice]]'s Generic Coupled T-Lines model and how to use them in your circuits.
== Introducing the Generic Coupled T-Lines Device ==
[[RF.Spice]]'s Generic Couple T-Lines can be accessed from '''Menu > Parts > [[Transmission Lines]] > Generic T-Lines > Generic Coupled T-Lines'''. It is a four-port device that models two parallel transmission line segments with equal lengths. Coupled lines have several applications in RF circuit design including directional couplers, hybrids and filters. Normally you would use the positive pins of the four ports and ground all the four negative pins. In [[RF.Spice A/D]], the generic coupled T-lines device is characterized by its even and odd mode impedances, Z0e and Z0o. The system characteristic impedance of the device is defined as the geometric mean of these two impedances: Z<sub>0s</sub> = √(Z<sub>0e</sub>Z<sub>0o</sub>). Even though real physical coupled line structures have slightly different effective permittivities, the average of these two is usually taken as the effective permittivity of overall transmission line. The figure below shows the property dialog of the Generic Coupled T-Lines device:
<table>
== Exciting the Even and Odd Modes of Coupled Lines ==
In the analysis of coupled [[Transmission Lines|transmission lines]], it is customary to decompose signals as the superposition of two orthogonal modes: the even and odd modes. In this part of the tutorial lesson, you will excite each of these modes. First, you will excite the even mode. Connect the two voltage sources VS1 and VS2 through the source resistors RS1 and RS2 to the positive pins of Port 1 and Port 3 of the coupled lines device, respectively. Connect the two load resistors RL1 and RL3 to the positive pins of Port 2 and Port 4 of the coupled lines device, respectively. Place voltage probe markers VP1 and VP2 at Ports 1 and 3, respectively.
<table>
</table>
[[File:RF92.png|thumb|550px|Assigning the real and imaginary parts of the voltage signals for the table plots.]]
Run an AC Sweep Test of this circuit according to the table below. In the AC Sweep Test Panel, check both "Graph" and "Table" checkboxes.
. Set the start and stop frequencies of the sweep to 500MHz and 3.5GHz, respectively, with a linear frequency step size of 10MHz.
{| border="0"
|-
| valign="bottomtop"|[[File:RF91.png|thumb-{|400pxclass="wikitable"|left-! scope="row"|Assigning the voltage magnitude signals for the graph plots.]]Start Frequency| valign500Meg|-! scope="bottomrow"|Stop Frequency[[File:RF92| 3.png5G|thumb-! scope="row"|400pxSteps/Interval|left10Meg|Assigning the real and imaginary parts of the voltage signals for the table plots.]]-! scope="row"| Interval Type| Linear|-! scope="row"| Preset Graph Plots| vm(vp1), vm(vp2)
|-
! scope="row"| Preset Table Plots
| vr(vp1), vi(vp1), vr(vp2), vi(vp2)
|}
After the completion of the AC sweep, you should get a voltage graph and a voltage data table similar to the ones shown below. Note from the real/imag parts table that the two input voltages are equal.
After <table><tr><td>[[File:RFTUT6 5.png|thumb|750px|Graph of the completion amplitude of the AC sweep, you should get a voltage graph voltages VP1 and data table similar to VP2 for the ones shown beloweven mode excitation. Note from the real]]</td></tr></imag parts table that the two input voltages are equal. >
<table>
<tr>
<td>
[[File:RFTUT6 6.png|thumb|400px|Real and imaginary parts of the voltages VP1 and VP2 for the even mode excitation.]]
</td>
</tr>
</table>
{| border="0"|-| valign="bottom"|[[File:RF73newNext, you will excite the odd mode.png|thumb|750px|left|Graph of In this case, you will connect just a single voltage source VS1 in series with the amplitude of source resistor RS1 between the voltages VP1 positive pins of Port 1 and VP2 for Port 3 of the even mode excitationcoupled lines device as shown in the opposite figure.]]| valign="bottom"|[[File:RF74.png|thumb|350px|left|Real Connect the two load resistors RL1 and imaginary parts RL3 to the positive pins of Port 2 and Port 4 of the voltages coupled lines device, respectively. Place voltage probes VP1 and VP2 for the even mode excitationat Ports 1 and 3, respectively.]]|-|}
<table>
<tr>
<td>
[[File:RF75.png|thumb|550px|Exciting the odd mode of a Generic Coupled T-Lines device.]]
</td>
</tr>
</table>
[[File:RF75Run an AC Sweep Test of this circuit with the same sweep settings as in the previous case.png|thumb|450px|Exciting After the odd mode completion of the AC sweep, you should get a Generic Coupled T-Lines devicevoltage graph and data table similar to the ones shown below. This time, from the real/imag parts table you can see that the two input voltages are 180° out of phase.]]
Next, you will excite the odd mode<table><tr><td>[[File:RFTUT6 7. In this case, you will connect just a single voltage source VS1 in series with the source resistor RS1 between the positive pins png|thumb|750px|Graph of Port 1 and Port 3 of the coupled lines device as shown in the opposite figure. Connect the two load resistors RL1 and RL3 to the positive pins of Port 2 and Port 4 amplitude of the coupled lines device, respectively. Place voltage probes voltages VP1 and VP2 at Ports 1 and 3, respectively. Run an AC Sweep Test of this circuit with for the same sweep settings as in the previous caseodd mode excitation. After the completion of the AC sweep, you should get a voltage graph and data table similar to the ones shown below. This time, from the real]]</td></tr></imag parts table you can see that the two input voltages are 180° out of phase. >
<table>
<tr>
<td>
[[File:RFTUT6 8.png|thumb|400px|Real and imaginary parts of the voltages VP1 and VP2 for the odd mode excitation.]]
</td>
</tr>
</table>
{| border="0"|-| valign="bottom"|[[File:RF76new.png|thumb|750px|left|Graph of the amplitude of the voltages VP1 and VP2 for the odd mode excitation.]]| valign="bottom"|[[File:RF77.png|thumb|350px|left|Real and imaginary parts of the voltages VP1 and VP2 for the odd mode excitation.]]|-|} == Coupled Line AnalysisMode Theory ==
The Generic Coupled T-Lines device is a four-port device that can easily be analyzed based on its impedance (Z) [[parameters]]. Consider the four-port being driven from Ports 1 and 3 in the even mode, with Ports 2 and 4 terminated in two identical loads Z<sub>L</sub>. The voltage at any point on either T-Line segment can be expressed as:
<math>V_{P1} = V_1(z=0) = -V_{P2} = -V_2(z=0) = V_o^+ e^{j\beta L} \left[ 1 + \Gamma_{L}^o e^{-2j\beta L} \right] </math>
The above results can clearly be observed on the voltage graphs of the previous part, which exhibit a standing wave pattern. Note that in this case, Γ<sub>L</sub><sup>e</sup> = (50-70)/(50+70) = -0.167 and Γ<sub>L</sub><sup>o</sup> = (50-30)/(50+30) = 0.25. As you already saw in Tutorial Lesson 1, the exponent βL = 2πL/λ<sub>g</sub> = 2π√(ε<sub>eff</sub>)(f.L)/c determines the frequency variation of the voltage.
For Z-parameter calculations, we open-circuit Ports 2 and 4 (Z<sub>L</sub> = ∞). In this case, Γ<sub>L</sub><sup>e</sup> = Γ<sub>L</sub><sup>o</sup> = 1, and the voltages at Ports 1 and 3 now reduce to:
|}
where Z<sub>S</sub> = (Z<sub>0e</sub> + Z<sub>0o</sub>)/2 and Z<sub>D</sub> = (Z<sub>0e</sub> - Z<sub>0o</sub>)/2.
== Verifying Your Simulation Results ==
The equations derived earlier for the probe voltages (VP1 and VP2) exhibit a standing wave pattern, which can clearly be observed in your node voltage graphs for both cases of even and odd mode excitations. As you already saw in Tutorial Lesson 1, the exponential term:
<math> e^{-j2\beta L} = e^{-j2 \frac{2\pi}{\lambda_g} L } = e^{-j2\pi f \frac{2L}{c} \sqrt{\epsilon_{eff}} } </math>
determines the frequency variation of the voltages, which have a periodic nature. Since ε<sub>eff</sub> = 1 in this case, the period is c/(2L) = 3×10<sup>8</sup>/(2×0.075) = 2GHz. You can also see from the voltage graphs that the spacing between to consecutive maxima or minima is 2GHz.
You can also verify that for the even mode excitation:
<math> |V_{P1}^e|_{max} = V_e^+ |\ 1+|\Gamma_L^e|\ | \\
|V_{P1}^e|_{min} = V_e^+ |\ 1-|\Gamma_L^e|\ | </math>
Similarly, for the odd mode excitation:
<math> |V_{P1}^o|_{max} = V_o^+ |\ 1+|\Gamma_L^o|\ | \\
|V_{P1}^o|_{min} = V_o^+ |\ 1-|\Gamma_L^o|\ | </math>
From the above equations, one can write:
<math> |\Gamma_L^e| = \frac{|V_{P1}^e|_{max} - |V_{P1}^e|_{min}}{|V_{P1}^e|_{max} + |V_{P1}^e|_{min}} </math>
<math> |\Gamma_L^o| = \frac{|V_{P1}^o|_{max} - |V_{P1}^o|_{min}}{|V_{P1}^o|_{max} + |V_{P1}^o|_{min}} </math>
Also note that in this case, Γ<sub>L</sub><sup>e</sup> = (50-70)/(50+70) = -0.167 and Γ<sub>L</sub><sup>o</sup> = (50-30)/(50+30) = 0.25. The table below summarizes the results obtained from the simulated data graphs:
{| border="0"
|-
| valign="top"|
|-
{| class="wikitable"
|-
! scope="col"| |V<sub>P1</sub><sup>e</sup>|<sub>max</sub>
! scope="col"| |V<sub>P1</sub><sup>e</sup>|<sub>min</sub>
! scope="col"| |V<sub>P1</sub><sup>o</sup>|<sub>max</sub>
! scope="col"| |V<sub>P1</sub><sup>o</sup>|<sub>min</sub>
! scope="col"| Simulated |Γ<sub>L</sub><sup>e</sup>|
! scope="col"| Analytical |Γ<sub>L</sub><sup>e</sup>|
! scope="col"| Simulated |Γ<sub>L</sub><sup>o</sup>|
! scope="col"| Analytical |Γ<sub>L</sub><sup>o</sup>|
|-
| 0.662V
| 0.5V
| 0.333V
| 0.209V
| 0.14
| 0.16
| 0.23
| 0.25
|}
<p> </p>
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