EMC concepts explained
Shunt Inductive Discontinuity along the Transmission Line and Transmission Lines in Parallel
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his article discusses the impact of a shunt inductive discontinuity along the line as well as the effect of feeding two transmission lines in parallel. The analytical results are verified through the HyperLynx simulations and laboratory measurements.
1.1 Reflections at the Shunt Inductive Discontinuity – Analysis
Consider the circuit shown in Figure 1.1, where the transmission line of length l has a shunt inductive discontinuity in the middle of the line at a location z = d.
Figure 1.1: Shunt inductive discontinuity along a transmission line
Note that the transmission line is matched at the source and at the load; it is also assumed that the initial current through the inductor is zero, iL (0– = 0).
When the switch closes at t = 0, a wave originates at z = 0, [1], with
and travels towards the discontinuity. When this wave arrives at the discontinuity (at the time t = T), the reflected waves, vr and ir are created. The transmission line immediately to the right of the discontinuity looks to the circuit on the left of the discontinuity like a shunt resistance equal to the characteristic impedance of the right line [2]. Figure 1.2 illustrates this.
Figure 1.2: Incident and reflected waves at the inductive discontinuity
The reflected current wave is related to the reflected voltage wave by
KVL and KCL at the discontinuity produce
Our initial goal is to determine the reflected voltage vr (t) at the location z = d, i.e., vr (d, t). The ultimate goal is to determine the total voltage at the discontinuity, vL (d, t).
The current iR (t) can be expressed as
From Eq. (1.2b), we obtain the inductor current as
Utilizing Equations (1.1b), (1.1c), and (1.3) in (1.4) we get
The differential v-i relationship for the inductor is
Using Equations (1.1a), (1.2a), and (1.5) in Eq. (1.6) we obtain
or
This differential equation needs to be solved for vr (t), for t > T, subject to the initial condition vr (t = T). Let’s determine this initial condition. Initially, the current through an inductor iL (T–) is zero. Thus, the inductor acts as an open circuit, and the incident wave sees the discontinuity as a matched load. Therefore, there is no reflection and
Eq. (1.8) can be written as
where
The solution of Eq. (1.10) was derived in [2] as
Utilizing Equations (1.9) and (1.11) in Eq. (1.12), we obtain
The total voltage across the discontinuity is
or
Equation (1.14b) predicts that at t = T, the voltage at the load rises from zero to VG/2 and then decays exponentially to z. Let’s verify these observations through simulations and measurements.
1.2 Reflections at the Inductive Discontinuity – Simulation
Figure 1.3 shows the HyperLynx schematic of the transmission line with an inductive discontinuity.
Figure 1.3: Inductive discontinuity – HyperLynx schematic
The simulation results are shown in Figure 1.4.
Note that the simulation results verify the analytical results.
Figure 1.4: Inductive discontinuity – Voltages at the source (z = 0 ) and the load (z = d )
1.3 Reflections at the Inductive Discontinuity – Measurements
The measurement setup is shown in Figure 1.5.
Figure 1.5: Inductive discontinuity – Measurement setup
The measurement results are shown in Figure 1.6.
Note that the measurement results verify the simulation and analytical results.
Figure 1.6: Inductive discontinuity – Measurement results
2.1 Feeding Transmission Lines in Parallel – Analysis
Consider the circuit shown in Figure 2.1, where the transmission line of length d is connected to the two other transmission lines.
Figure 2.1: Transmission lines in parallel
Note that load resistors are matched to the transmission lines, and all transmission lines have the same characteristic impedances. This makes the simulations and measurements easier to follow since there are no reflections at the loads.
When the switch closes at t = 0, a wave originates at z = 0 and travels towards the discontinuity. The transmission lines immediately to the right of the discontinuity look to the circuit on the left of the discontinuity like two resistances in parallel. Their values are equal to the corresponding values of their characteristic impedances. When the incident wave arrives at the discontinuity (at the time t = T), the reflected wave, vr and ir, is created, and we have a situation depicted in Figure 2.2.
Figure 2.2: Incident and reflected waves at the discontinuity
This part of the circuit is identical to the one discussed in [3] where the transmission line had a shunt resistive discontinuity. Thus, the total voltage across at the discontinuity is
and the total voltage at the source (after t = 2T) is
2.2 Feeding Transmission Lines in Parallel – Simulations
Figure 2.3 shows the HyperLynx schematic of the transmission line in parallel.
Figure 2.3: Transmission lines in parallel – HyperLynx schematic
The simulation results are shown in Figure 2.4.
Figure 2.4: Transmission lines in parallel – Voltages at the source (z = 0 ) and the discontinuity (z = d )
2.3 Feeding Transmission Lines in Parallel – Measurements
The measurement setup is shown in Figure 2.5.
Figure 2.5: Transmission lines in parallel – Measurement setup
The measurement results are shown in Figure 2.6.
Figure 2.6: Transmission lines in parallel – Measurement results
Note that the measurement results verify the simulation and analytical results.
References
- Adamczyk, B., “Transmission Line Reflections at a Resistive Load,” In Compliance Magazine, January 2017.
- Adamczyk, B., Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017.
- Adamczyk, B., “Transmission Line Reflections at a Shunt Resistive and Reactive Discontinuity Along the Line,” In Compliance Magazine, October 2022.
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Dr. Bogdan Adamczyk is professor and director of the EMC Center at Grand Valley State University (http://www.gvsu.edu/emccenter) where he regularly teaches EMC certificate courses for industry. He is an iNARTE certified EMC Master Design Engineer. Prof. Adamczyk is the author of the textbook “Foundations of Electromagnetic Compatibility with Practical Applications” (Wiley, 2017) and the upcoming textbook “Principles of Electromagnetic Compatibility with Laboratory Exercises” (Wiley 2022). He can be reached at adamczyb@gvsu.edu.