Transmission Line Reflections at a Shunt Resistive and Reactive Discontinuity Along the Line

This article discusses the reflections on a transmission line at a shunt resistive and capacitive discontinuity along the line. The analytical results are verified through the HyperLynx simulations and laboratory measurements.

1.1  Reflections at the Shunt Resistive Discontinuity – Analysis

Consider the circuit shown in Figure 1.1, where the transmission line of length l has a shunt resistive discontinuity in the middle of the line, at a location z = d.

Figure 1.1: Shunt resistive discontinuity along a transmission line

Note that the transmission line is matched at the source, and the resistive discontinuity and the load resistor values are equal to the characteristic impedance of the line.

When the switch closes at t = 0, a wave originates at z = 0, [1], and travels towards the discontinuity. At the time, t = T this wave arrives at the discontinuity. 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].

When this 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 1.2.

Figure 1.2: Reflection at the resistive discontinuity

The circuit in Figure 1.2 is equivalent to the one shown in Figure 1.3.

Figure 1.3: Equivalent circuit

The reflection coefficient at the discontinuity is

(1.1)

The reflected voltage is related to the incident voltage by the reflection coefficient as

 (1.2)

The total voltage at the discontinuity is

 (1.3)

Since, [1],

 (1.4) 

the total voltage at the discontinuity is

 (1.5)

The reflected voltage travels back to the source, arriving there at t = 2T. Since the source is matched to the transmission line, there is no reflection, and the total voltage at the source becomes

 (1.6)

1.2  Reflections at the Shunt Resistive Discontinuity – Simulations

Figure 1.4 shows the HyperLynx schematic of the transmission line with a shunt resistive discontinuity.

Figure 1.4: Shunt resistive discontinuity – HyperLynx schematic

The simulation results are shown in Figure 1.5.

Figure 1.5: Shunt resistive discontinuity – Voltages at the source (z = 0) and the load (z = d)

1.3  Reflections at the Shunt Resistive Discontinuity – Measurements

The measurement setup is shown in Figure 1.6.

Figure 1.6: Shunt resistive discontinuity – Measurement setup

The measurement results are shown in Figure 1.7. 

Figure 1.7: Shunt resistive discontinuity – Measurement results

Note that the measurement results verify the simulation and analytical results.

2.1  Reflections at the Shunt Capacitive Discontinuity – Analysis

Consider the circuit shown in Figure 2.1, where the transmission line of length l has a shunt capacitive discontinuity in the middle of the line at a location z = d.

Figure 2.1: Shunt capacitive discontinuity along a transmission line

Note that the load resistor value is equal to the characteristic impedance of the transmission line; it is also assumed that the initial voltage across the capacitor is zero, v(0) = 0.

When the switch closes at t = 0, a wave originates at z = 0, [1], and travels towards the discontinuity. 

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]. 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 capacitive discontinuity

This part of the circuit is identical to the one discussed in [3], where the transmission line was terminated with an RC load. Thus, the total voltage across the capacitive discontinuity is

 (2.1)

Equation (2.1) predicts that at t = T, the voltage at the discontinuity is zero and increases exponentially to VG  /2. Let’s verify these observations through simulations and measurements.

2.2  Reflections at the Shunt Capacitive Discontinuity – Simulation

Figure 2.3 shows the HyperLynx schematic of the transmission line with a capacitive discontinuity.

Figure 2.3: Capacitive discontinuity – HyperLynx schematic

The simulation results are shown in Figure 2.4.

Figure 2.4: Capacitive discontinuity – Voltages at the source (z = 0) and the load (z = d)

2.3  Reflections at the RC Load – Measurements

The measurement setup is shown in Figure 2.5.

Figure 2.5: Capacitive discontinuity – Measurement setup

The measurement results are shown in Figure 2.6. 

Figure 2.6: Capacitive discontinuity – Measurement results

Note that the measurement results verify the simulation and analytical results. 

References

  1. Adamczyk, B., “Transmission Line Reflections at a Resistive Load,” In Compliance Magazine, January 2017.
  2. Adamczyk, B. Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017. 
  3. Adamczyk, B., “Transmission Line Reflections at the RL and RC Loads,” In Compliance Magazine, January 2021.

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