# Transmission Line Reflections at a Resistive Load

Foundations

Consider a transmission line of length L driven by a constant voltage source VG with a source resistance RG, and terminated by a resistive load RL, as shown in Figure 1. ZC is the characteristic impedance of the transmission line and T is the time it takes for the voltage wave to travel from the source to the load.

Figure 1: Transmission line driven by a constant source and terminated by a resistive load

When the switch closes at t = 0 a forward voltage wave, V +, originates at z = 0 and travels toward the load. This is shown in Figure 2.

Figure 2: Forward voltage wave originates at the source and travels toward the load

The value of this voltage is

(1)

As this wave travels along the transmission line, the voltage along the line changes from 0 to V + and remains at that value (for now). At the time T (about 18 ns for a 12 ft RG58 cable) the voltage wave reaches the load and sets up a reflection, V . This is shown in Figure 3(a).

Figure 3: Forward wave arrives at the load (a) incident and reflected voltages (b) total voltage

The reflected voltage, V , is related to the incident voltage, V +, by

(2)

where ΓL is the load reflection coefficient:

(3)

The total voltage at the load is the sum of the incident and reflected voltages

(4)

The reflected voltage wave, V , now travels back to the source, as shown in Figure 3(b). As this wave travels back to the source, the voltage along the line changes from V + to V + + V . This wave reaches the source at the time 2T and sets up another reflection V -+. This is shown in Figure 4(a).

Figure 4: Reflected wave arrives at the source (a) incident, reflected and re-reflected voltages (b) total voltage

The voltage reflected at the source, V -+, is related to the incident voltage,V , by

(5)

(6)

where ΓS is the source reflection coefficient:

The total voltage at the source is now

(7)

The voltage wave, V -+ now travels forward to the load, as shown in Figure 4(b).

If the source is matched to the transmission line, i.e., R= ZC then the reflection coefficient at the source is zero and therefore there is no reflection at the source. Then the total voltage at the source upon the arrival of the reflected voltage is

(8)

and the reflection process terminates.

Verification

The experimental setup and the circuit model for reflection measurements is shown in Figure 5.

Figure 5: Experimental setup for the load reflection measurements

Its circuit model is shown in Figure 6.

Figure 6: Circuit model for the reflection measurements

A 2 Vpp (open-circuit voltage) pulse signal was sent from the function generator along the 12 ft coaxial cable (RG58) to the resistive load. The voltages at the source (VS) and at the load (VL) were measured using the oscilloscope probes. Four cases of the load values were considered as shown in Figure 5.

Calculations for each case were performed and are summarized in Table 1. The measurement results are shown in Figure 7.

 RL = ∞ Figure 7(a) RL = 22Ω Figure 7(b) RL = 47Ω Figure 7(c) RL = 47Ω Figure 7(d) Initial voltage at the source, V+ , z = 0, t = 0 1V 1V 1V 1V Load reflection coefficient, ΓL 1 -0.39 -0.03 0.624 Voltage reflected at the load, V– , z = L, t = T 1V -0.39V -0.03V 0.624V Total voltage at the load, V+ + V– , z = L, t = T 2V 0.61V 0.97V 1.624V Total voltage at the source, V+ + V– , z = 0, t = 2T 2V 0.61V 0.97V 1.624V

Table 1: Calculated values

Figure 7: Measurement results

The calculated and measured values, shown in Table 1, closely match, verifying the concept of the transmission line reflections.

Dr. Bogdan Adamczyk is a professor and the director of the EMC Center at Grand Valley State University (GVSU). He is also the founder and principal educator of EMC Educational Services LLC which specializes in EMC courses for the industry. Prof. Adamczyk is the author of the upcoming book “Foundations of Electromagnetic Compatibility with Practical Application” (Wiley, 2017). He can be reached at profbogdan@emcspectrum.com.

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).

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1. Wolfgang Leister