Transmission Line Reflections at a Resistive Load

Foundations

Consider a transmission line of length L driven by a constant voltage source V_G with a source resistance R_G, and terminated by a resistive load R_L, as shown in Figure 1. Z_C 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.

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.

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).

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

 (2)

where Γ_L 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).

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

 (5)

 (6)

where Γ_S 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_G = Z_C 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.

Its circuit model is shown in Figure 6.

A 2 V_pp (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 (V_S) and at the load (V_L) 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.

Table 1: Calculated values

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