Transmission Line Reflections at the RL and RC Loads

January 1, 2021

This article discusses the reflections on a transmission line terminated with either an RL or an RC load. The detailed analytical derivations are verified through the HyperLynx simulations and laboratory measurements.

1.1 Reflections at the RL Load – Analysis

Consider the circuit shown in Figure 1.1, where the transmission line of length d is terminated by an RL load. (Reflections at the purely inductive load are discussed in [1]).

Figure 1.1: RL termination of 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 current through the inductor is zero, i_L(0_) = 0.

When the switch closes at t = 0, a wave originates at z = 0, [2], with

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(1.1a)

(1.1b)

and travels towards the load. When this wave arrives at the load, (at the time t = T), the reflected waves, v_r and i_r are created. This is shown in Figure 1.2.

Figure 1.2: Incident and reflected waves at the RL load

The reflected current wave is related to the reflected voltage wave by

(1.1c)

KVL and KCL at the load produce

(1.2a)

(1.2b)

Our initial goal is to determine the reflected voltage v_r(t) at the location z = d, i.e., v_r(d,t). The ultimate goal is to determine the total voltage at the load, v_RL(d,t). From Eq. (1.2a) we obtain the inductor voltage as

(1.3)

The load resistor voltage can be obtained from

(1.4)

Using Eqns. (1.1a) and (1.4) in Eq. (1.3) produces

(1.5)

The differential v-i relationship for the inductor is

(1.6)

Utilizing Eqns. (1.2b) and (1.6) in Eq. (1.5) we get

(1.7)

Using Eq. (1.1b) and (1.1c) in Eq. (1.7) we have

(1.8)

Since V_G and Z_C are constant Eq. (1.8) reduces to

(1.9)

This differential equation needs to be solved for v_r(t), for t > T, subject to the initial condition v_r(t = T). Let’s determine this initial condition. Using Eqns. (1.1a) and (1.1c) in (1.2b) gives

(1.10)

Evaluating t = T it at , we get

(1.11)

Since the current through the inductor cannot change instantaneously, we have i_L(T), and thus

(1.12)

Now we are ready to solve Eq. (1.9), subject to the initial condition in Eq. (1.12):

(1.13)

First, let’s rewrite this equation in a standard form:

(1.14)

(1.15)

where

(1.16)

The solution of Eq. (1.16) was derived in [3] as

(1.17)

Utilizing Eqns. (1.12) and (1.16) in Eq. (1.17) we obtain

(1.18)

The total voltage across the RL load is

(1.19a)

(1.19b)

Equation (1.19b) predicts that at t = T, the voltage at the load rises from zero to V_G, and then decays exponentially to V_G/2. Let’s verify these observations through simulations and measurements.

1.2 Reflections at the RL Load – Simulation

Figure 1.3 shows the HyperLynx schematic of the transmission line terminated in an RL load.

Figure 1.3: RL load – HyperLynx schematic

The simulation results are shown in Figure 1.4.

Figure 1.4: RL load – Voltages at the source (*z = 0*) and the load (*z = d*)

1.3 Reflections at the RL Load – Measurements

The measurement setup and the results are shown in Figure 1.5.

The measurement results are shown in Figure 1.6.

Figure 1.6: RL load – Measurement results

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

2.1 Reflections at the RC Load – Analysis

Consider the circuit shown in Figure 2.1 where the transmission line of length d is terminated by an RC load. (Reflections at the purely capacitive load are discussed in [1]).

Figure 2.1: RC termination of 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_C(0_) = 0.

When the switch closes at t = 0, a wave originates at z = 0, with the initial voltage and current values given by Eqns. (1.1a) and (1.1b); this wave travels towards the load. When the wave arrives at the load, (at the time t = T), the reflected waves, v_r and i_r are created. This is shown in Figure 2.2.

Figure 2.2: Incident and reflected waves at the RC load

The reflected current wave is related to the reflected voltage wave by Eq. (1.1c). KVL and KCL at the load produce

(2.1a)

(2.1b)

Our initial goal is to determine the reflected voltage v_r(t) at the location z = d, i.e., v_r(d, t). The ultimate goal is to determine the voltage at the load, v_C(d, t). From Eq. (2.1b) we obtain the capacitor current as

(2.2)

The load resistor current can be obtained from

(2.3)

Using Eqns. (1.1b), (1.1c) and (2.3) in Eq. (2.2) produces

(2.4)

The differential v-i relationship for the capacitor is

(2.5)

Utilizing Eqns. (1.1a) and (2.4) in Eq. (2.5) we get

(2.6)

Since V_G is constant Eq. (2.6) reduces to

(2.7)

This differential equation needs to be solved for v_r(t), for t > T, subject to the initial condition v_r(t = T). Let’s determine this initial condition. Using Eq. (1.1a) in (2.1a) gives

(2.8)

Evaluating it at t = T, we get

(2.9)

Since the voltage across the capacitor cannot change instantaneously, we have v_C(T) = 0, and thus

(2.10)

Now we are ready to solve Eq. (2.7), subject to the initial condition in Eq. (2.10):

(2.11)

First, let’s rewrite this equation in a standard form:

(2.12)

This equation is in the form of Eq. (1.15), with

(2.13)

The solution of Eq. (2.13) is of the form presented in Eq. (1.17). Utilizing Eqns. (2.10) and (2.13) in Eq. (1.17) we obtain

(2.14)

The total voltage across the RC load is

(2.15a)

(2.15b)

Equation (2.15b) predicts that at t = T, the voltage at the load is zero and increases exponentially to V_G/2. Let’s verify these observations through simulations and measurements.