EMC concepts explained
Transmission Line Reflections at the RL and RC Loads
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his 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]).
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, iL(0_) = 0.
When the switch closes at t = 0, a wave originates at z = 0, [2], with
and travels towards the load. When this wave arrives at the load, (at the time t = T), the reflected waves, vr and ir are created. This is shown in Figure 1.2.
The reflected current wave is related to the reflected voltage wave by
KVL and KCL at the load produce
Figure 1.1: RL termination of a transmission line
Figure 1.2: Incident and reflected waves at the RL load
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 load, vRL(d,t). From Eq. (1.2a) we obtain the inductor voltage as
The load resistor voltage can be obtained from
Using Eqns. (1.1a) and (1.4) in Eq. (1.3) produces
The differential v-i relationship for the inductor is
Utilizing Eqns. (1.2b) and (1.6) in Eq. (1.5) we get
Using Eq. (1.1b) and (1.1c) in Eq. (1.7) we have
Since VG and ZC are constant Eq. (1.8) reduces to
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. Using Eqns. (1.1a) and (1.1c) in (1.2b) gives
Evaluating it at t = T, we get
Since the current through the inductor cannot change instantaneously, we have iL(T), and thus
Now we are ready to solve Eq. (1.9), subject to the initial condition in Eq. (1.12):
First, let’s rewrite this equation in a standard form:
or
where
The solution of Eq. (1.16) was derived in [3] as
Utilizing Eqns. (1.12) and (1.16) in Eq. (1.17), we obtain
The total voltage across the RL load is
or
Equation (1.19b) predicts that at t = T, the voltage at the load rises from zero to VG, and then decays exponentially to VG/2. Let’s verify these observations through simulations and measurements.
1.2 Reflections at the RL Load – Simulation
Figure 1.3 on page 42 shows the HyperLynx schematic of the transmission line terminated in an RL load.
The simulation results are shown in Figure 1.4.
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.
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]).
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, vC(0_) = 0.
Figure 1.3: RL load – HyperLynx schematic
Figure 1.4: RL load – Voltages at the source (z = 0) and the load (z = d)
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, vr and ir are created. This is shown in Figure 2.2.
The reflected current wave is related to the reflected voltage wave by Eq. (1.1c). KVL and KCL at the load 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 voltage at the load, vC(d, t).
Figure 1.5: RL load – Measurement setup
Figure 1.6: RL load – Measurement results
From Eq. (2.1b) we obtain the capacitor current as
The load resistor current can be obtained from
Using Eqns. (1.1b), (1.1c) and (2.3) in Eq. (2.2) produces
The differential v-i relationship for the capacitor is
Utilizing Eqns. (1.1a) and (2.4) in Eq. (2.5) we get
Since VG is constant Eq. (2.6) reduces to
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. Using Eq. (1.1a) in (2.1a) gives
Evaluating it at t = T, we get
Since the voltage across the capacitor cannot change instantaneously, we have vC(T) = 0, and thus
Now we are ready to solve Eq. (2.7), subject to the initial condition in Eq. (2.10):
First, let’s rewrite this equation in a standard form:
This equation is in the form of Eq. (1.15), with
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
The total voltage across the RC load is
or
Figure 2.1: RC termination of a transmission line
Figure 2.2: Incident and reflected waves at the RC load
Equation (2.15b) predicts that at t = T, the voltage at the load is zero and increases exponentially to VG/2. Let’s verify these observations through simulations and measurements.
2.2 Reflections at the RC Load – Simulation
Figure 2.3 shows the HyperLynx schematic of the transmission line terminated in an RC load.
The simulation results are shown in Figure 2.4.
Figure 2.3: RC Load – HyperLynx schematic
Figure 2.3: RC Load – HyperLynx schematic
Figure 2.4: RC load – Voltages at the source (z = 0) and the load (z = d)
Figure 2.5: RC load – Measurement setup
2.3 Reflections at the RC Load – Measurements
The measurement setup and the results are shown in Figure 2.5.
The measurement results are shown in Figure 2.6.
Note that the measurement results verify the simulation and the analytical results.
Figure 2.6: RC load – Measurement results
References
- Adamczyk, B., “Transmission Line Reflections at a Reactive Load,” In Compliance Magazine, December 2018.
- 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.
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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.