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# Reflection coefficient of traveling waves

2022/07/19

Author: Noyafa–CCTV Tester

Reflection Coefficient of Traveling Waves The reflection degree of traveling waves can be expressed by the ratio of the reflected voltage (current) to the incident voltage (current) at the impedance mismatch point where the reflection occurs. This ratio is called the reflection coefficient. Let the line wave impedance be Z1, and the equivalent impedance of the impedance mismatch point is Z2, see the reflection coefficient of the traveling wave. The reflection degree of the traveling wave can be calculated from the reflected voltage (current) at the impedance mismatch point where the reflection occurs and the incident voltage (current). This ratio is called the reflection coefficient. Assuming that the line wave impedance is Z1, and the equivalent impedance of the impedance mismatch point is Z2, as shown in Figure 2.5, the voltage reflection coefficient is:ρu=UfUi =(Z2-Z1)(Z2+Z1) (2.3) Reflection of traveling wave Assuming that the incident wave is a forward traveling wave, the relationship between incident voltage and current wave: ii=Ui/Z0 (2.4) and the corresponding reflection The wave is a reverse traveling wave, the relationship between the reflected voltage and the current wave: if = -UfZ0 (2.5) Deduced from equations 2.3, 2.4, and 2.5, the current reflection coefficient at the impedance mismatch point:ρi=ifii=-UfUi=-ρu It can be seen that the current reflection coefficient and the voltage reflection coefficient of the impedance mismatch point are equal in magnitude and opposite in sign.

The reflection coefficients in several cases are discussed below. 1. Open circuit When the cable has an open circuit point, or the traveling wave moves to the open circuit terminal of the cable, Z2→∞. According to Equation 2.3, since Z1 is much smaller than Z2, the effect of Z1 can be ignored, and the voltage reflection coefficient can be calculated:ρThe open circuit of u=1 causes the total reflection of the voltage (Fig. 2.6), and the voltage reflected wave has the same polarity as the incident wave.

The actual open-circuit point voltage is the sum of the incident voltage and the reflected voltage, so there is a voltage doubling phenomenon. Voltage reflection at the open-circuit terminal The current reflection coefficient of the open-circuit point is -1, the reflected current and the incident current are equal in magnitude and opposite in direction, and the actual open-circuit point current is the sum of the two, so it is zero. The current at the open-circuit point is zero and the voltage is doubled, which can be interpreted as the fact that after the traveling wave reaches the open-circuit point, the magnetic field energy carried by the current is all converted into the electric field energy represented by the line voltage.

2. Short circuit When there is a short circuit point in the cable, Z2=0, according to the formula 2.3, calculate the voltage reflection coefficient:ρThe reflected voltage at the short-circuit point of u=-1 is equal in magnitude to the incident voltage and opposite in direction (Figure 2.7), and its combined voltage is zero. Reflection at the short-circuit point The reflection coefficient of the current at the short-circuit point is +1, the reflected current is equal to the incident current, and the current at the short-circuit point doubles. The voltage at the short-circuit point is zero and the current is doubled, indicating that after the traveling wave reaches the short-circuit point, all the electric field energy is converted into the magnetic field energy.

3. A low-resistance fault occurs in the cable. When a low-resistance fault occurs in the middle of the cable, see Figure 2.8. The cables on both sides of the resistance are replaced by resistors whose size is equal to the wave impedance value Z0. The fault resistance Rf and the wave impedance value Z0 of the second cable The parallel connection constitutes the load impedance of the first cable, namely: Z2= RfZ0(Rf+Z0) Fault point voltage reflection coefficient: Pu=(Z2-Z1)(Z2+Z1)=-1/(1+2K) (2.6) where K=Rf/Z0. Equation 2.6 is particularly useful for analyzing the reflection of low-voltage pulses at the fault point. Equivalent circuit of cable low-resistance fault point 4. Inductance When the cable load is an inductance, as shown in Figure 2.9, the reflection coefficient is no longer a simple real number, but a time-varying quantity.

It can be deduced that the voltage reflection coefficient is: u=2e- t/τ - 1 (2.7) where τ=L/Z0, which is called the time constant, and L is the inductance value. t=0 ,ρu=1 t=τ,ρu=-0.26t→∞ ,ρu=-1 Figure 2.9 The reflection of the inductor can be seen. After the terminal is connected to the inductor, the voltage reflection coefficientρu will vary from +1 to -1 over time. Because when t=0, the voltage wave just reaches the end of the cable, because the current on the inductance cannot change abruptly, the inductance is equivalent to an open circuit, so the reflection coefficientρu=1; and t→∞When the current on the inductor enters a steady state, the voltage is zero, which is equivalent to a short circuit, soρu=-1, see Figure 2.10.

The time when the voltage reflection coefficient is zero, t0=τn2. Figure 2.10 Reflection Coefficient of Inductor 5. When the capacitor is terminated with a capacitor, Figure 2.11, the voltage reflection coefficient is derived: 11 Reflection of the capacitorρu=1-2e-t/τ where τ=Z0C, called time constant, C is the capacitance value. It can be seen that when the terminal is connected to the capacitor, the reflection coefficient changes from -1 to +1 with time. When t=0, the voltage on the capacitor cannot change abruptly, which is equivalent to a short circuit, so the reflection coefficientρu=-1, and when t→∞When , the voltage on the capacitor has stabilized, which is equivalent to an open circuit, so the reflection coefficient is 1, as shown in Figure 2.12.

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