Wave reflection and the reflection coefficient

It has already been mentioned that for a lossless line equation (7.8) applies. If the line is increased in length the inductance increases, and the capacitance increases because the capacitance area is greater. So overall the L/C ratio remains the same.

When the load and generator impedances are both equal to the characteristic impedance of the line, maximum energy is absorbed by the load. However, when a line is not terminated in its own impedance the load absorbs only a portion of the energy, reflecting the remainder back along the line. Standing waves of voltage and current are set up, their amplitude depending on the extent of the mismatch. The standing waves are smaller in amplitude than the generator signal because the resistance absorbs a portion of the energy, sending the remainder back as reflected waves.

It is instructive at this point to explain what is meant by a standing wave. A wave travelling along a transmission line consists of electric and magnetic components, and energy is stored in the magnetic field of the line inductance (y LI2) and the electric field of the line capacitance (-j CV2). This energy is interchanged between the magnetic and electric fields and causes the transmission of the electromagnetic energy along the line.

When a wave reaches an open-circuited termination the magnetic field collapses since the current is zero. The energy is not lost but is converted into electrical energy adding to that already caused by the existing electric field. Hence the voltage at the termination doubles, and this increased voltage causes the reflected wave which moves back along the line in the reverse direction.

The same reasoning can be applied to a short-circuited termination. In this case the electric field collapses at the termination and its energy changes to magnetic energy. This results in a doubling of the current.

Extreme resistance loading conditions are shown in Fig. 7.2. In Fig. 7.2(a) the line is terminated in an open circuit and, regardless of the length of line, the current and voltage conditions at the termination end are the same. In an open circuit the current is a minimum and so the impedance and voltage are maximum. In Fig. 7.2(b) the line termination is a short circuit resulting in maximum current, while the impedance and voltage are minimum.

Figures 7.3(a) and 7.3(b) show terminations in pure capacitance and pure inductance, respectively. In Fig. 7.3(a) all of the energy is reflected because a capacitor does not absorb energy. The relative phases of the waves shift according to the reactance and the line impedance. It is assumed in this case that the capacitive reactance equals the characteristic impedance simulating a phase condition of 45°.

At the termination end of the line the current and voltage arrive in phase, but due to the current through Zo and Xc in series, the current leads by 45° while the voltage lags by 45°. So although current and voltage arrive in phase they are reflected 90° out of phase, as shown by the standing waves.

The voltage standing wave is a minimum at a point A/8 from the termination end when Xc = Zo. When Xc is greater the arrangement acts more like an open circuit and the minimum voltage point moves further away from the termination end. When Xc is less than Zo it is more like a short circuit, and the voltage minimum is closer to the end.

In Fig. 7.3(b) the line is terminated in its inductance and the waveforms occur when XL = Zo. Conditions are similar to the capacitance case, except that the phase shifts occur in the opposite direction. The shift is again 45° but the waveforms at the termination end

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Responses

  • asmeret
    When reflection coefficient is equal to 1 of transmission line?
    2 years ago

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