• Electromagnetics I
  • Ch 3: Transmission Lines
  • Loc 3.12
  • Electromagnetics I
  • Ch 3
  • Loc 3.12

Voltage Reflection Coefficient

We now consider the scenario shown in Figure 3.12.1. Here a wave arriving from the left along a lossless transmission line having characteristic impedance

arrives at a termination located at

. The impedance looking into the termination is

, which may be real-, imaginary-, or complex-valued. The questions are: Under what circumstances is a reflection – i.e., a leftward traveling wave – expected, and what precisely is that wave?

The potential and current of the incident wave are related by the constant value of

. Similarly, the potential and current of the reflected wave are related by

. Therefore, it suffices to consider either potential or current. Choosing potential, we may express the incident wave as

(3.12.1)

where

is determined by the source of the wave, and so is effectively a “given.” Any reflected wave must have the form

(3.12.2)

Therefore, the problem is solved by determining the value of

given

,

, and

.

Considering the situation at

, note that by definition we have

(3.12.3)

where

and

are the potential across and current through the termination, respectively. Also, the potential and current on either side of the

interface must be equal. Thus,

(3.12.4)
(3.12.5)

where

and

are the currents associated with

and

, respectively. Since the voltage and current are related by

, Equation 3.12.5 may be rewritten as follows:

(3.12.6)

Evaluating the left sides of Equations 3.12.4 and 3.12.6 at

, we find:

(3.12.7)
(3.12.8)

Substituting these expressions into Equation 3.12.3 we obtain:

(3.12.9)

Solving for

we obtain

(3.12.10)

Thus, the answer to the question posed earlier is that

(3.12.11)
(3.12.12)

The quantity

is known as the voltage reflection coefficient. Note that when

,

and therefore

. In other words,

If the terminating impedance is equal to the characteristic impedance of the transmission line, then there is no reflection.

If, on the other hand,

, then

,

, and a leftward-traveling reflected wave exists.

Since

may be real-, imaginary-, or complex-valued,

too may be real-, imaginary-, or complex-valued. Therefore,

may be different from

in magnitude, sign, or phase.

Note also that

is not the ratio of

to

. The ratio of the current coefficients is actually

. It is quite simple to show this with a simple modification to the above procedure and is left as an exercise for the student.

Summarizing:

The voltage reflection coefficient

, given by Equation 3.12.12, determines the magnitude and phase of the reflected wave given the incident wave, the characteristic impedance of the transmission line, and the terminating impedance.

We now consider values of

that arise for commonly-encountered terminations.

Matched Load (

). In this case, the termination may be a device with impedance

, or the termination may be another transmission line having the same characteristic impedance. When

,

and there is no reflection.

Open Circuit. An “open circuit” is the absence of a termination. This condition implies

, and subsequently

. Since the current reflection coefficient is

, the reflected current wave is 180 out of phase with the incident current wave, making the total current at the open circuit equal to zero, as expected.

Short Circuit. “Short circuit” means

, and subsequently

. In this case, the phase of

is 180, and therefore, the potential of the reflected wave cancels the potential of the incident wave at the open circuit, making the total potential equal to zero, as it must be. Since the current reflection coefficient is

in this case, the reflected current wave is in phase with the incident current wave, and the magnitude of the total current at the short circuit non-zero as expected.

Purely Reactive Load. A purely reactive load, including that presented by a capacitor or inductor, has

where

is reactance. In particular, an inductor is represented by

and a capacitor is represented by

. We find

The numerator and denominator have the same magnitude, so

. Let

be the phase of the denominator (

). Then, the phase of the numerator is

. Subsequently, the phase of

is

. Thus, we see that the phase of

is no longer limited to be 0 or 180 , but can be any value in between. The phase of reflected wave is subsequently shifted by this amount.

Other Terminations. Any other termination, including series and parallel combinations of any number of devices, can be expressed as a value of

which is, in general, complex-valued. The associated value of

is limited to the range 0 to 1. To see this, note:

Note that the smallest possible value of

occurs when the numerator is zero; i.e., when

. Therefore, the smallest value of

is zero. The largest possible value of

occurs when

(i.e., an open circuit) or

(a short circuit); the result in either case is

. Thus,

 

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0

 
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