' 6.4 Reflections and Standing Waves
FIGURE 6.12 Standing waves on
open-circuited line
FIGURE 6.12 Standing waves on a short-circuited line
one-quarter wavelength away, zero again at a distance of one-half wave-length, and so on. In other words, the graph of current as a function of posi-tion for an open-circuited line looks just like the graph of voltage versus position for a short-circuited line. Similarly, the current on a short-circuited line has a maximum at the termination, a minimum one-quarter wave-length away, and another maximum at a distance of a half wavewave-length, just like the voltage curve for the open-circuited line.
What happens when the line is mismatched, but not so drastically as dis-cussed above? There will be a reflected wave, but it will not have as large an amplitude as the incident wave since some of the incident signal will be dis-sipated in the load. The amplitude and phase angle will depend on the load impedance compared to that at the line. The incident and reflected voltages are related by the coefficient of reflection:
Γ = V
Γ = coefficient of reflection Vr = reflected voltage
Vi= incident voltage It can be shown that
Γ = −
ZL = load impedance
Z0 = characteristic impedance of the line
In general,Γis complex, but for a lossless line it is a real number if the load is resistive. It is positive for ZL> Z0and negative for ZL< Z0. F or ZL=Z0, the reflection coefficient is of course zero. A positive real coefficient means that the incident and reflected voltages are in phase at the load.
When a reflected signal is present but of lower amplitude than the inci-dent wave, there will be standing waves of voltage and current, but there will be no point on the line where the voltage or current remains zero over the whole cycle. See Figure 6.13 for an example.
It is possible to define the voltage standing-wave ratio (VSWR or just SWR) as follows:
SWR V
= Vmax
min
(6.10)
where
SWR = voltage standing-wave ratio Vmax = maximum rms voltage on the line
Vmin = minimum rms voltage on the line
The SWR concerns magnitudes only and is thus a real number. It must be positive and greater than or equal to 1. For a matched line the SWR is 1 (some-times expressed as 1:1 to emphasize that it is a ratio), and the closer the line is to being matched, the lower the SWR. The SWR has the advantage of being easier to measure than the reflection coefficient, but the latter is more useful in many calculations. Since both are essentially measures of the amount of reflection on a line, it is possible to find a relationship between them.
The maximum voltage on the line occurs where the incident and re-flected signals are in phase, and the minimum voltage is found where they are out of phase. Therefore, using absolute value signs to emphasize the lack of a need for phase information, we can write
Vmax = Vi + Vr (6.11)
and
Vmin = Vi − Vr (6.12)
Combining Equations (6.10), (6.11), and (6.12), we get
(6.13)
=
A little algebra will show that |Γ| can also be expressed in terms of SWR:
Γ = −
For the special, but important, case of a lossless line terminated in a resis-tive impedance, it is possible to find a simple relationship between stand-ing-wave ratio and the load and line impedances. First, suppose that ZL> Z0. Then from Equation (6.13),
Use of the appropriate equation will always give an SWR that is greater than or equal to one, and positive.
EXAMPLE6.6
Y
A 50-Ωline is terminated in a 25-Ωresistance. Find the SWR.
SOLUTION
In this case, Z0> ZLso the solution is given by Equation (6.16).
SWR Z
The presence of standing waves causes the voltage at some points on the line to be higher than it would be with a matched line, while at other points the voltage is low but the current is higher than with a matched line. This sit-uation results in increased losses. In a transmitting application, standing waves put additional stress on the line and can result in failure of the line or of equipment connected to it. For instance, if the transmitter happens to be connected at or near a voltage maximum, the output circuit of the transmit-ter may be subjected to a dangerous overvoltage condition. This is especially likely to damage solid-state transmitters, which for this reason are often equipped with circuits to reduce the output power in the presence of an SWR greater than about 2:1.
Reflections can cause the power delivered to the load to be less than it would be with a matched line for the same source, because some of the power is reflected back to the source. Since power is proportional to the square of voltage, the fraction of the power that is reflected isΓ2, that is,
Pr = Γ2Pi (6.17)
where
Pr = power reflected from the load Pi = incident power at the load
Γ = voltage reflection coefficient
SometimesΓ2is referred to as the power reflection coefficient.
The amount of power absorbed by the load is the difference between the incident power and the reflected power, that is,
PL = Pi− Γ2Pi (6.18)
= Pi(1− Γ2)
EXAMPLE6.7
Y
A generator sends 50 mW down a 50-Ωline. The generator is matched to the line but the load is not. If the coefficient of reflection is 0.5, how much power is reflected and how much is dissipated in the load?
SOLUTION
The amount of power that is reflected is, from Equation (6.17):
Pr = Γ2Pi
= 0.52×50 mW
= 12.5 mW
The remainder of the power reaches the load. This amount is PL = Pi−Pr
= 50 mW−12.5 mW
= 37.5 mW
Alternatively, the load power can be calculated directly from Equa-tion (6.18):
PL = Pi(1− Γ2)
= 50 mW(1−0.52)
= 37.5 mW
X
Since SWR is easier to measure than the reflection coefficient, an expres-sion for the power absorbed by the load in terms of the SWR would be useful.
It is easy to derive such an expression by using the relationship betweenΓ and SWR given in Equation (6.13). The derivation is left as an exercise; the result is
P SWR
SWR P
L = i
+ 4
1 2
( ) (6.19)
EXAMPLE6.8
Y
A transmitter supplies 50 W to a load through a line with an SWR of 2. Find the power absorbed by the load.
SOLUTION
Reflections on transmission lines can cause problems in receiving appli-cations as well. For instance, reflections on a television antenna feedline can cause a double image or “ghost” to appear. In data transmission, reflections can distort pulses, causing errors.
Variation of Impedance Along a Line
A matched line presents its characteristic impedance to a source located any distance from the load. If the line is not matched, however, the imped-ance seen by the source can vary greatly with its distimped-ance from the load.
At those points where the voltage is high and the current low, the impedance is higher than at points with the opposite current and voltage characteris-tics. In addition, the phase angle of the impedance can vary. At some points, a mismatched line may look inductive, at others capacitive, and at a few points, resistive. Very near the load, the impedance looking into the line is close to ZL. This is one reason why transmission-line techniques need to be used only with relatively high frequencies and/or long lines: a line shorter than about one-sixteenth of a wavelength can usually be ignored.
The impedance that a lossless transmission line presents to a source var-ies in a periodic way. We have already noticed that the standing-wave pat-tern repeats itself every one-half wavelength along the line; the impedance varies in the same fashion. At the load and at distances from the load that are multiples of one-half wavelength, the impedance looking into the line is that of the load.
The impedance at any point on a lossless transmission line is given by the equation
where
Z = impedance looking from the source toward the load ZL = load impedance
Z0 = characteristic impedance of the line
θ = distance to the load in degrees (for example a quarter wavelength would be 90°)
Provided that cosθ ≠0, this simplifies to
Z Z Z jZ
These equations can be fairly tedious to work with, especially when ZLis complex. Most transmission-line impedance calculations are done using computers. Many of the computer programs give their results in the form of a Smith chart, which will be described in Appendix A.
Characteristics of Shorted and Open Lines
Though a section of transmission line that is terminated in an open or short circuit is useless for transmitting power, it can serve other purposes. Such a line can be used as an inductive or capacitive reactance or even as a resonant circuit. In practice, short-circuited sections are much more useful, because open-circuited lines tend to radiate energy from the open end.
The impedance of a short-circuited line can be found from Equation (6.21), by setting ZLequal to zero. The impedance looking toward the short circuit is
Z =jZ0tan θ (6.22)
Note that the impedance has no resistive component, since no power can be dissipated in this line. For short lengths (less than one-quarter wave-length or 90°), the impedance is inductive. At one-quarter wavewave-length, the line looks like a parallel-resonant circuit, with infinite impedance. The line is capacitive for lengths between one-quarter and one-half wavelength, since the tangent of the corresponding angle is negative. At a length of one-half wavelength, or 180°, tan θ is zero, and the line behaves like a series-resonant circuit having zero impedance. For longer lines the cycle repeats.
Figure 6.14 shows graphically how the impedance varies with length. It is easy to remember that a short length of shorted line is inductive by visual-izing the shorted end as a loop or coil hence an inductance.
The open-circuited line, on the other hand, is capacitive in short lengths. You can remember this by thinking of the two parallel lines at the
end of the transmission line as a capacitor. It is series-resonant at a length of one-quarter wavelength, inductive between one-quarter and one-half wave-length, and parallel-resonant at a length of one-half wavelength. For longer lines the cycle repeats. Figure 6.15 shows graphically how the impedance varies with length for this line.
Short transmission-line sections, called stubs, can be substituted for ca-pacitors, inductors or tuned circuits in applications where lumped-constant components (conventional inductors or capacitors) would be inconvenient or impractical. At VHFand UHFfrequencies the required values of induc-tance and capaciinduc-tance are often very small. It would be difficult to build a 1 pFcapacitor, for instance; the capacitance between its leads might well be more than that. In addition, physically small components are difficult to use where large amounts of power are involved, as in transmitters, because of
FIGURE 6.14