Table of Contents

**Standing Waves**

**Standing Waves:** When a signal is applied to a transmission line, it appears at the other end of the line sometime later because of the propagation delay. If a resistive load equal to the characteristic impedance of a line is connected at the end of the line, the signal is absorbed by the load, and power is dissipated as heat. If the load is an antenna, the signal is converted to electromagnetic energy and radiated into space.

If the load at the end of a line is an open circuit or a short circuit or has an impedance other than the characteristic impedance of the line, the signal is not fully absorbed by the load. When a line is not terminated properly, some of the energy is reflected from the end of the line and actually moves back up the line, toward the generator. This refl ected voltage adds to the forward or incident generator voltage and forms a composite voltage that is distributed along the line. This pattern of voltage and its related current constitute what is called a standing wave.

Standing waves are not desirable. The reflection indicates that the power produced by the generator is not totally absorbed by the load. In some cases, e.g., a short-circuited or open line, no power gets to the load because all the power is reflected back to the generator. The following sections examine in detail how standing waves are generated.

**The Relationship Between Reflections and Standing Waves**

Fig. 13-15 will be used to illustrate how reflections are generated and how they contribute to the formation of standing waves. Part (a) shows how a dc pulse propagates along a transmission line made up of identical LC sections. A battery (generator) is used as the input signal along with a switch to create an on/off dc pulse.

The transmission line is open at the end rather than being terminated in the characteristic impedance of the line. An open transmission line will, of course, produce a reflection and standing waves. Note that the generator has an internal impedance of Rg, which is equal to the characteristic impedance of the transmission line. Assume a transmission line impedance of 75 Ω and an internal generator resistance of 75 Ω. The 10 Ω supplied by the generator is, therefore, distributed equally across the impedance of the line and the internal resistance.

Now assume that the switch is closed to connect the generator to the line. As you know, connecting a dc source to reactive components such as inductors and capacitors produces transient signals as the inductors oppose changes in current while the capacitors oppose changes in voltages. Capacitor C1 initially acts as a short circuit when the switch is closed, but soon begins to charge toward the battery voltage through L1.

As soon as the voltage at point A begins to rise, it applies a voltage to the next section of the transmission line made up of C2 and L2. Therefore, C2 begins to charge through L2. The process continues on down the line until C4 charges through L4, and so on. The signal moves down the line from left to right as the capacitors charge. Lossless (zero-resistance) components are assumed, and so the last capacitor C4 eventually charges to the supply voltage.

For the purposes of this illustration, assume that the length of the line and its other characteristics are such that the time delay is 500 ns: 500 ns after the switch is closed, an output pulse will occur at the end of the line. At this time, the voltage across the output capacitance C4 is equal to 5 V or one-half of the supply voltage.

The instant that the output capacitance charges to its final value of 5 V, all current flow in the line ceases, causing any magnetic field around the inductors to collapse. The energy stored in the magnetic field of L4 is equal to the energy stored in the output capacitance C4. Therefore, a voltage of 5 V is induced into the inductor. The polarity of this voltage will be in such a direction that it adds to the charge already on the capacitor. Thus the capacitor will charge two times the applied 5-V voltage or 10 V.

A similar effect then takes place in L3. The magnetic field across L3 collapses, doubling the voltage charge on C3. Next, the magnetic field around L2 collapses, charging C2 to 10 V. The same effect occurs in L1 and C1. Once the signal reaches the right end of the line, a reverse charging effect takes place on the capacitors from right to left. The effect is as if a signal were moving from output to input. This moving charge from right to left is the reflection or reflected wave, and the input wave from the generator to the end of the line is the incident wave.

It takes another 500 ns for the reflected wave to get back to the generator. At the end of 1 μs, the input to the transmission line goes more positive by 5 V, for a total of 10 V. Fig. 13-15(b) shows the waveforms for the input, output, and reflected voltages with respect to time. Observing the waveforms, follow the previously described action with the closing of the switch at time t0. Because both the characteristic impedance of the line and the internal generator resistance is 75 Ω, one-half of the battery voltage appears at the input to the line at point A.

This voltage propagates down the line, charging the line capacitors as it goes until it reaches the end of the line and fully charges the output capacitance. At that time the current in the inductance begins to cease, with magnetic fields collapsing and inducing voltages that double the output voltage at the end of the line. Thus after 500 ns the output across the open end of the line is 10 V.

The reflection begins and now moves back down the line from right to left; after another 500 ns, it reaches the line input, the input to the line to jump to 10 V. Once the reflection stops, the entire line capacitance is fully charged to 10 V, as might be expected. The preceding description concerned what is known as an open-circuit load. Another extreme condition is a short-circuit load. For this situation, assume a short across C4 in Fig. 13-15(a).

When the switch is closed, again 5 V is applied to the input of the line, which is then propagated down the line as the line capacitors charge. Because the end of the line is short-circuited, inductor L4 is, in effect, the load for this line. The voltage on C3 is then applied to L4. At this point, the reflection begins. The current in L4 collapses, inducing a voltage which is then propagated down the line in the opposite direction.

The voltage induced in L4 is equal and opposite to the voltage propagated down the line. Therefore, this voltage is equal and opposite to the voltage on C3, which causes C3 to be discharged. As the reflection works its way back down the line from right to left, the line capacitance is continually discharged until it reaches the generator. It takes 500 ns for the charge to reach the end of the line and another 500 ns for the reflection to move back to the generator. Thus in a total of 1 μs, the input voltage switches from 5 to 0 V. Of course, the voltage across the output short remains zero during the entire time.

Open and shorted transmission lines are sometimes used to create special effects. In practice, however, the load on a transmission line is neither infinite nor 0 Ω; rather, it is typically some value in between. The load may be resistive or may have a reactive component. Antennas typically do not have a perfect resistance value. Instead, they frequently have a small capacitive or inductive reactance.

Thus the load impedance is equivalent to a series RC or RL circuit with an impedance of the form R ± jX. If the load is not exactly resistive and is equal to the characteristic impedance of the line, a reflection is produced, the exact voltage levels depending on the complex impedance of the load. Usually, some of the power is absorbed by the resistive part of the line; the mismatch still produces a reflection, but the reflection is not equal to the original signal, as in the case of a shorted or open load.

In most communication applications, the signal applied to a transmission line is an ac signal. This situation can be analyzed by assuming the signal to be a sine wave. The effect of the line on a sine wave is like that described above in the discussion based on an analysis of Fig. 13-15. If the line is terminated in a resistive load equal to the characteristic impedance of the line, the sine wave signal is fully absorbed by the load and no reflection occurs.

**Matched Lines**

Ideally, a transmission line should be terminated in a load that has a resistive impedance equal to the characteristic impedance of the line. This is called a matched line. For example, a 50-Ω coaxial cable should be terminated with a 50-Ω resistance, as shown in Fig. 13-16. If the load is an antenna, then that antenna should look like resistance of 50 Ω. When the load impedance and the characteristic impedance of the line match, the transmission goes smoothly and maximum power transfer—less any resistive losses in the line—takes place. The line can be any length. One of the key objectives in designing antenna and transmission line systems is to ensure this match.

Alternating-current voltage (or current) at any point on a matched line is a constant value (disregarding losses). A correctly terminated transmission line, therefore, is said to be flat. For example, if a voltmeter is moved down a matched line from generator to load and the RMS voltage values are plotted, the resulting wavelength versus voltage line will be flat (see Fig. 13-17). Resistive losses in the line would, of course, produce a small voltage drop along the line, giving it a downward tilt for greater lengths.

If the load impedance is different from the line characteristic impedance, not all the power transmitted is absorbed by the load. The power not absorbed by the load is reflected back toward the source. The power sent down the line toward the load is called forward or incident power; the power not absorbed by the load is called reflected power. The signal actually on a line is simply the algebraic sum of the forward and reflected signals.

Reflected power can represent a significant loss. If a line has a 3-dB loss end to end, only a 3-dB attenuation of the reflected wave will occur by the time the signal reaches the generator. What happens there depends upon the relative impedance of the generator and the line. Only part of the reflected energy is dissipated in the line. In cases in which the mismatch between load resistive impedance and line impedance is great, the reflected power can be high enough to actually cause damage to the transmitter or the line itself.

**Shorted Lines**

The condition for a short circuit is illustrated in Fig. 13-18. The graph below the transmission line shows the plot of voltage and current at each point on the line that would be generated by using the values given by a voltmeter and ammeter moved along the line. As expected in the case of a short at the end of a line, the voltage is zero when the current is maximum.

All the power is reflected back toward the generator. Looking at the plot, you can see that the voltage and current variations distribute themselves according to the signal wavelength. The fixed pattern, which is the result of a composite of the forward and reflected signals, repeats every half wavelength. The voltage and current levels at the generator are dependent on signal wavelength and the line length.

The phase of the reflected voltage at the generator end of the line depends upon the length of the line. If the line is some multiple of one-quarter wavelength, the reflected wave will be in phase with the incident wave and the two will add, producing a signal at the generator that is twice the generator voltage.

If the line length is some multiple of the one-half wavelength, the reflected wave will be exactly 180° out of phase with the incident wave and the two will cancel, giving a zero voltage at the generator. In other words, the effects of the reflected wave can simulate an open or short circuit at the generator.

**Open Lines**

Fig. 13-19 shows the standing waves on an open-circuited line. With an infinite impedance load, the voltage at the end of the line is maximum when the current is zero. All the energy is reflected, setting up the stationary pattern of voltage and current standing waves shown.

**Mismatched (Resonant) Lines**

Most often, lines do not terminate in a short or open circuit. Rather, the load impedance does not exactly match the transmission line impedance. Furthermore, the load, usually an antenna, will probably have a reactive component, either inductive or capacitive, in addition to its resistance. Under these conditions, the line is said to be resonant. Such a mismatch produces standing waves, but the amplitude of these waves is lower than that of the standing waves resulting from short or open circuits. The distribution of these standing waves looks like that shown in Fig. 13-20. Note that the voltage or current never goes to zero, as it does with an open or shorted line.

**Calculating the Standing Wave Ratio**

The magnitude of the standing waves on a transmission line is determined by the ratio of the maximum current to the minimum current, or the ratio of the maximum voltage to the minimum voltage, along the line. These ratios are referred to as the standing wave ratio (SWR).

SWR = I_{max}/I_{min}= V_{max}/V_{min}

Under the shorted and open conditions described earlier, the current or voltage minima are zero. This produces an SWR of infinity. It means that no power is dissipated in the load; all the power is reflected.

In the ideal case, there are no standing waves. The voltage and current are constant along the line, so there are no maxima or minima (or the maximum and minima are the same). Therefore, the SWR is 1.

Measuring the maximum and minimum values of voltage and current on a line is not practical in the real world, so other ways of computing the SWR have been devised. For example, the SWR can be computed if the impedance of the transmission line and the actual impedance of the load are known. The SWR is the ratio of the load impedance Zl to the characteristic impedance Z0, or vice versa.

For example, if a 75-Ω antenna load is connected to a 50-Ω transmission line, the SWR is 75/50 = 1.5. Since the standing wave is really the composite of the original incident wave added to the reflected wave, the SWR can also be defined in terms of those waves. The ratio of the reflected voltage wave Vr to the incident voltage wave Vi is called the reflection coefficient

The reflection coefficient provides information on current and voltage along the line. Also, T= reflected power/incident power.

If a line is terminated in its characteristic impedance, then there is no reflected voltage, so Vr = 0 and T= 0. If the line is open or shorted, then total reflection occurs. This means that Vr and Vi are the same, so T= 1. The reflection coefficient really expresses the percentage of reflected voltage to incident voltage. If G is 0.5, for example, the reflected voltage is 50 percent of the incident voltage, and the reflected power is 25 percent of the incident power [T^{2} = (0.5)^{2} = 0.254].

If the load is not matched but also is not open or short, the line will have voltage minima and maxima, as described previously. These can be used to obtain the reflection coefficient by using the formula

The SWR is obtained from the refl ection coeffi cient according to the equation

**Example 13.5** An RG-11/U foam coaxial cable has a maximum voltage standing wave of 52 V and a minimum voltage of 17 V. Find (a) the SWR, (b) the reflection coefficient, and (c) the value of a resistive load.

If the load matches the line impedance, then T = 0. The preceding formula gives an SWR of 1, as expected. With an open or shorted load, T= 1. This produces an SWR of infinity.

The refl ection coeffi cient can also be determined from the line and load impedances

For the example of an antenna load of 75 Ω and a coaxial cable of 50 Ω, the reflection coeffi cient is T= (75 – 50)/(75 + 50) = 25/125 = 0.2.

The importance of the SWR is that it gives a relative indication of just how much power is lost in the transmission line and the generator. This assumes that none of the reflected power is re-refl ected by the generator. In a typical transmitter, some power is reflected and sent to the load again.

The curve in Fig. 13-21 shows the relationship between the percentage of reflected power and the SWR. The percentage of reflected power is also expressed by the term return loss and is given directly in watts or decibels (dB). Naturally, when the standing wave ratio is 1, the percentage of reflected power is 0. But as a line and load mismatch grow, reflected power increases. When the SWR is 1.5, the percentage of reflected power is 4 percent. This is still not too bad, as 96 percent of the power gets to the load.

It is possible to compute reflected power Pr if given the SWR and the incident power Pl. Since T^{2} = P_{r} /P_{i}, then P_{r} T^{2}P_{i}. Knowing the SWR, you can compute t and then solve by using the preceding equation

One of the best and most practical ways to compute the SWR is to measure the forward power Pf and refl ected power Pr and then use the formula

Several good circuits have been invented to measure forward and reflected power. And commercial test instruments are also available that can be inserted into a transmission line and make these measurements. The data is read from a front panel meter or digital display. Then the figures are plugged into the formula above. Some sophisticated test instruments have a built-in embedded computer to make this calculation automatically and display the SWR value.

For example, assume that you measure a forward power of 35 W and reflected power of 7 W. The SWR is

For SWR values of 2 or less, the reflected power is less than 10 percent, which means that 90 percent gets to the load. For most applications this is acceptable. For SWR values higher than 2, the percentage of reflected power increases dramatically, and measures must be taken to reduce the SWR to prevent potential damage. Some solid-state systems shut down automatically if the SWR is greater than 2. The most common approach to reducing the SWR is to add or include a π, L, or T LC network to offset antenna reactance and other resistive components and to produce an impedance match. Antenna length can also be adjusted to improve the impedance match.

**Example 13.6** The line input to the cable in Example 13-5 is 30 W. What is the output power? (See Fig. 13-21; disregard attenuation due to length.) The percentage of reflected power with the SWR of 3.05 is about 25.62

**Transmission Lines as Circuit Elements**

The standing wave conditions resulting from open- and short-circuited loads must usually be avoided in working with transmission lines. However, with one-quarter and one-half wavelength transmissions, these open- and short-circuited loads can be used as resonant or reactive circuits.

**Resonant Circuits and Reactive Components**

Consider the shorted one-quarter wavelength (λ/4) line shown in Fig. 13-22. At the load end, voltage is zero and current is maximum. But one-quarter wavelength back, at the generator, the voltage is maximum and the current is zero. To the generator, the line appears as an open circuit, or at least a very high impedance. The key point here is that this condition exists at only one frequency, the frequency at which the line is exactly one-quarter wavelength. Because of this frequency sensitivity, the line acts as an LC tuned or resonant circuit, in this case, a parallel resonant circuit because of its very high impedance at the reference frequency.

With a shorted one-half wavelength line, the standing wave pattern is like that shown in Fig. 13-23. The generator sees the same conditions as at the end of the line, i.e., zero voltage and maximum current. This represents a short, or very low impedance. That condition occurs only if the line is exactly one-half wavelength long at the generator frequency. In this case, the line looks like a series resonant circuit to the generator.

If the line length is less than the one-quarter wavelength at the operating frequency, the shorted line looks like an inductor to the generator. If the shorted line is between one-quarter and one-half wavelength, it looks like a capacitor to the generator. All these conditions repeat with multiple one-quarter or one-half wavelengths of shorted lines.

Similar results are obtained with an open line, as shown in Fig. 13-24. To the generator, a one-quarter wavelength line looks like a series resonant circuit and a one-half wavelength line looks like a parallel resonant circuit, just the opposite of a shorted line.

If the line is less than the one-quarter wavelength, the generator sees a capacitance. If the line is between one-quarter and one-half wavelength, the generator sees an inductance. These characteristics repeat for lines that are some multiple of one-quarter or one-half wavelengths.

Fig. 13-25 is a summary of the conditions represented by open and shorted lines of lengths up to one wavelength. The horizontal axis is length, in wavelengths, and the vertical axis is the reactance of the line, in ohms, expressed in terms of the line’s characteristic impedance. The solid curves are shorted lines and the dashed curves are open-circuit lines.

If the line acts as a series resonant circuit, its impedance is zero. If the line is of such a length that it acts as a parallel resonant circuit, its impedance is near infinity. If the line is some intermediate length, it is reactive. For example, consider a shorted one-eighth wavelength line. The horizontal divisions represent one-sixteenth wavelength, so two of these represent one-eighth wavelength. Assume that the line has a characteristic impedance of 50 V.

At the one-eighth wavelength point on the left-hand solid curve is a reading of 1. This means that the line acts as an inductive reactance of 1 x Z_{0}, or 1 x 50 = 50 Ω. An open line about three-eighths wavelength would have the same effect, as the leftmost dashed curve in Fig. 13-24 indicates.

How could a capacitive reactance of 150Ω be created with the same 50-Ω line? First, locate the 150-Ω point on the capacitive reactance scale in Fig. 13-25. Since 150/50=3, the 150-Ω point is at XC = 3. Next, draw a line from this point to the right until it intersects with two of the curves. Then read the wavelength from the horizontal scale. A capacitive reactance of 150 Ω with a 50-Ω line can be achieved with an open line somewhat longer than 1⁄32 wavelength or a shorted line a bit longer than 9⁄32 wavelength.

**Stripline and Microstrip**

At low frequencies (below about 300 MHz), the characteristics of open and shorted lines discussed in the previous sections have little signifi cance. At low frequencies, the lines are just too long to be used as reactive components or as fi lters and tuned circuits. However, at UHF (300 to 3000 MHz) and microwave (1 GHz and greater) frequencies, the length of one-half wavelength is less than 1 ft; the values of inductance and capacitance become so small that it is diffi cult to realize them physically with standard coils and capacitors.

Special transmission lines constructed with copper patterns on a printed circuit board (PCB), called microstrip or stripline, can be used as tuned circuits, filters, phase shifters, reactive components, and impedance-matching circuits at these high frequencies.

A PCB is a flat insulating base made of fiberglass or some other insulating base material to which is bonded copper on one or both sides and sometimes in several layers. Tefl on or ceramic is used as the base for some PCBs in microwave applications. In microwave ICs, the base is often alumina or even sapphire. The copper is etched away in patterns to form the interconnections for transistors, ICs, resistors, and other components. Thus point-to-point connections with wire are eliminated. Diodes, transistors, and other components are mounted right on the PCB and connected directly to the formed microstrip or stripline.

**Microstrip**

A microstrip is a flat conductor separated by an insulating dielectric from a large conducting ground plane [Fig. 13-26(a)]. The microstrip is usually one-quarter or one-half wavelength long. The ground plane is the circuit common. This type of microstrip is equivalent to an unbalanced line. Shorted lines are usually preferred over open lines. Microstrip can also be made in a two-line balanced version [Fig. 13-26(b)].

The characteristic impedance of microstrip, as with any transmission line, is dependent on its physical characteristics. It can be calculated by using the formula

where Z = characteristic impedance

ε = dielectric constant

w = width of copper trace

t = thickness of the copper trace

h = distance between trace and ground plane (dielectric thickness) of dielectric

Any units of measurement can be used (e.g., inches or millimeters), as long as all dimensions are in the same units. (See Fig. 13-27.) The dielectric constant of the popular FR-4 fiberglass PC board material is 4.5. The value of ε for Tefl on is 3.

The characteristic impedance of microstrip with the dimensions h = 0.0625 in, w = 0.1 in, t = 0.003 in, and ε = 4.5 is

**Stripline**

Stripline is a flat conductor sandwiched between two ground planes (Fig. 13-28). It is more difficult to make than a microstrip; however, it does not radiate as a microstrip does. Radiation produces losses. The length is one-quarter or one-half wavelength at the desired operating frequency, and shorted lines are more commonly used than open lines.

The characteristic impedance of stripline is given by the formula

Fig. 13-29 shows the dimensions required to make the calculations.

Even tinier microstrip and strip lines can be made by using monolithic, thin-film, and hybrid IC techniques. When these are combined with diodes, transistors, and other components, microwave integrated circuits (MICs) are formed.

**Example 13.7** A microstrip transmission line is to be used as a capacitor of 4 pF at 800 MHz. The PCB dielectric is 3.6. The microstrip dimensions are h = 0.0625 in, w = 0.13 in, and t = 0.002 in. What are (a) the characteristic impedance of the line and (b) the reactance of the capacitor?

**Example 13.8** What is the length of the transmission line in Example 13-7? Refer to Fig. 13-25. Take the ratio of XC to Z0

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