Transmission Line Basics
The two primary requirements of a transmission line are that (1) the line introduces minimum attenuation to the signal and (2) the line not radiate any of the signals as radio energy. All transmission lines and connectors are designed with these requirements in mind.
Types of Transmission Lines
A parallel-wire line is made of two parallel conductors separated by a space of 1⁄2 into several inches. Fig. 13-1(a) shows a two-wire balanced line in which insulating spacers have been used to keep the wires separated. Such lines are rarely used today. A variation of the parallel line is the 300-Ω twin-lead type shown in Fig. 13-1(b), where the spacing between the wires is maintained by a continuous plastic insulator. Parallel-wire lines are rarely used today.
The most widely used type of transmission line is a coaxial cable, which consists of a solid center conductor surrounded by a dielectric material, usually a plastic insulator such as Tefl on [see Fig. 13-1(c)]. An air or gas dielectric, in which the center conductor is held in place by periodic insulating spacers, can also be used. Over the insulator is a second conductor, a tubular braid or shield made of fine wires. An outer plastic sheath protects and insulates the braid. Coaxial cable comes in a variety of sizes, from approximately 1⁄4 into several inches in diameter.
Twisted-pair cable, as the name implies, uses two insulated solid copper wires covered with insulation and loosely twisted together. See Fig. 13-1 (d). This type of cable was originally used in telephone wiring and is still used for that today. But it is also used for security system wiring of sensors and other equipment. And twisted-pair cable, as you saw in Chap. 12, is one of the most widely used types of wiring in local-area networks (LANs). It is generally known as an unshielded twisted-pair (UTP) cable. There are many grades of twisted-pair cable for handling low-frequency audio or high-frequency pulses. The size of wire, type of insulation, and tightness of the twist (twists per inch) determine its characteristics. It is available with an overall braid shield and is called shielded twisted-pair (STP) cable. The most common version contains four pairs within a common insulated tubing.
Balanced Versus Unbalanced Lines
Transmission lines can be balanced or unbalanced. A balanced line is one in which neither wire is connected to the ground. Instead, the signal on each wire is referenced to the ground. The same current flows in each wire with respect to the ground, although the direction of current in one wire is 180° out of phase with the current in the other wire. In an unbalanced line, one conductor is connected to the ground. The twisted-pair line [Fig. 13-1(d)] may be used in a balanced or an unbalanced arrangement, although the balanced form is more common.
The open-wire line has a balanced configuration. A typical feed arrangement is shown in Fig. 13-2(a). The driving generator and the receiving circuit are center-tapped transformers in which the center taps are grounded. Balanced-line wires offer signifi cant protection from noise pickup and cross talk. Because of the identical polarities of the signals on balanced lines, any external signal induced into the cable appears on both wires simultaneously but cancels at the receiver. This is called common-mode rejection, and noise reduction can be as great as 60 to 70 dB.
Fig. 13-2(b) shows an unbalanced line. Coaxial cables are unbalanced lines; the current in the center conductor is referenced to the braid, which is connected to ground. Coaxial cable and shielded twisted-pair cable provide significant but not complete protection from noise pickup or cross talk from inductive or capacitive coupling due to external signals. Unshielded lines may pick up signals and cross-talk and can even radiate energy, resulting in an undesirable loss of signal.
It is sometimes necessary or desirable to convert from balanced to the unbalanced operation or vice versa. This is done with a device called a balun, from “balanced-unbalanced.”
Wavelength of Cables
The two-wire cables that carry 60-Hz power line signals into homes are transmission lines, as are the wires connecting the audio output of stereo receivers to stereo speakers.
At these low frequencies, the transmission line acts as a carrier of the ac voltage. For these applications, the only characteristic of the cable of interest is a resistive loss. The size and electrical characteristics of low-frequency lines can vary widely without affecting performance. An exception is conductor size, which determines current-carrying capability and the voltage drop over long distances. The electrical length of conductors is typically short compared to 1 wavelength of the frequency they carry. A pair of current-carrying conductors are not considered to be a transmission line unless it is at least 0.1 λ long at the signal frequency. Cables used to carry RF energy are not simply resistive conductors but are complex equivalents of inductors, capacitors, and resistors.
Furthermore, whenever the length of a transmission line is the same order of magnitude as or greater than the wavelength of the transmitted signal, the line takes on special characteristics and requires more complex analysis. As discussed earlier, the wavelength is the length or distance of one cycle of an ac wave or the distance that an ac wave travels in the time required for one cycle of that signal. Mathematically, wavelength λ is the ratio of the speed of light to the frequency of the signal f: λ = 300,000,000/f, where 300 million is the speed of light, in meters per second, in free space or air (300,000,000 m/s ≈ 186,400 mi/s) and f is in hertz. This is also the speed of a radio signal. The wavelength of a 60-Hz power line signal is then
λ = 300,000,000/60 = 5 x 106 m
That’s an incredibly long distance—several thousand miles. Practical transmission line distances at such frequencies are, of course, far smaller. At radio frequencies, however, say 3 MHz or more, the wavelength becomes considerably shorter. The wavelength at 3 MHz is λ = 300,000,000/3,000,000= 100 m, a distance of a little more than 300 ft, or the length of a football field. That is a very practical distance. As the frequency gets higher, the wavelength gets shorter. At higher frequencies, the wavelength formula is simplified to λ = 300/f, where frequency is in megahertz. A 50-MHz signal has a wavelength of 6 m. By using feet instead of meters, the wavelength formula becomes λ = 984/f, where f is in megahertz (λ is now expressed in feet).
If the wavelength is known, frequency can be computed as follows:
f (MHz) = 300/λ (m) or f (MHz) = 984 /λ (ft)
The distance represented by a wavelength in a given cable depends on the type of cable. The speed in a cable can be anywhere from 0.5 to 0.95 times the speed of light waves (radio waves) in space, and the signal wavelength in a cable will be proportionally less than the wavelength of that signal in space. Thus the calculated length of cables is shorter than wavelengths in free space. This is discussed later.
Example 13.1 For an operating frequency of 450 MHz, what length of a pair of conductors is considered to be a transmission line? (A pair of conductors does not act as a transmission line unless it is at least 0.1 λ long.)
λ = 984/450 = 2.19 ft
0.1 λ = 2.19(0.1) = 0.219 ft (2.628 in)
Example 13.2 Calculate the physical length of the transmission line in Example 13-1 a 3⁄8 λ long.
3/8 λ = 2.19(3)/8 = 0.82 ft (9.84 in)
Most transmission lines terminate in some kind of connector, a device that connects the cable to a piece of equipment or to another cable. An ordinary ac power plug and outlet are basic types of connectors. Special connectors are used with parallel lines and coaxial cable. Connectors, ubiquitous in communication equipment, are often taken for granted. This is unfortunate because they are a common failure point in many applications.
Coaxial Cable Connectors
The coaxial cable requires special connectors that will maintain the characteristics of the cable. Although the inner conductor and shield braid could theoretically be secured with screws as parallel lines, the result would be a drastic change in electrical attributes, resulting in signal attenuation, distortion, and other problems. Thus coaxial connectors are designed not only to provide a convenient way to attach and disconnect equipment and cables but also to maintain the physical integrity and electrical properties of the cable.
The choice of a coaxial connector depends on the type and size of the cable, the frequency of operation, and the application. The most common types are the PL-259 or UHF, BNC, F, SMA, and N-type connectors.
The PL-259 connector is shown in Fig. 13-3(a); the internal construction and connection principles for the PL-259 are shown in Fig. 13-3(b). The body of the connector is designed to fit around the end of a coaxial cable and to provide convenient ways to attach the shield braid and the inner conductor. The inner conductor is soldered to a male pin that is insulated from the body of the connector, which is soldered or crimped to the braid. A coupler fits over the body; it has inner threads that permit the connector to attach to matching screw threads on a female connector called the SO-239. See Fig. 13-3c.
The PL-259, which is also referred to as a UHF connector, can be used up to low UHF values (less than 500 MHz), although it is more widely used at HF and VHF. It can accommodate both large (up to 0.5-in) and small (0.25-in) coaxial cable.
Another very popular connector is the BNC connector (Fig. 13-4). BNC connectors are widely used on 0.25-in coaxial cables for attaching test instruments, such as oscilloscopes, frequency counters, and spectrum analyzers, to the equipment being tested. BNC connectors are also widely used on 0.25-in coaxial cables in LANs and some UHF radios.
In BNC connectors, the center conductor of the cable is soldered or crimped to a male pin, and the shield braid is attached to the body of the connector. An outer shell or coupler rotates and physically attaches the connector to a mating female connector by way of a pin and cam channel on the rotating coupling [see Fig. 13-4(b)].
One of the many variations of BNC connectors is the barrel connector, which allows two cables to be attached to each other end to end, and the T coupler, which permits taps on cables [see Fig. 13-4(a) and (d)]. Another variation is the SMA connector, which uses screw threads instead of the cam slot and pin (Fig. 13-5). The SMA connector is characterized by the hexagonal shape of the body of the male connector. Like the BNC connector, it is used with a smaller coaxial cable.
The least expensive coaxial cable connector is the F-type connector, which is widely used for TV sets, VCRs, DVD players, and cable TV. The cable plug and its matching chassis jack are shown in Fig. 13-6. The shield of the coaxial cable is crimped to the connector, and the solid wire center conductor of the cable, rather than a separate pin, is used as the connection. A hex-shaped outer ring is threaded to attach the plug to the mating jack.
Another inexpensive coaxial connector is the well-known RCA phonograph connector (Fig. 13-7), which is used primarily in audio equipment. Originally designed over 60 years ago to connect phonograph pick-up arms from turntables to amplifi ers, these versatile and low-cost devices can be used at radio frequencies and have been used for TV set connections in the low VHF range.
The best-performing coaxial connector is the N-type connector (Fig. 13-8), which is used mainly on large coaxial cable at the higher frequencies, both UHF and microwave. N-type connectors are complex and expensive, but do a better job than other connectors in maintaining the electrical characteristics of the cable through the interconnections.
When the length of a transmission line is longer than several wavelengths at the signal frequency, the two parallel conductors of the transmission line appear as a complex impedance. The wires exhibit considerable series inductance whose reactance is significant at high frequencies. In series with this inductance is the resistance of the wire or braid making up the conductors, which includes inherent ohmic resistance plus any resistance due to skin effect. Furthermore, the parallel conductors form a distributed capacitance with the insulation, which acts as the dielectric. In addition, there is a shunt or leakage resistance or conductance (G)across the cable as the result of imperfections in the insulation between the conductors. The result is that to a high-frequency signal, the transmission line appears as a distributed low-pass filter consisting of series inductors and resistors and shunts capacitors and resistors [Fig. 13-9(a)]. This is called a lumped model of a distributed line.
In the simplified equivalent circuit in Fig. 13-9(b), the inductance, resistance, and capacitance have been combined into larger equivalent lumps. The shunt leakage resistance is very high and has a negligible effect, so it is ignored. In short segments of the line, the series resistance of the conductors can sometimes be ignored because it is so low as to be insignifi cant. Over longer lengths, however, this resistance is responsible for considerable signal attenuation. The effects of the inductance and capacitance are considerable, and in fact they determine the characteristics of the line.
An RF generator connected to such a transmission line sees an impedance that is a function of the inductance, resistance, and capacitance in the circuit—the characteristic or surge impedance Z0. If we assume that the length of the line is infinite, this impedance is resistive. The characteristic impedance is also purely resistive for a finite length of the line if a resistive load equal to the characteristic impedance is connected to the end of the line.
Determining Z0 from Inductance and Capacitance
For an infinitely long transmission line, the characteristic impedance Z0 is given by the formula Z0=√L/√C, where Z0 is in ohms, L is the inductance of the transmission line for a given length, and C is the capacitance for that same length. The formula is valid even for finite lengths if the transmission line is terminated with a load resistor equal to the characteristic impedance (see Fig. 13-10). This is the normal connection for a transmission line in any application. In equation form,
RL = Z0
If the line, load, and generator impedances are made equal, as is the case with matched generator and load resistances, the criterion for maximum power transfer is met.
An impedance meter or bridge can be used to measure the inductance and capacitance of a section of parallel line or coaxial cable to obtain the values needed to calculate the impedance. Assume, e.g., that a capacitance of 0.0022 μF (2200 pF) is measured for 100 ft. The inductance of each conductor is measured separately and then added, for a total of 5.5 μH. (Resistance is ignored because it does not enter into the calculation of characteristic impedance; however, it will cause signal attenuation over long distances.) The surge impedance is then
In practice, it is unnecessary to make these calculations because cable manufacturers
always specify impedance.
The characteristic impedance of a cable is independent of length. We calculated it by using a value of L and C for 100 ft, but 50 Ω is the correct value for 1 ft or 1000 ft. Note that the actual impedance approaches the calculated impedance only if the cable is several wavelengths or more in length as terminated in its characteristic impedance. Forline lengths less than 1.0 λ, characteristic impedance does not matter.
Most transmission lines come with standard fixed values of characteristic impedance. For example, the widely used twin-lead balanced line [Fig. 13-1(b)] has a characteristic impedance of 300 Ω. Open-wire line [Fig. 13-1(a)], which is no longer widely used, was made with impedances of 450 and 600 Ω. The common characteristic impedance of coaxial cable are 52, 53.5, 75, 93, and 125 Ω.
An important consideration in transmission line applications is that the speed of the signal in the transmission line is slower than the speed of a signal in free space.
The velocity of propagation of a signal in a cable is less than the velocity of propagation of light in free space by a fraction called the velocity factor (VF), which is the ratio of the velocity in the transmission line Vp to the velocity in free space Vc:
VF = Vp/Vc or VF = Vp/c
where Vc = c = 300,000,000 m/s.
Velocity factors in transmission lines vary from approximately 0.5 to 0.9. The velocity factor of a coaxial cable is typically 0.6 to 0.8. The open-wire line has a VF of about 0.9, and 300-V twin-lead line has a velocity factor of about 0.8.
Calculating Velocity Factor
The velocity factor in a line can be computed with the expression VF =1/√ε, where ε is the dielectric constant of the insulating material. For example, if the dielectric in a coaxial cable is Tefl on, the dielectric constant is 2.1 and the velocity factor is 1/√2.1 = 1/1.45 = 0.69. That is, the speed of the signal in the coaxial cable is 0.69 times the speed of light, or 0.69 x 300,000,000 =207,000,000 m/s (128,616 mi/s).
If a lossless (zero-resistance) line is assumed, an approximation of the velocity of propagation can be computed with the expression
Vp =1/√LC ft/s
where l is the length or total distance of travel of the signal in feet or some other units of length and L and C are given in that same unit. Assume, e.g., a coaxial cable with a characteristic impedance of 50 V and a capacitance of 30 pF/ft. The inductance per foot is 0.075 μH or 75 nH. The velocity of propagation per foot in this cable is
or 126,262 mi/s, or 204 x 106 m/s.
The velocity factor is then
Calculating Transmission Line Length
The velocity factor must be taken into consideration in computing the length of a transmission line in wavelengths. It is sometimes necessary to use a one-half or one-quarter wavelength of a specific type of transmission line for a specific purpose, e.g., impedance matching, filtering, and tuning.
The formula given earlier for one wavelength of a signal in free space is λ = 984/f. This expression, however, must be modified by the velocity factor to arrive at the true length of a transmission line. The new formula is
λ (ft) = 984 VF/f (MHz)
Suppose, e.g., that we want to find the actual length in feet of a quarter-wavelength segment of coaxial cable with a VF of 0.65 at 30 MHz. Using the formula gives λ = 984(VF/f ) = 984(0.65/30) = 21.32 ft. The length in feet is one-quarter of this or 21.32/4 = 5.33 ft.
The correct velocity factor for calculating the correct length of a given transmission line can be obtained from manufacturers’ literature and various handbooks.
Because the velocity of propagation of a transmission line is less than the velocity of propagation in free space, it is logical to assume that any line will slow down or delay any signal applied to it. A signal applied at one end of a line appears sometime later at the other end of the line. This is called the time delay or transit time for the line. A transmission line used specifically for the purpose of achieving delay is called a delay line.
Fig. 13-11 shows the effect of time delay on a sine wave signal and a pulse train. The output sine wave appears later in time than the input, so it is shifted in phase. The effect is the same as if a lagging phase shift were introduced by a reactive circuit. In the case of the pulse train, the pulse delay is determined by a factor that depends on the type and length of the delay line.
The amount of delay time is a function of a line’s inductance and capacitance. The opposition to changes in current offered by the inductance plus the charge and discharge time of the capacitance leads to a finite delay. This delay time is computed with the expression
td = √LC
where td is in seconds and L and C are the inductance and capacitance, respectively, per unit length of line. If L and C are given in terms of feet, the delay time will be per foot. For example, if the capacitance of a particular line is 30 pF/ft and its inductance is 0.075 μH/ft, the delay time is
A 50-ft length of this line would introduce 1.5 x 50 = 75 ns of delay.
Time delay introduced by a coaxial cable can also be calculated by using the formula
td = 1.016√ε ns/ft
where td is the time delay in nanoseconds per foot and ε is the dielectric constant of the cable.
For example, the total time delay introduced by a 75-ft cable with a dielectric constant of 2.3 is
td = 1.016√ε = 1.01612.3 = 1.016(1.517) = 1.54 ns/ft
for a total delay of 1.54(75) = 115.6 ns.
To determine the phase shift represented by the delay, the frequency and period of the sine wave must be known. The period or time T for one cycle can be determined with the well-known formula T = 1/f, where f is the frequency of the sine wave. Assume a frequency of 4 MHz. The period is
T = 1/4×106 = 250 x 10-9 = 250 ns
The phase shift of the previously described 50-ft line with a delay of 75 ns is given by
θ = 360 td/T = 360(75)/250 = 108°
Transmission line delay is usually ignored in RF applications, and it is virtually irrelevant in radio communication. However, in high-frequency applications where timing is important, transmission line delay can be significant. For example, in LANs, the time of transition of the binary pulses on a coaxial cable is often the determining factor in calculating the maximum allowed cable length.
Some applications require exact timing and sequencing of signals, especially pulses. A coaxial delay line can be used for this purpose. Obviously, a large roll of coaxial cable is not a convenient component in modern electronic equipment. As a result, artifi cial delay lines have been developed. These are made up of individual inductors and capacitors connected as a low-pass filter to simulate a distributed transmission line. Alternatively, a more compact distributed delay line can be constructed consisting of a coil of insulated wire wound over a metallic form. The coil of wire provides the distributed inductance and at the same time acts as one plate of a distributed capacitor. The metallic form is the other plate. Such artifi cial delay lines are widely used in TV sets,
oscilloscopes, radar units, and many other pieces of electronic equipment.
Transmission Line Specifications
Fig. 13-12 summarizes the specifications of several popular types of coaxial cable. Many coaxial cables are designated by an alphanumeric code beginning with the letters RG or a manufacturer’s part number. The primary specifications are characteristic impedance and attenuation. Other important specifications are maximum breakdown voltage rating, capacitance per foot, velocity factor, and outside diameter in inches. The attenuation is the amount of power lost per 100 ft of cable expressed in decibels at 100 MHz. Attenuation is directly proportional to cable length and increases with frequency. Detailed charts and graphs of attenuation versus frequency are also available so that users can predict the losses for their applications. Attenuation versus frequency for four coaxial cable types is plotted in Fig. 13-13. The loss is significant at very high frequencies. However, the larger the cable, the lower the loss. For purposes of comparison, look at the characteristics of the 300-V twin-lead cable listed in Fig. 13-12. Note the low loss compared to coaxial cable.
The loss in a cable can be significant, especially at the higher frequencies. In Example 13-3, the transmitter put 100 W into the line, but at the end of the line, the output power—the level of the signal that is applied to the antenna—was only 13.35 W. A major loss of 86.65 W dissipated as heat in the transmission line.
A 165-ft section of RG-58A/U at 100 MHz is being used to connect a transmitter to an antenna. Its attenuation for 100 ft at 100 MHz is 5.3 dB. Its input power from a transmitter is 100 W. What are the total attenuation and the output power to the antenna?
Several things can be done to minimize loss. First, every attempt should be made to find a way to shorten the distance between the transmitter and the antenna. If that is not feasible, it may be possible to use a larger cable. For the application in Example 13.3, the RG-58A/U cable was used. This cable has a characteristic impedance of 53.5 Ω, so any value near that would be satisfactory. One possibility would be the RG-8/U, with an impedance of 52 Ω and an attenuation of only 2.5 dB/100 ft. An even better choice would be the 9913 cable, with an impedance of 50 Ω and an attenuation of 1.3 dB/100 ft.
When you are considering the relationship between cable length and attenuation, remember that a transmission line is a low-pass filter whose cutoff frequency depends
both on distributed inductance and capacitance along the line and on length. The longer the line, the lower it’s cutoff frequency. This means that higher-frequency signals beyond the cutoff frequency are rolled off at a rapid rate.
This is illustrated in Fig. 13-14, which shows attenuation curves for four lengths of a popular type of coaxial cable. Remember that the cutoff frequency is the 3-dB down point on a frequency-response curve. If we assume that an attenuation of 3 dB is the same as a 3-dB loss, then we can estimate the cutoff frequency of different lengths of cable. The 3-dB down level is marked on the graph. Now, note the cutoff frequency for different lengths of cable. The shorter cable (100 ft) has the highest cutoff frequency,making the bandwidth about 30 MHz. The 200-ft cable has a cutoff of about 8 MHz, the 500-ft cable has a cutoff of approximately 2 MHz, and the 1000-ft cable has a cutoff of approximately 1 MHz. The higher frequencies are passed but are severely attenuated by the cable as it gets longer. It should be clear why it is important to use the larger, lowerloss cables for longer runs despite cost and handling inconvenience.
Finally, a gain antenna can be used to offset cable loss.
A 150-ft length of RG-62A/U coaxial cable is used as a transmission line. Find (a) the load impedance that must be used to terminate the line to avoid reflections, (b) the equivalent inductance per foot, (c) the time delay introduced by the cable, (d) the phase shift that occurs on a 2.5-MHz sine wave, and (e) the total attenuation in decibels. (Refer to Fig. 13-12.)