Chapter 13 Transmission Lines 13-1: Transmission-Line Basics 13-2: Standing Waves 13-3: Transmission Lines as Circuit Elements 13-4: The Smith Chart 13-1: Transmission-Line Basics 13-2: Standing Waves 13-3: Transmission Lines as Circuit Elements 13-4: The Smith Chart Transmission lines in communication carry telephone signals, computer data in LANs, TV signals in cable TV systems, and signals from a transmitter to an antenna or from an antenna to a receiver. Transmission lines are also circuits. Their electrical characteristics are critical and must be matched to the equipment for successful communication to take place. The two primary requirements of a transmission line are: 1. The line should introduce minimum attenuation to the signal. 2. The line should not radiate any of the signal as radio energy. Types of Transmission Lines Parallel-wire line is made of two parallel conductors separated by a space of ½ inch to several inches. The most widely used type of transmission line is the coaxial cable. It consists of a solid center conductor surrounded by a dielectric material, usually a plastic insulator such as Teflon. Twisted-pair cable uses two insulated solid copper wires covered with insulation and loosely twisted together. Figure 13-1: Common types of transmission lines. (a) Open-wire line. (b) Open-wire line called twin lead. (c) Coaxial cable (d) Twisted-pair cable. Balanced Versus Unbalanced Lines Transmission lines can be balanced or unbalanced. A balanced line is one in which neither wire is connected to ground. The signal on each wire is referenced to ground. In an unbalanced line, one conductor is connected to ground. Open-wire line has a balanced configuration. Figure 13-2: (a) Balanced line. (b) Unbalanced line Characteristic Impedance When the length of transmission line is longer than several wavelengths at the signal frequency, the two parallel conductors of the transmission line appear as a complex impedance. An RF generator connected to a considerable length of transmission line sees an impedance that is a function of the inductance, resistance, and capacitance in the circuit—the characteristic or surge impedance (Z0). l = 300,000,000 (m/s) / f(Hz) Figure 13-9 A transmission line appears as a distributed low-pass filter to any driving generator. (a) A distributed line with lumped components (b) Simplified equivalent circuit Figure 13-10 A transmission line whose load is resistive and equal to the surge impedance appears as an equal resistance to the generator. Zo = L C Velocity Factor 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). VF = Vp/Vc where Vp is the velocity in the transmission line and Vc the velocity in free space Time Delay Because the velocity of propagation of a transmission line is less than the velocity of propagation in free space, any line will slow down or delay any signal applied to it. A signal applied at one end of a line appears some time later at the other end of the line. This is called the time delay or transit time. A transmission line used specifically for the purpose of achieving delay is called a delay line. Figure 13-11: The effect of the time delay of a transmission line on signals. (a) Sine wave delay causes a lagging phase shift. (b) Pulse delay Transmission-Line Specifications Attenuation is directly proportional to cable length and increases with frequency. A transmission line is a low-pass filter whose cutoff frequency depends on distributed inductance and capacitance along the line and on length. It is important to use larger, low-loss cables for longer runs despite cost and handling inconvenience. A gain antenna can be used to offset cable loss. Example 13-3 A l65-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? Table 13-12 Table of common transmission line characteristics Figure 13-14: Attenuation versus length for RG-58A/U coaxial cable. Note that both scales on the graph are logarithmic Example 13-4 A 150-ft length of RG-62AIU 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 total attenuation in decibels. 13-1: Transmission-Line Basics 13-2: Standing Waves 13-3: Transmission Lines as Circuit Elements 13-4: The Smith Chart If the load on the line is an antenna, the signal is converted into electromagnetic energy and radiated into space. 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 at the end of the line is an open 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. Matched Lines A matched transmission line is one terminated in a load that has a resistive impedance equal to the characteristic impedance of the line. Alternating voltage (or current) at any point on a matched line is a constant value. A correctly terminated transmission line is said to be flat. The power sent down the line toward the load is called forward or incident power. Power not absorbed by the load is called reflected power. Figure 13-16: A transmission line must be terminated in its characteristic impedance for proper operation When a line is not terminated properly, some of the energy is reflected and moves back up the line, toward the generator. This reflected voltage adds to the forward or incident generator voltage and forms a composite voltage that is distributed along the line. The pattern of voltage and its related current forms a standing wave. Standing waves are not desirable. Figure 13-8 Standing waves on a shorted transmission line Figure 13-9 Standing waves on a open-circuit transmission line Figure 13-20 Transmission line with mismatched load and the resulting standing waves 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). Imax Vmax = SWR = Imin Vmin V V SWR 1 = max min = Vmax Vmin SWR 1 SWR = 1 = 1 1 pr 1 pr Pi Pi 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. a. SWR = Vmax / Vmin = 52 / 17 = 3.05 b. Γ = (Vmax – Vmin) / (Vmax + Vmin) = (52 -17) / (52 + 17) = 0.51 c. SWR = 3.05 Z0 = 75Ω SWR = Z0 /Zl or Zl /Z0 Zl = Z0 (SWR) = 75(3.05) = 228.75 Ω or Zl = Z0 / (SWR) = 75 / 3.05 = 24.59 Ω 13-1: Transmission-Line Basics 13-2: Standing Waves 13-3: Transmission Lines as Circuit Elements 13-4: The Smith Chart 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. Figure 13-25: Impedance and reactance variations of shorted and open lines for lengths up to one wavelength 13-1: Transmission-Line Basics 13-2: Standing Waves 13-3: Transmission Lines as Circuit Elements 13-4: The Smith Chart The Smith Chart is a sophisticated graph that permits visual solutions to transmission line calculations. Despite the availability of the computing options today, this format provides a more or less standardized way of viewing and solving transmission-line and related problems. Figure 13-30: The Smith chart The horizontal axis is the pure resistance or zero-reactance line. The point at the far left end of the line represents zero resistance, and the point at the far right represents infinite resistance. The resistance circles are centered on and pass through this pure resistance line. The circles are all tangent to one another at the infinite resistance point, and the centers of all the circles fall on the resistance line. Any point on the outer circle represents a resistance of 0 Ω. The R = 1 circle passes through the exact center of the resistance line and is known as the prime center. Values of pure resistance and the characteristic impedance of transmission line are plotted on this line. The linear scales printed at the bottom of Smith charts are used to find the SWR, dB loss, and reflection coefficient. The remainder of the Smith chart is completed by adding reactance circles All circles meet at the infinite resistance point Each circle represents a constant reactance point, with the inductivereactance circles at the top and the capacitive-reactance circles at the bottom Example 1 for Fig. 13-34. The operating frequency for a 24-ft piece of RG-58A/U coaxial cable is 140 MHz. The load is resistive, with a resistance of 93 Ω. What is the impedance seen by a transmitter? Example 2 for Fig. 13-36. An antenna is connected to the 24ft 53.5-Ω RG-58A/U coaxial cable. The load is 40 + j30 Ω. What impedance does the transmitter see?