# 2.1 General Considerations

In most electrical engineering curricula, the study of electromagnetics is preceded by one or more courses on electrical circuits. In this book, we use this background to build a bridge between circuit theory and electromagnetic theory. The bridge is provided by transmission lines, which is the topic of this chapter. By modeling transmission lines in the form of equivalent circuits, we can use Kirchhoff’s voltage and current laws to develop wave equations whose solutions provide an understanding of wave propagation, standing waves, and power transfer. Familiarity with these concepts facilitates the presentation of material in later chapters.

Although the notion of transmission lines may encompass all structures and media that serve to transfer energy or information between two points, including nerve fibers in the human body and fluids and solids that support the propagation of mechanical pressure waves, this chapter focuses on transmission lines that guide electromagnetic signals. Such transmission lines include telephone wires, coaxial cables carrying audio and video information to TV sets or digital data to computer monitors, microstrips printed on microwave circuit boards, and optical fibers carrying light waves for the transmission of data at very high rates.

Fundamentally, a transmission line is a two-port network, with each port consisting of two terminals, as illustrated in Fig. 2-1. One of the ports, the line’s sending end, is connected to a source (also called the generator). The other port, the line’s receiving end, is connected to a load. The source connected to the transmission line’s sending end may be any circuit generating an output voltage, such as a radar transmitter, an amplifier, or a computer terminal operating in transmission mode. From circuit theory, a dc source can be represented by a Thévenin-equivalent generator circuit consisting of a generator voltage V g in series with a generator resistance R g, as shown in Fig. 2-1. In the case of alternating-current (ac) signals, the generator circuit is represented by a voltage phasor V˜g and an impedance Z g.

The load circuit, or simply the load, may be a transmitting antenna in the case of radar, a computer terminal operating in the receiving mode, the input terminals of an amplifier, or any output circuit whose input terminals can be represented by an equivalent load impedance Z L.

# Figure 2-1 A transmission line is a two-port network connecting a generator circuit at the sending end to a load at the receiving end.

Figure 2 1 Full Alternative Text

# 2.1.1 The Role of Wavelength

In low-frequency circuits, circuit elements usually are interconnected using simple wires. In the circuit shown in Fig. 2-2, for example, the generator is connected to a simple RC load via a pair of wires. In view of our definition in the preceding paragraphs of what constitutes a transmission line, we pose the question: Is the pair of wires between terminals A A′ and terminals B B′ a transmission line? If so, under what set of circumstances should we explicitly treat the pair of wires as a transmission line as opposed to ignoring their presence altogether and treating the circuit as only an RC load connected to a generator V˜g? The answer to the first question is: Yes, the pair of wires does constitute a transmission line. The answer to the second question is: The factors that determine whether or not we should treat the wires as a transmission line is governed by the length of the line l and the frequency f of the signal provided by the generator. (As we will see later, the determining factor is the ratio of the length l to the wavelength λ of the wave propagating on the transmission line between the source at terminals A A′ and the load at terminals B B′.) If the generator voltage is cosinusoidal in time, then the voltage across the input terminals A A′ is

V A A′=V g(t)=V 0 cos ω t (V),(2.1)

where ω=2 π f is the angular frequency. If we assume that the current flowing through the wires travels at the speed of light, c=3×10 8 m/s, then the voltage across the output terminals B B′ will have to be delayed in time relative to that across A A′ by the travel delay time l/c. Thus, assuming no ohmic losses in the transmission line and ignoring other transmission-line effects discussed later in this chapter,

V B B′(t)=V A A′(t−l/c)=V 0 cos ω(t−l/c)=V 0 cos(ω t−ϕ 0),(2.2)

with

ϕ 0=ω l c(rad).(2.3)

# Figure 2-2 Generator connected to an RC circuit through a transmission line of length l.

Figure 2 2 Full Alternative Text

Thus, the time delay associated with the length of the line l manifests itself as a constant phase shift ϕ 0 in the argument of the cosine. Let us compare V B B′ with V A A′ at t=0 for an ultralow-frequency electronic circuit operating at a frequency f=1 kHz. For a typical wire length l=5 cm, Eqs. (2.1) and (2.2) give V A A′=V 0 and V B B′=V 0 cos(2 π f l/c)=0.999999999998 V 0. Hence, for all practical purposes, the presence of the transmission line may be ignored, and terminal A A′ may be treated as identical with B B′ so far as its voltage is concerned. On the other hand, had the line been a 20-km long telephone cable carrying a 1 kHz voice signal, then the same calculation would have led to V B B′=0.91 V 0, which is a deviation of 9%. Furthermore, had the frequency of the signal on the 5-cm long wire been 1.5 GHz instead of 1 kHz, the voltage at the end of the line would have been V B B′=0 ! Thus, the pair of wires would have a voltage V 0 at A A′ and zero at B B′ simultaneously. The determining factor is the magnitude of ϕ 0=ω l/c. From Eq. (1.27), the velocity of propagation u p of a traveling wave is related to the oscillation frequency f and the wavelength λ by

u p=f λ(m/s).

In the present case, u p=c. Hence, the phase delay is

ϕ 0=ω l c=2 π f l c=2 π l λ radians.(2.4)

Power loss on the line and dispersive effects may need to be considered as well.

This means that the shape of a rectangular pulse, which through Fourier analysis can be decomposed into many sinusoidal waves of different frequencies, gets distorted as it travels down the line because its different frequency components do not propagate at the same velocity (Fig. 2-3). Preservation of pulse shape is very important in high-speed data transmission not only between terminals but also across transmission line segments fabricated within high-speed integrated circuits. At 10 GHz, for example, the wavelength is λ=3 cm in air but only on the order of 1 cm in a semiconductor material. Hence, even lengths between devices on the order of millimeters become significant, and their presence has to be accounted for in the design of the circuit.

# Figure 2-3 A dispersionless line does not distort signals passing through it regardless of its length, whereas a dispersive line distorts the shape of the input pulses because the different frequency components propagate at different velocities. The degree of distortion is proportional to the length of the dispersive line.

Figure 2 3 Full Alternative Text

# 2.1.2 Propagation Modes

A few examples of common types of transmission lines are shown in Fig. 2-4. Transmission lines may be classified into two basic types:

- Transverse electromagnetic (TEM) transmission lines: Waves propagating along these lines are characterized by electric and magnetic fields that are entirely transverse to the direction of propagation. Such an orthogonal configuration is called a TEM mode. A good example is the coaxial line shown in Fig. 2-5. The electric field is in the radial direction between the inner and outer conductors, while the magnetic field circles the inner conductor, and neither has a component along the line axis (the direction of wave propagation). Other TEM transmission lines include the two-wire line and the parallel-plate line (both shown in Fig. 2-4). Although the fields present on a microstrip line do not adhere to the exact definition of a TEM mode, the nontransverse field components are sufficiently small (in comparison with the transverse components) that they may be ignored, thereby allowing the inclusion of microstrip lines in the TEM class. A common feature among TEM lines is that they consist of two parallel conducting surfaces.

  # Figure 2-4 A few examples of transverse electromagnetic (TEM) and higher-order transmission lines.

  Figure 2 4 Full Alternative Text

  # Figure 2-5 In a coaxial line, the electric field is in the radial direction between the inner and outer conductors, and the magnetic field forms circles around the inner conductor. The coaxial line is a transverse electromagnetic (TEM) transmission line because both the electric and magnetic fields are orthogonal to the direction of propagation between the generator and the load.

  Figure 2 5 Full Alternative Text

- Higher-order transmission lines: Waves propagating along these lines have at least one significant field component in the direction of propagation. Hollow conducting waveguides, dielectric rods, and optical fibers belong to this class of lines (Chapter 8).

Only TEM-mode transmission lines are treated in this chapter. This is because they are more commonly used in practice, and fortunately, less mathematical rigor is required for treating them than is required for lines that support higher-order modes. We start our treatment by representing the transmission line in terms of a lumped-element circuit model, and then we apply Kirchhoff’s voltage and current laws to derive a pair of equations governing their behavior, known as the telegrapher’s equations. By combining these equations, we obtain wave equations for the voltage and current at any location along the line. Solution of the wave equations for the sinusoidal steady-state case leads to a set of formulas that can be used to solve a wide range of practical problems. In the latter part of this chapter, we introduce a graphical tool known as the Smith chart, which facilitates the solution of transmission-line problems without having to perform laborious calculations involving complex numbers.