# 2.2 Lumped-Element Model

When we draw a schematic of an electronic circuit, we use specific symbols to represent resistors, capacitors, inductors, diodes, and the like. In each case, the symbol represents the functionality of the device rather than its shape, size, or other attributes. We shall do the same for transmission lines.

Thus, Fig. 2-6(a) may represent a coaxial line, a two-wire line, or any other TEM line.

# Figure 2-6 Regardless of its cross-sectional shape, a TEM transmission line is represented by the parallel-wire configuration shown in part (a). To obtain equations relating voltages and currents, the line is subdivided into small differential sections in part (b), each of which is then represented by an equivalent circuit in part (c).

(a) Parallel-wire representation

(b) Differential sections each Dz long

(c) Each section is represented by an equivalent circuit

Figure 2 6 Full Alternative Text

Drawing again on our familiarity with electronic circuits, when we analyze a circuit containing a transistor, we mimic the functionality of the transistor by an equivalent circuit composed of sources, resistors, and capacitors. We apply the same approach to the transmission line by orienting the line along the z direction, subdividing it into differential sections each of length Δ z (Fig. 2-6(b)), and then representing each section by an equivalent circuit, as illustrated in Fig. 2-6(c). This representation, often called the lumped-element circuit model, consists of four basic elements with values that henceforth will be called the transmission line parameters. These are:

- R′: The combined resistance of both conductors per unit length, in Ω/m,

- L′: The combined inductance of both conductors per unit length, in H/m,

- G′: The conductance of the insulation medium between the two conductors per unit length, in S/m, and

- C′: The capacitance of the two conductors per unit length, in F/m.

Whereas the four line parameters are characterized by different formulas for different types of transmission lines, the equivalent model represented by Fig. 2-6(c) is equally applicable to all TEM transmission lines. The prime superscript is used as a reminder that the line parameters are differential quantities whose units are per unit length.

Expressions for the line parameters R′, L′, G′, and C′ are given in Table 2-1 for the three types of TEM transmission lines diagrammed in parts (a) through (c) of Fig. 2-4. For each of these lines, the expressions are functions of two sets of parameters: (1) geometric parameters defining the cross-sectional dimensions of the given line and (2) the electromagnetic constitutive parameters of the conducting and insulating materials. The pertinent geometric parameters are:

- Coaxial line (Fig. 2-4(a)):

  a=outer radius of inner conductor, m b=inner radius of outer conductor, m

- Two-wire line (Fig. 2-4(b)):

  d=diameter of each wire, m D=spacing between wires’ centers, m

- Parallel-plate line (Fig. 2-4(c)):

  w=width of each plate, m h=thickness of insulation between plates, m

Appendix B contains tabulated values for these constitutive parameters for various materials. For the purposes of this chapter, we need not concern ourselves with the derivations leading to the expressions in Table 2-1. The techniques necessary for computing R′, L′, G′, and C′ for the general case of an arbitrary two-conductor configuration are presented in later chapters.

The lumped-element model shown in Fig. 2-6(c) reflects the physical phenomena associated with the currents and voltages on any TEM transmission line. It consists of two in-series elements, R′ and L′, and two shunt elements, G′ and C′. To explain the lumped-element model, consider a small section of a coaxial line, as shown in Fig. 2-7. The line consists of inner and outer conductors of radii a and b separated by a material with permittivity ε, permeability μ, and conductivity σ. The two metal conductors are made of a material with conductivity σ c and permeability μ c.

# Table 2-1 Transmission-line parameters R′, L′, G′, and C′ for three types of lines.

Table 2 1 Full Alternative Text

# Figure 2-7 Cross section of a coaxial line with inner conductor of radius a and outer conductor of radius b. The conductors have magnetic permeability μ c and conductivity σ c, and the spacing material between the conductors has permittivity ε, permeability μ, and conductivity σ.

Figure 2 7 Full Alternative Text

# Resistance R′

When a voltage source is connected across the terminals connected to the two conductors at the sending end of the line, currents flow through the conductors, primarily along the outer surface of the inner conductor and the inner surface of the outer conductor. The line resistance R′ accounts for the combined resistance per unit length of the inner and outer conductors. The expression for R′ is derived in Chapter 7 and is given by Eq. (7.96) as

R′=R s 2 π(1 a+1 b) (c o a x l i n e) (Ω/m),(2.5)

where R s, which represents the surface resistance of the conductors, is given by Eq. (7.92a) as

R s=π f μ c σ c(Ω).(2.6)

The surface resistance depends not only on the material properties of the conductors (σ c and μ c), but also on the frequency f of the wave traveling on the line.

# Inductance L′

Next, let us examine the line inductance L′, which accounts for the joint inductance of both conductors. Application of Ampère’s law in Chapter 5 to the definition of inductance leads to the following expression [Eq. (5.99)] for the inductance per unit length of a coaxial line:

L′=μ 2 π ln b a(c o a x l i n e)(H/m).(2.7)

# Conductance G′

The line conductance G′ accounts for current flow between the outer and inner conductors, made possible by the conductivity σ of the insulator. It is precisely because the current flow is from one conductor to the other that G′ appears as a shunt element in the lumped-element model. For the coaxial line, the conductance per unit length is given by Eq. (4.86) as

G′=2 π σ ln(b/a)(c o a x l i n e)(S/m).(2.8)

# Capacitance C′

The last line parameter on our list is the line capacitance C′. When equal and opposite charges are placed on any two noncontacting conductors, a voltage difference develops between them. Capacitance is defined as the ratio of the charge to the voltage difference. For the coaxial line, the capacitance per unit length is given by Eq. (4.127) as

C′=2 π ε ln(b/a)(c o a x l i n e)(F/m).(2.9)

All TEM transmission lines share the following useful relations:

and

If the insulating medium between the conductors is air, the transmission line is called an air line (e.g., coaxial air line or two-wire air line). For an air line, ε=ε 0=8.854×10−12 F/m , μ=μ 0=4 π×10−7 H/m, σ=0, and G′=0.

Concept Question 2-1: What is a transmission line? When should transmission-line effects be considered, and when may they be ignored?

Concept Question 2-2: What is the difference between dispersive and nondispersive transmission lines? What is the practical significance of dispersion?

Concept Question 2-3: What constitutes a TEM transmission line?

Concept Question 2-4: What purpose does the lumped-element circuit model serve? How are the line parameters R′, L′, G′, and C′ related to the physical and electromagnetic constitutive properties of the transmission line?