www.microwavejournal.com/articles/30170-transmission-lines

# Transmission Lines

#### From the book "Microwaves and Wireless Simplified" by Thomas S. Laverghetta by Artech House

July 31, 2018

The general definition of a transmission line is “a device that transfers energy from one point to another.” If that is the case, a wire connecting lead used in a DC circuit would satisfy that definition. For DC and low frequency applications (e.g., audio), this is adequate. Typically it is not called a transmission line but simply a wire. Regardless of terminology, it is a transmission line for such applications.

A simple wire lead does not work well for RF and microwave applications due to its large dimensions relative to a wavelength and losses due to the skin effect. To take into account the unique characteristics high frequency transmission lines, a more appropriate definition is “a device used to transfer energy from one point to another efficiently.” An efficient transfer of energy is one with a minimum amount of reflective loss (i.e. close to a perfect match, or VSWR = 1:1), as well as minimum amount of resistive loss due to mechanisms such as the skin effect. This is important at RF and microwave frequencies, because as frequency increases, energy lost in a transmission line is more difficult and costly to recover.

Figure 1 Equivalent circuit of a transmission line.

An equivalent circuit for a section of transmission line is shown in Figure 1. The representation is of only a single section of the transmission line, not the entire line. (The dashed lines on both sides of the figure indicate that there is more transmission line than is shown.) There are four parameters to consider: inductance (L), resistance (R), capacitance (C) and conductance (G). There is also the dielectric constant, ε, of the insulating material in the transmission line. Values for each of the four parameters are expressed in their appropriate units per unit length. The values are expressed per unit length because the equivalent circuit shown in Figure 1 represents only a section of the transmission line; that section length is the unit length.  The length can be expressed in any appropriate unit  (e.g., ft, m, cm). The value simply indicates what each parameter is for that length of line. The parameters are said to be distributed.

The inductance (microhenries/unit length) in a transmission line is due the current flowing in the metallic conductor. Alternating current (AC) flowing through the conductor produces a magnetic field. The field reaches its maximum at the maximum amplitude of the current flow. When the cycle reverses and begins to go in the reverse direction, the magnetic field collapses, creating an inductive force that produces a current that opposes the applied current. The arrangement of fields sets up an inductance on the transmission line that can be characterized as an inductance per unit length. Inductive reactance, which is the Ohmic result of inductance, increases with frequency and can cause problems for high frequency circuits. Thus, the inductance should generally be kept low for efficient operation.

The resistance (Ohms/unit length) shown in Figure 1 is also associated with the metallic conductor and current flow. Any time current flows through a metallic conductor, there is a loss due to finite resistance in the conductor. This obeys Ohm’s law, which requires a voltage drop across a resistance when current flows through it; thus, the loss in the conductor of a transmission line is caused by current flowing through its resistance causing a voltage drop.

The last two parameters are associated with the dielectric used in the transmission line. The first parameter is capacitance (farads/unit length). A capacitor has two conductive plates of a certain area separated by distance with a dielectric between them. In Figure 1, there are two plates, which are the upper and lower conductors of the transmission line. One plate is the center conductor and the other is the ground, or shield. These plates have an associated area per unit length separated by the dielectric in between, creating a capacitance per unit length.  Capacitive reactance, which is a result of the line capacitance, decreases with an increase in frequency and causes the propagating signals to be shorted to ground at certain frequencies. So, the distributed capacitance, as well as the distributed inductance and resistance of transmission lines should, in general, be minimized.

Conductance (siemens/unit length) is the amount of leakage through the dielectric. There is always a certain amount of conductance, because there is no such thing as a perfect dielectric. That means a certain amount of energy passing down the transmission line appears at the other conductor. It usually is a very small quantity, because many dielectrics are very good insulators in transmission line applications. Conductance is represented by a shunt resistor between the signal wire and the return wire (ground).

The term dielectric constant (ε) is the electrical property of the dielectric material that is between the conductors in a capacitor or transmission line. This may be air, Teflon or other material. It is a relative value equal to the ratio of the electric field in a vacuum divided by the electric field in the medium. The insertion of a dielectric medium between capacitor plates increases its capacitance or ability to store opposite charges on each plate. The dielectric constant of air is considered to be 1.

Figure 2 Lossless transmission line.

To further clarify the important characteristics of transmission lines, assume a perfect transmission line, that is, R = 0 and G = ∞. Those values indicate that the resistance of the conductor is so low that it can be ignored and that the dielectric is perfect (no leakage). The result of this assumption is shown in Figure 2, the circuit diagram for a lowpass filter, which is a component that, ideally, passes all electromagnetic energy below a certain frequency (its cutoff frequency) and attenuates everything above that frequency. With this characteristic, a transmission line has some frequency, depending on its individual characteristics, above which it is not useable. Operation above the cutoff frequency results in much higher loss in the transmission line than does operation within its specified frequency range. A transmission line, when used to connect circuits, should be nearly transparent or invisible. That is, the line physically and electrically connects circuit components but its effect on circuit operation is minimal.

## Types of Transmission Lines

#### Coaxial transmission lines

Figure 3 An example of return loss.

The word coaxial is defined as “having a common axis.” That is descriptive, but it does not do much for our understanding of transmission lines in a coaxial configuration. A coax line is defined as “a transmission line in which one conductor completely surrounds the other, the two being coaxial and separated by a continuous solid dielectric or by dielectric spacers.” Figure 3 is an end view of a coaxial transmission line that shows how those definitions apply.

There are three basic parameters. The first is ε, the dielectric constant of the material used to separate the two conductors. Often, that material is Teflon, which has an ε value of 2.1. Other common dielectrics are polystyrene (ε = 2.56) or polyethylene (ε = 2.26). The dielectric is an important part of the coaxial line both mechanically and electrically. It is important mechanically because it provides the support for the center conductor and separation (spacing) between the center conductor and the shield. It is important electrically because the type of dielectric determines the velocity of the electromagnetic wave traveling through the transmission line. The higher the dielectric constant, the slower the electromagnetic wave travels. The second parameter is the outside diameter, d, of the inner conductor, also referred to as the center conductor. The third parameter D, is the inside diameter of the outer conductor (usually the shield). These three parameters determine impedance, capacitance, inductance, attenuation and cutoff frequency.

The ratio D/d is a commonly used in characterizing coaxial transmission properties. For example, three common coaxial transmission lines, RG-58, RG-59 and RG-62, look basically alike because their outside diameters are similar; however, RG-58 is a 50 Ohm line, RG-59 is a 75 Ohm line and RG-62 is a 93 Ohm line. Since one of the factors that determines the impedance of a coaxial transmission line is the D/d ratio, and since the inside diameters of the outer conductors are basically are the same, there must be three different d values. In fact, that is the case. The RG-58 center conductor diameter is 0.031 in., RG-59 is 0.083 in. and RG-62 is 0.025 in.

There are commonly two types of coaxial transmission lines or cables: flexible and semirigid. Each has uses for specific applications. It is important to understand that every transmission line is a lowpass filter, so each type of coaxial cable has an upper frequency limit. This is true for all transmission lines, whether they are coaxial, distributed lines or even waveguides. All transmission lines have a cutoff frequency.

#### Flexible coaxial transmission lines

Figure 4 Coaxial cable.

The dictionary defines flexible as “able to bend without breaking,” an excellent definition for flexible coaxial cable. It is a cable that is able to bend in many different directions and that, if the minimum bend radius for the cable is not exceeded, will not break or affect the electrical parameters of the cable in any way. This type of transmission line is one that can be used for applications in which the line must be bent around corners to make the necessary connections. It also is an excellent transmission line for use in a laboratory environment, where many connect/disconnect operations are needed and where there is no standard way of connecting a cable to a piece of equipment.

A typical flexible transmission line (see Figure 4), consists of four sections: a center conductor (solid or stranded wire), a dielectric (usually Teflon), a braided outer conductor and an outer covering. This construction is used for RG-58, RG-59 and RG-62 cables (see Figure 5). We have already seen that the center conductors are different to make the different impedances of each of the cables. The dielectric is basically the same, as are the braid diameters and the outer coating. This is where confusion can occur, because the visible portions of the transmission lines are so similar.

Figure 5 Flexible coaxial cable construction.

The center conductor can be either a solid wire or a series of wires in a stranded configuration. This is the “hot” lead of the transmission line, where the signal is carried. Also, it is the determining factor for the inductance and the resistance in the equivalent circuit. The center conductor is usually solid copper or copper cladding over steel. Because it is a metal and carries a varying current, an inductance is set up and there is also resistance. The current is electromagnetic, that is, there is an electric and a magnetic component. These two components are 90 degrees out of phase. It is a complex current that cannot be measured with a simple ammeter.

One of the functions of the dielectric is to serve as a spacer separating the center conductor and the outer conductor, but it also determines the electrical parameters of the transmission line, such as the characteristic impedance, capacitance per foot, attenuation, cutoff frequency and velocity of the RF and microwave energy propagating through the transmission line.

The outer conductor serves two functions. It is a ground reference for the signal on the center conductor and is also a shield. The shield keeps external signals out and internal signals in. The braid construction may be single, double or triaxial (two braids separated by an insulator). The single-braid construction consists of bare, tinned or silverplated copper wires. The double braid consists of two single braids with no insulation between them. Triaxial construction consists of two single braids with a layer of insulation between them. The type and the degree of shielding needed for a particular application determines which scheme is best. It may depend, in whole or in part, on the environment, the type and amplitude of signals and/or cost factors. A variety of shielding types are available to fit most applications.

The outer coating provides environmental protection for the cable. Viewing the outer coating, several popular cables; RG-58, RG-59 and RG-62 cables may seem nearly identical (see Table 1), although they have distinctively different electrical characteristics.

 Cable Type Z (Ω) Outer-Coating Diameter (in.) Outer-Conductor Inner Diameter (in.) Inner-Conductor Outer Diameter (in.) RG-58 50 0.195 0.150 0.083 RG-59 75 0.242 0.191 0.031 RG-62 93 0.242 0.191 0.025

Flexible cables are available in a variety of different types and usually have an RG prefix (military designation for cables). The overall cable outside diameter can range from 0.078 (5/64) to over 1 in. Often, the size of the cable is dictated by the CW power handling requirement (continuous wave (CW) is the type of signal characterized by an uninterrupted AC sine wave). Other times, the size is dictated by such factors as installation area (does it need to bend around many corners?) or environmental conditions (both atmospheric and electrical). As with any choice, the application and its specifications dictate which cable to use. No single cable is suitable for every job.

These transmission lines are flexible and able to be bent around corners and into tight areas. However, even flexible cables have a minimum bend radius, that is, a radius beyond which bending the cables results in serious performance degradation; they cannot be tied in a knot and be expected to perform as well as when stretched out straight. Consult the cable data sheet to determine its minimum bend radius.

Figure 6 Flexible coaxial cable assembly with connectors (Source: Thomas S. Laverghetta, Modern Microwave Measurements and Techniques, Artech House, Norwood, Mass., 1988.)

Flexible coaxial cables have many applications, ranging from finished RF cables for equipment shipped to a customer to the typical 50 Ohm cables hanging in every electronics lab. The performance of the final, assembled flexible transmission line depends not only on the cable, itself, but also on the connectors placed on each end. There are many types of connectors that can be attached to a flexible cable. A great deal of care must be taken in the choice of a connector to go on the ends of a flexible cable. The wrong connector can completely cancel all the benefits a good cable choice.

Four typical ones are shown in Figure 6. The subminiature A (SMA) connector (Figure 6a) is used for many microwave applications and comes in a variety of configurations, such as with two- and four-hole flanges for attachment to a chassis. The one shown in Figure 6a is the type used for cable connection. Figure 6b is a BNC connector, a type seen frequently on flexible cables in electronic laboratories. They are good for low frequency applications. Higher frequency circuits and systems should not use BNC connectors because the ground connection is not adequate when the frequency increases. The TNC connector (Figure 6c) is an improved version of the BNC connector for use in higher frequency applications. It is actually a BNC connector with threads, which provides a much better ground connection. The N connector (Figure 6d) is a larger connector suitable for a wide variety of applications over a large range of frequencies. It also has threads for ground connection and can be fitted to a number of flexible cable types.

Figure 7 Coaxial cable connector types; SMA (a), BNC (b), TNC (c), N (d).

#### Semi-rigid coaxial transmission lines

The dictionary defines rigid as “not bending or being flexible, stiff.” Semi-rigid, on the other hand,  is something that is fairly solid but capable of being bent. At first sight, a semi-rigid coaxial transmission line (cable) appears rigid. It does, with some effort, however, bend to a specific bend radius and subsequently holds its shape (as if rigid). Once a semi-rigid cable has been bent into shape, however, it should not be re-straightened or re-bent. This will likely change the physical properties of the cable and thus change its electrical properties, as well. For example, multiple bend operations may induce a sharp corner in the center conductor center rather than a smooth bend. The impedance will be different at the corner, causing reflections that degrade performance. Semi-rigid cable, like flexible cable, has a minimum bend radius specified in the manufacturer’s data sheet.

Figure 8 Semi-rigid coaxial cable.

Semi-rigid cable consists of a solid center conductor, a solid dielectric and a solid outer conductor. Every part of a semi-rigid cable is solid, giving it its rigid appearance and characteristics. From Figures 4 and 7, you can see the similarities between semi-rigid and flexible cables. There is a coaxial construction for both transmission lines, i.e., a center conductor is surrounded by a dielectric with an outer conductor used for ground over the entire line. Figure 8 is a typical semi-rigid cable. The main difference between semi-rigid and flexible cable is the solid outer conductor for the semi-rigid cable. The solid conductor improves shielding and gives the cable greater rigidity. The solid outer conductor might be simply a copper sheath over the entire cable or it might be plated for specific applications. Often the cable is gold plated to reduce oxidation of the copper over time, adding to the performance, and to the price. Other cables may be tin-lead plated for the same purpose.

Flexible cable is designated by an RG number (such as RG-58 and RG-59), whereas semi-rigid is designated by a number only, that number being the outside diameter of the cable. Common sizes (in in.) of semi-rigid cable are 0.020, 0.034, 0.047, 0.056, 0.070, 0.085, 0.141, 0.215, 0.250 and 0.325. The three sizes most widely used are 0.085, 0.141 and 0.250. The large majority of semi-rigid cables used in equipment are 0.141 cables.

The other two common types of semi-rigid cable, 0.085 and 0.250, are used in applications where 0.141 may not be suitable. The 0.085 cable is used when there is very little room in which to maneuver, so the bends in the cable must be very small. This small-diameter cable fits this need, but at the price of higher attenuation than its 0.141 counterpart. The power-handling capability of the 0.085 cable also is less than that of the 0.141, which is understandable, since it is much smaller and there is less area to dissipate power.

The 0.250 semi-rigid cable is used primarily for higher power applications, because it has much more area to dissipate power. For comparison, consider the power-handling capability of the three sizes of cable at 1 GHz. The 0.085 cable is rated at 222 W, the 0.141 cable is rated at 600 W and the 0.250 cable is rated at 1200 W (each is CW power). The 0.250 cable also exhibits lower attenuation. At 1 GHz, the 0.085 cable has 0.187 dB/ft of attenuation, 0.141 cable has 0.116  dB/ft and 0.250 cable has 0.073 dB/ft. Its drawback is its size; it requires a lot of room to make a bend. Thus, 0.250 cable is used mainly for straight runs in high-power applications.

 Attenuation (dB/100 ft) Power (W) Frequency (GHz) 0.4 1.0 3.0 5.0 10.0 0.4 1.0 3.0 5.0 10.0 Cable Type 0.141 7.2 11.6 21.5 28.8 44.5 1000 600 310 230 160 RG-58 11 20 41 - - 75 44 22 15 -

Semi-rigid cable has many superior electrical properties over flexible cable. In all cases, it can handle more power and exhibit less loss than a flexible cable of comparable size. For example, a comparison of a length of RG-58 flexible cable with a length of 0.141 semi-rigid cable at five frequencies is shown in Table 2. These two cable types have similar outside diameters (the outer conductor of the RG-58 is 0.150 in.). Attenuation at 3 GHz, for example, is 0.215 dB/ft for 0.141 cable versus 0.41 dB/ft for RG-58. At that frequency, the 0.141 cable can handle 310 W, while the RG-58 cable can handle only 22 W. The RG-58 cable is also not specified at all for attenuation at 5 or 10 GHz and is not specified for power-handling capability at 10 GHz. Recall that all transmission lines have an upper frequency limits since they all act like lowpass filters. That is clearly indicated in Table 2 for the RG-58 cable.

The numbers in Table 2 might lead one to ask, “Why ever use flexible cables?” First, semi-rigid cables cost considerably more than flexible cables; an important consideration, especially in the commercial market. Second, for testing applications, semi-rigid cables are not very practical. Most tests require many connect/disconnect operations, which can put strain on the cables. Also, as the tests change in a lab, there are different positions for the cables that are formed much better with flexible cables. Third, in some finished products, the cables must meander through the chassis to various locations. Semi-rigid cables are not suitable for those applications in many cases.

Figure 9 Typical semi-rigid cable assemblies.

#### Strip transmission line (stripline)

The terms strip transmission line, stripline, tri-plate and sandwich line all refer to the same type of transmission line. The term most commonly used is stripline. Stripline is different from, yet also similar to, flexible and semirigid cables. As a matter of fact, stripline actually evolved from coaxial transmission line. The stripline shown in Figure 9 appears to be similar to a coaxial cable; it looks like a coaxial cable that has been flattened.

The coaxial structure shown in Figure 10a is a typical center conductor surrounded by a dielectric with a shield completely around the entire structure. Figure 10b is a side view of a stripline structure. If pressure is applied at the top and bottom of the coaxial structure in Figure 10a, the circular structure deforms into an oblong shape rather than a concentric circular form. The electric field lines go from the center conductor to the outside shields of the structure. With further pressure, the two ends eventually split and break. What remains is a top ground (outside shield), a bottom ground and a rectangular center conductor, the same structure shown in Figure 9.

Figure 10 Typical semi-rigid cable assemblies.

In both Figures 9 and 10, it can be seen that stripline actually is two circuit boards sandwiched together (hence the term sandwich line). There is a ground (copper) on the top and a ground on the bottom, with the actual circuit in between and surrounded by a dielectric medium. The circuit is what is termed the center conductor in Figure 10 and is the straight line running through the center of the device in Figure 9. In the coaxial structure of Figure 10, the center conductor is surrounded by a dielectric material. In the stripline structure, each circuit board  contains a dielectric layer, which is the same for both boards, that is, the center conductor is surrounded by the same dielectric, top and bottom. In its simplest form, a piece of stripline has a lower circuit board with a circuit etched on one side of a dielectric layer and a complete ground plane (covering the entire circuit board) on the other. The top board has a similar ground plane on one side of the dielectric layer and no metallization on the other side. The two boards are permanently bonded together with an adhesive to form a monolithic structure with two ground planes, separated by a virtually continuous dielectric interrupted only by the center conductor in between.

It is necessary to clarify the term ground plane. In conventional low-frequency, DC and digital circuits, a ground is a point on a chassis where wires carrying return current are terminated to ensure a common potential. In RF and microwave circuits, electromagnetic waves sent down transmission lines require an entire (continuous) plane or large area of ground rather than a single point. High frequency signals have periodicities that are perturbed by point grounds. That is why circuit boards have copper plating across their entire outer surfaces.

Figure 11 Stripline Evolution, coaxial cross-section
(a) to stripline cross-section (b).

Stripline is of uniform construction. Figure 9 depicts that uniformity, with the circuit being completely surrounded by dielectric on all sides. Because of that uniformity, stripline has a natural shielding effect on the circuit because it is completely enclosed, with metal on top and bottom. A stripline package also has metal on the ends and sides, resulting in a circuit within a complete metal box with excellent shielding.

An important parameter in stripline is the ground-plane spacing (GPS), shown in Figure 11. GPS is, as the name implies, is the spacing between the inner surfaces of the ground planes, designated as b.  It is used to calculate the width of transmission lines (w) that are etched on the circuit board and the spacing (s) between two transmission lines. The relationships between w, s and b are shown in Figure 12.

A close tolerance on circuit board dielectric thickness must be maintained to ensure that the widths and the spacings are as accurate as possible. Consider that the characteristic impedance of a strip transmission line is determined by the ratio w/b; and similarly, the coupling of energy between two transmission lines is determined by the s/b ratio. If the b dimension in the denominator is not tightly controlled it is difficult to maintain a desired characteristic impedance as well as a desired level energy coupling between circuits.

Figure 12 Ground plane spacing.

Stripline has become a popular medium for the construction of transmission lines and microwave circuit components. A drawback to using stripline for some applications is also touted as an advantage; it is a sealed enclosed structure. Although it offers good circuit shielding, there is limited access to circuit elements for test and troubleshooting. Because stripline packages contain laminated boards laminated boards and are often epoxy sealed, semiconductor devices such as diodes and transistors are generally not integrated into the structure, unless the integrated module is considered non-repairable.

Figure 13 Relationships of ground plane spacing (b), transmission line width (w), and spacing between lines (s).

For many applications, stripline is an excellent choice as a high frequency transmission line. Where there is a requirement for excellent shielding, strip transmission line should be considered. Also, stripline is popular for the construction of passive components, such as couplers, power dividers, filters and impedance transformers. Figure 13 shows a stripline circuit with the top dielectric and ground layers removed to show the center conductor patterns that form the components.

#### Microstrip

Microstrip transmission line solves the problem of inaccessibility that stripline poses. Microstrip, shown in Figure 14, is similar to stripline, except that there is no top dielectric or conductor layer. There is nothing but air on top of the circuitry and a dielectric material underneath with a bottom ground plane. The width of the transmission line is designated by w, t is the thickness of the copper circuit and b is the thickness of the dielectric. It can be seen that b is similar to that for stripline except that it represents only one thickness of material.

Figure 14 Stripline with the top dielectric and ground layers removed to expose circuit elements.

The material thickness of microstrip is as important as it is for stripline, and the relationships are similar. What is different in microstrip, however, is the dielectric constant. In stripline, the dielectric surrounding the center conductor is uniform. Stripline due to its symmetry is well-behaved. Microstrip, while it has a uniform dielectric material underneath its circuit element, has only air above it. The boundary between the dielectric material and the air has been the subject of numerous research projects and papers. The interface can be simplified if one assumes the existence of an effective dielectric constant, εeff, the resultant of the two dielectric constants (air above and the dielectric material below). The calculation takes into account the relative dielectric constant of the material, εr, and the dielectric constant of air, εo = 1.

The effective dielectric constant is used for all calculations where a dielectric constant term is used. Note that every time the impedance of a microstrip line changes, the effective dielectric constant changes as well. That is because the microstrip width changes with a change in impedance. The lower the impedance, the wider the line and vice versa. To calculate a filling factor for microstrip, which is a compensating factor for the difference in dielectric constant, the ratio w/b ratio is used. Thus, a change in impedance is a change in filling factor and, consequently, a change in effective dielectric constant. The relationship between impedance and effective dielectric constant is a point that sometimes escapes designers. This relationship must be taken into account many times in the design of a multisection power divider or directional coupler. Such devices require different impedance lines that are all one quarter-wavelength long.

Figure 15 Microstrip transmision line.

Microstrip transmission lines are probably the most common types of transmission lines found in today’s RF and microwave circuits. The wireless markets use many microstrip circuits that enable easy component placement and attachment on top of the circuit board. Manufacturers that use surface mount technology (SMT) components use microstrip because of the ease of construction. It is clearly easier to attach components to a circuit board that has all the transmission lines on top of the board and very visible (see Figure 15).

Figure 16 Microstrip assembly.

#### Coplanar waveguide

A representation of coplanar waveguide is shown in Figure 16. At first glance, it resembles microstrip construction. It has a single circuit board, like microstrip; it has the circuit traces on the top of the board, like microstrip; and it has air over the top of the circuit board, like microstrip. Looking at it a little closer, however, one sees some distinct differences.

In microstrip construction, there is a circuit trace on top of the board material of a certain width and thickness along with a complete ground plane on the reverse side of the board. In a coplanar waveguide, there is still a circuit trace on the top of the board that is a certain width and thickness, but there are also ground planes on both sides of the circuit trace and, as can be seen in Figure 16, there is no ground plane on the bottom of the circuit board. The ground plane on both sides of the circuit trace is where the transmission line structure gets its name. It is also possible to construct coplanar waveguide as it is shown in Figure 16 but with a complete ground plane on the reverse side. This ground-backed coplanar waveguide is shown in Figure 17 along with conventional coplanar waveguide.

Figure 17 Coplaner waveguide.

One of the properties of coplanar waveguide that makes it attractive for RF and microwave applications is the fact that both series and shunt elements can be attached to the transmission line with relative ease. In the case of microstrip or stripline, the fabrication of such elements can be difficult. With coplanar waveguide, the ground plane is adjacent to the center strip carrying the signal, and connections to it are straightforward.

#### Waveguide

Usually when we think of waveguide, we think of frequencies well up into the GHz range. We also picture rectangular pieces of hardware with flanges screwed together and that exhibit very low losses. It may be difficult to visualize how it works because there is no center conductor (as in coaxial transmission line) for energy to travel. This is because the energy travels similar to the way it propagates in free space, but is guided by multiple reflections off the waveguide walls.

While every transmission line is a waveguide since it provides a path to “guide” electromagnetic waves down the line, the term waveguide is commonly reserved for what might be called a hollow-pipe waveguide, or at least a waveguide in which two distinct conductors are not present. The “open space” of the waveguide is where the electromagnetic energy finds the path of least resistance and can propagate (or move) down the line to get to its destination. Waveguide is used at microwave frequencies (particularly at higher microwave frequencies) for two reasons: it is often easier to fabricate than coaxial lines; and it often exhibits less loss. Coaxial transmission lines require center conductors that are supported by a dielectric material; waveguide requires no center conductor, its dielectric is air.

Figure 18 Cross sections of coplanar waveguide (a), and ground-backed coplanar waveguide (b).

One way to describe a rectangular waveguide is to start from a two-wire transmission line as illustrated in Figure 18. The two-wire transmission line is just as the name implies, two wires separated by a dielectric (air in this case). It can be seen from the figure that two adjacent points (one on each wire) are extended to be one-quarter wavelength long, above and below the lines. The two points are then connected to form a short circuit (low impedance). Due to the periodic nature of the RF waveform, at quarter wavelength distance from the short circuit (at the connections to the transmission line) the short circuit is transformed to a high impedance. This is referred to as a quarter-wave stub. Since the short appears as a high impedance in the center of the structure, there is no effect on the transmission of power. If the number of stubs is increased infinity, while the spacing of adjacent stubs is reduced to zero, a rectangular waveguide is formed.

The two dimensions (a and b) are important. The a dimension cannot be less than one-half wavelength. This is because the guide is made up of two quarter-wavelength stubs separated by a small distance. Electromagnetic radiation with l/2 greater than or equal to 2a cannot propagate. The frequency for which 2a is exactly l/2 is called the cutoff frequency, fc. The a and b dimensions are related, as well; the b dimension is slightly more than half of the a dimension.

To understand nature of waveguide, one must visualize a wave (an electromagnetic wave) and not think about voltages and currents. The wave of interest in coaxial transmission lines, microstrip and stripline is the transverse electromagnetic wave (TEM). This is an electric wave and a magnetic wave propagating down the line alternately exchanging energy between electric and magnetic fields. These waves are 90° apart and propagate together with their fields transverse to the direction of propagation.

In conventional rectangular waveguide, either the electric field is transverse to the direction of propagation, i.e. transverse electric (TE) or the magnetic field is transverse to the direction of propagation, i.e., transverse magnetic (TM). In addition to the TE or TM designation, subscripts are used to describe the electric and magnetic field configuration. The general representation is TEmn and TMmn where the subscript “m” indicates the number of half-wave variations of the electric (or magnetic) field along the “a” (wide) dimension of the guide and the subscript “n” is the number of half-wave variations of the electric (or magnetic) field in the “b” (narrow) dimension. The most common mode of operation is the TE10 mode.

The term characteristic impedance is used in describing other forms of transmission lines. For waveguide there is a corresponding term called characteristic wave impedance. This is a representation of the ratio of the electric field and the magnetic field at a certain point in the waveguide. Where the characteristic impedance of a coaxial line is an important parameter, the characteristic wave impedance for waveguide is seldom determined or used, if it is available, since it only characterizes the guide at a certain point. The only place where it might be of value is where the waveguide interfaces with other components in a circuit.

## Summary

The transmission lines covered in this section illustrate the need for special structures for RF and microwave use. With these high frequencies, it is no longer possible to think of an interconnection between two circuits as simply a piece of wire that is wrapped or soldered to make the connection. There is a requirement for a low-loss connection for use in this range, and that low-loss connection is the transmission line.