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Authorized translation from the original German language published by Carl Hanser Verlag GmbH & Co. KG, Munich.FRG.

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RF and microwave engineering : fundamentals of wireless communications / Frank Gustrau.

1. Radio circuits. 2. Microwave circuits. 3. Wireless communication systems–Equipment and supplies. I. Title.

This textbook aims to provide students with a fundamental and practical understanding of the basic principles of radio frequency and microwave engineering as well as with physical aspects of wireless communications.

In recent years, wireless technology has become increasingly common, especially in the fields of communication (e.g. data networks, mobile telephony), identification (RFID), navigation (GPS) and detection (radar). Ever since, radio applications have been using comparatively high carrier frequencies, which enable better use of the electromagnetic spectrum and allow the design of much more efficient antennas. Based on low-cost manufacturing processes and modern computer aided design tools, new areas of application will enable the use of higher bandwidths in the future.

If we look at circuit technology today, we can see that high-speed digital circuits with their high data rates reach the radio frequency range. Consequently, digital circuit designers face new design challenges: transmission lines need a more refined treatment, parasitic coupling between adjacent components becomes more apparent, resonant structures show unintentional electromagnetic radiation and distributed structures may offer advantages over classical lumped elements. Digital technology will therefore move closer to RF concepts like transmission line theory and electromagnetic field-based design approaches.

Today we can see the use of various radio applications and high-data-rate communication systems in many technical products, for example, those from the automotive sector, which once was solely associated with mechanical engineering. Therefore, the basic principles of radio frequency technology today are no longer just another side discipline, but provide the foundations to various fields of engineering such as electrical engineering, information and communications technology as well as adjoining mechatronics and automotive engineering.

The field of radio frequency and microwave covers a wide range of topics. This full range is, of course, beyond the scope of this textbook that focuses on the fundamentals of the subject. A distinctive feature of high frequency technology compared to classical electrical engineering is the fact that dimensions of structures are no longer small compared to the wavelength. The resulting wave propagation processes then lead to typical high frequency phenomena: reflection, resonance and radiation. Hence, the centre point of attention of this book is wave propagation, its representation, its effects and its utilization in passive circuits and antenna structures.

What I have excluded from this book are active electronic components—like transistors—and the whole spectrum of high frequency electronics, such as the design of amplifiers, mixers and oscillators. In order to deal with this in detail, the basics of electronic circuit design theory and semiconductor physics would be required. Those topics are beyond the scope of this book.

If we look at conceptualizing RF components and antennas today, we can clearly see that software tools for Electronic Design Automation (EDA) have become an essential part of the whole process. Therefore, various design examples have been incorporated with the use of both circuit simulators and electromagnetic (EM) simulation software. The following programs have been applied:

As the market of such software products is ever changing, the readers are highly recommended to start their own research and find the product that best fits their needs.

At the end of each chapter, problems are given in order to deepen the reader's understanding of the chapter material and practice the new competences. Solutions to the problems are being published and updated by the author on the following Internet address:

Finally, and with great pleasure, I would like to say thank you to my colleagues and students who have made helpful suggestions to this book by proofreading passages or initiating invaluable discussions during the course of my lectures. Last but not the least I express gratitude to my family for continuously supporting me all the way from the beginning to the completion of this book.

This chapter provides a short overview on widely used microwave and RF applications and the denomination of frequency bands. We will start out with an illustrative case on wave propagation which will introduce fundamental aspects of high frequency technology. Then we will give an overview of the content of the following chapters to facilitate easy orientation and quick navigation to selected issues.

1.1 Radiofrequency and Microwave Applications

Today, at home or on the move, every one of us uses devices that employ wireless technology to an increasing extent. Figure 1.1 shows a selection of wireless communication, navigation, identification and detection applications.

Figure 1.1 (a) Examples of wireless applications (b) RF components and propagation of electromagnetic waves.

In the future we will see a growing progression of the trend of applying components and systems of high frequency technology to new areas of application. The development and maintenance of such systems requires an extensive knowledge of the high frequency behaviour of basic elements (e.g. resistors, capacitors, inductors, transmission lines, transistors), components (e.g. antennas), circuits (e.g. filters, amplifiers, mixers) including physical issues such as electromagnetic wave propagation.

High frequency technology has always been of major importance in the field of radio applications, recently though RF design methods have started to develop as a crucial factor with rapid digital circuits. Due to the increasing processing speed of digital circuits, high frequency signals occur which, in turn, create demand for RF design methods.

In addition, the high frequency technology's proximity to electromagnetic field theory overlaps with aspects of electromagnetic compatibility (EMC). Setups for conducted and radiated measurements, which are used in this context, are based on principles of high frequency technology. If devices do not comply with EMC limits in general a careful analysis of the circumstances will be required to achieve improvements. Often, high frequency issues play a major role here.

Table 1.1 shows a number of standard RF and microwave applications and their associated frequency bands [1–3]. The applications include terrestrial voice and data communication, that is cellular networks and wireless communication networks, as well as terrestrial and satellite based broadcasting systems. Wireless identification systems (RFID) within ISM bands enjoy increasing popularity among cargo traffic and logistics businesses. As for the field of navigation, GPS should be highlighted, which is already installed in numerous vehicles and mobile devices. Also in the automotive sector, radar systems are used to monitor the surrounding aresa or serve as sensors for driver assistance systems.

1.2 Frequency Bands

For better orientation, the electromagnetic spectrum is divided into a number of frequency bands. Various naming conventions have been established in different parts of the world, which often are used in parallel. Table 1.2 shows a customary classification of the frequency range from 3 Hz to 300 GHz into eight frequency decades according to the recommendation of the International Telecommunications Union (ITU) [4].

Figure 1.2a shows a commonly used designation of different frequency bands according to IEEE-standards [5]. The unsystematic use of characters and band ranges, which has developed over the years, can be regarded as a clear disadvantage. A more recent naming convention according to NATO is shown by Figure 1.2b [6, 7]. Here, the mapping of characters to frequency bands is much more systematic. However, the band names are not common in practical application yet.

Figure 1.2 Denomination of frequency bands according to different standards. (a) Denomination of frequency bands according to IEEE Std. 521–2002 (b) Denomination of frequency bands according to NATO.

A number of legal foundations and regulative measures ensure fault-free operation of radio applications. Frequency, as a scarce resource, is being divided and carefully administered [8, 9]. Determined frequency bands are allocated to industrial, scientific and medical (ISM) applications. These frequency bands are known as ISM bands and are shown in Table 1.3. As an example, the frequency range at 2.45 GHz is for the operation of microwave ovens and WLAN systems. A further frequency band reserved for wireless non-public short-range data transmission (in Europe) uses the 863 to 870 MHz frequency band [10], for example for RFID applications.

1.3 Physical Phenomena in the High Frequency Domain

We will now take a deeper look at RF engineering through two examples that introduce wave propagation on transmission lines and electromagnetic radiation from antennas.

1.3.1 Electrically Short Transmission Line

As a first example we consider a simple circuit (Figure 1.3a) with a sinusoidal (monofrequent) voltage source (internal resistance R_I), which is connected to a load resistor R_A = R_I by an electrically short transmission line. Electrically short means that the transmission length ell

of the line is much shorter than the wavelength λ, that is ell

λ. In vacuum–or approximately air–electromagnetic waves propagate with the speed of light c₀.

where ε_r is the relative permittivity and μ_r is the relative permeability of the medium. Typical values for a practical coaxial line would be ε_r = 2 and μ_r = 1, resulting in a speed of light of c ≈ 2.12 · 10⁸ m/s on that line. Given–as an example–a frequency of f = 1 MHz we get a wavelength of λ₀ = 300 m in free space and λ = 212 m on the previously discussed line. A transmission line of ell

= 1 m would then be classified as electrically short ( ell

λ). For simplicity¹, we assume further on that the load resistance R_A equals the internal resistance R_I of the source.

Figure 1.3 Network with voltage source, transmission line and load resistor. Transmission line is electrically short in (a), (b) and electrically long in (c), (d).

Alternatively, electrically short can be expressed by the propagation time τ a signal needs to pass through the entire transmission line. Assuming that electromagnetic processes spread with the speed of light c, the transmission of a signal from the start through to the end of a line requires a time span τ

If the time span τ needed for a signal to travel through the whole line is substantially smaller than the cycle time T of its sinusoidal signal, it seems as if the signal change appears simultaneously along the whole line. Signal delay is thus surely negligible.

A transmission line is defined as being electrically short, if its length ell

is substantially shorter than the wavelength λ of the signal's operating frequency ( ell

λ) or–in other words–if the duration of a signal travelling from the start to the end of a line τ (delay time) is substantially shorter than its cycle time T (τ

T).

Let us have a look at Figure 1.3b where the current changes slowly in a sinus-like pattern. The term slowly refers to the period T that we assume to be much greater than the propagation time τ along the line. The sine wave starts at t = 0 with a value of zero and reaches its peak after a quarter of the time period (t = T/4). Again after half the time period (t = T/2) it passes through zero and reaches a negative peak at t = 3T/4. This sequence repeats periodically. Since signal delay τ can be omitted compared to the time period T, the signal along the line appears to be spatially constant. According to the voltage divider rule the voltage along the line equals just half of the value of the voltage source u₀(t). The input voltage u_in(t) and the output (load) voltage u_A(t) are–at least approximately–equal.

1.3.2 Transmission Line with Length Greater than One-Tenth of Wavelength

In the next step, we significantly increase the frequency f, so that the line is no longer electrically short. We choose the value of the frequency, such that the line length will equal ell

= 5/4 · λ = 1.25λ (Figure 1.3c). Now signal delay τ compared to period duration T must be taken into consideration. In Figure 1.3d we can see how far the wave has travelled at times of t = T/4, t = T/2 and so forth. The voltage distribution is no longer spatially constant. After t = 5/4T the signal reaches the end of the line.

If the transmission line is not electrically short, the voltage along the line will not show a constant course any longer. On the contrary, a sinusoidal course illustrates the wave-nature of this electromagnetic phenomenon.

Also we can see that the electric voltage u_A(t) at the line termination is no longer equal to that at the line input voltage u_in(t). A phase difference exists between those two points.

In order to fully characterize the transmission line effects, a transmission line must be described by two additional parameters along with its length: (a) the characteristic impedance Z₀ and (b) the propagation constant γ. Both must be taken into account when designing RF circuits.

In our example we used a characteristic line impedance Z₀ equal to the load and source resistance (Z₀ = R_A = R_I). This is the most simple case and is often applied when using transmission lines. However, if the characteristic line impedance Z₀ and terminating resistor R_A are not equal to each other, the wave will be reflected at the end of the line. Relationships resulting from these effects will be looked at in Chapter 3 which deals in detail with transmission line theory.

1.3.3 Radiation and Antennas

Now let us take a look at a second example. Here we have a geometrically simple structure (Figure 1.4a), which consists of a rectangular metallic patch with side length ell

arranged above a continuous metallic ground plane. Insulation material (dielectric material) is located between both metallic surfaces. Two terminals are connected to feed the structure.

Figure 1.4 Electrical characteristic of a geometrical simple structure: (a) geometry and (b) imaginary part of admittance.

The geometric structure resembles that of a parallel plate capacitor, which has a homogeneous electrical field set up between the metal surfaces. Therefore, we see capacitive behaviour ((Figure 1.4b) Admittance Y = jωC) at low frequency values (geometrical dimensions are significantly below wavelength ( ell

λ)). By further increasing the frequency, we can observe resonant behaviour due to the unavoidable inductance of feed lines.

At high frequency levels a completely new phenomenon can be observed: with the structure's side length approaching half of a wavelength ( ell

≈ λ/2), electromagnetic energy will be radiated into space. Now the structure can be used as an antenna (patch antenna).

This example clearly illustrates that even a geometrically simple structure can display complex behaviour at high frequency levels. This behaviour cannot yet be described by common circuit theory and requires electromagnetic field theory.

1.4 Outline of the Following Chapters

The last two examples have given us some insight into the fact that problems involving RF cannot simply be treated with conventional methods, but need a toolset adjusted to the characteristics of RF technology. Chapters 2 to 8 therefore give an in-depth insight into how best to solve RF-problems and show the methods we commonly apply.

First, the principles of electromagnetic field theory and wave propagation are reviewed in Chapter 2, in order to understand the mechanisms of passive high-frequency circuits and antennas. The mathematical formulas used in this chapter mainly serve the purpose of illustrating mathematical derivations and are not intended for further calculations. Nowadays, in work practice, modern RF circuit and field simulation software packages provide approximate solutions based on the above mentioned theories. Nonetheless, an engineer needs to understand these mathematical foundations in order to evaluate such given solutions of different commercial software products with respect to their plausibility and accuracy.

Transmission lines are a major and important component in RF circuits. The simple structure of a transmission line may be used in a variety of very different applications. Chapter 3 will therefore deal with the detailed relationships of voltage and current waves on transmission lines. Calculations in this context can be easily followed and form a safe foundation for treating the ever-occurring issue of transmission lines. This chapter gives a short introduction to the Smith chart as a common instrument of presentation and design in RF technology.

Chapter 4 continues with the characterization of transmission line structures and provides a deeper insight into technically important line types such as coaxial lines, planar transmission lines and hollow waveguide structures. It also gives an overview on common mode and differential mode signals on three conductor lines, since they play a major role in the circuit design of filters and couplers.

Chapter 5 introduces scattering parameters, with which the behaviour of RF circuits is characterized. Scattering parameters link wave quantities at different ports of RF circuits. A great advantage of scattering parameters compared to impedance and admittance parameters (which are preferably used when frequency levels are low) is that they can be directly measured by vector network analysers even at high frequencies.

The basic principles provided in Chapters 2 to 5 now pave the way for the description of more complex practical passive RF circuitry, which is the focus of Chapter 6. It is shown that a thoughtful interconnection of transmission lines may create matching circuits, filters, power splitters or couplers. The aim here is to get to know different important design methods and to apply them to examples and case studies, rather than focusing on their mathematical derivation. The examples given are processed with RF circuit and field simulators and thus show how to employ these tools. Also, as a side issue (as it is not the main purpose of this book), a short preview on electronic circuits and their basic characteristics is given.

In radio communication, an antenna provides the link between aerial radio waves in free space and signals on transmission lines. Therefore, Chapter 7 deals with technically important parameters, which serve to describe the radiation behaviour of antennas. To deepen knowledge, we will mathematically deduce the functioning of a basic antenna element (Hertzian dipole). Further on, we take a look at important practical single and array antenna structures and test various design rules on some illustrative examples.

Finally, when it comes to evaluating wireless systems, it is not enough to only consider the antenna alone. Instead, interference from the surroundings (environment) on the wave propagation between antennas must also be taken into account. Chapter 8 thus introduces basic propagation phenomena and their effects on signal transmission. The book concludes with a short review of empirical and physical path loss models.

References

2. Golio M, Golio J (2008) The RF and Microwave Handbook, Second Edition. CRC Press.

4. ITU (2000) ITU-R Recommendation V.431: Nomenclature of the Frequency and Wavelength Bands Used in Telecommunications. International Telecommunication Union.

5. IEEE (2002) IEEE Std 521-2002 Standard Letter Designations for Radar-Frequency Bands. IEEE.

6. Macnamara T (2010) Introduction to Antenna Placement and Installation. John Wiley & Sons.

7. Meinke H, Gundlach FW (1992) Taschenbuch der Hochfrequenztechnik. Springer.

8. Bundesnetzagentur (2008) Frequenznutzungsplan gemaess §54 TKG ueber die Aufteilung des Frequenzbereichs von 9 kHz bis 275 GHz auf die Frequenznutzungen sowie ueber die Festlegungen fuer diese Frequenznutzungen. Bundesnetzagentur.

9. CEPT/ECC (2009) The European Table of Frequency Allocations and Utilizations in the frequency range 9 kHz to 3000 GHz. European Conference of Postal and Telecommunications Administrations, Electronic Communications Committee.

10. Bundesnetzagentur (2005) Allgemeinzuteilung von Frequenzen fuer nichtoeffentliche Funkanwendungen geringer Reichweite zur Datenuebertragung; Non-specific Short Range Devices (SRD). Bundesnetzagentur Vfg 92.

¹ The reason for this determination will become clear when we discuss the fundamentals of transmission line theory in Chapter 3.

In this chapter we will recall the basic electric and magnetic field quantities. We start with the static—non time-varying—case and highlight the relation between electromagnetic field quantities and circuit variables like voltage and current. A full description of time-varying electromagnetic fields is given by Maxwell's equations in combination with additional boundary conditions. Finally we discuss important solutions of Maxwell's equations that are necessary for the understanding of high-frequency behaviour of technical components like transmission lines and antennas.

Our discussion on electromagnetic theory is limited to fundamental aspects. More detailed treatments on this topic can be found in [1–6].

2.1 Electric and Magnetic Fields

In this section we introduce mathematical formulas and physical relations for the static case in order to get a visual understanding of electric and magnetic field quantities.

2.1.1 Electrostatic Fields

2.1.1.1 Field Strength and Voltage

Historically it has been known for a long time that electric charges are the origin of electrical phenomena. Charges produce forces upon each other. We distinguish between positive and negative charges. Charges of opposite signs attract each other whereas charges of same sign push each other away. The absolute value of the Coulomb force F_C between two point charges¹ Q₁ and Q₂, separated by a distance r, is given by

The direction of the Coulomb force is defined by a straight line through both charges (see Figure 2.1a and 2.1b). In the case of three or more charges we can add up the vectors of the forces by the principle of superposition (vector sum in Figure 2.1c).

Figure 2.1 Coulomb forces between (a) two positive charges, (b) two charges of opposite polarity and (c) three charges of equal polarity.

Charges exist in discrete values (integer values of elementary charge e = 1.602 · 10⁻¹⁹ C). However, in practical (macroscopic) engineering configurations the number of charges is so huge that we can treat charge as a continuous quantity.

The abovementioned point-charges are assumed to be located at a single point in space. In the case of a continuous distribution of charges in space we introduce a new term called electric charge density eta

. The total charge Q in a certain volume V is then expressed as

Now we will replace the concept of direct forces between distant charges by a new concept of an electric field that emanates from a charge and leads to interaction with other charges. We look at the electric field strength images/c02_I0003.gif

of charge Q₁ at the point in space where a charge Q₂ is located. Let us assume a small positive test charge Q₂. The electric field images/c02_I0004.gif

is given by the force images/c02_I0005.gif

upon the test charge Q₂ divided by the test charge Q₂ itself. Hence, E₁ becomes independent of Q₂.

The vector of the electric field strength images/c02_I0007.gif

points in the direction of the force images/c02_I0008.gif

. Mathematically the electric field is simply the ratio of images/c02_I0009.gif

and Q₂. Interestingly we can interpret images/c02_I0010.gif

as a vector field that exists in the space surrounding charge Q₁, even if no test charge Q₂ is present. We can easily measure the electric field distribution of charge Q₁ by moving a test charge Q₂ around charge Q₁.

The distribution of the electric vector field of a positive charge Q₁—located at the origin of a spherical coordinate system—is directly derived from Coulomb's law in Equation 2.1 as

A vector field is conveniently visualized by field lines. Figure 2.2 shows field line representations of different charge configurations. Field lines represent direction and amplitude of the vector fields. The direction of the vector field is oriented tangentially to the field lines, and the density of field lines is a measure of the absolute value (denser field lines meaning higher field amplitude).

Figure 2.2 Electric field lines for (a) a positive point charge, (b) between two positive charges, (c) between two charges of opposite polarity and (d) in a parallel-plate capacitor (with fringing fields at the edges).

Looking at the field distributions in Figure 2.2 we notice that the electric field lines always² start at positive charges and end at negative ones. If only positive or negative charges are present the field lines extend to infinity. Therefore, we consider a positive charge as a source of the electric field and a negative charge as a sink.

Field lines of the electrostatic field are open field lines, that is they start at positive and end at negative charges. The electrostatic vector field is a curl-free or conservative field.

Consider a charge Q₂ that moves through the electric field images/c02_I0012.gif

of a stationary charge Q₁ from point images/c02_I0013.gif

. During movement along the path from images/c02_I0015.gif

the charge Q₂ experiences a force images/c02_I0017.gif

. From physics we know that work images/c02_I0018.gif

is done.

The dot product between force images/c02_I0020.gif

and differential path vector images/c02_I0021.gif

indicates that work is maximum for a path tangential to the field lines. A path perpendicular to the force yields zero work. In Equation 2.5 the constant charge Q₂ can be taken outside the integral sign. We interpret the integral over the electric field strength as a new electric quantity called voltage U.

The electric field is a vector field that is defined for any point in space images/c02_I0023.gif

. The voltage U is defined between two points images/c02_I0024.gif

and

and therefore it is an integral or circuit quantity. The voltage is no field quantity.

If we define point images/c02_I0026.gif

as a fixed reference point images/c02_I0027.gif

we derive a new field quantity very similar to the voltage. The electric potential ϕ is given by

The electric potential is a scalar field, that is, for any point in space images/c02_I0029.gif

is given by a single (scalar) value. In the electrostatic case the electric potential ϕ represents the electric voltage between the point images/c02_I0030.gif

and the reference point images/c02_I0031.gif

As discussed earlier the electrostatic field images/c02_I0032.gif

is a conservative field. From mathematics we know that conservative fields can be expressed as a gradient³ of a scalar field. Interestingly the scalar field that leads to the electric field is the negative value of the electric potential

Equation 2.8 shows the definition of the gradient operator in Cartesian coordinates. Definitions of the gradient operator in other coordinate systems are listed in Appendix A.1.

The gradient operator on the scalar electric potential results in an electric vector field. The gradient points in the direction of maximum (positive) change in the scalar potential function. The gradient is normal to an equipotential surface (i.e. surface with constant potential). This feature of the gradient is useful when optimization algorithms are considered: in order to find a local maximum of a function it is most efficient to proceed in the direction of maximum change.

The definition of the potential ϕ in Equation 2.7 is ambiguous. There is an infinite number of potential functions since the position of the reference point is not uniquely defined. However the electric field—defined by a force on a charge—is unique. Potentials with different reference points only differ in an additive constant. In Equation 2.8 we see that the electric field is calculated from partial derivatives of the scalar potential. Hence, an additional constant has no impact on the resulting value.

2.1.1.2 Polarization and Relative Permittivity

Up to now we have considered charges in free space. In the presence of non-conducting, dielectric materials new expressions can be conveniently defined.

Let us look at a parallel-plate capacitor. The capacitor consists of two ideally conducting plates and a separating dielectric material. In Figure 2.3a the capacitor is filled with vacuum. The charges on both plates are equal in magnitude but have opposite signs: Q+ on the upper plate and Q− on the lower plate. If we assume the lateral extensions of the plates are much larger than the spacing between them we can neglect the fringing fields at the edges and regard the electric field as homogeneous. For the air-filled parallel-plate capacitor in Figure 2.3a we can easily calculate the voltage between the plates from Equation 2.6 as U₀ = E₀d, where d is the distance between the plates. If we now put an isolating material (dielectric) between the plates (Figure 2.3b), we observe that the new voltage U_M is reduced U_M < U₀. If we remove the dielectric medium the former voltage U₀ is restored. The process is reversible.

Figure 2.3 Understanding polarization in a dielectric medium: (a) air-filled parallel-plate capacitor with initial electric field E₀, (b) dielectric medium between plates gives rise to an opposing field E_P (c) reduced electric field strength E_M inside the dielectric medium, (d) dipoles in random order without external electric field and (e) dipoles aligned by external field.

In order to understand the macroscopic effect of reduced voltage when a dielectric is placed inside the capacitor, we will now take a look at the microscopic composition of dielectric materials. We first consider a material that consists of polar particles, that is the molecules represent electric dipoles (Figure 2.3d). Without the external field the dipoles are in random order. Adjacent charges cancel each other out. There is no macroscopic net effect.

If we switch on the external field E₀ the dipoles align with the field due to Coulomb forces (Figure 2.3e). In the inner part of the material adjacent charges cancel out. However, at the upper and lower surface of the dielectric material a charge density remains. The surface charge density gives rise to a secondary field E_P that is opposed to the external field. Inside the material we observe a diminished field E_M

This effect is known as polarization.⁴ The amplitude of polarization, that is the strength of the opposing field E_P depends on the microscopic structure of the material.

We have shown that materials with molecules that form electric dipoles can be polarized, that is the dipoles align with the field. It can be shown that even materials with non-dipole molecules can be polarized. Let us consider a simple model of a charge-neutral atom with a positive nucleus in the centre and surrounding negative electrons. The centre of the positive and negative charges is now located at the same point in space. If we put such a material into an external electric field E₀ the field exerts opposing forces upon positive and negative charges. This results in a displacement of the charges and induces a dipole moment. Hence we observe a polarizing effect too.

Let us go back now to our initial experiment where a dielectric medium inside the parallel-plate capacitor led to a reduced voltage U_M

In order to get rid of the description of microscopic effects we focus on the net macroscopic effect: the dielectric material reduces the voltage that we measure and this is as a result of the reduced field E_M in the medium. Therefore, we introduce a new term denoted as relative permittivity: the relative permittivity ε_r is the ratio between the unperturbed external field E₀ and the reduced field E_M due to polarization

The relative permittivity is a dimensionless quantity equal to or larger than one. Table 2.1 lists values for common engineering materials. Vacuum has a relative permittivity of one, for air the value is slightly higher (ε_r = 1.0006) [3] but can be considered to be one for nearly all engineering applications.

Table 2.1 Relative permittivity ε_r and loss tangent tanδ_ε of common dielectric materials in their typical frequency range

There are materials with anisotropic properties, that is the relative permittivity that we measure with our parallel-plate capacitor configuration depends on the orientation of the dielectric material in the capacitor. In this case we can describe the polarizing effect by a matrix rather than by a single value. In this book we will limit our discussion to isotropic materials.

Polarization and time-varying fields

We will close our discussion of microscopic phenomena by looking at time-varying fields. Let us connect our parallel-plate capacitor to a harmonic voltage source. The charges now switch polarity periodically. The dipoles of the dielectric material change their orientation with the same frequency as the field. Above a certain frequency the dipoles cannot follow the external field due to their inertia and the effect of polarization is reduced. Therefore, we expect a decrease of the relative permittivity with frequency.

A model for the frequency dependence ε_r(f) is given by Debye formulas where f is the frequency of the external field. A detailed discussion is beyond the scope of this text and can be found elsewhere [13]. Fortunately most RF engineering materials have nearly constant relative permittivity values in their technical frequency range of usage.

Another important effect that occurs when a time-varying field is applied is heating of the material. On a microscopic scale the molecules constantly align with the changing field and their movement produces thermal energy (polarization losses). We will see in Section 2.2.4 that this heat loss can be described macroscopically by a term denoted loss tangent. For the purpose of reference we include the loss tangent for important materials in Table 2.1 [8, 9].

2.1.1.3 Electric Flux Density

Using the permittivity and the electric field strength we define the electric flux density images/c02_I0037.gif

as a new vector. The letter D comes from the alternate notation of the electric flux density which is displacement.

For isotropic materials (ε_r is a scalar) the electric flux density images/c02_I0039.gif

and the electric field strength images/c02_I0040.gif

are proportional and point in the same direction. The displacement images/c02_I0041.gif

is an important vector, as we will see when we discuss Gauss' law in Section 2.2. The unit of images/c02_I0042.gif

is C/m² (Charge per area). Integrating the electric flux density images/c02_I0043.gif

over an area A gives us the electric flux Ψ_e

2.1.1.4 Electric Energy and Capacitance

Energy is a common and general concept in physics. In an electric field distribution energy can be stored. The electrostatic energy density is defined as

where

is the electric flux density and images/c02_I0047.gif

the electric field strength. The latter term in Equation 2.14 is valid for isotropic materials and most convenient for calculations later in this book. The unit of the energy density is Joule/m³.

If we integrate the energy density over a volume V we get the electric energy stored in that volume.

Let us assume that the electric field is created by a capacitor with a capacitance C and a voltage U between its electrodes. In this case the energy can be written by using voltage and capacitance as

Hence, a capacitor is a device which stores electric energy and its capacitance is a measure of how much energy can be stored for a given voltage.

Capacitance in general is defined as the ratio between charge on the electrodes and voltage between them.

For a parallel-plate capacitor with plate area A, plate distance d and dielectric material (relative permittivity ε_r) between the electrodes the capacitance is

2.1.2 Steady Electric Current and Magnetic Fields

2.1.2.1 Current Density, Power Density and Resistance

In the previous sections we discussed electrostatic fields, that is stationary fields from non-moving charges. Now we consider charges that move in space. We will see that moving charges give rise to magnetic fields.

Let us first introduce some useful terms for the description of moving charges. In Figure 2.4a we observe charges ΔQ that move through an area A for a given time interval Δt. The current I then gives the amount of charge ΔQ per time interval Δt.

Current—like voltage—is an integrated quantity (not a field quantity), because it depends on an area of observation.

Figure 2.4 (a) Conduction current I as a result of charges passing through a cross-section area A and (b) current density images/c02_I0297.gif .

3GPP	Third Generation Partnership Project
Al₂O₃	Alumina
Balun	Balanced-Unbalanced
CAD	Computer Aided Design
DC	Direct Current
DFT	Discrete Fourier Transform
DUT	Device Under Test
EM	ElectroMagnetic
EMC	ElectroMagnetic Compatibility
ESR	Equivalent Series Resistance
FDTD	Finite-Difference Time-Domain
FEM	Finite Element Method
FR4	Glass reinforced epoxy laminate
GaAs	Gallium arsenide
GPS	Global Positioning System
GSM	Global System for Mobile Communication
GTD	Geometrical Theory of Diffraction
GUI	Graphical User Interface
HPBW	Half Power Beam Width
ICNIRP	International Commission on Non-Ionizing Radiation Protection
IFA	Inverted-F Antenna
ISM	Industrial, Scientific, Medical
ITU	International Communications Union
LHCP	Left-Hand Circular Polarization
LHEP	Left-Hand Elliptical Polarization
LNA	Low-Noise Amplifier
LOS	Line of Sight
LTE	Long Term Evolution
LTI	Linear Time-Invariant
MIMO	Multiple-Input Multiple-Output
MMIC	Monolithic Microwave Integrated Circuits
MoM	Method Of Moments
NA	Network Analyser
NLOS	Non Line of Sight
PA	Power Amplifier
PCB	Printed Circuit Board
PEC	Perfect Electric Conductor
PML	Perfectly Matched Layer
PTFE	Polytetraflouroethylene
Radar	Radio Detection and Ranging
RCS	Radar Cross-Section
RF	Radio Frequency
RFID	Radio Frequency Identification
RHCP	Right-Hand Circular Polarization
RHEP	Right-Hand Elliptical Polarization
RMS	Root Mean Square
SAR	Specific Absorption Rate
SMA	SubMiniature Type A
SMD	Surface Mounted Device
TEM	Transversal Electromagnetic
UMTS	Universal Mobile Telecommunication System
UTD	Uniform Theory of Diffraction
UWB	Ultra-WideBand
VNA	Vector Network Analyser
VSWR	Voltage Standing Wave Ratio
WLAN	Wireless Local Area Network

A	Area (m²)
A_dB	Attenuation (dB)
	Magnetic vector potential (Tm)
A_eff	Effective antenna area (m²)
A	ABCD matrix (matrix elements have different units)
	Magnetic flux density (magnetic induction) (T; Tesla)
B	Bandwidth (Hz; Hertz)
BW	Bandwidth (angular frequency) (1/s)
c	Velocity of a wave (m/s)
C	Capacitance (F; Farad)
C(φ, ϑ)	Radiation pattern function (dimensionless)
C′	Capacitance per unit length (F/m)
D	Directivity (dimensionless)
	Electric flux density (C/m²)
	Electric field strength (V/m)
f	Frequency (Hz)
f_c	Cut-off frequency (Hz)
	Force (N; Newton)
	Coulomb Force (N)
	Lorentz Force (N)
G	Conductance (1/Ω = S)
G	Gain (dimensionless)
G	Green's function (1/m)
G′	Conductance per unit length (S/m)
	Magnetic field strength (A/m)
H	Hybrid matrix (matrix elements have different units)
I	Current (A; Ampere)
I	Identity matrix (dimensionless)
j	Imaginary unit (dimensionless)
	Electric current density (A/m²)
	Surface current density (A/m)
k	Coupling coefficient (dimensionless)
k	Wavenumber (1/m)
k_c	Cut-off wavenumber (1/m)
	Wave vector (1/m)
, L	Length (m)
L	Inductance (H; Henry)
L	Pathloss (dimensionless)
L′	Inductance per unit length (H/m)
p	Power density (W/m³)
P	Power (W; Watt)
P_antenna	Accepted power (W)
P_inc	Incoming power (W)
P_rad	Radiated power (W)
Q	Charge (C; Coulomb)
Q	Quality factor (dimensionless)
r	Radial coordinate (m)
R	Resistance (Ω)
R_DC	Resistance for steady currents (Ω)
R_ESR	Equivalent series resistance (Ω)
R_RF	Resistance for radio frequencies (Ω)
R_rad	Radiation resistance (Ω)
R′	Resistance per unit length (Ω/m)
s_kl	Scattering parameter (dimensionless)
S	Scattering matrix (dimensionless)
	Poynting vector (W/m²)
	Average value of Poynting vector (W/m²)
t	Time (s; second)
T	Period (s)
tanδ	Loss tangent (dimensionless)
U	Voltage (V; Volt)
	Velocity (m/s)
v_gr	Group velocity (m/s)
v_ph	Phase velocity (m/s)
V	Volume (m³)
w_e	Electric energy density (J/m³)
W_e	Electric energy (J; Joule)
w_m	Magnetic energy density (J/m³)
W_m	Magnetic energy (J)
x, y, z	Cartesian coordinates (m)
Y	Admittance (S; Siemens)
Y	Admittance matrix (S)
Z_A	Load impedance (Ω)
Z_F	Characteristic wave impedance (Ω)
Z_F0	Characteristic impedance of free space (Ω)
Z_in	Input impedance (Ω)
Z₀	Characteristic line impedance (Ω)
	Port reference impedance (Ω)
Z_0,cm	Common mode line impedance (Ω)
Z_0,diff	Differential mode line impedance (Ω)
Z_0e	Even mode line impedance (Ω)
Z_0o	Odd mode line impedance (Ω)
Z	Impedance matrix (Ω)

α	Attenuation coefficient (1/m)
β	Phase constant (1/m)
δ	Skin depth (m)
Δ	Laplace operator (1/m²)
ε = ε₀ε_r	Permittivity (As/(Vm))
ε_r	Relative permittivity (dimensionless)
ε_r,eff	Effective relative permittivity (dimensionless)
η	Radiation efficiency (dimensionless)
η_total	Total radiation efficiency (dimensionless)
γ	Propagation constant (1/m)
λ	Wavelength (m)
λ_W	Wavelength inside waveguide (m)
μ = μ₀μ_r	Permeability (Vs/(Am))
μ_r	Relative permeability (dimensionless)
∇	Nabla operator (1/m)
φ	Phase angle (rad)
φ	Azimuth angle (rad)
ϕ	Scalar electric potential (V)
φ₀	Initial phase (rad)
Ψ_e	Electric flux (C)
Ψ_m	Magnetic flux (Wb, Weber) (Vs)
ρ	Volume charge density (C/m³)
ρ_S	Surface charge density (C/m²)
σ	Conductivity (S/m; Siemens/m)
σ	Radar cross-section (m²)
ϑ	Elevation angle (rad)
ϑ_iB	Brewster angle (rad)
ϑ_ic	Critical angle (rad)
ω	Angular frequency (1/s)

μ₀	4π · 10⁻⁷ Vs/(Am)	Permeability of free space
ε₀	8.854 · 10⁻¹² As/(Vm)	Permittivity of free space
c₀	2.99792458 · 10⁸ m/s	Speed of light in vacuum
e	1.602 · 10⁻¹⁹ C	Elementary charge
Z_F0	120 π Ω ≈ 377 Ω	Characteristic impedance of free space

Cellular mobile telephony
GSM 900	Global System for Mobile Communication	880 … 960 MHz
GSM 1800	Global System for Mobile Communication	1.71 … 1.88 GHz
UMTS	Universal Mobile Telecommunications System	1.92 … 2.17 GHz
Tetra	Trunked radio	440 … 470 MHz
Wireless networks
WLAN	Wireless local area network	2.45 GHz, 5 GHz
Bluetooth	Short range radio	2.45 GHz
Navigation
GPS	Global Positioning System	1.2 GHz, 1.575 GHz
Identification
RFID	Radio-Frequency Identification	13.56 MHz, 868 MHz,
		2.45 GHz, 5 GHz
Radio broadcasting
FM	Analog broadcast transmitter network	87.5 … 108 MHz
DAB	Digital Audio Broadcasting	223 … 230 MHz
DVB-T	Digital Video Broadcasting - Terrestrial	470 … 790 MHz
DVB-S	Digital Video Broadcasting - Satellite	10.7 … 12.75 GHz
Radar applications
SRR	Automotive short range radar	24 GHz
ACC	Adaptive cruise control radar	77 GHz

Frequency range	Denomination
3 … 30 kHz	VLF - Very Low Frequency
30 … 300 kHz	LF - Low Frequency
300 kHz … 3 MHz	MF - Medium Frequency
3 … 30 MHz	HF - High Frequency
30 … 300 MHz	VHF - Very High Frequency
300 MHz … 3 GHz	UHF - Ultra High Frequency
3 … 30 GHz	SHF - Super High Frequency
30 … 300 GHz	EHF - Extremely High Frequency

13.553 … 13.567 MHz	26.957 … 27.283 MHz
40.66 … 40.70 MHz	433.05 … 434.79 MHz
2.4 … 2.5 GHz	5.725 … 5.875 GHz
24 … 24.25 GHz	61 … 61.5 GHz
122 … 123 GHz	244 … 246 GHz