## Circular Waveguide

The analysis of a circular waveguide shown in Figure 3.8 is best carried out using the cylindrical coordinate system. The principle of analysis is similar to that of the rectangular waveguide. The solutions of the longitudinal magnetic fields of the TE wave modes where Jn is the Bessel function of the order n. From the boundary condition Hr (r a) 0, it follows that dHrIdr(r a) 0, and further that jn (kca) 0, in which the apostrophe stands for derivative. From this we get where p 'nm is the m th...

## Oscillators

An oscillator is a circuit that transfers dc power to RF power 5-8 . The generated RF signal may be sinusoidal or distorted because of containing harmonic components. Important characteristics of an oscillator are its frequency and tuning range, output power, frequency stability, and spectral purity. An oscillator may be modeled either as a feedback circuit providing a nonzero output voltage for zero-input voltage or as a one-port circuit having negative resistance. In this text we use the...

## Fundamental Concepts of Antennas

That means that the properties of an antenna are similar both in the transmitting mode and in the receiving mode. For example, if a transmitting antenna radiates to certain directions, it can also receive from those directions the same radiation pattern applies for both cases. The reciprocity does not apply if nonreciprocal components such as ferrite devices or amplifiers are integrated into the antenna. Also, a link between two antennas is reciprocal The total...

## Propagation via Ionosphere

The highest layers of the Earth's atmosphere are called the ionosphere, because they contain plasma, which is ionized gas (free electrons and ions). The ionosphere extends from 60 to 1,000 km. Below 60 km the ionization is insignificant because the solar ionizing radiation is getting weaker due to absorption in the higher layers, and because recombination of plasma is fast due to high density of molecules. Above 1,000 km the density of molecules is too low for a significant phenomenon. It is...

## Polarization of a Plane Wave

Electromagnetic fields are vector quantities, which have a direction in space. The polarization of a plane wave refers to this orientation of the electric field vector, which may be a fixed orientation (a linear polarization) or may change with time (a circular or elliptical polarization). The electric field of a plane wave can be presented as a sum of two orthogonal components E (E ux + E 2 uy ) f jkz (2.72) where ux and Uy are the unit vectors in the x and y direction, respectively. In a...

## Interpretations of Maxwells Equations

Maxwell's equations may be presented in words as follows I The electric flux (surface integral of the electric flux) through a closed surface is equal to the total charge within the volume confined by the surface. II The magnetic flux (surface integral of the magnetic flux) through any closed surface is zero. III The line integral of the electric field along a closed contour is equal to the negative time derivative of the magnetic flux through the closed contour. IV The line integral of the...

## Dipole and Monopole Antennas

Wire antennas are popular at frequencies below 1 GHz. The dipole antenna is the most often used wire antenna. It is a straight wire, which is usually split in the middle so that it can be fed by a transmission line. It can be thought that a dipole shown in Figure 9.7 consists of current elements in a line. The far field is calculated by summing the fields produced by the current elements, that is, by integrating (9.20) where l is the length of the dipole. It can be assumed that the current...

## Filters

A resonator with two couplings passes through signals having frequencies near the resonance frequency in other words, it acts as a bandpass filter. A hollow metal waveguide acts as a highpass filter, because it has a cutoff frequency that depends on the dimensions. In general a filter is a two-port, which prevents propagation ofundesired signals while desired signals pass it. In an ideal case, there is no insertion loss in the passband, but the attenuation in the stopband is infinite. Depending...

## Maxwells Equations

Maxwell's equations relate the fields (E and H) and their sources (p and J) to each other. The electric field strength E and the magnetic flux density B may be considered the basic quantities, because they allow calculation of a force F, applied to a charge, q, moving at a velocity, v, in an electromagnetic field this is obtained using Lorentz's force law The electric flux density D and the magnetic field strength H take into account the presence of materials. The electric and magnetic...

Electromagnetic waves propagate in a vacuum with the speed of light, c 299,792,458 m s or about 3 X 108 m s. The electric and magnetic fields of a plane wave oscillate in phase and are perpendicular to each other and to the direction of propagation. The frequency of oscillation is f, and the wavelength is A c f Electromagnetic waves also may be considered to behave like particles of zero rest mass. The radiation consists of quanta, photons that have an energy of W hf where h 6.6256 X 1034 Js is...

## Transmission Lines and Waveguides

Transmission lines and waveguides carry signals and power between different devices and within them. We can form many kinds of components, such as directional couplers and filters, by connecting sections of transmission lines or waveguides (see Chapters 6 and 7). Usually, lines consisting of two or more conductors are called transmission lines, and lines or wave-guiding structures having a single metal tube or no conductors at all are called waveguides. However, the use of these terms is not...

## Reflection and Transmission at a Dielectric Interface

Let us consider a plane wave that is incident at a planar interface of two lossless media, as illustrated in Figure 2.7. The wave comes from medium 1, which is characterized by 1 and 1, to medium 2 with 2 and 2. The planar interface is at z 0. The angle of incidence is d1 and the propagation vector k 1 is in the xz-plane. Part E from the incident field is reflected at an angle d and part E2 is transmitted through the interface and leaves at an angle of 62. Figure 2.7 Reflection and transmission...

## S21 s3

Figure 5.6 Simplification of flow graphs (a) branches in series (b) branches in parallel (c) self-loop (d) duplication of node with duplication of feeding branch and (e) duplication of node with duplication of leaving branch. A self-loop connected to a node can be eliminated, if the gain of the branch(es) feeding the node is divided by 1 minus the gain of the loop, because from V2 S21 Vi + S22 V2 it follows A node can be duplicated by duplicating the branch(es) feeding the node, Figure 5.6(d),...

## Radio Waves as a Part of the Electromagnetic Spectrum

Electromagnetic waves cover a wide range of frequencies or wavelengths, as shown in Figure 1.1. The classification is based mainly on the sources of Figure 1.1 Electromagnetic spectrum. radiation. Boundaries of the ranges are not sharp, since different sources may produce waves in overlapping ranges of frequencies. The wavelengths of radio waves range from thousands of kilometers down to 0.1 mm. The frequency range is from a few hertz up to 3 THz. The waves having shorter wavelengths or higher...

Figure 9.5 shows a short current element at the origin. The element of a length dz along the z-axis carries an alternating sinusoidal current 10, which is constant along the element. This kind of current element is also called Figure 9.5 shows a short current element at the origin. The element of a length dz along the z-axis carries an alternating sinusoidal current 10, which is constant along the element. This kind of current element is also called Figure 9.5 Radiating current element. the...

The voltages and currents of a low-frequency circuit can be defined uniquely. In case of a high-frequency circuit this is not necessarily true. Voltages and currents can be defined uniquely only for a transmission line that carries a pure TEM mode. For example, the voltage and current of a rectangular waveguide can be defined in several ways. In such a line, the electric and magnetic fields and the power propagating in the waveguide are more fundamental quantities than the voltage and current....

## History of Radio Engineering from Maxwell to the Present

The Scottish physicist and mathematician James Clerk Maxwell (1831-1879) predicted the existence of electromagnetic waves. He combined Gauss' law for electric and magnetic fields, Ampere's law for magnetic fields, and the Faraday-Henry law of electromagnetic induction, and added displacement current to Ampere's law. He formulated a set of equations, which he published in 1864. These equations showed the interrelation of electric and magnetic fields. Maxwell proposed that visible light is formed...

## Microstrip Directional Couplers

Let us consider two parallel microstrip lines that are placed so close to each other that the fields of the lines couple to each other. Then a wave propagating in one line can excite a wave in the other line. The fields of a coupled line can be presented as a superposition of an even mode and an odd mode, shown in Figure 6.9. In case of the even mode the currents of the lines are equal and in the same direction in case of the odd mode they are equal but in the opposite directions. A coupled...

A radio transmitter must produce a signal that has enough power, has generally a very accurate frequency, and has a clean enough spectrum so that the transmitter does not disturb users ofother radio systems. Information to be transmitted, the baseband signal, is attached to a sinusoidal carrier signal by modulating the carrier amplitude, frequency, or phase either analogically or digitally (see Section 11.3). Low-power transmitters are usually based on a semiconductor device, a transistor or...

## Monolithic Microwave Circuits

Circuits consisting of microstrip lines, lumped passive elements (resistors, inductors, and capacitors), and semiconductor diodes and transistors may be integrated (connected without connectors) and be made very small. If components are soldered or bonded on a microstrip circuit, we call it a hybrid circuit. If a circuit is integrated directly on the surface of a semiconductor substrate, it is called a monolithic integrated circuit. Up to about 2 GHz the monolithic circuits are made on Si at...

## V1 c f

From the boundary conditions for the fields in a metal waveguide it follows that the transverse magnetic field cannot have a normal component at the boundary, that is, n H 0. It follows from 3.7 and 3.8 that dHz Idx 0, when x 0 or a, and dHz Idy 0, when y 0 or b. Also, the tangential component of the electric field must vanish at the boundary or n X E 0 Ex x, 0 0, Ex x, b 0, Ey 0, y 0, Ey a, y 0. It results from these boundary conditions that k i a nn and k2 b mn, where n 0, 1, 2, . . . and m...

## Power Dividers

The T-junctions shown in Figure 6.2 are simple power dividers. However, all the ports of a lossless three-port circuit cannot be matched. We can prove this by considering the properties of the scattering matrices. A passive, reciprocal, and matched three-port would have a scattering matrix, as As we stated in Chapter 5, the scattering matrix of a lossless circuit is unitary, from which it follows that S1212 S1312 1 S1212 S 2312 1 S1312 S 2312 1 Figure 6.2 T-junctions a a waveguide junction b a...

## Reflection from a Mismatched Load

In the following analysis we will use the transmission line model and use voltages and currents. In principle we could, of course, use electric and magnetic fields, but the advantage of using voltages and currents is that the characteristic impedance of a line is always by definition the ratio of the voltage and current, but not always directly the ratio of the electric and magnetic field see 3.51 . In impedance matching it is the characteristic impedance of the line that matters, not the wave...

## Propagation as a Ground Surface Wave

If the transmitting and receiving antennas are close in comparison to a wavelength to the ground, the wave propagates bound to the ground, as a Figure 10.16 Wave propagation via the ionosphere. Figure 10.16 Wave propagation via the ionosphere. surface wave. The electric field strength of the wave decreases rapidly as the distance from the surface increases. At low frequencies the attenuation of such a ground wave is small, and the wave can propagate beyond the horizon thousands of kilometers,...

## Reflector Antennas

Reflector antennas are used as high-gain, narrow-beam antennas in fixed radio links, satellite communication, radars, and radio astronomy. A parabolic reflector antenna is the most common of reflector antennas. Figure 9.26 a shows a parabolic antenna fed from the primary focus. The equation of the surface is where F is the focal length. The rays coming from the focal point are converted parallel by the reflector or vice versa. A more physical interpretation is that the fields radiated by the...

## Horn Antennas

An open waveguide end operates as a simple antenna. It has a broad, unsym-metrical beam and a rather large impedance mismatch. A much better antenna, a horn antenna, is obtained by widening the waveguide end, as shown in Figure 9.23. H-plane, E-plane, and pyramidal horns are fed from a rectangular waveguide. An H-plane horn is widened along the broad side of the waveguide, an E-plane horn along the narrow side. A pyramidal horn is broadened in both directions. The distribution of the aperture...

## Bending Refraction of Radio Waves in Troposphere

The refraction index n yje of the troposphere fluctuates over time and location. In normal conditions the refraction index decreases monotonically versus altitude, because the air density decreases. Because a phenomenon of this kind is a weak function of altitude, it causes slow bending of the ray. Fast changes in the refraction index cause scattering and reflections. Turbulence, where temperature or humidity differs strongly from those of the surroundings, gives rise to scattering. Reflections...

## Waveguide Directional Couplers

We may make a directional coupler by placing side by side two rectangular metal waveguides having coupling holes in the common wall 1 . Figure 6.7 shows a simple directional coupler having two holes d Ag 4 apart in the broad wall of the waveguides. Let us assume that the coupling factor of a single hole is Bf in the forward direction and By in the backward direction. Figure 6.7 Two-hole waveguide directional coupler. For example, if a wave is applied to port 1 and its field is equal to 1 in the...

## Scattering Matrix of a Directional Coupler

Let us next derive the scattering matrix of an ideal directional coupler. Because ports 1 and 3 as well ports 2 and 4 are isolated from each other, S13 S24 0. Let us assume that ports 1 and 2 are matched, or S11 S22 0. Due to the reciprocity Sj Sj . Thus, the scattering matrix is The scattering matrix of a lossless circuit is unitary. From rows 1 and 4 we get S14 S44 0 and from rows 2 and 3 S23 S33 0. Because S14 and S23 are nonzero, parameters S33 and S44 must be zero, and ports 3 and 4 must...

## Fields in Media

In the above equations, the permittivity e and permeability l represent the properties of the medium. A medium is homogeneous if its properties are constant, independent of location. An isotropic medium has the same properties in all directions. The properties of a linear medium are independent on field strength. In a vacuum, e e0 8.8542 x 10 F m and 0 10 H m. In other homogeneous media, e ere0 and xr 0, where the dielectric constant er , that is, the relative permittivity, and the relative...

## Design of Microwave Filters

At microwave frequencies we have two major problems in realizing the synthesized filters. First, good lumped elements do not exist instead we must use distributed elements. The frequency behavior of the distributed elements is more complicated than that of the lumped ideal ones, which makes the filter synthesis difficult. However, design of a narrow-band filter is easy, because over a narrow bandwidth many distributed elements may be modeled by ideal inductors and capacitors. Second, the filter...

## Environment and Propagation Mechanisms

The troposphere is the lowest part of the atmosphere, where all weather phenomena occur. It extends on the poles to about 9 km and on the equator to about 17 km. The troposphere is inhomogeneous and constantly changing. Temperature, pressure, humidity, and precipitation affect the propagation of radio waves. In the troposphere the radio waves attenuate, scatter, refract, and reflect the amplitude and phase of the received signal may fluctuate randomly due to multipath propagation the...

## Tropospheric Attenuation

At frequencies above a few gigahertz, the attenuation due to atmospheric absorption and scattering must be taken into account. This attenuation can be divided into two parts attenuation due to clear air and attenuation due to precipitation raindrops, hail, and snow flakes and fog. Attenuation of the clear air is mainly due to resonance states of oxygen O2 and water vapor H2O molecules. An energy quantum corresponding to the resonance frequency may change the rotational energy state of a gas...

## Microstrip Line

A microstrip line consists of a metal strip on one side and a ground plane on the other side of a substrate, as shown in Figure 3-13- The substrate is made of a low-loss dielectric material such as polytetrafluoroethylene Teflon , aluminum oxide alumina , or quartz. A pure TEM wave mode can propagate in a microstrip line only if all fields are in the same medium. Then the solution for the field can be derived from Laplace's equation. In a case where the nonstatic fields are in two different...