# MICROWAVE BY LIAO PDF

Microwave. Devices and Circuits. Third Edition. SAMUEL Y. LIAO. Professor of Electrical Engineering. California State University, Fresno. Iii PRENTICE HALL. Download Microwave Devices and Circuits By Samuel Y. Liao – An ideal text and a ready reference on the latest in microwave electronic technology, this book. Microwave Devices by Samuel y Liao - Ebook download as PDF File .pdf) or read book online.

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Microwave Devices and Circuits by Samuel Y Liao (3rd Edt).pdf - Ebook download as PDF File .pdf) or read book online. Microwave. Devices and Circuits Third Edition. SAMUEL Y. LIAO Professor of Electrical Engineering California State University, Fresno. Iii PRENTICE HALL. Microwave Devices and Circuits Samuel Liao - Free ebook download as PDF File .pdf), Text File .txt) or read book online for free. It can be used to read.

With no loss in generality.

The four statements can be proved by applying Faraday's law. The tangential components of magnetic field intensity are discontinuous at the boundary by an amount equal to the surface-current density on the boundary.

There are four basic rules for boundary conditions at the surface between two different materials: Ampere's law. The normal components of magnetic flux density are continuous across the boundary. The tangential components of electric field intensity are continuous across the boundary. The normal components of electric flux density are discontinuous at the boundary by an amount equal to the surface-charge density on the boundary.

Gauss's law. In medium 1 the fields are the sum of an incident wave plus a reflected wave. This situation is shown in Fig. The simplest reflection problem is that of a uniform plane wave normally incident on a plane boundary between two dielectric media with no surface-charge density and surface-current density.

So the boundary equations are E. In medium 2 there are transmitted waves: E Is in the Plane of Incidence. The incident power density minus the transmitted power density would yield the reflected power density as Oblique-incidence reflection. The plane of incidence is defined by the di- rection of propagation and the line normal to the boundary. Whenever a wave is incident obliquely on the boundary surface between two media.

Transmitted wave z Figure Reflection and transmission of oblique incidence. The terms horizontal and vertical po- larization refer to the phenomenon of waves from horizontal and vertical antennas.

The linearly polarized uniform plane waves with E lying in and H normal to the plane of incidence are im- pinging obliquely on a boundary between two lossless dielectric materials as shown in Fig.

The polarization of a wave is an extremely useful concept for computing elec- tromagnetic power flow. For example. The wave impedance can be expressed in terms of the reflection coefficient of. This is From Eq. As Fig. For guided waves in waveguides. This is well known as Snell's law. Eo sin 8. If medium 2 is free space and medium I is a nonmagnetic dielectric. The preceding two equations are known as Fresnel's formulas for E in the plane of incidence.

Then the transmission coefficient is given by cos 0. If His in the plane of incidence.. H Is in the Plane of Incidence. The electron-density distribution of each layer varies with the time of day. Figure shows the wave components of electromag- netic wave from a nondirectional antenna to a receiving station.. Although important in many communication systems. For vertical incidence. The reflected wave is reflected from the earth in such a way as to reach the receiver.. During the day the electron density N is approximately 10 12 electrons per cubic meter at an altitude between 90 and km.

This region extends from about 50 km above the earth to several earth radii and has different layers designated as C. The surface wave is a wave diffracted. The ionosphere is that region of the earth's atmosphere in which the con- stituent gases are ionized by solar radiation. Hz This means that a microwave of frequency Fer will be reflected back to the earth if the electron density is equal to or higher than the required maximum electron den- sity Nmax electrons per cubic meter.

The electron density determines the reflection and refraction of microwaves. The ground wave is further divided into a direct wave. The E and Flayers have a permanent existence. The sky wave reaches the receiving station after reflection from the iono- sphere. Energy radiated from the nondirectional antenna of Fig.

When power radiates from the transmitting antenna. Receiving antenna antenna Transmitter Receiver Gr G. It can easily be found from the stan-. It is the only wave considered in this book. This compo- nent is important at broadcast frequencies. The direct wave travels a nearly straight path from the transmitting antenna to the re- ceiving station.

Pr Figure Electromagnetic energy transmission between two antennas. The termfree space will be used to denote vacuum or any other medium having essentially the same charac- teristics as vacuum. This component must be considered in blind-landing systems in which ranges of less than 2 km are important. If the received power is expressed in terms of decibels. It is simply a consequence of the fact that. It should be noted that the free-space attenuation is entirely different from the dissipative attenuation of a medium such as atmosphere that absorbs energy from the wave.

It does not imply that a higher-frequency wave decreases in magni- tude more rapidly than a lower-frequency wave.. The factor 47TR 2 in Eq. Hence the intrinsic impedance for a lossless dielectric can be ex- pressed in terms of air.

The presence of a loss in the medium introduces wave dispersion by con- ductivity. Dispersion makes a general solution in the time domain impossible except by Fourier expansion methods.

There are three types of lossy media: Thus only solutions for the frequency domain or steady state will be given.

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The electric and magnetic wave equations in the frequency domain as shown in Eqs. Table lists the conductivities of materials. Expressed mathematically.. The energy transmitted by the wave traveling through the medium will decrease continuously as the wave propagates because ohmic losses are present..

Bakelite insulator approx. Vertical polarization. Seawater conductor 4 approx. Mica insulator approx. Cast iron conductor J06 approx. Sulfur insulator approx. I x Zinc conductor 1. Glass insulator approx. Graphite conductor approx. Paraffin insulator approx.. From Fig. Tellurium conductor 5 x approx. Carbon conductor 3 x approx. Mercury conductor Nichrome conductor Constantan conductor 2 x Silicon steel conductor 2 x German silver conductor 3 x Lead conductor 5 x Tin conductor 9 x Phosphor bronze conductor Brass conductor I.

Seawater is a good example. At some low frequencies the. The reflectivity of a good conductor for electric fields in horizontal polarization as shown in Eq. This result indicates that copper is a perfect reflector for electromagnetic waves. The magnitudes of reflectivity of copper for vertical and horizontal polariza- tions are computed by Eqs.

This phase relationship is shown in Fig. When the conductivity cannot be neglected.. If the loss tangent is very small-that is. Horizontal polarization A wave incident on the boundary at an angle equal to or greater than the critical angle will be totally reflected. For total reflection L l we set. For vertical polarization. Hence the total reflection occurs only if the wave propagates from a dense medium into a less dense medium.

The reflectivity of a lossy dielectric for electric fields in horizontal polarization as shown in Eq. This is because the value of sin. Therefore the reflectivity of a lossy dielectric in vertical polarization is given by f.

The reflections of electromagnetic waves by such lossy dielectric materials as sea- water. Figures to show. The conductivities a and relative dielectric constants E. October The metallic-film coatings on a plastic substrate are used in such applications as windshields on airplanes or auto- mobiles. Gen- erally the coated metallic film should have a high melting point. Liao [3]. If the film thickness is very large compared to the electron mean-free-path.

When the film thickness is on the order of the electron mean- free-path. The measured values of the refractive index n and the extinction index k of thin metallic-film coatings deposited in a vacuum [6] are tabulated in Table The extinction in- dex is related to the exponential decay of the wave as it passes through a medium. Most optical plastics are suitable as substrate materials for a dome window and for metallic-film applications.

Table lists the values of the refractive index n of several nonabsorbing plastic substrate materials in common use [6]. Figure Absorption and reflection of film coating on plastic substrate. From electro- magnetic plane wave theory in the far field. Absorption loss A. As described in Section When a radio wave propagates from a medium of high intrinsic impedance into a medium of low intrinsic impedance.

Good conductors. Microwave radiation attenuation by a metallic-film coating on substrate consists of three parts [7]: For very electrically thin film. The reflection loss R due to the multiple boundaries of the substrate glass coated with a metallic film can be analyzed by means of the energy-transmission theory see Eq. Reflection loss R. Thus the correction term of Eq. Using the multireflection and transmission theory. Figure shows light transmittance.

The reflection loss between the substrate and the air is small and. The incidence and the emergence media are dielectrics of refractive index no. At room temperature the properties of bulk gold are Conductivity: A gold film. Its surface. Thus the total attenuation loss is the same in both cases. Surface resistance. From the electromagnetic theory of luminous transmission in transparent media. Figure shows graphi- cally the microwave radiation attenuation versus the surface resistance of the gold- film coating Substituting the values of the surface resistances for gold films in Eq..

Microwave radiation attenuation. The surface resistance is decreased as the thickness of the gold film is increased. According to the Fuchs-Sondheimer theory. Figure shows surface resistances of gold film in ohms per square against the thicknesses of gold film from 10 to A Light transmittance T and light reflection loss R of a gold-film coating on a plastic glass are computed by using Eqs..

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Figure illustrates the relationship of light transmittance versus wave- length for a given surface resistance of gold film.. Optimum condition. The surface resistance of a metallic film decreases as the thickness of the film coating increases.

For the visible-light region. The results are presented graphically in Fig. The refractive index nz of the nonabsorbing plastic glass is taken as 1. Light transmittance. Then from the values of T and R absorption loss A and total attenuation L are calculated.

When the absorption loss in the substrate material is considered.

This relation- ship for the visible-light region is shown in Fig. The refractive index no of air or vacuum is unity. The data agree with Hawthorne's conclusion [ Figure Light transmittance T and light attenuation loss L versus wave- length A with film thickness t as parameter for gold film.

From Eq.. De- termine: Calculation of a Gold-Film Coating A gold film of 80 A is coated on a plastic substrate with a refractive index of 1. The gold-film surface resistance in ohms per square b.. The light transmittance T for red light of 0. The microwave attenuation in decibels c. At room temperature the properties of bulk silver and bulk copper are Silver Conductivity: The surface resistances of silver and copper films are computed by using Eqs.

Figure plots the surface resistances of silver and copper films. Liao [II]. The surface resistance. November Light transmittance T and light reflection loss R of silver-film and copper-film coatings are computed by using Eqs.. The refractive index no of air or vac- uum is unity. Fig- ure shows the microwave radiation attenuation versus the surface resistance of silver-film or copper-film coating. The refractive index n2 of the nonabsorbing plastic glass is taken as The values of the refractive index n and the extinction index k of the silver-film and copper-film coatings deposited in a vacuum for the light- frequency range are taken from Table The relationship is illustrated in Fig.

The light transmittance is increased as the surface re- sistance is increased. The results are illustrated in Figs. Silver film Surface resistance R. Optimum Condition. From the values of light transmittance T and light reflection loss R. Substitution of the values of the surface resis- tances of silver or copper films in Eq. Copper film 0.

L per cent ' ''-. Film thickness I- 40 Computation of a Copper-Film Coating A copper film of 60 A is coated on a plastic substrate with a refractive index of 1. The microwave attenuation in decibels c The copper-film surface resistance in ohms per square b. San Antonio. Design of a gold film on a glass substrate for maximum light transmittance and RF shielding effectiveness.

Electromagnetic shielding with transparent coated glass. IEEE Trans. Engineering Electromagnetics. Chapter 2.

MIT Press. Reftectivities of electromagnetic waves by seawater. The conductivity of thin metallic films according to the electron theory of metals. Light transmittance and microwave attenuation of a gold-film coating on a plastic substrate. Department of the Navy. March Problems in shielding electrical and electronic equipment. The light transmittance T is estimated from Fig.. Chapters 10 and McGraw-Hill Book Com- pany..

RF shielding effectiveness and light transmittance of copper or silver coat- ing on plastic substrate. The mean-free-path of electrons in metals.. China Lake. Electromagnetic Energy Transmission and Radiation. Advances in Physics.. Shielding theory and practice.

June A report for the Naval Weapons Center. Foundations for Microwave Engineering. Naval Air Development Center. Light transmittance and RF shielding effectiveness of a gold film on a glass substrate. August Neglect the reflections from the back face of the slab. The electric field of a plane wave propagating in free space is given in complex nota- tion by Where Ux and Uy are unit vectors in the x and y directions of a right-hand coordinate system.

If the electron density is assumed to be 10 14 electrons per cubic meter. Start from Eq. Electromagnetic Waves and Radiating Systems. Derive an expression for the relative dielectric constant of this medium. Marcel Dekker. Determine the pseudo-Brewster angle tfJ in terms of v. Verify Eq. The conductivity r: Determine the type of wave polarization linear.. It is assumed the medium is a nonmagnetic insulator. The reflection and refraction of microwave propagating in the ionosphere are deter- mined by the electron density in the ionosphere.

In which direction is the wave propagating? The wave is striking normally to and propagating through the slab. If the incident wave is not entirely vertically polarized there will be some reflection but the reflected wave will be entirely of horizontal polarization.

Express the magnetic field intensity H of the propagating wave. Calculate the average power flow per square meter in the direction of the propaga- tion.

Classical and Modern Theory and Appli- cations. At the Brewster angle there is no reflected wave when the incident wave is vertically polarized. Find the frequency of the propagating signal. Chapters 4. Determine the relationship between the phase and group velocities..

A shield is made of copper with a thickness of 1 mm. If a uniform plane wave is normally incident upon the copper shield. A radar transmitter has an output power of kW average. Problems 59 The propagation constant c. Repeat Problem for silver film. Write a complete FORTRAN program to compute the magnitudes of reflectivity in horizontal polarization against a grazing angle for seawater.

Determine the pseudo-Brewster angle if! Calculate the surface conductivity. The intrinsic impedance b. Use FI0.

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The frequency varies from 0. The electric intensity of the transmitted wave A uniform plane wave is incident normally from air onto the surface of seawater.

Bulk gold has a conductivity of 4. Copper has a conductivity of 5. The wavelength varies from to. Silver has a conductivity of 0. The phase velocity The electric intensity of the incident wave is x I0. Calculate the pseudo-Brewster angles for seawater. Refer to Problem for specifications. The electric intensity of the reflected wave b.

Write a complete FORTRAN program to compute the magnitudes of reflectivity in ver- tical polarization against a grazing angle of seawater. Print the outputs in three columns such as fre- quency GHz.

Dry ground has a conductivity of 5 x I0. Use F Hollerith format for character outputs. Print the outputs in column form with proper head-letters and units. Refer to Table for the values of n and k. Start from Eqs. The refractive index n of the nonab- sorbing plastic glass is I. Derive Eq. The refractive index n and extinction index k for aluminum film are tabulated in Table P Refer to Problem 2. The values of the refractive index n and the extinction index k of a gold-film coating on a nonabsorbing plastic glass de- posited in a vacuum are listed in Table Repeat Problem for a copper-film coating on a nonabsorbing plastic glass for the wavelengths from to The refractive index n of the nonabsorbing plastic glass is 1.

Repeat Problems and for dry ground. Repeat Problems and for dry sand. Repeat Problem for a silver-film coating on a nonabsorbing plastic glass for the wavelengths from to A with an increment of A each step. Repeat Problems and for concrete cement. J index n index k A. These inductances and capacitances are distributed along the length of a conductor. If a line is properly matched to its characteristic impedance at each terminal. Thus microwave transmission lines can be analyzed in terms of voltage.

If the spacing between the lines is smaller than the wavelength of the transmitted signal. Since the wave- length is short in comparison to the physical length of the line. This is not true for a long transmission line over a wide range of frequen- cies. The solution of Maxwell's equations involves three space variables in addition to the time variable. The distributed-circuit method. In ordinary circuit theory it is assumed that all impedance elements are lumped constants. Frequencies of operation are so high that inductances of short lengths of con- ductors and capacitances between short conductors and their surroundings cannot be neglected.

Based on uniformly distributed-circuit theory. By Kirchhoff's voltage law. GI 1ic i z. The parameters are expressed in their respective names per unit length.

The voltage and current on the line are the functions of both position z and time t. The phasors give the magnitudes and phases of the sinusoidal function at each position of z.

The instantaneous line voltage and current can be expressed as v z. The term involving e-jf3z shows a wave traveling in the positive z direction.

The only dif- ference is that the transmission-line equations are one-dimensional. V-eazejf3z In Eq. If a lossless transmission line used for microwave frequencies has an air dielectric and contains no ferromagnetic materials. That is. Line Characteristic Impedance and Propagation Constant A transmission line has the following parameters: From Eq Solution a Figure shows a transmission line terminated in an impedance Ze.

V-e'Ye Zo The ratio of the voltage to the current at the receiving end is the load impedance. It is usually more convenient to start solving the transmission-line problem from the re- ceiving rather than the sending end. If the load impedance is equal to the line characteristic impedance. For a loss- less line. Note that Jc is the phase angle between the incident and reflected voltages at the receiving end.

It is usually called the phase angle of the reflection coefficient. V-eyc Ze. The general solution of the reflection coefficient at any point on the line. The generalized reflection coefficient is defined as From Fig. For a lossy line. Irc I: It is evident that re will be zero and there will be no reflection from the re- ceiving end when the terminating impedance is equal to the characteristic impedance.

The reflection. Thus a terminating impedance that differs from the characteristic impedance will create a reflected wave traveling toward the source from the termina- tion.

Otherwise it is called an improperly terminated line. This can be expressed as 1. As described earlier. The letter T represents the transmission coefficient. According to the principle of conservation of energy. Load Figure Power transmission on a line. Ol Solution a. Compute a the reflection coefficient.

Then the voltage-wave equation can be expressed as This is called the equation of the voltage standing wave.. By doing so and substituting the proper values of f3z in the equation.

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The distance between any two successive maxima or minima is one-half wave- length.. The refractive index n of the nonabsorbing plastic glass is 1. The refractive index n and extinction index k for aluminum film are tabulated in Table P Derive Eq.

## Microwave Devices and Circuits Samuel Liao

The refractive index n of the nonabsorbing plastic glass is I. Refer to Table for the values of n and k. Repeat Problems and for dry sand. Start from Eqs. The values of the refractive index n and the extinction index k of a gold-film coating on a nonabsorbing plastic glass deposited in a vacuum are listed in Table Refer to Problem 2. Repeat Problems and for concrete cement. Repeat Problem for a copper-film coating on a nonabsorbing plastic glass for the wavelengths from to Thus microwave transmission lines can be analyzed in terms of voltage.

In ordinary circuit theory it is assumed that all impedance elements are lumped constants. This is not true for a long transmission line over a wide range of frequencies. The solution of Maxwell's equations involves three space variables in addition to the time variable. Frequencies of operation are so high that inductances of short lengths of conductors and capacitances between short conductors and their surroundings cannot be neglected.

If the spacing between the lines is smaller than the wavelength of the transmitted signal. Since the wavelength is short in comparison to the physical length of the line. If a line is properly matched to its characteristic impedance at each terminal.

## Microwave Devices by Samuel y Liao

These inductances and capacitances are distributed along the length of a conductor. The distributed-circuit method. By Kirchhoff's voltage law. The parameters are expressed in their respective names per unit length. Based on uniformly distributed-circuit theory. The phasors give the magnitudes and phases of the sinusoidal function at each position of z.

The instantaneous line voltage and current can be expressed as v z. The voltage and current on the line are the functions of both position z and time t. The term involving e-jf3z shows a wave traveling in the positive z direction. The only difference is that the transmission-line equations are one-dimensional. If a lossless transmission line used for microwave frequencies has an air dielectric and contains no ferromagnetic materials. Solution a. Line Characteristic Impedance and Propagation Constant A transmission line has the following parameters: That is.

It is usually more convenient to start solving the transmission-line problem from the receiving rather than the sending end. If the load impedance is equal to the line characteristic impedance. V-e'Ye Zo The ratio of the voltage to the current at the receiving end is the load impedance. Figure shows a transmission line terminated in an impedance Ze.

The generalized reflection coefficient is defined as From Fig. It is evident that re will be zero and there will be no reflection from the receiving end when the terminating impedance is equal to the characteristic impedance.

It is usually called the phase angle of the reflection coefficient. For a lossless line. V-eyc Ze. For a lossy line. The general solution of the reflection coefficient at any point on the line. Note that Jc is the phase angle between the incident and reflected voltages at the receiving end. Microwave Transmission Lines 68 The reflection coefficient. Irc I: As described earlier. The reflection.

Thus a terminating impedance that differs from the characteristic impedance will create a reflected wave traveling toward the source from the termination. According to the principle of conservation of energy.

The letter T represents the transmission coefficient. Otherwise it is called an improperly terminated line. This can be expressed as 1. Load Power transmission on a line. Compute a the reflection coefficient.

The maximum and minimum values of Eq. Then the voltage-wave equation can be expressed as This is called the equation of the voltage standing wave. By doing so and substituting the proper values of f3z in the equation. The distance between any two successive maxima or minima is one-half wavelength. A further study of Eq.. When the positive wave and the negative wave have equal amplitudes that is. The voltage and current may be expressed as real functions of time and space: The voltage nodes and current nodes are interlaced a quarter wavelength apart.

## Microwave Devices and Circuits, 3rd Edition

Voltage Max. The points of zero current are called the current nodes. On a lowloss line the ratio remains fairly constant. The standing-wave ratio cannot be defined on a lossy line because the standing-wave pattern changes markedly from one position to another.

When the standing-wave ratio is unity. That IS. It should be noted that since the standing-wave ratios of voltage and current are identical. The standing-wave ratio of a pure traveling wave is unity and that of a pure standing wave is infinite. The ratio of the maximum of the standing-wave pattern to the minimum is defined as the standing-wave ratio.

Since the reflected wave is defined as the product of an incident wave and its reflection coefficient. Figure shows the pure-standing-wave patterns of the phasor of Eqs. As a result of Eq. The curve in Fig. Zo ZQ z. Sending end -.

Zo e-Y'] Zo e-ye It is a tedious task to solve Eqs. These equations can be simplified by replacing the exponential factors with either hyperbolic functions or circular functions.

Zo e-yd] e- d. Zo eyz Z Z. Y e Then substituting these results in Eqs. Zo e-yd The line impedance at the sending end can also be found from Eq. Zs tanh yz Similarly. Rearrangement of Eq. ZoZs cosh yz. Zo tanh yz Z Zo cosh yz. Measure the sending-end impedance with the receiving end short-circuited and record the result: Determination of characteristic impedance..

A common procedure for determining the characteristic impedance and propagation constant of a given transmission line is to take two measurements: An examination of Eqs. Zmax or Zmin is repeated: Measure the sending-end impedance with the receiving end open-circuited and record the result: Since Vmax and Vmin are separated by a quarter-wavelength.

For every interval of a half-wavelength distance along the line. Microwave Transmission Lines 80 Chap. The line is assumed to be 2. The line is energized by a generator which has an output impedance of 50 n and an open-circuit output voltage of 30 V rms. Then 27T. The instantaneous power delivered to the load Solution a. The magnitude of the instantaneous load voltage c. The input impedance b. By simple rotation of the chart. A number of impedance charts have been designed to facilitate the graphic solution of transmission-line problems.

To see how a Smith chart works.

The reflection coefficient at any other location along a line as shown in Eq. JX and fc z. The solution of such problems is tedious and difficult because the accurate manipulation of numerous equations is necessary.

To simplify their solution. The chart is applicable to the analysis of a lossless line as well as a lossy line. The purpose of this section is to present the graphic solutions of transmission-line problems by using the Smith chart.

The most popular chart is that developed by Phillip H. Smith [l]. From Eqs. The Smith chart consists of a plot of the normalized impedance or admittance with the angle and magnitude of a generalized complex reflection coefficient in a unity circle.

Basically all the charts are derived from the fundamental relationships expressed in the transmission equations. All constant resistance circles are plotted in Fig. Figure Constant f circles and electrical-length radials f3d.

Equation also describes a family of circles. Figure Constant reactance x circles. The lower half of the diagram represents. I max. The constant r and constant x loci form two families of orthogonal circles in the chart.

For admittance the constant r circles become constant g circles. Since the normalized admittance y is a reciprocal of the normalized impedance z.

When a normalized impedance z is located on the chart. The normalized impedance or admittance is repeated for every half wavelength of distance. The characteristics of the Smith chart are summarized as follows: This is evident since 1. The horizontal radius to the left of the chart center corresponds to Vmin.

The horizontal radius to the right of the chart center corresponds to Vmax. The distances are given in wavelengths toward the generator and also toward the load. The magnitude of the reflection coefficient is related to the standing-wave ratio by the following expression: The use of the Smith chart is illustrated in the following examples. Move a distance from the point at 0. Read 0. Suppose that the characteristic impedance of the line Ro is 50 fl.

The location intersected by the circle at the right portion of the real axis indicates the SWR. Impedance Determination with Short-Circuit Minima Shift The location of a minimum instead of a maximum is usually specified because it can be determined more accurately.

Figure shows the diagram for the example. Make a standing-wave circle with the center at 1. When the line is shorted. Microwave Transmission Lines 88 Chap. Determine the load impedance. When the load is shorted. Shmtod Diagram for Example A line terminated in its characteristic impedance has a standing-wave ratio of unity and transmits a given power without reflection. The distance between two successive minima is one-half wavelength. Matching devices are necessary to flatten the line..

A "flat" line is nonresonant. In transmission-line problems matching means simply terminating the line in its characteristic impedance. In circuit theory. Usually the input impedance to the antenna itself is not equal to the characteristic impedance of the line. Matching a transmission line has a special meaning. A complete matched transmission-line system is shown in Fig.

Standing waves lead to increased losses and frequently cause the transmitter to malfunction. When the line is loaded. A common application of RF transmission lines is the one in which there is a feeder connection between a transmitter and an antenna. Move a distance of 0. Draw a line from this point to the center of the chart. This condition is sometimes referred to as a conjugate match.

At every point the impedances looking in opposite directions are conjugate. The stub must be located at that point on the line where the real part of the admittance. At audio frequencies an ironcored transformer is almost universally used as an impedance-matching device. Matching can be tried first on the load side to flatten the line. Occasionally an iron-cored transformer is also used at radio frequencies. Since the matching problems involve parallel connections on the transmission line.

Short-circuited sections are preferable to open-circuited ones because a good short circuit is easier to obtain than a good open circuit. In a practical transmission-line system. Because of the variable loads. If Zo is real. Compute the normalized load admittance and enter it on the Smith chart see Fig. Otherwise The stub length is then adjusted so that its susceptance just cancels out the susceptance of the line at the junction.

The characteristic impedance of the stub is fl. Find the stub position closest to the load and length so that a match is obtained. The admittances must be converted to normalized values for matching on the Smith chart.

The length of the fixed section is usually one-eighth. Double-stub devices consist of two short-circuited stubs connected in parallel with a fixed length between them. Microwave Transmission Lines 92 Chap. The stub that is nearest the load is used to adjust the susceptance and is located at a fixed wavelength from the constant conductance.

It is clear that the stub and the portion of the line from the load to the junction are in parallel. If an inductive stub is required. Then Eq. Since the characteristic impedance of the stub is different from that of the line. Then the admittance of the line at the second stub as shown in Fig. If the positions and lengths of the stubs are chosen properly.

Compute the normalized load impedance ze and enter it on the chart as shown in Fig. The first stub is placed at 0. What terminations are forbidden for matching the line by the double-stub device? Solution I. Determine the length of the short-circuited stubs when the match is achieved. At the junction Move ye for a distance of 0. Now Y11 must be on this spacing circle.

So it is hard to describe a definite procedure for solving the double-matching problems. Normally the solution of a double-stub-matching problem can be worked out backward from the load toward the generator. The flexible coaxial lines are available in different types. The dielectric material used in these coaxial lines is polyethylene. The loss per unit length for foam polyethylene is even appreciably less than the equivalent solid polyethylene. In some coaxial cables.

Particularly for the RG series. It can be seen from Fig. That is In quite a few practical matching problems. Their diameters vary from 0. Since that time many modifications and new designs for microwave connectors have been proposed and developed. The connector provides repeatable connections and has a very low voltage standing-wave ratio VSWR. The BNC can accept flexible cables with diameters of up to 6. It is now the most commonly used connector for frequencies under 1 GHz.

Its VSWR is extremely low. TNC is not shown. The function of the thread is to stop radiation at higher frequencies. The connector is seldom used above 24 GHz because of higher-order modes. Seven types of microwave coaxial connectors are described below see Fig. The main application of SMA connectors is on components for microwave systems.

Maury Microwave also has an MPC series available. The connector operates very well at frequencies up to about 4 GHz. The connector is manufactured by Sealectro Corporation and can accept flexible cables with diameters of up to 3. The standardization of coaxial connectors during World War II was mandatory for microwave operation to maintain a low reflection coefficient or a low voltage standingwave ratio VSWR.

The connector provides a coupling mechanism without male or female distinction and is the most repeatable connecting device used for very accurate SO-ohm measurement applications. Either the male or female end of this ohm connector can mate with the opposite type of SMA connector. The APC J f Type Microwa ve coaxial connectors.

S male a APC Transmission line calculator. Electronic Transmission Technology: A series published at intervals by the Kay Electric Co It is The reflection coefficient at a point 2 m away from the load Smith Charts-Their Development and Use. The VSWR on the line b. Microwave Transmission Lines 98 Chap. A lossless line has a characteristic impedance of 75 0 and is terminated in a load of. The reflection coefficient at the load b.

The power transmitted by the line Modern Transmission Line Theory and Applications. The transition coefficient at the load c. A lossless transmission line has a characteristic impedance of 50 0 and is terminated in a load of 0.

The propagation constant of the line is 0. The maximum voltage Vmax and minimum voltage Vmin on the line c. Microwave Engineering: Passive Circuits.

The magnitude of a voltage wave incident to the line is 20 V rms.

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An improved transmission line calculator. A transmission line has a characteristic impedance of 0 and is terminated in a load of The line is m long and is terminated in its characteristic impedance. A coaxial line with a solid polyethylene dielectric is to be used at a frequency of 3 GHz. Problems 99 0. One stub is at the load.

Show that the sendingend voltage is equal to the output voltage of the generator. Two possible solutions. A lossless line has a characteristic impedance of 50 0 and is loaded by The sending end is energized by a generator which has an open-circuit output voltage of Vg rms and an interval impedance equal to the characteristic impedance of the line. A lossless transmission line has a characteristic impedance of 0 and is operated at a frequency of 10 GHz.

Calculate the receiving-end power at the load. Its characteristic impedance Zo is 50 0 and its attenuation constant a is 0. The magnitude of the sending-end voltage and of the receiving-end voltage b. The sending-end power and the receiving-end power c. The sending-end impedance b. The magnitude of the receiving-end voltage c. The propagation constant of the wave along the line c.

The reflection coefficient at the load e. Determine the distance in centimeters from the load to the place where the stub should be located. Find the length of the stub in centimeters.

The power delivered to the load. Determine the magnitude of the receiving-end voltage. The wavelengths of the line It is proposed to use a short-circuited stub to match a pure resistor load to the line.

The signal frequency is 4 GHz. Find the sending-end impedance. The line is assumed to be 2i wavelengths long. The velocity factor which is defined as the ratio of phase velocity over the velocity of light in free space is 0.

The characteristic impedance of the line in both rectangular form and polar form b. The line is energized by a generator which has an open-circuit output voltage of 20 V rms and output impedance of 75 0. The reactance which. A lossless transmission line is terminated in an open circuit. The sending-end impedance if the line is assumed a quarter-wavelength long A lossless transmission line has a characteristic impedance of l 00 0 and is terminated in a load of 75 0.

The line is energized by a generator of 20 V rms with an internal resistance of 50 0.

The observed standing-wave ratio on the line is 5. A quarter-wave lossless line has a characteristic impedance of 50 0 and is terminated in a load of 0. The frequency is tuned at 3 GHz. Determine the lengths in wavelength of the two short-circuited stubs when a match is achieved. A generator. The line is 0. Locate and crosshatch the forbidden region of the normalized admittance for possible match.

Microwave Transmission Lines One stub is located at the load. The reactances contributed by the stub b. The characteristic resistances of the lossless line and stubs are fl. The lengths of the two shorted double-stub tuners [Note: There are two sets of solutions. The stub may be placed at a location closest to the antenna.

Determine the load Ze. Find the magnitude and the phase of the current flowing through the end of the stub. A single-stub tuner is to match a lossless line of fl to a load of The frequency is 3 GHz.

A lossless transmission line connected to a TV set has a characteristic impedance of fl. The problem is to design a shorted stub with the same characteristic impedance to match the antenna to the line. Determine the susceptance contributed by the stub. Find the lengths in A of two shorted stubs.

Determine the magnitude and the phase of the voltage across the stub location. The load current is measured to be 2 A. A lossless transmission line has a characteristic impedance of fl and is terminated by an impedance Z1. A double-stub matching line is shown in Fig. The reception is assumed to be Channel 83 at a frequency of 0.

Find the distance in centimeters between the antenna and the point where the stub is placed. Determine the length in meters of the short-circuited stub. Find the distance in meters from the load to the tuning stub. The observed standing-wave ratio on the line is 6.

Calculate the length in centimeters of the stub. The standing-wave ratio along the line is 2. Find l' 1 and d which provide a proper match. To match the load to the line. Find the load impedance. A matched transmission line is shown in Fig. Determine the minimum distance between the load and matching section. A lossless transmission line has a characteristic impedance Zo of 0 and is loaded by an unknown impedance.

With the line and load properly matched determine the VSWR on the section of line between the stubs. Problems A lossless transmission line has a characteristic impedance of 0 and is loaded by an unknown impedance. In waveguides the electric and magnetic fields are confined to the space within the guides. Thus no power is lost through radiation. At frequency range X band from 8. The dominant mode in a particular guide is the mode having the lowest cutofffrequency.

It is possible to propagate several modes of electromagnetic waves within a waveguide. It is advisable to choose the dimensions of a guide in such a way that. Otherwise the electromagnetic energy with a frequency below the cutoff frequency for that particular mode will be attenuated to a negligible value in a relatively short distance. These modes correspond to solutions of Maxwell's equations for particular waveguides. The process of solving the waveguide problems may involve three steps: Varian and S.

In R. Kompfner invented the helix-type traveling-wave tube TWT. In the past two decades, however, microwave solid-state devices—such as tunnel diodes, Gunn diodes, transferred electron devices TEDs , and avalanche transit-time devices have been developed to perform these functions. The conception and subsequent development of TEDs and avalanche transit-time devices were among the outstanding technical achievements.

Rid- ley and T. Watkins in and C. Hilsum in independently predicted that the transferred electron effect would occur in GaAs gallium arsenide.

In J. Figure shows a typical microwave system. In order to design a microwave system and conduct a proper test of it, an ade- quate knowledge of the components involved is essential. Besides microwave devices, the text therefore describes microwave components, such as resonators, cav- ities, microstrip lines, hybrids, and microwave integrated circuits. The meter-kilogram-second units the International System of Units are used throughout unless otherwise indicated. Table lists the most commonly used MKS units.

The physical constants commonly used in the text are listed in Table Many engineering computations use the absolute temperature in degrees Kelvin, and therefore a temperature conversion is necessary to convert the temperatures from either centigrade or Fahrenheit to Kelvin scale. It can be seen that From Eq. The unit of work or energy is called the electron volt eV , which means that if an electron falls through a potential of one volt, its kinetic energy will increase 1 eV.

Equation 1- is valid in regions in which there is space charge as well as in regions that are free of charge. The direction of the force is perpendicular to the plane of both v and B. This type of magnetic force is analogous to the problem of a mass tied to a rope and twirled around with constant velocity.

The force in the rope remains constant in magnitude and is always directed toward the center of the circle and thus is perpen- dicular to the motion. At any point on the circle the outward centrifugal force is equal to the pulling force. This means that faster-moving electrons or particles traverse larger circles in the same time that a slower-moving particle moves in a smaller circle. This very important result is the operating basis of such microwave devices as magnetic-focusing apparatus.

In microwave devices and circuits, however, only one dimension is involved in most cases. So the equations of motion become simple and can easily be solved. An ex- ample may show how to solve some of the preceding equations. Example Electron Motion in an Electromagnetic Field The inner cylinder of radius a is the cathode and the outer shell with radius b is the anode.

Solution 1. Write the equations of motion for electrons in cylindrical coordinates. From b Application of the boundary conditions: The cutoff voltage is e a2 2 V0. Van Nostrand Company, Princeton, N. Prentice- Hall, Inc. Determine which electrode the electron will strike. Compute the kinetic energy of the electron in electronvolts eV when it strikes the electrode. A circular cavity is constructed of a center conductor and an outer conductor.

The in- ner center conductor has a radius of 1 cm and is grounded. The electrons are emit- ted from the cathode at the center conductor and move toward the anode at the outer conductor. Here electromagnetic plane waves are described in detail. The principles of electromagnetic plane waves are based on the relationships between electricity and magnetism.

A plane wave has a plane front, a cylindrical wave has a cylindrical front, and a spherical wave has a spherical front. The front of a wave is sometimes referred to as an equiphase surface.

This type of wave is known as the transverse electromagnetic TEM wave. If only the transverse electric wave exists, the wave is called TE-mode wave. That means there is no component of the electric wave in the direction of propagation.

In TM modes only the transverse magnetic wave exists. Alternatively, J0 is often the current on a surface or in a region [1]. In most cases, however, the current density source J0 may not exist. H J. V X H Substitution of Eq. Re represents the real part of the complex power, and the asterisk indicates the complex conjugate. The left-hand side of Eq.

Furthermore, let Pi,. A uniform plane wave is a wave whose magnitude and phase are both constant. A spherical wave in free space is a uniform plane wave as observed at a far distance. Its equi- phase surfaces are concentric spheres, expanding as the wave travels outward from the source, and its magnitude is constant.

Electromagnetic waves in free space are typical uniform plane waves. Thus a uniform plane wave is a transverse electromag- netic wave or a TEM wave. A nonuniform plane wave is a wave whose amplitude not phase may vary within a plane normal to the direction of propagation. Figure shows uniform electric and magnetic plane waves in rectangular coordinates.

There are four basic rules for boundary conditions at the surface between two different materials: The normal components of electric flux density are discontinuous at the boundary by an amount equal to the surface-charge density on the boundary.

The simplest reflection problem is that of a uniform plane wave normally incident on a plane boundary between two dielectric media with no surface-charge density and surface-current density.

This situation is shown in Fig. In medium 2 there are transmitted waves: E Is in the Plane of Incidence. The linearly polarized uniform plane waves with E lying in and H normal to the plane of incidence are im- pinging obliquely on a boundary between two lossless dielectric materials as shown in Fig.The voltage and current may be expressed as real functions of time and space: The light transmittance T is estimated from Fig.

November This phase relationship is shown in Fig. The connector is manufactured by Sealectro Corporation and can accept flexible cables with diameters of up to 3. Grace in bioengineering, Kathy in electrical engineering, Gary in electronics engineering, and Jeannie in teachers education, for their valuable collective contributions. Commonly used microwave junctions include such waveguide tees as the E-. Besides microwave devices, the text therefore describes microwave components, such as resonators, cav- ities, microstrip lines, hybrids, and microwave integrated circuits.

The distances are given in wavelengths toward the generator and also toward the load. H Is in the Plane of Incidence.

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