MICROVE DEVICES

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Book:	MICROVE DEVICES

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Date:	Wednesday, 4 June 2025, 9:02 PM

Description

To equip students with knowledge on microwaves communication and devices and their applications.

1. Introduction
2. Transmission Lines
3. Further Transmission Lines Characteristics
- 3.1. Impedance Matching
- 3.2. Standing Wave and Standing Wave Ratio
4. Microwave Waveguides

1. Introduction

Topic 1

Introduction

Microwave essentially means very short wave. Microwaves are a form of energy in the electromagnetic (EM) spectrum. The EM spectrum runs from DC voltage to light and beyond as shown in figure 1.0.1.

Figure 1.0.1: Electromagnetic spectrum

The microwave frequency spectrum is usually taken to extended form 1 GHZ to 30 GHZ.This corresponds to wavelengths from 1 cm to 30 cm. The main reason why we have to go in for microwave frequency for communication is that lower frequency band (Radio Frequency Spectrum) became congested and demand for point to point communication continued to increase.

Microwaves being EM waves, have two components: the electrical (red) and the mag netic (blue) (Fig.1.0.2). The two components travel perpendicular to each other. The magnetic component allows us to use magnets and ferrite materials to affect wave behavior.

Microwaves have a wide rage of applications depending on their frequencies as shown in 1

Figure 1.0.2: Electromagnetic Wave

figure 1.0.3.

Figure 1.0.3: Applications of Microwaves depending on their frequencies

1.1 Microwave Devices

In the late 1930s it became evident that as the wavelength approached the physical di mensions of the vacuum tubes, the electron transit angle, interelectrode capacitance, and lead inductance appeared to limit the operation of vacuum tubes in microwave frequencies.

In 1935 A. A. Heil and 0. Heil suggested that microwave voltages be generated by using transit-time effects together with lumped tuned circuits. In 1939 W. C. Hahn and G.

F. Metcalf proposed a theory of velocity modulation for microwave tubes. Four months later R. H. Varian and S. F. Varian described a two-cavity klystron amplifier and oscillator by using velocity modulation.

In 1944 R. Kompfner invented the helix-type traveling-wave tube (TWT). Ever since then the concept of microwave tubes has deviated from that of conventional vacuum tubes as a result of the application of new principles in the amplification and generation of microwave energy.

Historically microwave generation and amplification were accomplished by means of velocity-modulation theory. In the past two decades, however, microwave solid-state devices-such as tunnel diodes, Gunn diodes, transferred electron devices (TEDs), and ava lanche transit-time devices have been developed to perform these functions. The concep tion and subsequent development of TEDs and avalanche transit-time devices were among the outstanding technical achievements.

B. K. Ridley and T. B. Watkins in 1961 and C. Hilsum in 1962 independently predicted that the transferred electron effect would occur in GaAs (gallium arsenide). In 1963 J. B. Gunn reported his "Gunn effect." The common characteristic of all microwave solidstate devices is the negative resistance that can be used for microwave oscillation and amplific ation.

The progress of TEDs and avalanche transit-time devices has been so swift that today they are firmly established as one of the most important classes of microwave solid-state devices.

1.2 Microwave Systems

A microwave system normally consists of a transmitter subsystem, including a microwave oscillator, waveguides, and a transmitting antenna, and a receiver subsystem that includes a receiving antenna, transmission line or waveguide, a microwave amplifier, and a receiver. Figure 1.2.1 shows a typical microwave system.

In order to design a microwave system and conduct a proper test of it, an adequate knowledge of the components involved is essential. Besides microwave devices, the text therefore describes microwave components, such as resonators, cavities, microstrip lines, hybrids, and microwave integrated circuits.

Figure 1.2.1: Microwave System

1.3 Microwaves Unit of Measure

Microwave measures can be expressed in different units, such as the CGS (centimeter gram-second) unit, MKS (meter-kilogram-second) unit, or another unit. The meter-kilogram second units (the International System of Units) are used throughout unless otherwise in dicated.

Figure 1.3.1: Microwave Units of measure

2. Transmission Lines

Topic 2

Transmission Lines

Conventional two-conductor transmission lines are commonly used for transmitting mi crowave energy. If a line is properly matched to its characteristic impedance at each ter minal, its efficiency can reach a maximum.

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.

Frequencies of operation are so high that inductances of short lengths of conductors and capacitances between short conductors and their surroundings cannot be neglected. These inductances and capacitances are distributed along the length of a conductor, and their ef fects combine at each point of the conductor.

Since the wavelength is short in comparison to the physical length of the line, distributed parameters cannot be represented accurately by means of a lumped-parameter equivalent circuit.

Thus microwave transmission lines can be analyzed in terms of voltage, current, and im pedance only by the distributed-circuit theory. If the spacing between the lines is smaller than the wavelength of the transmitted signal, the transmission line must be analyzed as a waveguide.

2.1 Transmission Line Equations and Solution

2.1.1 Transmission Line Equation

A transmission line can be analyzed either by the solution of Maxwell’s field equations or by the methods of distributed-circuit theory. The solution of Maxwell’s equations involves three space variables in addition to the time variable. The distributed-circuit method, how ever, involves only one space variable in addition to the time variable.In this section the latter method is used to analyze a transmission line in terms of the voltage, current, imped ance, and power along the line.

Based on uniformly distributed-circuit theory, the schematic circuit of a conventional two conductor transmission line with constant parameters R, L, G, and C is shown in figure 2.1.1.

Figure 2.1.1: Elementary section of a transmission line

The parameters are expressed in their respective names per unit length, and the wave propagation is assumed in the positive z direction.

Using Kirchhoff’s voltage law, the summation of the voltage drops around the central loop is given by

∂t+v(z,t) + ∂ v(z,t)

v(z,t) = i(z,t)R∆z+L∆z∂ i(z,t)

∂ z∆z (2.1.1)

Rearranging this equation, dividing it by ∆z, and then omitting the argument (z, t), we

obtain

−∂ v∂ z= Ri+L∂ i∂t(2.1.2)

Using Kirchhoff’s current law, the summation of the currents at point B in figure 2.1.1 can 6

be expressed as

i(z,t) = v(z+∆z,t)G∆z+C∆z∂ v(z+∆z,t)

∂t+i(z+∆z,t) (2.1.3)

v(z,t) + ∂ v(z,t) ∂t∆z

G∆z+C∆z∂∂t v(z,t) + ∂ v(z,t)

∂t∆z

+i(z,t) + ∂ i(z,t)

∂ z∆z

By rearranging the above current equation, dividing it by ∆z, omitting (z, t), and assuming

∆z equal to zero, we have

−∂ i∂ z= Gv+C∂ v∂t(2.1.4)

Differentiating equation 2.1.2 with respect to z and 2.1.4 with respect to t and combining the results, the final transmission-line equation in voltage form is

−∂2v

∂ z2= RGv+ (RC +LG)∂ v∂t+LC∂2v

∂t2(2.1.5)

Also, by differentiating equation 2.1.2 with respect to t and 2.1.4 with respect to z and combining the results, the final transmission-line equation in current form is

∂ z2= RGi+ (RC +LG)∂ i∂t+LC∂2i

−∂2i

∂t2(2.1.6)

All these transmission-line equations are applicable to the general transient solution. The voltage and current on the line are the functions of both position z and time t.

The instantaneous line voltage and current can be expressed as

v(z,t) = Re V(z)ejωt(2.1.7)

i(z,t) = Re I(z)ejωt(2.1.8)

The factors V(z) and V(z) are complex quantities of the sinusoidal functions of position z on the line and are known as phasors. The phasors give the magnitudes and phases of the sinusoidal function at each position of z, and they are expressed as

V(z) = V+e−γz +V−eγz(2.1.9)

I(z) = I+e−γz +I−eγz(2.1.10)

where γ = α + jβ is the propagation Constant, V+ and I+ indicate amplitudes in the pos itive z direction, V− and I− complex amplitudes in the negative z direction, α is the atten

uation constant in nepers per unit length, and β is the phase constant in radians per unit length.

Substituting jω for ∂∂tin equations 2.1.2, 2.1.4, 2.1.5 and 2.1.6 and divide each by ejωt, the transmission-line equations in phasor form of the frequency domain becomes

dz= −ZI (2.1.11)

dz= −YV (2.1.12)

d2V

dz2= γV (2.1.13)

d2I

dz2= γI (2.1.14)

where the following substitution have been made

Z = R+ jωL (ohms per unit length) (2.1.15)

Y = G+ jωC (mhos per unit length) (2.1.16)

γ =√ZY = α +β j (Propagation constant) (2.1.17)

For a lossless line, R = G = 0, and the transmission-line equations are expressed as

dz= −jωLI (2.1.18)

dz= −jωCV (2.1.19)

d2V

dz2= −ω2LCV (2.1.20)

d2I

dz2= −ω2LCI (2.1.21)

NB: Equations 4.2.5 and 4.2.6 for a transmission line are similar to equations of the elec tric and magnetic waves, respectively. The only difference is that the transmission-line equations are one-dimensional.

2.1.2 Solutions of the Transmission Lines

One possible solution for equation 4.2.5 is

V(z) = V+e−γz +V−eγz = V+e−αze−jβz +V−eαzejβz(2.1.22) 8

The term involving e−jβzshows a wave traveling in the positive z direction, and the term with the factor ejβzis a wave going in the negative z direction. The quantity βz is called the electrical length of the line and is measured in radians.

Similarly one possible solution for equation 4.2.6 is

I(z) = YoV+e−γz −V−eγz = Yo V+e−αze−jβz −V−eαzejβz (2.1.23)

In equation 4.2.7, the characteristic impedance of the line is defined as Zo =1Yo≡rZY=sR+ jωL

G+ jωC= Ro ± jXo (2.1.24)

At microwave frequencies R <<< ωL and G <<< ωC. By using the binomial expan sion, the propagation constant can be expressed as

γ =√ZY =p(R+ jωL)(G+ jωC)

(jω)2LC

1+R jωL

1+G jωC

= jω√LC 1+12R jωL

1+12G jωC

= jω√LC 1+12 R

jωL+G

=12

L+G

rL C

jωC

+ jω√LC

From the above equations the attenuation and phase constants are, respectively, given by

α =12

L+G

rL C

(2.1.25)

and

β = jω√LC (2.1.26)

Similarly, the characteristic impedance is found to be Zo =1Yo≡rZY=sR+ jωL

G+ jωCu

C(2.1.27)

From equation 2.1.26, the phase velocity is

νp =ωβ=1

√LC(2.1.28)

The product of LC is independent of the size and separation of the conductors and depends only on the permeability µ, and permittivity of ε of the insulating medium.

If a lossless transmission line used for microwave frequencies has an air dielectric and contains no ferromagnetic materials, free-space parameters can be assumed. Thus the numerical value of equation 2.1.28 for air-insulated conductors is approximately equal to the velocity of light in vacuum. That is

νp =ωβ=1

√LC=1

√µoεo= c = 3.0×108 m/s (2.1.29)

When the dielectric of a lossy microwave transmission line is not air, the phase velocity is smaller than the velocity of light in vacuum and is given by

νε =1

√µε=c

√µrεr(2.1.30)

In general, the relative phase velocity factor can be defined as

Velocity Factor =actual phase velocity

velocity of light in a vacuum

νr =νεc=1

√µrεr(2.1.31)

A low-loss transmission line filled only with dielectric medium, such as a coaxial line with solid dielectric between conductors, has a velocity factor on the order of about 0.65.

Example:

A transmission line has the following parameters;R = 2 Ω/m, G = 0.5 mhos/m, f = 1 GHz, L = 8 nH/m and C = 0.23 pF. Calculate: (a) the characteristic impedance; (b) the propagation constant.

Solution

(a) the characteristic impedance

Zo =

R+ jωL

G+ jωC= 179.44+ j26.50 10

(b) the propagation constant

γ =p(R+ jωL)(G+ jωC) = 0.051+ j0.273

2.2 Reflection and Transmission Coefficients

2.2.1 Reflection Coefficient

Figure 2.2.1 shows a transmission line terminated in an impedance Zl. It is usually more convenient to start solving the transmission-line problem from the receiving rather than the sending end, since the voltage-to-current relationship at the load point is fixed by the load impedance. The incident voltage and current waves traveling along the transmission line

Figure 2.2.1: Transmission line terminated in a load impedance.

are given by

V(z) = V+e−γz +V−eγz(2.2.1)

I(z) = I+e−γz +I−eγz(2.2.2)

in which the current wave can be expressed in terms of the voltage by I(z) = V+

Zoe−γz −V+

Zoeγz(2.2.3)

If the line has a length of l, the voltage and current at the receiving end become

Vl = V+e−γl +V−eγl(2.2.4)

Zoe−γl −V+

Il =V+

Zoeγl(2.2.5) 11

The ratio of the voltage to the current at the receiving end is the load impedance. i.e. Zl =VlIl= ZoV+e−γl +V−eγl

V+e−γl −V+eγl(2.2.6)

The reflection coefficient Γ, is defined as

Reflection Coefficient(Γ) = reflected voltage or current

Incident voltage or current

Γ =VrVi=−Ir

Ii(2.2.7)

Solving equation 2.2.6 for the ratio of the reflected voltage at the receiving end, which is V−eγl, to the incident voltage at the receiving end, which is V+eγl, the result is the reflection coefficient at the receiving end:

Γ =V−e−γl

V+e−γl=Zl −Zo

Zl +Zo(2.2.8)

If the load impedance and/or the characteristic impedance are complex quantities, as is usually the case, the reflection coefficient is generally a complex quantity that can be expressed as

Γ = |Γ|eiθl(2.2.9)

where |Γ| is the magnitude and not greater than unity. i.e. |Γ| ≤ 1. θlis the phase angle between the incident and reflected voltages at the receiving end. It is usually called the phase angle of the reflection coefficient.

2.2.2 Transmission Coefficient

A transmission line terminated in its characteristic impedance Zo is called a properly ter minated line. Otherwise it is called an improperly terminated line. As described earlier, there is a reflection coefficient Γ at any point along an improperly terminated line. Ac cording to the principle of conservation of energy, the incident power minus the reflected power must be equal to the power transmitted to the load. This can be expressed as

1−Γ2l =Zo

ZlT2(2.2.10)

Where T is the transmission coefficient defined as

Transmission Coefficient(T) = Transmitted voltage or current

Incident voltage or current=Vtr

Vi=−Itr

Figure 2.2.2: Power transmission on a line.

Figure 2.2.2 shows the transmission of power along a transmission line where Pinc is the incident power, Pre f the reflected power, and Ptr the transmitted power. Let the traveling waves at the receiving end be

V+e−γl +V−eγl = Vtre−γl(2.2.11)

Zoe−γl −V+

V+

Zoeγl =Vtr

Zle−γl(2.2.12)

Multiplication of equation 2.2.12 by Zl and substitution of the result in equation 2.2.11 yield

Γl =V−e−γl

V+e−γl=Zl −Zo

Zl +Zo(2.2.13)

which on substitution of the resulting equation in 2.2.10 we get

T =Vtr

V+=2Zl

Zl +Zo(2.2.14)

The power carried by the two waves in the side of the incident and reflected waves is

Pcr = Pinc −Pre f =

V+e−αl 2

2Zo−

V−eαl 2

2Zo(2.2.15)

The power carried to the load by the transmitted waves is

Ptr =

Vtre−αl 2

2Zl(2.2.16)

Setting Pcr=Ptr and using equations 2.2.13 and 2.2.14, we get

T2 =Zl

Zo(1−Γ)2l(2.2.17)

This relation verifies the previous statement that the transmitted power is equal to the dif ference of the incident power and reflected power. Example:

A certain transmission line has a characteristic impedance of 75+ j0.01Ω and is ter minated in a load impedance of 70+ j50Ω. Compute (a) the reflection coefficient; (b) the transmission coefficient.

Solution

(a) the reflection coefficient

Zl +Zo=(70+ j50)Ω−(75+ j0.01)Ω

Γ =Zl −Zo

(b) the propagation constant T =2Zl

(70+ j50)Ω+ (75+ j0.01)Ω= 0.08+ j0.32

Zl +Zo=2(70+ j50)Ω

(70+ j50)Ω+ (75+ j0.01)Ω= 1.08+ j0.32 14

Topic 3

Further Transmission Lines Characteristics

3.1 Standing Wave and Standing Wave Ratio

3.1.1 Standing Wave

The general solutions of the transmission-line equation consist of two waves traveling in opposite directions with unequal amplitude as discussed in subsection 2.1.2. Equation 2.1.22 can be written as

V(z) = V+e−αze−jβz +V−eαzejβz(3.1.1)

= V+e−αz[cos(βz)− jsin(βz)] +V−eαz[cos(βz) + jsin(βz)]

=V+e−αz +V−eαz cos(βz)− jV+e−αz −V−eαz sin(βz)

With no loss in generality it can be assumed that V+e−αzand V−eαzare real. Then the voltage-wave equation can be expressed as

Vs = V0e−jφ(3.1.2)

This is called the equation of the voltage standing wave, where

V0 =

V+e−αz +V−eαz 2cos2(βz)− jV+e−αz −V−eαz 2sin2(βz)o12(3.1.3)

3. Further Transmission Lines Characteristics

3.1 Standing Wave and Standing Wave Ratio

3.1.1 Standing Wave

3.1. Impedance Matching

3.3 Impedance Matching

Impedance matching is very desirable with radio frequency (RF) transmission lines. Stand ing waves lead to increased losses and frequently cause the transmitter to malfunction. A line terminated in its characteristic impedance has a standing-wave ratio of unity and trans mits a given power without reflection. Also, transmission efficiency is optimum where there is no reflected power. A "flat" line is non-resonant; that is, its input impedance al ways remains at the same value Zo when the frequency changes.

Matching a transmission line has a special meaning, one differing from that used in circuit theory to indicate equal impedance seen looking both directions from a given terminal pair for maximum power transfer. In circuit theory, maximum power transfer requires the load impedance to be equal to the complex conjugate of the generator. This condition is some times referred to as a conjugate match. In transmission-line problems matching means simply terminating the line in its characteristic impedance.

A common application of RF transmission lines is the one in which there is a feeder con nection between a transmitter and an antenna. Usually the input impedance to the antenna itself is not equal to the characteristic impedance of the line. Furthermore, the output im pedance of the transmitter may not be equal to the Zo of the line. Matching devices are necessary to flatten the line. A complete matched transmission-line system is shown in figure 3.3.1.

For a low-loss or lossless transmission line at radio frequency, the characteristic impedance Zo of the line is resistive. At every point the impedances looking in opposite directions are

Figure 3.3.1: Matched transmission-line system.

conjugate. If Zo is real, it is its own conjugate. Matching can be tried first on the load side to flatten the line; then adjustment may be made on the transmitter side to provide maximum power transfer.

3.3.1 Single Stub Matching

Although single-lumped inductors or capacitors can match the transmission line, it is more common to use the susceptive properties of short-circuited sections of transmission lines. Short-circuited sections are preferable to open-circuited ones because a good short circuit is easier to obtain than a good open circuit.

For a lossless line with Yg = Y0, maximum power transfer requires Y11 = Y0, where Y11 is the total admittance of the line and stub looking to the right at point 1 as shown in the figure 3.3.2.

The stub must be located at that point on the line where the real part of the admittance, looking toward the load, is Y0. In a normalized unit Y11 must be in the form

y11 = yd ±ys = 1

if the stub has the same characteristic impedance as that of the line. Otherwise Y11 = Yd ±Ys = Y0

The stub length is then adjusted so that its susceptance just cancels out the susceptance of the line at the junction.

Figure 3.3.2: Single-stub matching.

3.3.2 Double Stub Matching

Since single-stub matching is sometimes impractical because the stub cannot be placed physically in the ideal location, double-stub matching is needed. Double-stub devices consist of two short-circuited stubs connected in parallel with a fixed length between them. The length of the fixed section is usually one-eighth, three-eighths, or five-eighths of a wavelength.

The stub that is nearest the load is used to adjust the susceptance and is located at a fixed wavelength from the constant conductance unity circle (g = 1) on an appropriate constant-standing-wave-ratio circle. Then the admittance of the line at the second stub as shown in figure 3.3.3 is

y22 = yd2 ±ys2 = 1

Y22 = Yd2 ±Ys = Y0

In these two equations it is assumed that the stubs and the main line have the same charac teristic admittance. If the positions and lengths of the stubs are chosen properly, there will be no standing wave on the line to the left of the second stub measured from the load.

3.2. Standing Wave and Standing Wave Ratio

3.1 Standing Wave and Standing Wave Ratio

3.1.1 Standing Wave

V(z) = V+e−αze−jβz +V−eαzejβz(3.1.1)

= V+e−αz[cos(βz)− jsin(βz)] +V−eαz[cos(βz) + jsin(βz)]

=V+e−αz +V−eαz cos(βz)− jV+e−αz −V−eαz sin(βz)

With no loss in generality it can be assumed that V+e−αzand V−eαzare real. Then the voltage-wave equation can be expressed as

Vs = V0e−jφ(3.1.2)

This is called the equation of the voltage standing wave, where

V0 =

V+e−αz +V−eαz 2cos2(βz)− jV+e−αz −V−eαz 2sin2(βz)o12(3.1.3)

which is called the standing-wave pattern of the voltage wave or the amplitude of the

standing wave, and

φ = arctan

V+e−αz +V−eαz

V+e−αz −V−eαztan(βz) 15

(3.1.4)

which is called the phase pattern of the standing wave.

The maximum and minimum values of equation 3.1.3 can be found as usual by differen tiating the equation with respect to βz and equating the result to zero. By doing so and substituting the proper values of βz in the equation, we find that;

(i) The maximum amplitude is

Vmax = V+e−αz +V−eαz = V+e−αz(1+|Γ|)

this occurs when βz = nπ, where n = 0,±1,±2,......

(ii) The minimum amplitude is

Vmin = V+e−αz −V−eαz = V+e−αz(1−|Γ|)

this occurs when βz = (2n−1)π/2, where n = 0,±1,±2,......

(iii) The distance between any two successive maxima or minima is one-half wavelength,

since

βz = nπ z =nπβ=nπ

2π/λ= nλ2n = 0,±1,±2,......

then z1 =λ2. It is evident that there are no zeros in the minimum.

Similarly,

Imax = I+e−αz +I−eαz = I+e−αz(1+|Γ|)

Imin = I+e−αz −I−eαz = I+e−αz(1−|Γ|)

The standing-wave patterns of two oppositely traveling waves with unequal amp litude in lossy or lossless line are shown in figures 3.1.1 and 3.1.2

(iv) When V+ 6= 0 and V− = 0, the standing-wave pattern becomes V0 = V+e−αzand when V+ = 0 and V− 6= 0, the standing-wave pattern becomes V0 = V−eαz

(v) When the positive wave and the negative wave have equal amplitudes (i.e.,V+e−αz = V−eαz) or the magnitude of the reflection coefficient is unity, the standing-wave pattern with a zero phase is given by

Vs = 2V+e−αzcos(βz) (3.1.5)

which is called a pure standing wave. Similarly for the current we have Is = −j2YoV+e−αzsin(βz) (3.1.6)

Figure 3.1.1: Standing-wave pattern in a lossy line.

Figure 3.1.2: Voltage standing-wave pattern in a lossless line.

Equations 3.1.5 and 3.1.6 show that the voltage and current standing wavesare 90o out of phase along the line. The points of zero current are called the current nodes. The voltage nodes and current nodes are interlaced a quarter wavelength apart.

The voltage and current may be expressed as real functions of time and space: vs(z,t) = Re Vs(z)ejωt = 2V+e−αzcos(βz) cos(ωt) (3.1.7)

is(z,t) = Re is(z)ejωt = 2Y0e−αzcos(βz) cos(ωt) (3.1.8)

The amplitudes of equations 3.1.7 and 3.1.8 vary sinusoidally with time; the voltage is a maximum at the instant when the current is zero and vice versa as shown in the figure 3.1.3

Figure 3.1.3: Pure standing waves of voltage and current.

3.1.2 Standing Wave Ratio

Standing waves result from the simultaneous presence of waves traveling in opposite dir ections on a transmission line. The ratio of the maximum of the standing-wave pattern to the minimum is defined as the standing-wave ratio (ρ).i.e.,

Standing Wave Ratio(ρ) = Maximum voltage or current

Minimum voltage or current=|Vmax|

|Vmin|=|Imax|

|Imin|

The standing-wave ratio results from the fact that the two traveling-wave components of 18

the wave add in phase at some points and subtract at other points. The standing-wave ratio of a pure traveling wave is unity and that of a pure standing wave is infinite.

NB:When the standing-wave ratio is unity, there is no reflected wave and the line is called a flat line. The standing-wave ratio cannot be defined on a lossy line because the standing wave pattern changes markedly from one position to another. On a lowloss line the ratio remains fairly constant, and it may be defined for some region. For a lossless line, the ratio stays the same throughout the line.

Since the reflected wave is defined as the product of an incident wave and its reflection coefficient, the standing-wave ratio ρ is related to the reflection coefficient Γ by

ρ =1+|Γ|

1−|Γ|(3.1.9)

and vise versa

|Γ| =ρ −1

ρ +1(3.1.10)

The plot in figure shows the relationship between reflection coefficient |Γ| and standing wave ratio ρ. Since |Γ| ≤ 1, the standing-wave ratio is a positive real number and never

Figure 3.1.4: Standing wave ratio against reflection coefficient

less than unity, ρ ≥ 1 and the magnitude of the reflection coefficient is never greater than unity. Example:

A transmission line has a characteristic impedance of 50+ j0.01Ω and is terminated in a load impedance of 73− j42.5mega. Compute (a) the reflection coefficient; (b) the

standing wave ratio.

Solution

(a) the reflection coefficient

Γ =Zl −Zo

Zl +Zo=(73− j42.5)Ω−(50+ j0.01)Ω

(73− j42.5)Ω+ (50+ j0.01)Ω= 0.377

(b) the propagation constant

ρ =1+|Γ|

1−|Γ|= 2.21

3.2 Line Impedance and Admittance

3.2.1 Line Impedance

The line impedance of a transmission line is the complex ratio of the voltage phasor at any point to the current phasor at that point. It is defined as

Z =V(z)

I(z)(3.2.1)

Figure 3.2.1 shows a diagram for a transmission line. The voltage or current along a Figure 3.2.1: Diagram of a transmission line

line is the sum of the respective incident wave and reflected wave in the line. i.e., V(z) = Vinc +Vre f = V+e−γz +V−eγz(3.2.2)

I(z) = Iinc +Ire f = Y0V+e−γz −V−eγz (3.2.3)

At the sending end z = O; thus equations 3.2.2 and 3.2.3 becomes

IsZs = V+ +V−

IsZ0 = V+ −V−

when these to equations are solved simultaneously, we get

V+ =Is2(Zs +Z0) (3.2.4)

V+ =Is2(Zs −Z0) (3.2.5)

Substituting the value of V+ and V− in equations 3.2.2 and 3.2.3 we get that, V(z) = Is2 (Zs +Z0)e−γz + (Zs −Z0)eγz (3.2.6)

I(z) = Is 2Z0

(Zs +Z0)e−γz −(Zs −Z0)eγz (3.2.7)

Then the line impedance at any point z from the sending end in terms of Zs and Zo is

expressed as

Z = Z0(Zs +Z0)e−γz + (Zs −Z0)eγz

(Zs +Z0)e−γz −(Zs −Z0)eγz(3.2.8)

At z = l the line impedance at the receiving end in terms of Zs and Zo is given by Zl = Z0(Zs +Z0)e−γl + (Zs −Z0)eγl

(Zs +Z0)e−γl −(Zs −Z0)eγl(3.2.9)

Equations 3.2.8 and 3.2.9 are simplified by replacing the exponential factors with either hyperbolic functions or circular functions. The hyperbolic functions are obtained from

e±γl = cosh(γz)±sinh(γz) (3.2.10)

Substitution of the hyperbolic functions in equation 3.2.8 yields the line impedance at any point from the sending end as,

Z0 cosh(γz)−Zs sinh(γz)= Z0Zs −Z0 tanh(γz)

Z = Z0Zs cosh(γz)−Z0 sinh(γz)

Z0 −Zstanh(γz)(3.2.11)

For a lossless line, γ = jβ; and by using the following relationships between hyperbolic 21

and circular functions

sinh(jβz) = jsin(βz) and cosh(jβz) = j cos(βz) (3.2.12)

the impedance of a lossless transmission line (Zo = Ro) can be expressed in terms of the circular functions:

Z = R0Zs cos(βz)− jR0 sin(βz)

R0 cos(βz)− jZs sin(βz)= R0Zs − jR0 tan(βz)

R0 − jZstan(βz)(3.2.13)

The normalized impedance of a transmission line is defined as

|z| =ZZ0=1+Γ

1−Γ(3.2.14)

The normalized impedance for a lossless line has the following significant features i. The maximum normalized impedance is

R0=|Vmax|

zmax =Zmax

R0Imin=1+|Γ| 1−|Γ|= ρ

Here zmax is a positive real value and it is equal to the standing-wave ratio ρ at the location of any maximum voltage on the line.

ii. The minimum normalized impedance is

zmin =Zmin

R0=|Vmin|

R0Imax=1−|Γ|

1+|Γ|=1ρ

Here zmin is a positive real number also but equals the reciprocal of the standing wave ratio at the location of any minimum voltage on the line.

iii. For every interval of a half-wavelength distance along the line, zmax or zmin is

repeated:

zmax(z) = zmax

and

zmin(z) = zmin

z±λ2

iv. Since Vmax and Vmin are separated by a quarter-wavelength, zmax is equal to the reciprocal of zmin for every λ/4 separation:

zmax

z±λ2 =1 zmin(Z)

4. Microwave Waveguides

Microwave Waveguides

A waveguide consists of a hollow metallic tube of a rectangular or circular shape used to guide an electromagnetic wave. Waveguides are used principally at frequencies in the mi crowave range; inconveniently large guides would be required to transmit radio-frequency power at longer wavelengths.

In waveguides the electric and magnetic fields are confined to the space within the guides. Thus, no power is lost through radiation, and even the dielectric loss is negligible, since the guides are normally air-filled. However, there is some power loss as heat in the walls of the guides, but the loss is very small.

It is possible to propagate several modes of electromagnetic waves within a waveguide. These modes correspond to solutions of Maxwell’s equations for particular waveguides. A given waveguide has a definite cutoff frequency for each allowed mode.

If the frequency of the impressed signal is above the cutoff frequency for a given mode, the electromagnetic energy can be transmitted through the guide for that particular mode without attenuation. Otherwise the electromagnetic energy with a frequency below the cutoff frequency for that particular mode will be attenuated to a negligible value in a relat ively short distance.

The dominant mode in a particular guide is the mode having the lowest cutoff frequency. It is advisable to choose the dimensions of a guide in such a way that, for a given input signal, only the energy of the dominant mode can be transmitted through the guide.

The process of solving the waveguide problems may involve three steps: 27

i. The desired wave equations are written in the form of either rectangular or cyl indrical coordinate systems suitable to the problem at hand.

ii. The boundary conditions are then applied to the wave equations.

iii. The resultant equations usually are in the form of partial differential equations in either time or frequency domain. They can be solved by using the proper method.

4.1 Rectangular Waveguides

A rectangular waveguide is a hollow metallic tube with a rectangular cross section. The conducting walls of the guide confine the electromagnetic fields and thereby guide the electromagnetic wave. A number of distinct field configurations or modes can exist in waveguides.

When the waves travel longitudinally down the guide, the plane waves are reflected from wall to wall. This process results in a component of either electric or magnetic field in the direction of propagation of the resultant wave; therefore the wave is no longer a transverse electromagnetic (TEM) wave.

Figure 4.1.1 shows that any uniform plane wave in a lossless guide may be resolved into Transverse electromagnetic (TE) and Transverse Magnetic (TM) waves When the

Figure 4.1.1: Plane wave reflected in a waveguide

wavelength λ is in the direction of propagation of the incident wave, there will be one 28

component λn in the direction normal to the reflecting plane and another λp parallel to the plane given as

λn =λ

cosθ(4.1.1)

and

λp =λ

sinθ(4.1.2)

where θ is the angle of incidence and λ is the wavelength of the signal in unbounded me dium.

A plane wave in a waveguide resolves into two components: one standing wave in the direction normal to the reflecting walls of the guide and one traveling wave in the direc tion parallel to the reflecting walls. In lossless waveguides the modes may be classified as either transverse electric (TE) mode or transverse magnetic (TM) mode.

In rectangular guides the modes are designated TEmn or TMmn· The integer m denotes the number of half waves of electric or magnetic intensity in the x direction, and n is the number of half waves in the y direction if the propagation of the wave is assumed in the positive z direction.

4.1.1 Solutions of Wave Equations in Rectangular Coordinates A rectangular coordinate system is shown in figure 4.1.2

Figure 4.1.2: Rectangular waveguide

The electric and magnetic wave equations in frequency domain are given by ∇2~E = γ2~E (4.1.3)

∇2H~ = γ2H~ (4.1.4)

where γ =pjω(σ + jωε) = α + jβ. These are called the vector wave equations.The rectangular components of E or H satisfy the complex scalar wave equation or Helmholtz equation.

∇2ψ = γ2ψ (4.1.5)

∂ x2 +∂2

where ∇2 =∂2

∂ y2 +∂2 ∂ z2.

Equation 4.1.5 is solved by the method of separation of variables, the solution is assumed in the form of

ψ = X(x)YZ(z) (4.1.6)

where the variables are functions of x, y and z only respectively.

substitution of equation 4.1.6 to equation 4.1.5 and dividing it by equation 4.1.6 we get; d2X

dx2+1Yd2Y

dy2+1Zd2Z

dz2= γ2(4.1.7)

Since the sum of the three terms on the left-hand side is a constant and each term is in dependently variable, it follows that each term must be equal to a constant. Let the three terms be k2x;, k2y;, and k2z;, respectively; then the separation equation is given by

−k2x −k2x −k2x = γ2

The general solution of each differential equation in equation 4.1.7

d2X

dx2= k2xX (4.1.8)

d2Y

dy2= k2yY (4.1.9)

d2Z

dz2= k2z Z (4.1.10)

will be of the form

X = Asin(kxx) +Bcos(kxx) (4.1.11)

Y = Csin(kyy) +Dcos(kyy) (4.1.12)

X = E sin(kzz) +F cos(kzz) (4.1.13)

The total solution of the Helmholtz equation in rectangular coordinates is ψ = {Asin(kxx) +Bcos(kxx)} Csin(kyy) +Dcos(kyy) (4.1.14) {E sin(kzz) +F cos(kzz)}

The propagation of the wave in the guide is conventionally assumed in the positive z dir ection. It should be noted that the propagation constant γg in the guide differs from the intrinsic propagation constant γ of the dielectric. Let

γ2g = γ2 +k2x +k2y = γ2 +k2c(4.1.15)

where kc =

k2x +k2yis the cut-off wave number.

For a lossless dielectric, γ2 = −ω2µε. Thus, q

γg = ±

ω2µε −k2c(4.1.16)

There are three cases for the propagation constant γg in the waveguide.

i. Case I: There will be no wave propagation (evanescence) in the guide if ω2µε = k2cand γg = 0. This is the critical condition for cutoff propagation. The cutoff frequency is expressed as

fc =1 2π√µε

k2x +k2y

ii. Case II: The wave will be propagating in the guide if ω2µε > k2cand

γg = ±jβg = ±jω√µε

1−

fc f

This means that the operating frequency must be above the cutoff frequency in order for a wave to propagate in the guide.

iii. Case III: The wave will be attenuated if ω2µε < k2cand

γg = ±αg = ±ω√µε31

s fc

f−1

This means that if the operating frequency is below the cutoff frequency, the wave will decay exponentially with respect to a factor of −αgz and there will be no wave propagation because the propagation constant is a real quantity.

Therefore the solution to the Helmholtz equation in rectangular coordinates is given by ψ = {Asin(kxx) +Bcos(kxx)} Csin(kyy) +Dcos(kyy) e−jβgz(4.1.17)

4.2 TE Modes in Rectangular Waveguides

Figure 4.2.1 shows a rectangular waveguide of cross-section a × b along the z-axis The

Figure 4.2.1: Coordinates of rectangular waveguide

TEmn modes in a rectangular guide are characterized by Ez = 0. i.e., the z component of the magnetic field, Hz, must exist in order to have energy transmission in the guide. Consequently, from a given Helmholtz equation,

∇2H~ z = γ2H~ z (4.2.1)

a solution of the form

Hz =

Am sin

mπx a

+Bm cos

mπx a

Cn sin

nπy b

+Dn cos

nπy b

e−jβgz

(4.2.2)

will be determined in accordance with the given boundary conditions, where kx =mπaand ky =nπbare replaced.

For a lossless dielectric, Maxwell’s curl equations in frequency domain are ∇×~E = −jωµH~ (4.2.3)

∇×H~ = jωε~E (4.2.4)

Equations 4.2.3 and 4.2.4 have their components as,

∂ y−∂Ey

∂Ez ∂Ex

∂ z= −jωµHx

∂ z−∂Ez

∂ x= −jωµHy

∂Ey

∂ x−∂Ex

∂ y= −jωµHz

∂ y−∂Hy

∂Hz ∂Hx

∂ z= jωεEx

∂ z−∂Hz

∂ x= jωεEy

∂Hz

∂ x−∂Hx

∂ y= jωεEz

substituting ∂∂ z = −jβg and Ez = 0 the above equations are simplified βgEy = −ωµHx (4.2.5)

βgEx = −ωµHy (4.2.6)

∂Ey

∂ x−∂Ex

∂ y= −jωµHz (4.2.7)

∂Hz

∂ y+ jβgHy = jωεEx (4.2.8)

jβgHx −∂Hz

∂ x= jωεEy (4.2.9)

∂ x−∂Hx

∂Hz

∂ y= 0 (4.2.10)

Solving the above six equations we get the TE-mode field equations of

Ex = −jωµ k2c

∂Hz

∂ y(4.2.11)

Ey =jωµ k2c

∂Hz

∂ x(4.2.12)

Ez = 0 (4.2.13)

Hx = −jβg k2c

Hy = −jβg k2c

∂Hz

∂ x(4.2.14) ∂Hz

∂ y(4.2.15)

Hz = Equation 4.2.2 (4.2.16)

where k2c = ω2µε = β2g has been replaced.

Differentiating equation 4.2.2 w.r.t x and y and then substituting equations 4.2.11 through 4.2.16 will yield a set of field equations. The boundary conditions are applied to the newly found field equations in such a manner that either the tangent E field or the normal H field vanishes at the surface of the conductor.

Since Ex = 0, then ∂Hz

∂ y = 0 at y = 0,b, then Cn = 0. Also for Ey = 0, then ∂Hz

∂ x = 0 at

y = 0,a, then Am = 0.

It is generally concluded that the normal derivative of Hz must vanish at the conducting surface i.e.,∂Hz

∂n= 0 (4.2.17)

at the guide walls. Therefore the magnetic field in the positive z direction is given by

Hz = H0z cos

mπx a

cos

nπy b

e−jβgz(4.2.18)

where H0zis the constant amplitude. Substituting equation 4.2.18 in equations 4.2.11 to 4.2.16 yields TEmn field equations as

Ex = E0x cos

mπx a

sin

nπy b

e−jβgz(4.2.19)

Ey = E0y sinHx = H0x sin

mπx a

cos

nπy b

e−jβgz(4.2.20)

e−jβgz(4.2.21)

Hy = H0y cos

mπx a

sin

nπy b

e−jβgz(4.2.22)

where m and n are equal to 0,1,2,3..... with m = n = 0 excepted. 34

The cutoff wave number kc for the TEmn modes, is given by

kc =

r mπ a

2+ nπb 2= ωc√µε

The cutoff frequency fc for the TEmn modes, is given by

fc =1

2√µε

The phase constant βg is expressed as

r m a

2+ nb 2

βg = ω√µε

1−

fc f

The phase velocity in the positive z direction for the TEmn modes νg =ωβg=νp

1−

fcf 2

where νp = √1µε is the phase velocity in an unbounded dielectric. The characteristic wave impedance of TEmn modes in the guide is

Hy= −Ey

Hx=ωµ

Zg =Ex

βg=η

qµ

1−

fcf 2

where η =

εis the intrinsic impedance in an unbounded dielectric. The wavelengthλg

in the guide for the TEmn modes is given by λg =λ

1−

fcf 2

where λ =vpfis the wavelength in an unbounded dielectric.

The cutoff frequency is a function of the modes and guide dimensions, the physical size of the waveguide will determine the propagation of the modes. Whenever two or more modes have the same cutoff frequency, they are said to be degenerate modes. In a rectangular guide the corresponding TEmn and TMmn modes are always degenerate.

The mode with the lowest cutoff frequency in a particular guide is called the dominant 35

mode. The dominant mode in a rectangular guide with a < b is the TE10 mode. Each mode has a specific mode pattern (or field pattern).

It is normal for all modes to exist simultaneously in a given waveguide. The situation is not very serious, however. Actually, only the dominant mode propagates, and the higher modes near the sources or discontinuities decay very fast.

Example:

An air-filled rectangular waveguide of inside dimensions 7 x 3.5 cm operates in the dominant TE10 mode as shown in figure 4.2.2.

(a) Find the cutoff frequency.

(b) Determine the phase velocity of the wave in the guide at a frequency of 3.5 GHz. (c) Determine the guided wavelength at the same frequency.

Figure 4.2.2: Rectangular waveguide

Solution

(a) Find the cutoff frequency.

fc =c2a=3×108 m/s

2×7×10−2 m= 2.14 GHz

(b) Determine the phase velocity of the wave in the guide at a frequency of 3.5 GHz.

νg =c

fcf 2=3×108 m/s

1−

1−2.14 3.5

2= 3.78×108 m/s

λg = rλo 1−

1−2.14 3.5

2= 10.8 cm

4.3 TM Modes in Rectangular Waveguides

The TMmn modes in a rectangular guide are characterized by Hz = 0. i.e., the z compon ent of the electric field, Ez, must exist in order to have energy transmission in the guide. Consequently, from a given Helmholtz equation as

∇2~Ez = γ2~Ez (4.3.1)

A solution of the Helmholtz equation is in the form of

Ez =

Am sin

mπx a

+Bm cos

mπx a

Cn sin

nπy b

+Dn cos

nπy b

e−jβgz(4.3.2)

which must be determined in accordance with the given boundary conditions. The proced ures for doing so are similar to those used in finding the TE-mode wave.

The boundary conditions on Ez require that the field vanishes at the waveguide walls, since the tangent component of the electric field Ez is zero on the conducting surface. This requirement is that for Ez = 0 at x = 0,a, then Dn = 0. and for Ez = 0 at y = 0,b, then ∂Hz

∂ x = 0 at y = 0,a, then Am = 0.Thus the solution in equation 4.3.2 reduces to

Ez = E0z sin

where m and n equals to 1,2,3,4,......

mπx a

sin

nπy b

e−jβgz(4.3.3)

If either m = 0 or n = 0, the field intensities all vanish. So there is no TM01 or TM10 mode in a rectangular waveguide, which means that TE10 is the dominant mode in a rectangular waveguide for a > b. For Hz = 0, the field equations, after expanding ∇×H = jωεE, are

given by

∂Ez

∂ y+ jβgEy = −jωµHx (4.3.4)

∂Ez

∂ x+ jβgEx = jωµHy (4.3.5)

∂Ey

∂ x+∂Ex

∂ y= 0 (4.3.6)

βgHy = ωεEx (4.3.7)

−βgHx = ωεEy (4.3.8)

∂Hy

∂ x+∂Hx

∂ y= jωεEz (4.3.9)

Solving equations 4.3.4 through 4.3.9 simultaneously the resultant field equations for TM

modes are

Ex = −jβg k2c

Ey = −jβg k2c

∂Ez

∂ x(4.3.10) ∂Ez

∂ y(4.3.11)

Ez = Eq.4.3.3 (4.3.12)

Hx =jωε k2c

∂Ez

∂ y(4.3.13)

Hy = −jωε k2c

∂Ez

∂ x(4.3.14)

Hz = 0 (4.3.15)

where β2g −ω2µε = −k2cis replaced.

Differentiating equation 4.3.3 w.r.t x and y and then substituting equations 4.3.10 through 4.3.15 will yield a set of field equations. The TMmn mode field equations in rectangular waveguides are

Ex = E0x cos

mπx a

sin

nπy b

e−jβgz(4.3.16)

Ey = E0y sinHx = H0x sin

mπx a

cos

nπy b

e−jβgz(4.3.17)

e−jβgz(4.3.18)

Hy = H0y cos

where m and n are equal to 0,1,2,3.....

mπx a

sin

nπy b

e−jβgz(4.3.19)

Some of the TM-mode characteristic equations are identical to those of the TE modes, but some are different i.e.

fc =1 2√µε

r m a

2+ nb 2

βg = ω√µε

1−

fc f

νg =νp

Zg =βg

1−

fcf 2 fc

ωε= η

1−

λg =λ

1−

MICROVE DEVICES

Description

Table of contents

1. Introduction

2. Transmission Lines

3. Further Transmission Lines Characteristics

3.1. Impedance Matching

3.2. Standing Wave and Standing Wave Ratio

4. Microwave Waveguides