B.E. Sem. VII [EXTC] Fundamentals of Microwave …vidyalankar.org/file/engg_degree/prelim_paper_soln/SemVII/EXTC/... · Fundamentals of Microwave Engineering Prelim Question Paper

1113/Engg/BE/Pre Pap/2013/EXTC/Soln/M Wave 1

Fig.: Cavity Magnetron

Vidyalankar B.E. Sem. VII [EXTC]

Fundamentals of Microwave Engineering Prelim Question Paper Solution

MAGNETRONS It is an oscillator working at high microwave power. Unlike the electrons behaving in a klystron, the electrons in magnetron are in contact with RF field for a greater duration of time. Hence magnetrons provide higher efficiency. There are 3 types of magnetrons i) ve resistance type (< 500 MHz) ii) Cyclotron frequency type (> 100 MHz) iii) Travelling Wave/Cavity type Path (a) travel straight Path (b) application of rotational force Path (c) grazing the anode surface Path (d) back heating of cathode All the above explanation is for a static case in the absence of the RF field in the cavity of magnetron. The cavity magnetron shown in fig (a) has 8 cavities that are tightly coupled to each other. In general Ncavity tightly coupled system will have N resonant frequencies or modes. Each mode is characterized by resonant frequency of each cavity and phase of oscillation relative to the adjacent cavity. For the sustained oscillations the total phase sift around the ring of the cavity resonator is 2n whee n is an integer. Thus the phase shift between two adjacent cavities is given by

n2 n

N

where n = 0, 1, 2, 3, …, N/2, indicates the nth mode of oscillation.

1. (a)

Vidyala

nkar

Vidyalankar : B.E. Microwave


Due to excitation of the anode cavities by RF noise voltage in biasing circuit, the RF field lines are fringed out of the slot to the space between the anode and cathode. The accelerated electrons in the trajectory, when retarded by this RF field, transfer energy from the electron to the cavities to grow RF oscillations till the system RF losses balance the RF oscillations for stability. MICROSTRIP LINES Microstrip line is an unsymmetrical strip line that is nothing but a parallel plate transmission line. It has dielectric substrate whose one face is metallised ground and other face has thin conducting strip of thickness 't' and width 'w'. As compared to strip line microstrip line doesn't have upper ground plane. Some time shielding is present. Advantages of Microstrip Lines: There are certain advantages of microstrip lines over strip lines, coaxial lines, and waveguides. i) Complete conductor pattern may be deposited and processed on a single dielectric substrate

which is supported by a single metal ground plane. Thus fabrication costs would be substantially lower than strip lines, coaxial or waveguide circuits.

ii) Due to the planar nature of the microstrip structure, both package and unpackaged semiconductor chip can be conveniently attached to the microstrip element.

iii) Also there is an easy access to the top surface making it easy to amount passive or active discrete devices and also for making minor adjustments after the circuit has been fabricated. This also allows access for probing and measurement purposes.

Disadvantages of Microstrip lines: i) Due to the openness of the microstrip structure, they have higher radiation losses or

interference due to nearby conductors. These can be reduced by choosing thin substrates with high dielectric constants.

ii) Because of proximity of the airdielectric, air interface with the microstrip conductor at the interface, a discontinuity in the electric and magnetic fields is generated. This results in a microstrip configuration that becomes a mixed dielectric transmission structure with impure TEM modes propagating. This makes the analysis complicated.

k DIAGRAMS AND WAVE VELOCITIES When studying the passband and stopband characteristics of a periodic structure, it is useful to plot the propagation constant, , versus the propagation constant of the unloaded line, k (or ). Such a graph is called a k diagram, or Brillouin diagram ( after L. Brillouin, a physicist who studied wave propagation in periodic crystal structures) .

1. (b)

1. (c)

Slope= vg/C

Slope=vp/C

kc

k

k = Operating

point

Propagation

0

Fig. 1 : k diagram for a waveguide mode.

cutoff Vidyala

nkar

Prelim Question Paper Solution


The k diagram can be plotted from (cos d = cos b

2 sin ), which is the dispersion relation

for a general periodic structure. In fact, a k diagram can be used to study the dispersion characteristics of many types of microwave components and transmission lines. For instance, consider the dispersion relation for a waveguide mode:

= 2 2ck k

or k = 2 2ck … (1)

where kc is the cutoff wavenumber of the mode, k is the freespace wavenumber, and is the propagation constant of the mode. Relation (1) is plotted in the k diagram of Figure 1. For values of k kc , there is no real solution for , so the mode is nonpropagating . For k kc , the mode propagates, and k approaches for large values of (TEM propagation). The k diagram is also useful for interpreting the various wave velocities associated with a dispersive structure. The phase velocity is

p = k

c

… (2)

which is seen to be equal to c ( speed of light ) times the slope of the line from the origin to the operation point on the k diagram. The group velocity is

g = d dk

cd d

… (3)

which is the slope of the k curve at the operating point. Thus, referring to Figure 1, we see that the phase velocity for a propagating waveguide mode is infinite at cutoff and approaches c ( from above ) as k increases. The group velocity, however, is zero at cutoff and approaches c ( from below ) as k increases. MEASUREMENT OF IMPEDANCE USING REFLECTOMETER The reflectometer indicates magnitude of impedance but not the phase angle, whereas a slotted line waveguide measurement gives both. A typical set up for reflectometer technique is shown in Figure below where two directional couplers are used to sample the incident power Pi and the reflected power Pr from load. Both the directional couplers are identical (except their direction). The magnitude of the reflection coefficient can be directly obtained on the reflectometer from which impedance can be calculated.

From reflectometer reading we have, r

i

P

P

1. (d)

Reflectometer = r

i

P 100refl. power

Inc.power P 100

Forward detector

Reverse detector

. . source

Pad Forward D.C.

20 dB Load 2LReverse D.C.

20 dB Pi

(Directional Coupler)

iP 100 cP 100

Pr

Vidyala

nkar



Knowing we can calculate VSWR and impedance by using the relations

g

g

z z1S and

1 z z

where zg is the known wave impedance and z is unknown impedance. Due to directional property of the couplers, there will be no interference between forward and reverse waves. The input power is kept to a low level by means of pad. The reflectometer accuracy is greatest at low VSWR (i.e. low reflection coefficient). In a source free linear isotropic homogeneous region, Maxwells curl equation in phasor form are : E = jH H = j E

taking curl of equation (1), E = j ( H ) = j (j E ) = 2E E = ( E ) 2 E E = 0 {source free region} Hence 2 E = 2 E 2 E + 2 E = 0 Similarly, 2 H + 2H = 0 General Plane Wave Solutions In free space, the Helmholtz equation for E can be written as

2 20E k E =

2 2 2202 2 2

E E Ek E

x y z

= 0, … (2)

and this vector wave equation holds for each rectangular component of E :

2 2 2

2i i i0 i2 2 2

E E Ek E

x y z

= 0, … (3)

where the index i = x, y or z. This equation will now be solved by the method of separation of variables, a standard technique for treating such partial differential equations.

The method begins by assuming that the solution to (3) for, say, Ex, can be written as a product of three functions for each of the three coordinates:

Ex(x,y,z) = f(x) g(y) h(z) … (4) Substituting this form into (3) and dividing by fgh gives

20

f g hk

f g h

= 0, … (5)

where the double primes denote the second derivative. Now the key step in the argument is to recognize that each of the terms in (5) must be equal to a constant, since they are independent of each other. That is, f / f is only a function of x, and the remaining terms in (5) do not depend on x, so f/f must be a constant, and similarly for the other terms in (5). Thus, we define three separation constants, kx, ky, and kz, such that

f / f = 2xk ; g / g = 2

yk ; h / h = 2zk ;

or 2

2x2

d fk f 0;

dx

22y2

d gk g 0;

dy

22z2

d hk h 0

dz … (6)

Combining (5) and (6) shows that

2 2 2 2x y z 0k k k k … (7)

2. (a) (1)

Vidyala

nkar



The partial differential equation of (3) has now been reduced to three separate ordinary

differential equation in (6). Solutions to these equations are of the form yx jk yjk xe ,e , and

zjk ze , respectively. The terms with + signs result in waves traveling in the negative x, y, or z direction, while the terms with signs result in waves traveling in the positive direction. Both solutions are possible and are valid; the amount to which these are excited is dependent on the source of the fields. We will select a plane wave traveling in the positive direction for each coordinate, and write the complete solution for Ex as

Ex(x, y, z) = x y zj(k x k y k z)Ae ,

… (8)

where A is an arbitrary amplitude constant. Now define a wavenumber vector k as k = x y zˆ ˆ ˆk x k y k z = 0 ˆk n … (9)

Then from (67) k = k0, and so n̂ is a unit vector in the direction of propagation. Also define

a position vector as r = ˆ ˆ ˆx x y y z z ; … (10)

then (8) can be written as

Ex (x, y, z) = jk. rAe … (11)

Solutions to (3) for Ey and Ez are, of course, similar in form to Ex of (11), but with different amplitude constants :

Ey (x, y, z) = jk rBe , … (12)

Ez (x, y, z) = jk rCe … (13)

The x, y, and z dependences of the three components of E in (11)(13) must be the same (same kx, ky, kz), because the divergence condition that

E = yx zEE E

x y z

= 0

must also be applied in order to satisfy Maxwells equations, which implies that Ex, Ey, and Ez must each have the same variation in x, y, and z. (Note that the solutions in the preceding section automatically satisfied the divergence condition, since Ex was the only component of E, and Ex did not vary with x). This condition also imposes a constraint on the amplitudes A, B, and C, since if

0E = ˆ ˆ ˆAx By Cz , we have

E = jk r0E e ,

and E = jk r jk r0 0(E e ) E e = jk r

0jk E e 0 , where vector identity was used. Thus, we must have 0k E = 0, … (14)

which means that the electric field amplitude vector 0E must be perpendicular to the direction

of propagation, k. This condition is a general result for plane waves and implies that only two of the three amplitude constants, A, B, and C, can be chosen independently.

The magnetic field can be found from Maxwells equation, E = 0j H … (15) to give

H = jk r0

0 0

j jE (E e )

= jk r

00

jE e

= jk r0

0

jE ( jk)e

= jk r00

0

kn̂ E e

Vidyala

nkar



= jk r0

0

1n̂ E e

= 0

1n̂ E

… (16)

where vector identity was used in obtaining the second line. This result shows that the magnetic field intensity vector H lies in a plane normal to k, the direction of propagation, and that H is perpendicular to E . See figure 6 for an illustration of these vector relations. The

quantity 0 = 0 0/ = 377 in (16) is the intrinsic impedance of free-space.

The time-domain expression for the electric field can be found as

(x, y,z, t) = Re { j tE(x, y,z)e }

= Re { jk r j t0E e e }

= 0E cos(k r t), … (17)

assuming that the amplitude constants, A, B, and C contained in 0E are real. If these constants are not real, their phases should be included inside the cosine term of (17).

Consider a General Lossy medium Here E = jH H = j E + E Here following similar approach, we will get

2 E + 2 1 E

= 0

we define complex propagation constant = + j

= j 1 j

We assume an electric field E in x-direction only such that there is no variation in y and z directions

0y z

Hence the Hemholtzz equation reduces to

2

x 2x2

EE

x

= 0

Which has solution

Ex(z) = E+ ze + E ze The positive travelling wave then has propagation factor of form

ze = z j ze e which in time domain is of form

ze cos (t bz)

x

y z

E

H

n̂

Fig. 1 : Orientation of the E, H, and k = 0 ˆk n vectors for a general plane wave

Vidyala

nkar



This represents a wave travelling in +ve z-direction.

vp =

and = 2

The associated magnetic field can be calculated as

Hy = xj E

z

= z zj(E e E e )

As with Lossless case

(wave impedance)

= j

=

in lossless case

There are 4 modes of operation of Gunn Diode 1) Gunn oscillator mode 2) Stable amplification mode 3) LSA oscillator mode 4) Gas circuit oscillator mode 1) Gunn Oscillation mode : This mode is defined in the region where the product of frequency

multiplied by length is about 107 cm/s and the product of doping multiplied by length is greater than 1012 cm2. In this region the device is unstable because of cyclic formation of either the accumulation layer or the high-field domain. In a circuit with relatively low impedance the device operates in the high-field domain mode and the frequency of oscillation is near the intrinsic frequency. When the device is operated in a relatively high-Q cavity and coupled properly to the load, the domain is quenched or delayed (or both) before nucleating. In this case, the oscillation frequency is almost entirely determined by the resonance frequency of the cavity and has a value of several times the intrinsic frequency.

2) Stable Amplification mode : This mode is defined in the region where the product of

frequency times length is about 107 cm/s and the product of doping times length is between 1011 and 1012 cm2.

3) LSA Oscillation mode : This mode is defined in the region where the product of frequency

times length is about 107 cm/s and the quotient of doping divided by frequency is between 2 104 and 2 105.

4) Bias-circuit oscillation mode : This mode occurs only when there is either Gunn or LSA

oscillation, and it is usually at the region where the product of frequency times length is too small to appear in the figure. When a bulk diode is biased to threshold, the average current suddenly drops as Gunn oscillations begin. The drop in current at the threshold can lead to oscillations in the bias circuit that are typically 1 kHz to 100 MHz.

Single Stub Impedance Matching on Lines The input impedance of an open circuited or short circuited lossless line is purely reactive. Such a section is usually connected across the line at a convenient point and cancels the negative part of impedance at this point seen looking towards the load. Such a section is called impedance matching stub. Since the stub is connected in shunt across the main transmission line, it becomes possible to use the stub of desired length and construction without any restriction imposed by the main line. Usually, the length of matching stub is kept within quarter wavelength so that the stub remains particularly lossless at all high frequencies. A short circuited stub of length less than / 4 offers inductive reactance at the input while an open circuited stub of length less than /4 offers capacitive reactance at the input. Thus depending upon the requirements either the short circuited or the open circuited stub may be used.

2. (b)

3. (a) Vidyala

nkar



Fig. : Connection of stub on line

Since the Z0 of a line is usually a pure resistance, the impedance of the load, as seen by the line (towards load), should be resistance of the same value. i.e. R0 (or conductance = 1/R0) If such a match does not attain, energy will be reflected from the load. Standing waves will be set up. Energy transfer effectiveness will be less than optimum. Because the matching stubs are connected in parallel with the main transmission line, it is easier to consider the analysis in term of admittance rather than impedance. The method of connecting stub is shown in figure. In the figure the short circuited stub has been connected at a distance d1 from the receiving end. When the line is terminated in impedance ZL ( Z0), reflections take place. The input impedance of the line seen towards the load at a point of voltage maximum will be S Z0 and at a point of voltage minimum will be Z0/ S, where S is the standing wave ratio and Z0 is the characteristic impedance of the lossless line. Hence at some point intermediate between a voltage maximum and a voltage minimum; the input impedance of the line can be expected to be equal to Z0. If this point is at a distance d1 from load, stub is connected at this point. The input admittance of the line at d1 before insertion of the stub can be written as

0

1Y jB

R

The value of susceptance B may be positive or negative depending upon the toad impedance ZL. This susceptance B is required to be eliminated in order to provide impedance matching and thus making the line smooth. Now if a stub having input susceptance B is connected at a distance d1 from load, the total input admittance at d1 is

t0 0

1 1Y jB jB

R R

Z = Z0 at d1 The length d2 is selected such that the input susceptance of the stub is equal to jB. The lengths d1 and d2 are given by

1

v1

cosd

4

and 12

Sd tan

2 S 1

where = reflection angle |v| = magnitude of reflection coefficient. S = VSWR Usually a short circuited stub is preferred to an open circuited stub for the following reasons: i) It is easy to provide short circuit or zero resistance across any pair of terminals by simply

connecting a large plate of conducting material, where as it is difficult to keep perfect open circuit condition requiring very high insulation resistance across the stub terminals. This becomes more difficult at high frequencies involved in stub matching.

ii) The short circuited stub has lower loss of energy due to radiation because a large metal plate can always be used at the shorted end not allowing any radiation of energy.

iii) The same stub can be used for matching at different frequencies or for different loads by varying the length of stub by means of a sliding short circuited bar.

Vidyala

nkar



Disadvantages of Single Stub Matching i) The single stub matching system is useful for a fixed frequency only because, as the

frequency changes, the location of the stub will have to be changed. The change of the susceptance of the stub does not however, present any problem, because the shorting plug may be moved to the required position. So single stub matching is a narrow band system.

ii) Another disadvantage of the single stub is that, for final adjustment the stub has to be moved along the line slightly. This is possible only in open wire lines and therefore, on coaxial lines, single stub matching may become inaccurate in practice, though it reduces the reflection losses to a considerable extent.

KURODA’S IDENTITIES The four Kuroda identities use redundant transmission line sections to achieve a more practical microwave filter implementation by performing any of the following operations: Physically separate transmission line stubs Transform series stubs into shunt stubs, or vice versa Change impractical characteristic impedances into more realizable ones The additional transmission line sections are called unit elements and / 8 long at c ; the unit elements are thus commensurate with the stubs used to implement the inductors and capacitors of the prototype design. The four identities are illustrated in Table 1, where each box represents a unit element, or transmission line, of the indicated characteristic impedance and length (/8atc). The inductors and capacitors represent shortcircuit and opencircuit stubs, respectively. We will prove the equivalence of the first case.

3. (b)

4. (a)

Fig. 1 : Richard’s transformation. (a) For an inductor to a shortcircuited stub. (b) For a capacitor to an opencircuited stub.

jBc jBc

(b)

jXL jXL

(a)

Vidyala

nkar



The two circuits of identity (a) in Table 1 can be redrawn as shown in Figure 35; we will show that these two networks are equivalent by showing that their ABCD matrices are identical. From Table 1, the ABCD matrix of a length of transmission line with characteristic impedance Z1 is

A BC D

= 1 1

2

1 1

cos jZ sin 1 j Z1

j jsin cos 11Z Z

… (1)

where tan . Now the opencircuited shunt stub in the first circuit in Figure 1 has an

impedance of jZ2 cot = jZ2/ , so the ABCD matrix of the entire circuit is

Fig. 1 : Equivalent circuits illustrating Kuroda identity (a) in Table 1.

Table 7 : The four Kuroda Identities (n2 = 1 + Z2/Z1)

Vidyala

nkar



L

A BC D

= 1

2

2 1

1 0 1 j Z1

j j1 1 1Z Z

= 1

2 12

1 2 2

1 j Z1

Z1 1j 11 Z Z Z

… (2a)

The shortcircuited series stub in the second circuit in Figure 1 has an impedance of j(Z1/n2)

tan 21j Z / n , so the ABCD matrix of the entire circuit is

R

A BC D

=

12 1

22

2

2

Z1 j j Z

11nnj n 11 0 1

Z

= 1 22

22 2 1

2 2

j1 Z Z

1 nZj n1 1

Z Z

… (2b)

The results in (2a) and (2b) are identical if we choose n2 = 1+Z2 / Z1. The other identities in Table 1 can be proved in the same way. (i) Directional couplers The most common use for directional coupler is to tap-off a sample of input power for

use in signal monitoring circuits. The sample can be used to determine various parameters of circuit like power level, frequency and can be integrated into feedback loop to achieve required system spaces.

Hybrid coupler is used in amplifier design. Hybrid coupler can be configured as an attenuator (ii) Magic Tee Can be used as a power combiner. Also can be configured as power divider. Can be configured as an Isolator. (iii) Isolators Isolate Device Under Test (DUT) from sensitive RF signals. A greater isolation can be provided using dual junction isolator. (iv) Circulators Can be configured as an inexpensive duplexer. When one port of 3 port circulator is terminated in a matched load, it can be used as an

ISOLATOR. (v) Attenuators Used to add stability to circuit by reducing amplitude and thus keeping power levels

under control. Used to remove noise from circuit.

4. (b)

Vidyala

nkar



CIRCULAR WAVEGUIDES A circular waveguide is tabular, circular conductor. Fig. shows a circular waveguide with a circular cross section of radius a and length z. The electric & magnetic wave equation in frequency domain are given by

2

2 2E

H

… (A)

Equation (A) are called vector wave equation. Propagation of TE modes in circular waveguide: It is commonly assumed that the waves in circular waveguide are propagating in position zdirection. For TE modes EZ = 0 but HZ 0 Equation A becomes 2HZ = 2HZ.

In cylindrical co-ordinates:

2HZ = 2 2

2z z zz2 2 2

H H H1 1r. H

r r r r z

Replacing 2

22z

we get,

2

2 2Z ZZ2

H H1 1r. H 0

r r r r

i.e. 2

2z zz2 2

H H1 1r h H 0

r r r r

… (1)

Here we observe that Hz is a function of r and . Using the method of separation of variables, the solution is assumed as, ZH R. … (2) where, R = a function of r only = a function of only Putting (2) in (1)

22

2 2

R1 1r R h R.

r r r r

i.e. 2

22 2

R Rr h R

r r r r

Dividing by R

2

22 2

1 R 1 dr h 0

rR r dr r d

Multiplying by r2

2

2 22

r d dR 1 dr h r 0

R dr dr d

… (3)

5. (a)

1 2 3

z

y

r a

x

z

Vidyala

nkar



Terms (1) & (3) are functions of r only and term (2) is a function of only

Let 2

22

1 d

d

n … (4)

Putting in equation (3)

2 2 2r d dRr h r 0

R dr dr

n

Multiplying this by R.

2 2 2d dRr r h r n R 0

dr dr

… (5)

This is a Bessel’s equation of order n. The Bessel’s equation is a second order equation & there should be two linearly independent solutions for each value of n. The general solution of equation (5) can be written as R = Cn Jn (hr) + Dn Nn (hr) … (6) where, Jn (hr) is the Bessel function of the first kind of nth order with an argument (Kc r) & Nn(hr) is the Bessel function of the second kind of nth order. Equation (4) is a second order differential equation of , the solution of it is = An sin (n) + Bn cos (n) … (7)

Putting (6) & (7) in equation (2) we get

HZ = [Cn Jn (hr) + Dn Nn (hr) ] [An sin (n ) + Bn cos (n )] gjB ze

… (8) where it is assumed that the guide is lossless. The distinctive property of Bessel function of the second kind of all orders is that they become infinite when the argument is zero. When we study wave propagation in circular waveguide, our region of interest includes the axis where r = 0. Since the infinite field is physically impossible, the solution HZ in equation (8) cannot contain Nn (hr) term. This means that the coefficient Dn must be zero for all n. Thus for wave mode problems inside a circular waveguide there is no need to concerned with the Nn (hr) But in case of coaxial waveguide the region of interest does not include r = 0. Hence in that case both Jn (hr) are present in the solution of R in equation (6). By making use of trigonometry the term

An sin (n) + Bn cos (n) = 2 2 1 nn n

n

AA B cos n tan

B

= Fn cos n.

where n = 1 n

n

An tan

B

& Fn = 2 2

n nA B

HZ = gjB zn n nC J hr F cos n .e

= gjB .zoz nH .J h.r cos n .e

where Hz = Cn Fn Now we obtain other field components (Er, E, Hr, H) in terms of Hz. for this we use Maxwell’s equation.

X E B H

XH D E

Vidyala

nkar



In cylindrical coordinates

r z

Z

r z

1 1a a a

r r X E r

E rE E

= z r z rr z

z

1 E E E 1 Ea rE a a rE

r dz r r r

Substituting = z

i.e. zr

E1E j H

r

… (i)

zr

Ej H

r

… (ii)

rz

E1 1rE j H

r r r

… (iii)

Similarly

zr

H1H j E

r

… (iv)

zr

HH j E

r

… (v)

rz

H1 1rH j E

r r r

… (vi)

+0.5

0.5

1.0

N0(hr) N1(hr)

N2(hr)

J0(hr)

J1(hr) J2(hr)

1 2 3 4 5 6 7 8 9 10

3.832 7.016 10.1

0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

Vidyala

nkar



From (iv)

Zr

Hj 1H E

r

Putting in (ii)

Zr r

Hj 1E j H j E

r

2

zr

HjE

r

or 2

Zr

Hh jE

r

zr 2

Hj 1E

rh

… (vii)

Putting in equation (ii)

zr 2

Hj 1H E

j j rh

i.e. z2

H1H

rh

… (viii)

From (1)

rH Eh

Putting it in (5)

zHE j E

j r

i.e. 2

zHE j

j r

or 2 2

zHE

j r

or 2

zHhE

j r

or z2

HjE

rh

… (ix)

Putting in (i) we get,

rH Ej

z2

Hj

j rh

i.e. zr 2

HH

rh

… (x)

& we already have

zE 0 &

gjB zz oz nH H J (hr)cos n e

… (xi)

Vidyala

nkar



Boundary condition The boundary conditions require that 1) The component of the electric field E which is

the tangential to the inner surface of the circular waveguide at r = a must vanish.

2) The r component of the magnetic field Hr, which is normal to the inner surface of r = a, must vanish.

i.e E = 0 at r = a z

r 0

H0

r

… using (ix)

or Hr = 0 at r = a z

r a

H0

r

… using (x)

Thus, gj z'zoz n

HH J hr cosn

r

at r = a

gjB z'zoz n

r a

HH J ha cosn e 0

r

… (9)

'nJ (ha) = 0 … (10)

where 'nJ (ha) indicates the derivative of Jn w.r.t. (ha).

Since the 'nJ are oscillatory functions, the '

nJ (ha) are also oscillatory functions. An infinite sequence of values of (ha) is also oscillatory functions. An infinite sequence of values of (ha) satisfies equation (10). The mth root of this equation is denoted by Pnm given by Pnm = a h The points, which are the roots of equation (10) correspond to the maxima & minima of the curves Jn(h a) as shown in fig. Pnm values for TEnm modes in circular waveguide are obtained from the following table.

m n = 0 n = 1 2 3 1 3.832 1.841 3.054 4.201 2 7.016 5331 6.706 8.015 3 10.173 8.536 9.969 11.346

Here, n represents the no. of full cycles of field variation in one revolution through 2 radians of m represents the no. of zeros of E i.e. Jn(ah) along the radial of the waveguide but zero on the axis is excluded if it exists. The permissible value of h can be written as

nmP 'h

a

Putting the value of Hz in equation for field we get the complete field equations of the TEnm modes in circular waveguide as

gj zzr 0z n2 2

Hj 1 hj 1E H J hr sin n ne

r rh h

i.e. gj z' nmr or n

P 'E E J r sin n e

a

gj z' nmo n

P 'E E J r cos n e

a

Ez = 0

r = a

E

Hr

Vidyala

nkar



gj z'z nmr oz n2 2

H P 'H H J r cos(n e

r ah h

gj zo ' nmn

E P 'J r cos(n e

2g a

gj zor nmn

E P 'H J r sin n e

2g a

gj znmz oz n

P 'H H J r cos n e

a

Where 2g = r

r

EE

H H

PHASE SHIFTER Many applications require phase shift to be introduced between two given position in a waveguide system. A phase shifter is a twoport passive device that produces a variable change in phase of the wave transmitted through it. since phase shift constant β is inversely proportional to guide wavelength g, the magnitude of g could be changed to obtain variable amount of phase shift. A fixed changed to obtain variable amount phase shift can be obtained by inserting dielectric rods across the diameters of a circular waveguide or by reducing wider dimension of a rectangular waveguide. The physical construction of phase shifter is same as that of a vane attenuator. It consist of a dielectric slab or vane specially shaped to minimize reflection effects. This slab is inserted through a longitudinal slot cut along the wider dimension of waveguide. The dielectric slab is made up of low loss material with r > 1. As the slab is moved nearer to centre, the microwave signal moves more slowly. From the electric field distribution it is seen that the signal is distorted from sinusoidal to that indicated because of dielectric slab. If the vane is inserted deeper, there is more change in the medium & there is greater phase shift. The amount of phase shift is maximum when the slab is at centre & minimum when it is adjacent to the wall of the waveguide if the vane is placed such that vane’s inside dimension is parallel to the direction of flux lines. In this phase shifter, the phase shift in continuous & hence it is also called as analog phase shifter. PARAMETRIC AMPLIFIER A parametric amplifier is the one that uses a nonlinear reactance (capacitive or Inductive) or a time varying reactance for the purpose of amplification.

The parametric devices basically depend on the possibility of increasing the energy of the signal at one frequency by supplying energy at some other frequency.

5. (b)

6. (a)

Vidyala

nkar



Fundamentals Consider a simple tank circuit in which capacitor plates position can be varied mechanically so as to change the separation distance between them,

We know, Q

VC

& 0 1ACd

To obtain amplification, the capacitor plates are pulled apart when the charge & the voltage

are at their maximum. At zero voltage, the capacitor plates are brought back to their original separation & this

require no expenditure of energy as the electric field is zero. The voltage & charge now swing to their wave maximum at which plates are once again pulled

apart & the process can be continued at each maximum and minimum of voltage & hence a signal builds up i.e. each time the plates are pulled apart, energy is added to the signal.

As amplitude builds up, it requires more & more force to separate the plates. It is to be noted that in the present case, energy is added twice per cycle i.e. circuit is pumped at twice the signal frequency.

Amplification Mechanism of Parametric Amplifier In a paramp. (i.e. parametric amplifier), the pump (source) generator acts as a local oscillators & varactor diode C(t) as the mixer as shown in fig below The signal frequency fS & pump frequency fP are mixed in a nonlinear capacitor C(t) to

generate voltages at fundamental frequencies fP & fS as well as the sum and difference frequencies mfP nfs across C(t). An output voltage at f0 is obtained across the load resistance RL. The o/p circuit which does not require external excitation is called the Idler circuit.

The o/p (or idler) frequency f0 in idler circuit is expressed a, the sum & difference of frequencies fS & fP i.e. f0 = mfP + nfS where m & n are +ve integers from 0 to . If f0 > fS, the device is called Parametric Up converter. If f0 < fS, the device is called parametric Down converter.

Vidyala

nkar



Applications Due to advantage of low noise amplification parametric amplifiers are extensively used in Long Range Radar Satellite Ground Station Radio telescopes Artificial satellites Microwave ground waves communications Radio astronomy CHEBYSHEV LOWPASS FILTER PROTOTYPE For an equalripple lowpass filter with a cutoff frequency c = 1, the power loss ratio is PLR = 2 2

N1 k T ( ) … (1)

where 1 + k2 is the ripple level in the passband. Since the Chebyshev polynomials have the property that

NT 0 = 0 for N odd,1 for Neven,

equation (1) shows that the filter will have a unity power loss ratio at = 0 for N odd, but a power loss ratio of 1 + k2 at = 0 for N even. Thus, there are two cases to consider, depending on N.

For the twoelement filter, the power loss ratio is given in terms of the component values. We see that T2(x) = 2x2 1, so equating (1) to get

2 4 21 k (4 4 1) = 2 2 2 2 2 2 2 2 2 411 1 R R C L 2LCR L C R

4R … (2)

which can be solved for R, L, and C if the ripple level (as determined by k2) is known. Thus, at = 0 we have that

2k = 21 R

4R

or R = 2 21 2k 2k 1 k (for N even). … (3)

6. (b)

c

1

Fig. 1 : Attenuation versus normalized frequency for maximally flat filter prototypes. Vidyala

nkar



Equating coefficients of 2 and 4 yields the additional relations

4k2 = 2 2 21L C R

4R

24k = 2 2 2 21R C L 2LCR

4R

which can be used to find L and C. Note that (3) gives a value for R that is not unity, so there will be an impedance mismatch if the load actually has unity (normalized)

Table 1 : Element values for EqualRipple LowPass Filter Prototypes (g0 = 1, c = 1, N = 1 to 10, 0.5 dB and 3.0 dB ripple)

0.5 dB Ripple N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

1 0.6986 1.0000 2 1.4029 0.7071 1.9841 3 1.5963 1.0967 1.5963 1.0000 4 1.6703 1.1926 2.3661 0.8419 1.9841 5 1.7058 1.2296 2.5408 1.2296 1.7058 1.0000 6 1.7254 1.2479 2.6064 1.3137 2.4758 0.8696 1.9841 7 1.7372 1.2583 2.6381 1.3444 2.6381 1.2583 1.7372 1.0000 8 1.7451 1.2647 2.6564 1.3590 2.6964 1.3389 2.5093 0.8796 1.9841 9 1.7504 1.2690 2.6678 1.3673 2.7239 1.3673 2.6678 1.2690 1.7504 1.0000 10 1.7543 1.2721 2.6754 1.3725 2.7392 1.3806 2.7231 1.3485 2.5239 0.8842 1.9841

3.0 dB Ripple

N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

1 1.9953 1.0000 2 3.1013 0.5339 5.8095 3 3.3487 0.7117 3.3487 1.0000 4 3.4389 0.7483 4.3471 0.5920 5.8095 5 3.4817 0.7618 4.5381 0.7618 3.4817 1.0000 6 3.5045 0.7685 4.6061 0.7929 4.4641 0.6033 5.8095 7 3.5182 0.7723 4.6386 0.8039 4.6386 0.7723 3.5182 1.0000 8 3.5277 0.7745 4.6575 0.8089 4.6990 0.8018 4.4990 0.6073 5.8095 9 3.5340 0.7760 4.6692 0.8118 4.7272 0.8118 4.6692 0.7760 3.5340 1.0000 10 3.5384 0.7771 4.6768 0.8136 4.7425 0.8164 4.7260 0.8051 4.5142 0.6091 5.8095

impedance; this can be corrected with a quarterwave transformer, or by using an additional filter element to make N odd. For odd N, it can be shown that R = 1. (This is because there is a unity power loss ratio at = 0 for N odd.) Tables exist for designing equalripple lowpass filters with a normalized source impedance and cutoff frequency ( '

c 1 ), and can be applied to either of the ladder circuits of Figure 25. This design data depends on the specified passband ripple level; Table 1 lists element values for normalized lowpass filter prototypes having 0.5 dB or 3.0 dB ripple, for N = 1 to 10. Notice that the load impedance gN+1 1 for even N. Linear Phase LowPass Filter Prototypes Filters having a maximally flat time delay, or a linear phase response, can be designed in the same way, but things are somewhat more complicated because the phase of the voltage transfer function is not as simply expressed as is its amplitude.

Vidyala

nkar



LORENTZ RECIPROCITY THEOREM Reciprocity is a general concept that occurs in many areas of physics and engineering, and

the reader may already be familiar with the reciprocity theorem of circuit theory. Here we will derive the Lorentz reciprocity theorem for electromagnetic fields in two different forms.

This theorem is use to obtain general properties of network matrices representing microwave circuits and to evaluate the coupling of waveguides from current probes and loops, and the coupling of waveguides through apertures. There are a number of other important uses of this powerful concept.

Consider the two separate sets of sources, 1J , 1M and 2 2J , M , which generate the fields 1 1E , H ,

and 2 2E , H , respectively, in the volume V enclosed by the closed surface S, as shown in figure 1.

Maxwells equations are satisfied individually for these two sets of sources and fields, so we can write

7. (a)

c

1

(a)

Fig. 2 : Attenuation versus normalized frequency for equalripple filter prototypes. (a) 0.5 dB ripple level. (b) 3.0 dB ripple level.

c

1

(b)

Vidyala

nkar



1E = 1 1j H M …(1a)

1H = 1 1j E J …(1b)

2E = 2 2j H M …(2a)

2H = 2 2j E J …(2b)

Now consider the quantity 1 2 2 1(E H E H ) , which can be expanded using vector identity to give 1 2 2 1 1 2 2 1 2 1 1 2J E J E M H M H …(3) Integrating over the volume V, and applying the divergence theorem given

1 2 2 1V(E H E H )dv = 1 2 2 1S

(E H E H ) ds …(3)

= 2 1 1 2 1 2 2 1V(E J E J H M H M )dv

Equation (3) represents a general form of the reciprocity theorem, but in practice a number of special situations often occur leading to some simplification. We will consider three cases.

Case 1 : S encloses no sources. Then 1 2 1 2J J M M 0, and the fields 1 1E ,H , and 2E , 2H are source-free fields. In this case, the right-hand side of (3) vanishes with the result that

1 2 2 1S SE H ds E H ds …(4)

Case 2 : S bounds a perfect conductor: For example, S may be the inner surface of a closed, perfectly conducting cavity. Then the surface integral of (3) vanishes, since 1 1E H .

1 2ˆ ˆn (n E ) H (by vector identity), and 1n̂ E is zero on the surface of a perfect conductor

(similarly for 2E ). The result is

1 2 1 2V(E J H M )dv = 2 1 2 1V

(E J H M )dv …(5)

This result is analogous to the reciprocity theorem of circuit theory. In words, this result states that the system response 1E or 2E is not changed when the source and observation points are

interchanged. That is, 2E (caused by 2J ) at 1J is the same as 1E (caused by 1J ) and 2J .

Case 3 : S is a sphere at infinity. In this case, the fields evaluated on S are very far from the sources and so can be considered locally as plane waves. Then the impedance relation

ˆH n E / applies to (3) to give

1 2 2 1 ˆ(E H E H ) n = 1 2 2 1ˆ ˆ(n E ) H (n E ) H

= 1 2 2 11 1

H H H H 0,

so that the result of (5) is again obtained. This result can also be obtained for the case of a closed surface S where the surface impedance boundary condition applies.

S 1J

1M

2J2M

n̂

V1 1E , H

2 2E , H

Fig. 1: Geometry for the Lorentz reciprocity theorem

Vidyala

nkar



IMPATT DIODE IMPATT diode is called as “Impact Avalanche Transit Time” diode. It is called so because it uses the avalanche and transit time properties of the semiconductor structure. The principle of operation of an IMPATT diode is based on the process of Avalanche multiplication and Impact ionization. Working As shown in the figure below, IMPATT is a diode, the junction being between the p+ and n layers

An extremely high voltage gradient (400 KV/cm) is applied to the IMPATT diode eventually resulting in a very high current. Such a high potential gradient back biasing the diode causes a flow of minority carriers across the junction.

Let a RF ac voltage be superimposed on the top of a high d. c. voltage. Increased velocity of electrons and holes result in additional electrons and holes by knocking them out of crystal structure by so called ‘Impact ionization’

These additional carriers continue the process at the junction and it now results into an avalanche.

When the original dc voltage was just at the threshold, this voltage will be exceeded during the whole of RF positive cycle and the avalanche current multiplication will be taking place during this entire time.

This avalanche multiplication process is not instantaneous but it takes a time such that the current pulse maximum at the junction occurs at instant when RF voltage across the diode is zero and going negative. A 90 phase shift between voltage and current has then been achieved.

7. (b)

Vidyala

nkar



The current pulse as shown in the figure above is situated at the junction. Due to applied reverse biased, this pulse moves towards the cathode with a drift velocity dependent on the presence of high dc field.

The thickness of highly doped region is adjusted such that time taken for current pulse to move from V = 0 to V = negative maximum of RF cycle is exactly 90.

Hence voltage and current are 180 out of phase and a dynamic RF negative resistance comes to existence.

Hence IMPATT diode is useful both as an oscillator and as an amplifier. The resonant frequency of IMPATT diode is given by ,

f = dV

2L

where Vd = Carrier drift velocity L = Length of the drift space charge region. Efficiency of IMPATT diode is given as

= ac aa

dc dd

P IVP IV

RICHARD’S TRANSFORMATION The transformation,

= p

tan tan

… (1)

maps the place to the plane, which repeats with a period of p/ 2 . This

transformation was introduced by P. Richard to synthesize an LC network using openand shortcircuited transmission lines. Thus, if we replace the frequency variable with , the reactance of an inductor can be written as LjX = j L jL tan … (2a) and the susceptance of a capacitor can be written as jBc = jC = jC tan … (2b)

These results indicate that an inductor can be replaced with a shortcircuited stub of length and characteristic impedance L, while a capacitor can be replaced with an opencircuited stub of length and characteristic impedance 1/C. A unity filter impedance is assumed.

Cutoff occurs at unity frequency for a lowpass filter prototype; to obtain the same cutoff frequency for the Richard’stransformed filter, (1) shows that = 1 tan

Which gives a stub length of = / 8 , where is the wavelength of the line at the cutoff frequency, c. At the frequency 0 = 2c . the lines will be / 4 long, and an attenuation pole will occur. At frequencies away from c , the impedances of the stubs will no longer match the original lumpedelement impedances, and the filter response will differ from the desired prototype response. Also, the response will be periodic in frequency, repeating every 4c.

In principle, then, the inductors and capacitors of a lumpedelement filter design can be replaced with shortcircuited and opencircuited stubs. Since the lengths of all the stubs are the same

c/ 8at , these lines are called commensurate lines.

MULTICAVITY KLYSTRON Gains of about 10 to 20 dB are typical with two-cavity tubes. A higher overall gain can be achieved by connecting several two cavity tubes in cascade, feeding the output of each of the tubes to the input of the succeeding one. Instead, multiple number of cavities can be used as in a multicavity klystron shown in Fig. below.

7. (c)

7. (d)

Vidyala

nkar



Here, each of the intermediate cavities act as a buncher with the passing electron beam inducing an enhanced RF voltage than the previous cavity. With four cavities, power gains of around 50 dB can be easily achieved. The cavities can all be tuned to the same frequency (synchronous tuning) for narrow band operation. Bandwidth can be improved by stagger tuning of cavities upto about 80 MHz with reduction in gain (to about 45 dB). This stagger tuning is employed in UHF klystrons for TV transmitter output tubes and in satellite earth station transmitters as power amplifiers at 6 GHz.

Vidyala

nkar

Documents

B.E. Sem. VII [EXTC] Fundamentals of Microwave …vidyalankar.org/file/engg_degree/prelim_paper_soln/SemVII/EXTC/... · Fundamentals of Microwave Engineering Prelim Question Paper