Microwaves and Radar - Navodaya Institute of and Radar Time: 3 hrs. Max. Marks: ... where ZY ZY2 ZY...

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Transcript of Microwaves and Radar - Navodaya Institute of and Radar Time: 3 hrs. Max. Marks: ... where ZY ZY2 ZY...

  • December 2012Fifth Semester B.E. Degree Examination (10EC54)

    Microwaves and RadarTime: 3 hrs. Max. Marks: 100

    Note: 1. Answer any FIVE full questions, selecting at least two questions from each part.

    PART - A

    1. a. Derive transmission line equations by the method of distributed circuit theory. (09 Marks)Ans:

    v(z, t)

    i(z, t)

    i(z, t) zz = (R +jL)z

    i(z,+z, t)

    Gz Cz

    i(z+z, t)

    v(z + z, t)

    z

    The change in voltage w.r.t the distance z along the line is given by

    iv Ri z L z

    t

    v iRi L

    z t

    = +

    = +

    v vAs z 0,

    z z

    v iRi L

    z t

    = + (1)

    Similarly change in current w.r.t z is given byv

    i G zv C zt

    i vGV C

    z t

    = +

    = +

    i iAgain as z 0,

    z z

    3-Micro waves and RadarDecember 2012.indd 473-Micro waves and RadarDecember 2012.indd 47 8/13/2013 1:21:26 PM8/13/2013 1:21:26 PM

  • December 2012 Microwave and RadarSet 2 -

    i VGV C

    z t

    = +

    (2)

    Differentiating eqn (1) w.r.t z partially, we get2

    2

    v i iR L

    z z t z

    = + (3)

    Using (2) in (3)2

    2

    v v VR GV C L GV C

    z t t t

    = + + +

    ( )2 2

    2 2

    v v vRGV RC GL LC

    z t z

    = + + +

    (4)

    = + + +

    2 2

    2 2

    i i iSimilarly, RGi (RC GL) LCz t t

    (5)

    Redenote the V & I in sinusodial formV = V (z, t) = V(z) ejt (6)i = i (z, t) = I (z) ejt (7)differentiating eqn (6) twice w.r.t z we get

    2 2j t

    2 2

    V d v(z)e

    z dz =

    (8)

    differentiating eqn (6) w.r.t. t we get

    ( )j tv v(z) e jt

    =

    (9)

    differentiating again w.r.t. t

    ( )2

    2j t2

    vV(z) e j

    t =

    ( )2

    2 j t2

    vv z e

    t =

    (10)

    Using eqn (8) to eqn (10) in eqn (4) we get

    ( ) ( ) ( )( )2

    j t j t j t 2 j t2

    d V(z)e RGV(z)e RC GL V z j e LC V z e

    dz = + + +

    ( ) ( )

    ( ) ( )

    = + +

    = + + +

    22

    2

    d v(z) RG j RC GL LC V zdz

    R G j C j L G j C V(z)

    ( )( )2

    2

    d V(z)R j L G j C V(z)

    dz = + + (11)

    Let Z = R + jL, Y = G + jC (12)Z series impedance/unit lengthy series admittance/unit length

    2

    2

    d V(z)ZY V(z)

    dZ= (13)

    3-Micro waves and RadarDecember 2012.indd 483-Micro waves and RadarDecember 2012.indd 48 8/13/2013 1:21:30 PM8/13/2013 1:21:30 PM

  • Microwave and Radar December 2012 Set 2 -

    or 2

    2

    d V(z)ZY V(z) 0

    dZ= = (14)

    Similarly, 2

    2

    d I(z)ZY I(z) 0

    dZ = (15)

    Transmission line equations are

    = = dv dIZI, YVdz dz

    (16)2 2

    2 22 2

    d v d Iv, I

    dz dz= = (17)

    2where ZY ZY

    ZY j

    = =

    = = +

    1. b. A single stand turner is to match a lossless line of 100 to a load of (800 + j300) . The frequency is 3GHz.

    i) Find the distance in meters from the load to the tuning stub ii) Determine the length in meters of the short-circuited stub (06 Marks)Ans: Given Data,

    Zo = 400ZL = 800 + j 300 l = ?d = ?Normalization:

    L

    o

    1

    Z 800 j3002 j0.75

    Z 400

    z 2 j 0.75

    += = +

    = +

    Procedure

    a) Plot the normalized zL on smith chart & name that point as 'A'.

    b) Draw a line from centre to zL point, i.e 0 to A

    c) With OA as radius & 'O' as centre draw a circle & name it as VSWR circle

    d) Draw a line diametrically opposite to A (zL) point to meet the VSWR circle point A (y point)

    e) Extend A to periphery, let that point be B

    f) Locate the intersection point of VSWR circle & G = 1 circle and name this point as C. Draw a line from O to C as OC & extend it to periphery. Let this point be C.

    To nd 'd':

    g) Move from point B to C in the clockwise direction (towards generator), so,

    8

    9

    dBC 0.1853

    c 3 10where 0.1m

    f 3 10

    = =

    = = =

    3-Micro waves and RadarDecember 2012.indd 493-Micro waves and RadarDecember 2012.indd 49 8/13/2013 1:21:30 PM8/13/2013 1:21:30 PM

  • December 2012 Microwave and RadarSet 2 -

    d = 0.1853

    d = 0.1853 0.1

    d = 0.01853mTo nd l:

    h) Find the value of susceptance at point C. i.e + 0.85

    3-Micro waves and RadarDecember 2012.indd 503-Micro waves and RadarDecember 2012.indd 50 8/13/2013 1:21:31 PM8/13/2013 1:21:31 PM

  • Microwave and Radar December 2012 Set 2 -

    To compensate this, stub is to be inserted at the point of 0.85 susceptance, mark this point as D, on the periphery. From the short circuit end move towards D.

    i.e PD 0.1308 0.1308 0.1m

    0.01308m

    = = =

    =

    ll

    l

    1. c. De ne re ection co-ef cient & derive an expression for re ection co-ef cient at load in terms of load impedance. (05 Marks)

    Ans: It is de ned as the ratio of amplitudes of re ected voltage to incident voltage at receiving end & is denoted by K.V(Z) = V+ e

    z + V ez (1)

    From de nition z

    z

    V eK

    V e

    ++

    =

    We know that

    zV V(Z) V e V e + == = +

    l ll l (2)

    L z lo o

    V e V eI I(z) |

    Z Z

    +

    == = l l

    (3)

    l

    l

    eL o

    V V e VZ Z

    I V e V e

    +

    + += =

    l

    l l

    l

    (4)

    Using componendo, dividendo method we get

    L o

    L o

    L O

    L O

    L O

    L O

    Z Z V e V e V e V e

    Z Z V e V e V e V e

    Z ZV eK

    V e Z Z

    Z ZK

    Z Z

    + +

    + +

    ++

    + += + + +

    = =

    +

    =

    +

    l l l l

    l l l l

    l

    l

    2. a. Using the Helmhotz equation, derive the eld equations or TE modes in rectangular waveguides. (09 Marks)

    Ans: The TEmn modes in a rectangular guide are characterized by EZ = 0From a given helmholtz equation 2Hz =

    2Hz

    Z

    b

    x

    y

    a

    3-Micro waves and RadarDecember 2012.indd 513-Micro waves and RadarDecember 2012.indd 51 8/13/2013 1:21:31 PM8/13/2013 1:21:31 PM

  • December 2012 Microwave and RadarSet 2 -

    A solution in the form of

    j z2 m m n n

    m x m x n y n yH A sin B cos C sin D cos e

    a a b b = + +

    (1)

    where Kx = m/a & ky = n/b are replaced.For a lossless dielectric, Maxwell's curl equations in frequency domain are E = jH (2) H = j !" In rectangular coordinates, their components are

    yzx

    EEj H

    y z

    =

    (4)

    x zy

    E Ej H

    z x

    =

    (5)

    y xz

    E Ej H

    x y

    =

    (6)

    yzx

    HHj E

    y z

    =

    (7)

    x zy

    H Hj E

    z x

    =

    (8)

    y xz

    H Hj E

    x y

    =

    (9)

    with substitution g zj & E 0,equationssimplified toz = =

    g Ey = Hx (10)

    g Ex = Hy (11)

    y y

    z

    E Ej H

    x y

    =

    (12)

    zg y x

    Hj H j E

    y

    + =

    (13)

    zg x y

    Hj H j E

    x

    =

    (14)

    y xH H

    0x y

    =

    (15)

    Solving these equations from (10) to (15) for Ex, Ey, Hx & Hy in terms of Hz will give the TE mode eld equations in rectangular waveguides as

    zx 2

    c

    HjE

    K y

    =

    (16)

    zy 2

    c

    HjE

    K x

    + =

    (17)

    3-Micro waves and RadarDecember 2012.indd 523-Micro waves and RadarDecember 2012.indd 52 8/13/2013 1:21:31 PM8/13/2013 1:21:31 PM

  • Microwave and Radar December 2012 Set 2 -

    Ez = 0

    g zx 2

    c

    j HH

    K x

    =

    (18)

    g zy 2

    c

    j HH

    K y

    =

    (19)

    j gZz m m n n

    m x n x n y n yH A sin B cos C sin D cos e

    a a b b = + +

    (20)

    2 2 2c gwhere k =

    Differentiating equation (20) w.r.t x & y then substituting results in (16) to (19) yield set of equations.

    zx n

    zy m

    z

    HSince E 0, 0 at y 0, Hence C 0

    y

    HSince E 0, then 0,at x 0 HenceA 0

    x

    HIt is concluded that, 0

    x

    = = = =

    = = = =

    =

    Magnetic eld in the +ve z direction is given by

    gj Z

    z oz

    m x n yH H cos cos e

    a b =

    (21)

    substituting equation (21) in (16) to (19) gives TEMN eld equations

    gj Z

    x ox

    m x n yE E cos sin e

    a b =

    (22)

    gj Z

    y oy

    m x n yE E sin cos e

    a b =

    (23)

    Ez = 0

    gj Z

    x ox

    m x n yH H sin cos e

    a b = (24)

    gj Z

    y oy

    m x n yH H cos sin e

    a b =

    (25)

    2. b. With a neat sketch, explain the 4 port microwave circulator & also obtain the s-matrix. (10 Marks)Ans:

    Port 4

    Port 2

    Port 3 Port 1

    3-Micro waves and RadarDecember 2012.indd 533-Micro waves and RadarDecember 2012.indd 53 8/13/2013 1:21:32 PM8/13/2013 1:21:32 PM

  • December 2012 Microwave and RadarSet 2 -

    Microwave circulator is a multiport waveguide junction in which the wave can ow only from the nth port to the (n + 1)th port in one direction.

    Four port CirculatorOne type of 4-port circulator is a combination of two 3dB side hole directional coupler & a rectangular waveguide with two non reciprocal phase shifters as shown below.

    Port 43 = 0

    0

    1 = 1800 2 = 180

    0