analysis of linear tapered waveguide by - · pdf file220 dwari,chakraborty,andsanyal sections....

Progress In Electromagnetics Research, PIER 64, 219–238, 2006

ANALYSIS OF LINEAR TAPERED WAVEGUIDE BYTWO APPROACHES

S. Dwari, A. Chakraborty, and S. Sanyal

Department of Electronics and Electrical Communication EngineeringIndian Institute of TechnologyKharagpur-721302, India

Abstract—This paper presents the analysis of linear taperedwaveguide. Voltage-standing-wave-ratio (VSWR) is obtained fromtransmission matrix of the taper waveguide. Taper section is dividedinto number of section having uniform length. Transmission matrix oftaper waveguide is found by multiplication of transmission matrix ofeach section. Transmission matrix of each section is obtained as theproduct of three matrices. One is of the initial length of transmissionline, second one is due to discontinuity and third one is of the finallength of transmission line. Transmission matrix of discontinuityis obtained by two methods. One is by equivalent circuit of stepdiscontinuity and another is by moment method. The results are seento be in good agreement with [1, 2] and [3].

1. INTRODUCTION

In a tapered section the dimension of waveguide varies smoothly andfor this reason there are possibilities of providing a transition fromone impedance level to another. Ref. [4] and then [5] analyzed taperline by dividing it into a number of sections of equal length into thedirection of propagation. Ref. [1] found the transmission matrix of eachsection as the product of five matrices and multiplied the matrices ofall sections.

This paper proposed that the transmission matrix of each sectionis obtained as the product of three matrices. They are contributedby initial length of transmission line, discontinuity and the finallength of transmission line. Transmission matrix of discontinuity isobtained by two procedures. They are by equivalent circuit of stepdiscontinuity and by moment method. The final transmission matrixof tapered waveguide is obtained by multiplication of matrices of all

220 Dwari, Chakraborty, and Sanyal

sections. Voltage-standing-wave-ratio (VSWR) is obtained from finaltransmission matrix.

2. TRANSMISSION MATRIX OF TAPER SECTION

The linear tapered waveguide is shown in Figure 1 and it can beconsidered as if it is formed with huge number of step discontinuities.Let the number of sections be N and the length of taper be L.Let 2a0, 2b0 are initial broad and narrow dimensions respectively and2a1, 2b1 are respective final broad and narrow dimensions.

Figure 1. (a) Tapered waveguide of length L connecting uniformwaveguides of dimensions 2a0, 2b0 and 2a1, 2b1. (b) Taper waveguideof Figure 1(a) divided into sections. (c) Expanded view of nth section.

The nth section is blown up and shown in Figure 1(c). Eachsection consists of an initial length of waveguide, a step discontinuityand a final length of waveguide. Each of the constituents can beexpressed by a transmission matrix of order 2.

For nth section initial and final broad dimensions are

2a0n = 2a0 +2(a1 − a0)

N(n− 1) (1)

2a1n = 2a0 +2(a1 − a0)

Nn (2)

Progress In Electromagnetics Research, PIER 64, 2006 221

Narrow wall dimensions are

2b0n = 2b0 +2(b1 − b0)

N(n− 1) (3)

2b1n = 2b0 +2(b1 − b0)

Nn (4)

The matrix of each section is the product of three matrices. Letmatrix [T1] is for initial length, [T3] is of final length and [T2] is fordiscontinuity. So transmission matrix of each section is given by

[T ] = [T1][T2][T3] (5)

The transmission matrices of all the sections are multiplied to obtainthe final transmission matrix of the taper.

Suppose d = LN is the length of each section, k0n is the propagation

constant of initial length and k1n is the propagation constant of finallength of nth section. Then [T1] can be written as

[T1] =

[ejk0n

d2 0

0 e−jk0nd2

](6)

[T3] can be written as

[T3] =

[ejk1n

d2 0

0 e−jk1nd2

](7)

Let Γ1n be the reflection co-efficient in the forward direction atdiscontinuity of nth section and Γ2n be the reflection co-efficient inthe backward direction at discontinuity of the nth section. Then [T2]can be written as

[T2] =1

1 + Γ1n

[1 −Γ2n

Γ1n 1 + Γ1n + Γ2n

](8)

To obtain [T2], reflection co-efficients should be determined. Reflectioncoefficient can be determined by two ways. One is using equivalentcircuit representing step discontinuities shown in Figure 2 as describedin [6]. Second is by using method of moment.

3. DETERMINATION OF CIRCUIT PARAMETERS OFEQUIVALENT CIRCUIT FOR STEP DISCONTINUITIES

Discontinuity of each section of linear taper waveguide is stepdiscontinuity. This step discontinuity can be represented by equivalent


Figure 2. Equivalent circuit of discontinuity.

Figure 3. Cross section of waveguide junction for step discontinuityat narrow dimension.

circuit as shown in Figure 2. Following [6] the circuit parameters of theequivalent circuit can be obtained. From circuit parameters reflectioncoefficients can be calculated. Evaluation of circuit parameters ofequivalent circuit for step discontinuities (1) in narrow dimension onlyand (2) in both narrow & broad dimension are as follows:

3.1. Evaluation of Circuit Parameters of Equivalent Circuitfor Step Discontinuities in Narrow Dimension

Consider the waveguide junction as shown in Figure 3. Let junction isat z = 0. Waveguide of dimension 2a× 2b extends up to z = − ∝ andwaveguide of dimension 2a × 2(b +W ) extends upto z = + ∝. Thedimensions are such that only dominant mode can propagate in eachsection. Let there be an incident wave from −z direction.

In the region z < 0 transverse fields are

E−t =

(e−jβz + Γejβz

) V0

1 + Γe0 +

∑i

Vieαizei

H−t = Y −

0

(e−jβz − Γejβz

) V0

1 + Γh0 −

∑i

YiVieαizhi (9)

where ei, hi are the mode vectors, αi are the cutoff mode-attenuationconstants, Yi are the characteristics admittances, and Γ is the reflectioncoefficient for the dominant mode. The subscript 0 signifies the


dominant mode. It is assumed that at z =∝ matched load is there. Soin the region z > 0

E+t = V0e

−jβz e0 +∑i

Vie−αiz ei

H+t = Y +

0 V0e−jβzh0 +

∑i

YiVie−αizhi (10)

where the carets are used to differentiate the above parameters fromthere z < 0 counterparts. At junction region∫

z=0

∫E+ ×H+ · ds =

∫z=0

∫E− ×H− · ds (11)

So

Y +0 V

20 +

∑i

YiV2i =

1 − Γ1 + Γ

Y −0 V

20 −

∑i

YiV2i (12)

Relative admittance observed from z < 0 is1 − Γ1 + Γ

=Y

Y −0

=G

Y −0

+ jB

Y −0

(13)

Y0 is real and Yi are imaginary. For real Vi and Vi

jB

Y −0

=

∑i

YiV2i +

∑i

YiV2i

Y −0 V

20

(14)

G

Y −0

=Y +

0 V20

Y −0 V

20

(15)

For equivalent circuit of Figure 2, with matched condition at z =∝, itis evident that

G

Y −0

= n2Y+0

Y −0

(16)

From Eq. (15) and Eq. (16)

n2 =V 2

0

V 20

(17)

For the junction shown in Figure 3, the dominant mode vectors [6] are

�e0 = uy

√1

2absin

π

2a(x+ a) (18)

�e0 = uy

√1

2a(b+W )sin

π

2a(x+ a) (19)


Assumed tangential electric field in the aperture

�Eat = uy sin

π

2a(x+ a) (20)

Therefore following [6]:

V0 =a∫

x=−a

b∫y=−b

1√2ab

sin2 π(x+ a)2a

dxdy =√

2ab (21)

V0 =a∫

x=−a

b∫y=−b

1√2a(b+W )

sin2 π(x+ a)2a

dxdy =

√2a

b+Wb (22)

Therefore

V0

V0=

√b

b+W(23)

n2 =b

b+W(24)

This is the transformation ratio of the transformer of equivalent circuit.Now the first summation in the numerator of Equation (14) is zero andsecond summation is related to aperture susceptance Ba by∑

i

YiV2i = j|V |2Ba = j4b2Ba (25)

Now

V0 =a∫

x=−a

b∫y=−b

sinπ(x+ a)

2a1√2ab

sinπ(x+ a)

2adxdy =

√2ab (26)

SoB =

2Bab

a(27)

3.1.1. Determination of Aperture Susceptance Ba

Let Ex = 0 and Ey = f(x, y) be known over the cross section at z = 0.For TE to x mode at z > 0 scalar potential function is:

ψ =∞∑

m=1

∞∑n=0

Amn sinmπ

2a(x+ a) cos

nπ

2(b+W )(y+ b+W )e−γmnz (28)


where Amn are mode amplitudes and γmn are the mode propagationconstants. In particular Ey at z = 0 is given by

Ey|z=0 =∞∑

m=1

∞∑n=0

γmnAmn sinmπ

2a(x+a) cos

nπ

2(b+W )(y+b+W ) (29)

This is the form of double Fourier series: a sine series in x and a cosineseries in y. It is thus evident that γmnAmn are fourier coefficients ofEy, or

γmnAmn = Emn =εn

2a(b+W )

a∫−a

(b+W )∫−(b+W )

Ey|z=0 sinmπ

2a(x+ a)

× cosnπ

2(b+W )(y + b+W )dxdy (30)

where εn = 1 for n = 0 and εn = 2 for n > 0 (Numann’s number). TheAmn and hence the field are now evaluated. The z directed complexpower at z = 0 is

P =∫

z=0

∫ (�E × �H∗

)· uzds

= −a∫

−a

(b+W )∫−(b+W )

[EyH∗x]z=0 dxdy

=a∫

−a

(b+W )∫−(b+W )

[∑m,n

Emn sinmπ(x+ a)

2acos

nπ(y + b+W )2(b+W )

]

∑p,q

k2 −(pπ

2a

)2

jωµγ∗pqE∗pq sin

pπ(x+ a)2a

cosqπ(y + b+W )

2(b+W )

dxdy

(31)

This reduces to

P =∞∑

m=1

∞∑n=0

(Y0)∗mn|Emn|22a(b+W )εmn

(32)


Assume

Ey|z=0 =

sin

π(x+ a)2a

−b < y < b0 |y| > b

(33)

Only non zero amplitudes are:

E10 = γ10A10 =b

b+W(34)

E1n = γ1nA1n=2nπ

[sin

nπ

b+W

(b+

W

2

)− sin

(nπW

2(b+W )

)](35)

Therefore

P =2ab2

b+W

(Y0)∗10 + 8

∞∑n=1

(Y0)∗1n

sin(nπb

b+W

)cos

(nπ

2

)nπb

b+W

2 (36)

Now

(Y0)10 =k2 −

(π

2a

)2

ωµβ=

√1 −

(fcf

)2

η(37)

(Y0)1n =k2 −

(π

2a

)2

−jωµα =j4(b+W )(Y0)10

λg

√√√√n2 −(

4(b+W )λg

)2(38)

Therefore

P ∗

|V |2 = (Y0)10

a

2(b+W )+ j

16aλg

∞∑n=1

sin2

(nπb

2(b+W )

)cos2

(nπ

2

)(nπb

b+W

)2√√√√n2−

(4(b+W )λg

)2

(39)


Figure 4. Cross section of waveguide junction for step discontinuityat both broad and narrow dimension.

where V = 2b. The imaginary part of this is the aperture susceptanceBa.

Ba =16aλgZ0

∞∑n=1

sin2

(nπb

2(b+W )

)cos2

(nπ

2

)(nπb

b+W

)2√√√√n2−

(4(b+W )λg

)2(40)

3.2. Evaluation of Circuit Parameters of Equivalent Circuitfor Step Discontinuities at Both Broad and NarrowDimension

Consider the waveguide junction as shown in Figure 4. Let junction isat z = 0. Waveguide of dimension 2a× 2b extends upto z = − ∝ andwaveguide of dimension 2(a+W1) × 2(b+W ) extends upto z = + ∝.The dimensions are such that only dominant mode can propagatein each section. Let there be an incident wave from −z direction.Following similar procedure as described earlier it can be shown:

n2 =16(a+W1)3ab

π2(b+W )W 21 (2a+W1)2

cos2πa

2(a+W1)(41)

B =2Bab

a(42)

Ba =∞∑

m=1

∞∑n=0

4a2λ2

[k2−

(mπ

2(a+W1)

)2]

(b+W )

n2π6b2η(a+W1)

1

1−(ma

a+W1

)2

2


Figure 5. Cross section of waveguide junction for step discontinuityat narrow dimension.

sin2

(mπ

2

)cos2

(nπ

2

)cos2

(mπa

2(a+W1)

)sin2

(nπb

2(b+W )

)√(

mλ

2(a+W1)

)2

+(

nλ

2(b+W )

)2

− 1

(43)

4. REFLECTION COEFFICIENT FOR STEPDISCONTINUITIES BY METHOD OF MOMENT

Method of moment can be used for the analysis of waveguidediscontinuities. Evaluation of reflection coefficient using momentmethod for step discontinuities (1) in narrow dimension only and (2)in both narrow & broad dimension are as follows:

4.1. Reflection Coefficient for Step Discontinuities inNarrow Dimension by Method of Moment

Consider the waveguide junction as shown in Figure 5. Let junctionis at z = 0. Waveguide of dimension 2a × 2b1 extends up to z = − ∝and waveguide of dimension 2a × 2b2 extends upto z = + ∝. It isassumed that b2 > b1. The dimensions are such that only dominantmode can propagate in each section. Let there be an incident wavefrom −z direction. In the present analysis, the following assumptionsare made:

• the x component of the electric field at the plane of the apertureis ignored;

• only the x component of the magnetic field at the aperture planeis considered;

• the excitation in the feed waveguide is TE10 dominant wavewith the incident electric field y-directed and uniform in thesame direction, the assumption of uniform y-directed electric


field having no component in the x-direction follows as a naturalconsequence;

• the electric field Ey in the aperture is assumed to vary only in thex-direction and is constant in the y-direction.

At the region of window the tangential component of the magneticfield on both sides should be identical. In this analysis two sourcesproducing these fields — the source in the waveguide exciting theTE10 mode and the magnetic current source at the discontinuity areconsidered. Using the principle of superposition, the x components ofthe magnetic field at the plane of discontinuity are derived.

When aperture is shortened (no magnetic current source) the xcomponent of magnetic fields are denoted by HI/1

x and HII/1x on guides

with narrow dimensions 2b1 & 2b2 respectively.When the generator is shortened the x component of magnetic

fields are denoted byHI/2x andHII/2

x on guides with narrow dimensions2b1 & 2b2 respectively.

At the plane of discontinuity using principle of superposition it ispossible to write

HI/1x +HI/2

x = HII/1x +HII/2

x (44)

Electric field at plane of discontinuity is considered as

�Edis = uy

M∑p=1

E′pe

′p, p = 1, 2, . . . . . . ,M. (45)

Where basis function e′p, (p = 1, 2, . . . . . . ,M) are defined by

e′p =

sin

pπ(x+ a)2a

,−a < x < a−b1 < y < b1

0, otherwise(46)

As in [6] internally scattered electric field is given by

�Escat =∑m

∑n

V emn�e

emn + V m

mn�emmn (47)

Magnetic fields are related to the electric fields as follows:

Hex = Y e

mn1Eey

Hmx = Y m

mn1Emy

}for wave propagating in −z direction. (48)

Hex = −Y e

mn2Eey

Hmx = −Y m

mn2Emy

}for wave propagating in +z direction. (49)


Where Y emn1 & Y m

mn1 are the characteristic admittances of TEmn &TMmn modes for the waveguide of −z direction Y e

mn2 & Y mmn2 are the

characteristic admittances of TEmn & TMmn modes for the waveguideof +z direction.

For waveguide of −z direction modal vectors are given by

�e emn =1π

√ab1εmεn

(mb1)2 + (na)2

[ux

(nπ

2b1

)cos

{mπ

2a(x+ a)

}

× sin{nπ

2b1(y + b1)

}− uy

(mπ

2a

)sin

{mπ

2a(x+ a)

}

× cos{nπ

2b1(y + b1)

}](50)

�emmn = − 2π

√ab1

(mb1)2 + (na)2

[ux

(mπ

2a

)cos

{mπ

2a(x+ a)

}

× sin{nπ

2b1(y + b1)

}+ uy

(nπ

2b1

)sin

{mπ

2a(x+ a)

}

× cos{nπ

2b1(y + b1)

}](51)

Similarly for waveguide of +z direction modal vectors are given by

�e emn =1π

√ab2εmεn

(mb2)2 + (na)2

[ux

(nπ

2b2

)cos

{mπ

2a(x+ a)

}

× sin{nπ

2b2(y + b2)

}− uy

(mπ

2a

)sin

{mπ

2a(x+ a)

}

× cos{nπ

2b2(y + b2)

}](52)

�emmn = − 2π

√ab2

(mb2)2 + (na)2

[ux

(mπ

2a

)cos

{mπ

2a(x+ a)

}

× sin{nπ

2b2(y + b2)

}+ uy

(nπ

2b2

)sin

{mπ

2a(x+ a)

}

× cos{nπ

2b2(y + b2)

}](53)

where εm, εn are the Neumann’s number satisfying

εi =

{1 for i = 02 otherwise

(54)


Following [6] modal voltages of waveguide of −z direction are given by

V emn =

∫aperture

∫�Edis · �e emnds

=mb1 sin(nπ)

nπ

√ab1εmεn

(mb1)2+(na)2

M∑p=1

E′p

[sin(p+m)ππ(p+m)

− sin(p−m)ππ(p−m)

]

(55)

V mmn =

∫aperture

∫�Edis · �emmnds

=2a sin(nπ)

π

√ab1

(mb1)2+(na)2

M∑p=1

E′p

[sin(p+m)ππ(p+m)


]

(56)

Similarly modal voltages of waveguide of +z direction are given by

V emn =

mb2nπ

√ab2εmεn

(mb2)2 + (na)2

M∑p=1

E′p

[sin(p+m)ππ(p+m)


]

×[sin

nπ(b2 + b1)2b2

− sinnπ(b2 − b1)

2b2

](57)

V mmn =

2aπ

√ab2

(mb2)2 + (na)2

M∑p=1

E′p

[sin(p+m)ππ(p+m)


]

×[sin

nπ(b2 + b1)2b2


2b2

](58)

NowHI/1

x = 2H incx = −2Y0 cos

πx

2a(59)

Where H incx is the x component of magnetic field due to incident TE10

mode and Y0 is the characteristic admittance of the line due to TE10

mode.Now

HII/1x = 0 (60)

HI/2x = −

∑m

∑n

M∑p=1

E′p[sin c(p+m)π−sin c(p−m)π]

[ab1

(mb1)2+(na)2

]

×(sin c(nπ))

[m2b1εmεnY

emn1

2a+

2n2a

b1Y mmn1

]


× sin{mπ

2a(x+ a)

}cos

{nπ

2b1(y + b1)

}(61)

HII/2x =

∑m

∑n

M∑p=1

E′p [sin c(p+m)π−sin c(p−m)π]

[ab2

(mb2)2+(na)2

]

×(

1nπ

) [m2b2εmεnY

emn2

2a+

2n2a

b2Y mmn2

]

×[sin

nπ(b2 + b1)2b2


2b2

]

× sin{mπ

2a(x+ a)

}cos

{nπ

2b2(y + b2)

}(62)

Therefore continuity equation will be

HI/1x +HI/2

x = HII/2x (63)

The weighting function Wq is selected to be the same form as the basisfunction.

Wq =

sin

qπ(x+ a)2a

,−a < x < a−b1 < y < b1

0, otherwise(64)

where q = 1, 2, . . . . . . ,M .Inner multiplication is defined as

< H,Wq >=∫

Aperture at discontinuity

∫H ·Wq dxdy (65)

So ⟨HI/1

x ,Wq

⟩+

⟨HI/2

x ,Wq

⟩=

⟨HII/2

x ,Wq

⟩(66)

orL1 + L2 = L3 (67)

Therefore

L1 =4Y0ab1π

[1

q + 1cos

π

2(2q + 1) +

1q − 1

cosπ

2(2q − 1)

](68)

L2 = 2M∑p=1

E′p

[∑m

∑n

[m2b1εmεnY

emn1

2a+

2n2aY mmn1

b1

]

×[

a2b21(mb1)2+(na)2

](sin(nπ))2 [sin c(p+m)π−sin c(p−m)π]


× [sin c(q +m)π − sin c(q −m)π]] (69)

L3 = −2M∑p=1

E′p

[∑m

∑n

[m2b2εmεnY

mmn2

2a+

2n2aY mmn2

b2

]

×[

a2b22(mb2)2 + (na)2

](1nπ

)2 [sin

nπ(b2+b1)2b2

−sinnπ(b2−b1)

2b2

]2

× [sin c(p+m)π−sin c(p−m)π] [sin c(q+m)π−sin c(q−m)π]

]

(70)

To solve the unknown coefficients E′p(p = 1, 2, . . . . . . ,M), Equation

(67) can be written in the matrix form for all p and q as below:

[A1] + [A2][E′p

]= [A3]

[E′p

](71)

or,[E′p

]= {[A3] − [A2]}−1 [A1] (72)

Ei = Incident electric field = cosπx

2a(73)

ER = Reflected electric field

= − cosπx

2a−

M∑p=1

E′p [sin c(p+ 1)π − sin c(p− 1)π] sin

π(x+a)2a

(74)

Reflection coefficient is given by

Γ1 = −1 −M∑p=1

E′p [sin c(p+ 1)π − sin c(p− 1)π] (75)

Similarly if there be an incident wave from +z direction then

Ei = Incident electric field = cosπx

2a(76)

ER = Reflected electric field

= − cosπx

2a− b1b2

M∑p=1

E′p [sin c(p+1)π − sin c(p−1)π] sin

π(x+a)2a

(77)


Figure 6. Cross section of waveguide junction for step discontinuityat both broad and narrow dimension.

Reflection coefficient is given by

Γ2 = −1 − b1b2

M∑p=1

E′p [sin c(p+ 1)π − sin c(p− 1)π] (78)

4.2. Reflection Coefficient for Step Discontinuities at BothBroad and Narrow Dimension by Method of Moment

Consider the waveguide junction as shown in Figure 6. Let junction isat z = 0. Waveguide of dimension 2a1 × 2b1 extends up to z = − ∝and waveguide of dimension 2a2 × 2b2 extends upto z = + ∝. It isconsidered that a1 > a2 and b2 > b1. The dimensions are such thatonly dominant mode can propagate in each section. Following similarprocedure as described in previous section it can be shown:

If there be an incident wave from −z direction then reflectioncoefficient:

Γ1 = −1 − a2

a1

M∑p=1

E′p

[cos

π(p+ 1)2

sin cπ(pa1 + a2)

2a1

− cosπ(p− 1)

2sin c

π(pa1 − a2)2a1

](79)

If there be an incident wave from +z direction then reflectioncoefficient:

Γ2 = −1 − b1b2

M∑p=1

E′p [sin c(p+ 1)π − sin c(p− 1)π] (80)


Figure 7. VSWR versus number of sections for tapering in narrowdimensions.

5. RESULTS

From final transmission matrix of taper waveguide reflection co-efficient and the VSWR are calculated. For tapering in narrowdimension considering 2a0 = 5.81 cm, 2b0 = 1.20 cm, 2a1 = 5.81 cm,2b1 = 2.91 cm, f = 3.96 GHz, and L = 4 cm, VSWR is plottedagainst number of sections N in Figure 7 using equivalent circuitrepresentation of step discontinuities. For large number of sectionsVSWR is converged. In convergence region the plot of VSWR withtaper length is shown in Figure 8 along with previously publishedresults.

Calculating reflection coefficient of step discontinuities by momentmethod, for tapering in narrow dimension with same dimensionalparameter as above, the plot of VSWR against length is shownin Figure 9 along with previously published results. Similarly fortapering in both dimension with 2a0 = 0.90 inch, 2b0 = 0.40 inch,2a1 = 0.75 inch, 2b1 = 0.60 inch, f = 8.7 GHz and L = 2.85 inch,VSWR is plotted against number of sections N in Figure 10 usingequivalent circuit representation of step discontinuities. In convergenceregion the plot of VSWR with frequency is shown in Figure 11 alongwith previously published results.

Calculating reflection coefficient of step discontinuities by momentmethod, for double taper section with same dimensional parameter asabove, the plot of VSWR against frequency is shown in Figure 12 alongwith previously published results.


Figure 8. VSWR versus taper length of a taper at a fixedfrequency of 3.96 GHz. A- Theoretical curve by Johnson’s [3] method.B- Theoretical curve by chakraborty & Sanyal’s [1] method. C-Experimental points as reported by Matsumaru [2]. D- By equivalentcircuit.

Figure 9. VSWR versus taper length of a taper at a fixedfrequency of 3.96 GHz. A- Theoretical curve by Johnson’s [3] method.B- Theoretical curve by chakraborty & Sanyal’s [1] method. C-Experimental points as reported by Matsumaru [2]. D- By momentcircuit.


Figure 10. VSWR versus number of sections for double taper section.

Figure 11. VSWR versus frequency for a linear double taper. A-Theoretical curve by Johnson’s [3] method. B- Theoretical curveby chakraborty & Sanyal’s [1] method. C- Experimental points asreported by Johnson [3]. D- By equivalent circuit.

Figure 12. VSWR versus frequency for a linear double taper. A-Theoretical curve by Johnson’s [3] method. B- Theoretical curveby chakraborty & Sanyal’s [1] method. C- Experimental points asreported by Johnson [3]. D- By moment circuit.


6. CONCLUSION

The results obtained in the presented methods are to be in goodagreement with results of [1, 2] and [3]. Here it is assumed that onlyTE10 mode is propagating along the tapered line. Even if other modesare generated they are not supported by the structure.

REFERENCES

1. Chakraborty, A. and G. S. Sanyal, “Transmission matrix of alinear double taper in rectangular waveguides,” IEEE Trans.Microw. Theory Tech., Vol. MTT-28, No. 6, 577–579, Jun. 1980.

2. Matsumaru, K., “Reflection coefficient of E-plane taperedwaveguides,” IRE Trans. Microw. Theory Tech., Vol. MTT-6,143–149, Apr. 1958.

3. Johnson, R. C., “Design of linear double tapers in rectangularwaveguides,” IRE Trans. Microw. Theory Tech., Vol. MTT-7,374–378, July 1959.

4. Ghosh, R. N., Microwave Circuit Theory and Analysis, Ch. 12,352–355, McGraw-Hill, New York, 1963.

5. Mallick, A. K. and G. S. Sanyal, “Electromagnatic wavepropagation in a rectangular waveguide with sinusoidally varyingwidth,” IEEE Trans. Microw. Theory Tech., Vol. MTT-26, 243–249, Apr. 1978.

6. Harrington, R. F., Time Harmonic Electromagnetic Fields,McGraw-Hill, New York, 1961.

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