Parallel Plate Waveguide

4- A parallel-plate waveguide having plate width and plate separation 1 cm is filled with Teflon ( 2.08rε = ). The waveguide is oriented along the z-direction. The center conductor of a coaxial cable is extended into space between the plates in the z-direction, exciting a mode(s) propagating in the z-direction with field components 0zE ≠ . The source does not excite TEM mode. a) What type of mode(s) are propagating? TM modes are propagating b) Write down the dispersion equation.

( )2

22 2

0 02.08 100

nk n

d

πβ ω µ ε π

= − = −

c) What is the usable frequency range for single-mode propagation?

( ) ( )

( ) ( )

st

nd

is TEM mode

is1 TM mode

GHz, GHz

is 2 TM mode

GHz GHz

usable freq range is fr

2(1)

22 (1) (1)

0 0 2

0 0

2(1)

22 (2) (2)

0 0 2

0 0

0

1

1002.08 100 65.32 10.90

22.08

2

2002.08 100 130.64 , 20.80

22.08

c

c c

c

c c

n

n

f

n

f

π ωω µ ε π ω

πω µ ε

π ωω µ ε π ω

πω µ ε

=

=

= ⇒ = = = =

=

= ⇒ = = = =

om10.9 GHz to 20.8GHz

d) What happens at sufficiently high frequencies (outside if the range obtained in part (b))? At frequencies above 20.8 GHz additional higher order modes will propagate.

4-(a) İçi hava dolu paralel plaka bir dalga kılavuzunun 5 GHz’te hiçbir TE ve TM modunun uyarılmaması için gereken plaka aralığı ne olmalıdır? b) Plakanın içi 9rε = dielektrigi ile dolu olursa simdi bu frekansta hangi TE ve TM modları uyarılır?

4- (a) Determine the height d of an air filled parallel-plate wave-guide so that the signals at

all frequencies below 5 GHz will propagate in the zero order mode ( 0 0TE ,TM ) but no higher order mode (TE ,TM , 1,2, 3,,,,n n n = ) will propagate for frequencies below 5 GHz.

(b) If the air dielectric in this guide is changed to a ceramic material of relative permittivity 9rε = , determine which modes can propagate at this operating frequency of 5 GHz.

9

1

0 0

1

8

1 1 2

2 2

a) 5GHz, 1 5 10 3cm2 2 2

b) 9, 100 , real 100 0.03 3

So only 1,2modeswill propagate (TE ,TE,TM,TM)

3 10(TE ,TM ) 1.6666

2 2 2 3 10 3

(TE ,TM

cn c

r

r

c

r r

c

n nc cf f n d

d d d

kdk n n n

c

n

nc cf GHz

d d

f

µ ε

ω εε ω µε π β π

π

ε ε −

= = → = = → × = → =

= = = = > ⇒ → × > → >

=

×= = = =

× × ×8

2

8

3 3 2

8

4 4 2

2 2 3 10) 3.3333

2 2 2 3 10 3

3 3 3 10(TE ,TM ) 5

2 2 2 3 10 3

4 4 3 10(TE ,TM ) 6.6666

2 2 2 3 10 3

r r

c

r r

c

r r

nc cGHz

d d

nc cf GHz

d d

nc cf GHz

d d

ε ε

ε ε

ε ε

−

−

−

× ×= = = =

× × ×

× ×= = = =

× × ×

× ×= = = =

× × ×

4- A parallel-plate waveguide has plate separation a =1 cm and is filled with teflon, having relative permittivity εr = 2.1 . a) Determine the max operating frequency such that only the TEM mode will propagate.

GHz1

0 0

110.35

2 2C

r r

cf f

a aε ε µ ε≤ = = =

b) If the operating wavelength is λ =5 mm how many and which modes will propagate?

1 1 2 2 3 3

GHz

GHz

GHz

GHz

GHz

GHz 7 modes will propagate (TEM, TE , TM , TE , TM , TE , TM )

0

1

2

3

4

1 2 3

0 0

141.4

0 0

1 10.35

2 20.7

3 31.05

4 41.4

, , 41.4

r r r

C

C

C

C

C

C C C

c cf V f

m f

m f

m f

m f

m f

f f f f

λε ε µ ε λ ε

= = = → = =

= → =

= → =

= → =

= → =

= → =

< = ⇒

6- A parallel-plate waveguide with plate separation a = 2 mm is partially filled with two dielectrics with relative permittivities εr1 = 4 and εr2 = 2.1. At a certain frequency, it is found that the TM1 mode propagates through the guide without suffering any reflective loss at the dielectric interface. Find this frequency. Hint: Remember Brewster’s angle?

5-The fields for parallel plate waveguide is given as

2 2

2 2

z z z zx x

c c

z z z zy y

c c

j E H j E HE H

k x y k y x

j E H j E HE H

k y x k x y

β ωµ ωµ β

β ωµ ωµ β

∂ ∂ ∂ ∂ = − + = − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = − + = − − ∂ ∂ ∂ ∂

a) For TM waves in a parallel plate waveguide derive Ez. Assume jγ β= .

b) What are the expressions for Ex and Hy for the parallel plate guide?

5- The cutoff frequency of the TE2 and TM2 modes in a parallel-plate waveguide is 100 GHz. If the plate separation is 1mm, determine the dielectric constant of the medium between the plates.

2

2

2

29

2c r

cr r

c c cf

afa aε

ε ε

= = → = =

1- 0 < f < 2 GHz arasında TEM modları propagate edecek şekilde parallel plaka dalga kılavuzu tasarlanacaktır. Plakalar arasındaki dielektrik malzemenin dielektrik sabiti 9’dur. a) Plakalar arasındaki mesafe en çok ne kadar olmalıdır? b) Bu mesafe 2.1 cm, ve f = 10 GHz ise hangi TEn and TMn modları propagate eder. 1- A parallel plate waveguide is to be designed so that only TEM modes can propagate in the frequency range 0 < f < 2 GHz. The relative permittivity and permeability of the material between the plates is 9 and 1, respectively. a) What is the maximum allowed spacing between the parallel plate waveguides plates?

b) If the plate spacing is 2.1 cm, and f = 10 GHz, what TEn and TMn modes will propagate?

1- True or False: a. The cutoff frequency for TEM waves is f = 0. True b. All waveguides have a TEM wave. False c. Waves below cutoff have no physical meaning. False d. In general, as the permittivity of the material filling a waveguide increases, the cutoff frequency of a given mode increases. False e. Phase velocity, must be less than or equal to the speed of light. False

Rectangular Waveguide 1- Boyutları a = 2.3 cm ve b = 1.2 cm olan ve TE10 modunda içi hava dolu dikdörtgensel bir dalga kılavuzundaki maksimum elektrik alan genliği 62 10 V m× ise bu kılavuzun taşıyacağı maksimum gücü hesaplayın. Kılavuzdaki frekans TE10 kesim frekansının 1.3 katıdır. 1- Assume an X-band air-filled rectangular waveguide of dimensions a=2.3cm and b=1.2cm. The operating mode is dominant TE10 mode. Calculate the maximum power that can be transmitted in the waveguide if the electric field in air is 62 10 V m× . The operating frequency is 30 percent higher than the cut-off frequency for the TE10 mode.

10

10 10

22

10 0 02

60 0 TE2 2 2

22

20

0 0TE

TE mode 0, sin sin ,

120 1202 10 , 590

1 ( ) 1 (1 1.3)

sin2 2

z z

x y

c

b ay

avy x

TE

jE E H x e jE x e h

h a a a a

E Hh a f f

E EP dxdy

γ γωµ π π π π

ωµ π π πη

π

η η

− −

= =

= = − = − =

= = × = = = Ω − −

= =∫ ∫10

20

0 0TE

350.8 kW2 2

b a

y x

x E abdxdy

a η= =

= = ∫ ∫

6- 10 GHz’te TE10 modunda çalışan dikdörtgensel dalga kılavuzunun boyutları 0.9a = in ve

0.4b = in’tir. A şağıdaki şekilde görüldüğü gibi bu kılavuzun sol tarafı hava sağ tarafı ise dielektrik sabiti 2.2 olan malzeme ile doludur. Ara yüzeydeki yansıma katsayısını hesaplayın. 6- An X-band rectangular waveguide having dimensions 0.9a = in and 0.4b = in, carries the TE10 mode at 10 GHz. Suppose part of this guide is filled with a lossless dielectric with a dielectric constant of 2.2, as shown below. Find the reflection coefficient at the interface between the two sections. Assume only the dominant mode propagates in each section.

(1) (1) 0

2 2

0(2) (2)

2 2

(2) (1)

(2) (1)

1206.56 GHz, 500

2 1 ( ) 1 (6.56 10)

120 2.24.42 GHz, 283

2 1 ( ) 1 (6.42 10)

283 5000.277

283 500

c

c

TE

c

r

TE

r c

TE TEin

TE TE

cf Z

a f f

cf Z

a f f

Z Z

Z Z

η π

η ε π

ε

= = = = = Ω− −

= = = = = Ω− −

− −Γ = = = −

+ +

1- An air filled rectangular waveguide having the dimensions a=2.286 cm and b=1.016 cm carries the dominant TE10 mode at 10 GHz. a) Find the cutoff frequency, guide wavelength, the phase velocity, and the wave impedance for this mode.

0

2

8 0TE2 2

3cm6.56GHz, 4 cm

2 0.7751 ( )

1204 10 m s, 500

0.775 0.7751 ( ) 1 ( )

c g

c

p

c c

cf

a f f

c cV Z

f f f f

λλ

η π

= = = = ≅−

= = = × = = = Ω− −

b) What is the new wave impedance for the dominant mode if the waveguide is now filled with a material with 2rε = ?

0

TE 2

6.56GHz 266.64.64GHz, 301

2 2 0.886 0.8861 ( )

r

c

r c

cf Z

a f f

η εη

ε= = = = = = = Ω

−

2- The transverse electric field in the dominant TE10 mode of a rectangular waveguide is given by 0 sin( )yE E x aπ= , where a is the guide width and b is the guide height, as usual.

The wave travels in the (+) z direction. Denoting the wave impedance by TE 0Z ωµ β= ,

a) Write an expression for the transverse magnetic field xH , in terms of 0E .

b) Although there is also a z-component of magnetic field, it does not contribute to power flow in the waveguide. Write an expression for the power density ( 2W m ) in the waveguide

(ie. the Poynting vector aveS ). c) Using your answer for (b), derive an expression for the total average power (in Watts)

passing through a rectanguilar waveguide. You will need 2 1 12 4sin sin2x dx x x= −∫

d) Consider now an air filled rectangular waveguide having the dimensions a=2.286 cm and b=1.016 cm operating in the dominant TE10 mode at 10 GHz. What is the maximum power that can be transmitted through this waveguide without inducing dielectric breakdown, which occurs at a field strength of 610 V m .

0

TE 0

220

TE

22 2 2

2 20 0 0

0 0 0TE TE TE

2

sin

1 1Re sin

2 2

1sin sin

2 2 4

6.56GHz, 1 ( ) 158 rad m2

y

x

ave

a b

aveS

c c

E E xH

Z a

E x

Z a

E x b E a ab EP d dydx udu

Z a Z Z

cf f f

a c

π

π

β π

ωµ

π

π

π

ωβ

∗

− = = −

= × =

= ⋅ = = =

= = = − =

∫∫ ∫ ∫ ∫

S E H

S S

TE max, 500 116kWZ Pωµ

β= = Ω ⇒ =

2-TE modunda uyarılan içi dielektrik malzeme dolu bir dikdörtgensel dalga kılavuzunun boyutları a = 5 cm ve b = 3 cm’dir. Bu kılavuzun içindeki elektrik alanın x bileşeninin ifadesi

1036cos(40 )sin(100 )sin(2.4 10 52.9 ) (V m)xE x y t zπ π π π= − × − ’dir.

a) mod indislerini bulun. b) ?rε = c) bu modun kesim frekansını bulun 2- A TE wave propagating in a dielectric filled rectangular waveguide of unknown permittivity has dimensions a = 5 cm and b = 3 cm. If the x-component of its electric field is given by 1036 cos(40 )sin(100 )sin(2.4 10 52.9 ) (V m)xE x y t zπ π π π= − × − :

a) Determine the mode number.

02

02

10

( , ) cos sin

Re ( , ) cos sin sin( )

36 cos(40 )sin(100 )sin(2.4 10 52.9 )

40

x

j t j zxx

x

j n m nE x y H x y

h b a b

n m nE E x y e H x y t z

h b a b

if E x y t z

mm

a

ω β

ωµ π π π

ωµ π π πω β

π π π π

ππ

−

= = = − −

= − × −

= → = TE232, 100 3n

nb

ππ= → = →

b) Determine the relative permittivity of the material in the guide.

2 2210 2 2

2 222

2

2.4 10 rad s, 52.9 rad m,

2.25

r r

r

m nk h

c a b

c m n

a b

ω µ ε π πω π β π β

π πε β

ω

= × = = − = − −

= + + =

,

c) Determine the cut-of frequency.

2 2

23

1 2 3( ) 10.77GHz

2cf

a b

π π

π µε

= + =

d) Determine the expression for yH .

TE 2 223

377 1569.9

1 ( ) 1 (10.77 12)

x

y r

EZ

H f f

η

ε= = = = Ω

− −

2- Consider a rectangular waveguide of dimensions 1 3a = cm and 1 4b = cm at an operating frequency of 80 GHz. a) Write down expressions for all six magnetic field and electric field intensity components for the TEz modes.

rad cm rad cm

1

2

3

4

5

92 2 2

8

cos sin

sin cos

sin cos

cos sin

cos cos

3 , 4

, 0,1,2,3,.......

0

2 80 10, 16

3 10

j z

x x y

j z

y x y

j z

x x y

j z

y x y

j z

z x y

x y

x y

E C k x k ye

E C k x k ye

H C k x k ye

H C k x k ye

H C k x k ye

m nk m k n

a bm n

m n

k k k kc

β

β

β

β

β

π ππ π

ω πβ

−

−

−

−

−

=

=

=

=

=

= = = =

=

= ≠

× ×= − − = = =

×rad m76

b) Which TEz modes can propagate in the waveguide?

102 2

, 10 ,

2 2

11 ,

01 ,

20 ,

3 10 cm s45GHz

2 2 (1 3)cm

1 175GHz

2

60GHz

90GHz

cm n cm n

cm n

cm n

cm n

c m nf TE f

a b

cTE f

a b

TE f

TE f

× = + → → = = ×

→ = + =

→ =

→ =

c) For the lowest orderTEz mode that is below cutoff, how far from a source in the z

direction will the amplitude of the electric field have dropped by a factor of 1 e ?

20 ,

2 22 2 2 2 2 2

,

2 2

8.64 1

90GHz

(16.75 rad cm) (18.8 rad cm) 8.64Np cm

10.12 cm 1.2mm

8.64

cm n

cm n x y

j z z

TE f

m nb k k k k k k

a b

j

e e e zβ

π π

− − −

→ =

= ± − = ± − − = ± − −

= ± − = ±

= = ⇒ = = =E ∼

9- Two rectangular waveguides are joined end-to-end. The waveguides have identical dimensions, where a =2b. One guide is air-filled, the other is filled with a dielectric characterized by εr . Determine the range of values of εr such that single-mode operation can be simultaneously ensured in both guides at some frequency.

1- The cutoff frequency of the TE10 mode is 10 GHz, whereas the cutoff frequency of the TE03 mode is 60 GHz. The waveguide is filled with glass dielectric of 04ε ε= . a) Calculate the a and b dimensions and the cutoff frequency of the TE10 mode in air.

b) For an air filled waveguide of the same dimensions as calculated in part (a), calculate the number of possible modes if the excitation frequency is 65 GHz.

Allowed modes

in GHz10 01 11 21 20 30

( ) 20 40 44.7 56.57 40 60c

TE TE TE TE TE TE

f

7- 11TE modunda kayıpsız ici hava dolu bir dikdörtgensel dalga kılavuzunun boyutları

7.214a = cm ve 3.404b = cm olsun. a) Kesim frekansını bulun. b) 11TE modu için 1.1 cf f=

frekansındaki kılavuzlanan dalganın boyu nedir? c) operasyon frekansı 9 GHz ise dalga kılavuzunda hangi modlar bulunur? 6- For the 11TE mode of an air-filled rectangular waveguide, 7.214a = cm, and 3.404b = cm:

a) Calculate the cut-of frequency. b) calculate the guide wavelength if 111.1 ( )cf f TE= .

c) Find the propagating (allowed) modes if the operating frequency is 9 GHz.

2 2

2 2 2

a) ( 0.07214) ( 0.03404) 4.8725 GHz2

2b) 0.1343 m

(1.1 ) ( 0.07214) ( 0.03404)

c

g

c

cf

c m n

π ππ

πλ

ω π π

= + =

= =− −

c) All other modes are above 9.0 GHz. Allowed modes

in GHz10 01 11 11 21 21 20 02 30 31 31 40

( ) 2.08 4.41 4.87 4.87 6.06 6.06 4.16 8.81 6.23 7.64 7.64 8.32c

TE TE TE TM TE TM TE TE TE TE TM TE

f

8- A rectangular waveguide is filled with polyethylene ( 02.5ε ε= ) and operates at 24 GHz. If

the cutoff frequency of a certain TE mode is 16 GHz, find the group velocity and the intrinsic impedance of the mode.

82 2 2 8

0

2 2 2

3 101 ( ) 1 ( ) 1 (16 24) 1.491 10 m/s

2.25

120337.2

1 ( ) 1 ( ) 2.25 1 (16 24)

g c c

r

TE

c r c

cu u f f f f

f f f f

ε

η η πη

ε

×′= − = − = − = ×

′= = = = Ω

− − −

8- A lossless air-filled rectangular waveguide (7.214a = cm, 3.404b = cm) operates in the

11TM mode at 1.1 cf f= . Determine the cutoff frequency, operating frequency, propagation

constant, and the guide wavelength of this mode.

( )

( )

22 2 -1

11 11 11 11

2

11

1 1 4.88 GHz, 1.1 (TE ) 5.37 GHz, 1 46.7 m2

1 , 2 5.61 cm, 13.5 cm

c c c

g c g

cf a b f f jk f f j

f f k

γ

λ λ λ π λ

= + = = = = − =

′ ′= − = = =

5- Determine the resonant frequency of the dominant cavity mode for an air filled rectangular waveguide cavity with dimensions a = 4 cm, b = 2 cm, and c = 7 cm.

2- In an air-filled rectangular waveguide with a = 2.286 cm and b = 1.016 cm, the y-component of the TE mode is given by

102 3sin cos sin(10 10 ) V myE x y t z

a b

π ππ β

= × −

a) Find the mode operation m = 2 and n = 3 b) Find the propagation constant

c) Find Hx

3- A section of X-band waveguide with dimensions a = 2.286 and b = 1.106 cm has perfectly conducting wall and is filled with a lossy dielectric (σd = 367.5 µS/m, εr = 2.1, µr = 1.) Find a) the dominant mode

b) the attenuation factor at 5 GHz

c) the range of operating frequencies where only the dominant mode can exist

2- A rectangular waveguide with a=2.5cm and b=1cm is operating below 15.1GHz. a) How many TE and TM modes can the waveguide support if the guide is filled with a medium characterized by σ=0,ε=4εo, µ=µo?

The cutoff frequency for the mn mode is: 2 21( ) ( )

2cmnf m a n b

µε= +

A waveguide can propagate if fcmn<f b) Calculate the cut-off frequency of each of these modes.

c) Calculate the phase constant, phase velocity and wave impedance for the TE10 and TM11 modes at the operating frequency of 15 GHz. for the TE10

for the TM11

7- An air-filled rectangular waveguide has dimensions a = 2.25 cm and b =1 cm. a) Find the frequency range for single-mode operation.

b) List all the modes that can propagate at 22 GHz.

c) Redesign the waveguide so that the transmitted power is maximized while the single-mode frequency range is retained. Find the maximum power the newly designed waveguide can transmit at 10 GHz. The breakdown in air occurs for fields stronger than 2 MV/m. Use a safety factor of 10.

8- A rectangular waveguide filled with a dielectric characterized by εr = 2 is to be constructed for single-mode operation at 3 GHz. The operating frequency is to be 20% higher than the cutoff frequency for the lowest mode, while being 20% lower than the cutoff frequency for the next higher mode. Calculate the guide dimensions, a and b.

1- For an air filled rectangular waveguide with inner dimension 0.9 inch by 0.4 inch cm the excitation frequency is 9 GHz. Calculate , , ,c gk k β λ for TE10 mode. Re do it for the same

waveguide filled with dielectric of 4rε = . (1 inch = 2.54 cm).

2 2 9

2 28

2

9

8

2

2

0.9 2.54 10 , 0.4 2.54 10 , 9 10

1 3 10 16.16

2 0.9 2.54 102

2 9 10188.5

3 10

137.430.9 2.54 10

1 ( ) 129.05

20.049

4

c

c

c

g

r c

a b f GHz

m nf GHz

a b

k

k

f f

m

f

π π

π µε

πω µε

π

β ω µε

πλ

β

ε

− −

−

−

= × × = × × = ×

× = + = = × × × ×

= = =×

= =× ×

= − =

= =

=8

2

9

8

92

8

3 103.28

2 0.9 2.54 10 22 9 10 2

376.993 10

137.43

2 9 10 21 (3.28 9) 351.06

3 102

0.018

c

g

GHz

k

k

m

π

πβ

πλ

β

−

×= =

× × × ×× × ×

= =×

=

× × ×= − =

×

= =

9- İçi hava dolu 6 GHz te çalışan TE modundaki dikdörtgen bir dalga kılavuzundaki elektrik alan aşağıda verilmiştir. Bu dalga kılavuzunda hangi TE mod indisleri bulunmaktadır? Bu moda ait kesim frekansını ve "wave impedans" değerini hesaplayın. 9- In a air filled rectangular waveguide, a TE mode operating at 6 GHz has

25 sin cos sin( 12 ) V/m

y

x yE t z

a b

π πω

= − .

a) Determine the mode of operation, cutoff frequency and the wave impedance.

221

2 2

2 2

22, 1, , 1 ( ) , 40

1 ( ) 6 1 (12 40 ) 5.973 GHz

1203978

1 ( ) 1 (5.973 6.000)

c

c

dTE

c

fm n TE f f

c

f f

f f

πβ β β ω µε π

β β π

η πη

′= = = − = = =

′= − = − =

= = = Ω− −

b) Determine xH

02

02

sin cos

5 sin cos

sin cos 1.267 sin cos

z

ys

z

ys

z

xs

j m m nE H x y e

a a bh

m nE x y e

a b

j m m n m nH H x y e x y

a a b a bh

γ

γ

γ

ωµ π π π

π π

β π π π π π

−

−

−

= − =

= = sin( ) mA/mt zω β −

2- An air-filled rectangular waveguide is used for the transmission of a microwave signal of frequency 4.5 GHz. The guide cross-sectional dimensions are 15 cm and 6 cm. a) Calculate the number of propagating modes at the operating frequency.

b) If the fundamental mode is used for signal transmission, find its guide wavelength, phase velocity and group velocity.

c) For the first non-propagating mode calculate its attenuation constant in dB/cm at the operating frequency.

3- Design a rectangular air-filled wave guide for mono-mode transmission of the frequency band 8.0 GHz – 12.0 GHz. a) To include appropriate safety margins, the dimensions of the wave guide have to be selected such that the lowest frequency of the band is 25% above the cutoff frequency of the fundamental mode, and the highest frequency of the band is 25% lower than the cutoff frequency of the next higher-order mode.

b) Assume next that the designed wave guide is filled with a dielectric characterized by ε=2.56 ε0 and µ=µ0. Determine all the modes propagating in the wave guide and list the corresponding cutoff frequencies.

c) The wave guide in (a) will be used for the propagation of a modulated narrow-band signal of bandwidth 20 MHz. If the modulation frequency is the center frequency of the operating band of the waveguide, calculate the time delay for the propagation of the signal over a distance of 10 m.

3- The lowest three cut-off frequencies of a rectangular waveguide are 10TE (3GHz),

01TE (4 GHz), and 20TE (6 GHz).

a) How many modes may propagate at 5 GHz? 3 modes ( 10TE , 01TE , and 11TE )

b) Now assume the waveguide is filled with a dielectric with 9rε = . What are the cutoff frequencies for the 10TE , 01TE , and 20TE modes?

c) Assume for this part that the width in the x-direction is 2 m and the height in the y-direction is 1 m. write all the field components for the 34TE modes in terms of the constants

0xE , 0yE , 0zE , 0xH , 0xH , 0xH and zk be sure to indicate any fields which are zero.

5- An air-dielectric L-band rectangular waveguide has a/b =2 and a dominant-mode cutoff frequency of 0.908 GHz. If the measured wavelength is 40 cm, find the operating frequency, guide dimensions and the phase constant.

10TE cm

GHz

2 28

9

2 2 2 22 2 2 2

2 2 22

2 8 22 9 2

2 2

2

1 3 1016.52

2 2 (2)(0.908 10 )2

1 ( )

(3 10 )(0.908 10 ) 1.178

(0.40)

c

c

c

g c

c

c

m n c cf a

a b a f

f f c cf f

f ff f

cf f

mk

a

µε

λλ λ λ

λ

λ

β

× → = + = → = = = × − = → = = ⇒ = + −

×∴ = + = × + =

= −

rad m

2 2

2 221 ( ) 1 ( ) 15.72c

c c c

n

b

fk f f f f

c

πβ

− =

= − = − =

Circular Waveguide 1- Bakırdan yapılan içi hava dolu dairesel kesitli bir dalga kılavuzunun yarıçapı a =1.5 cm olup TM11 modunda çalışmaktadır. Kılavuz içindeki dalga boyu 4.366 cm olarak ölçülmüştür. Çalışma frekansını ve propagasyon sabiti γ'yı hesaplayın. 1- An air-filled circular waveguide, made of copper, has a =1.5 cm and is operated in the TM11 mode. The guide wavelength was measured and found to be 4.366 cm. Find the frequency of operation and the propagation constant γ. k’th zeros of the n’th order Bessel functions are given below.

0 1 2 3 4( ) ( ) ( ) ( ) ( )

1 2.4048 3.8317 5.1356 6.3802 7.5883

2 5.5201 7.0156 8.4172 9.7610 11.0647

3 8.6537 10.1735 11.6198 13.0152 14.3725

4 11.7915 13.3237 14.7960 16.2235 17.6160

5 14.9309 16.4706 17.9598 19.4094 20.8269

k J x J x J x J x J x

GHz rad m

GHz

GHz

S m

11

811

2 2 2 2

0 0

2 2 2 2 2 2

0 0 11

7

(3.832)(3 10 ) 2( ) 12.1976 , 143.912

2 2

1. 1 ( ) (143.912) 1 ( ) 14

2. (143.912) ( ) 14

5.8 10 , = ,

c TM

g

c c

c

s

s c

p uf

a a

k f f f f f

k k p a f

kRfR j

a

πβ

π π λ

β ω µ ε

β ω µ ε

π µσ γ α β α α

σ βη

×= = = = =

= − ⇒ = − ⇒ = = − ⇒ = − ⇒ =

= × ⇒ = + = = Np m0.0111

0.0111 143.912jγ

=

⇒ = +