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1: sc=1, sc2=1 Surface: Electric field intensity, y component (V/m)

 

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Figure 1: Parallel plate transmission line, Ey -component at f = 850 MHz.

1: sc=1, sc2=1 Arrow Surface: Electric field

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0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 2: Parallel plate transmission line, E field at f = 850 MHz.

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COMM.RF.430 Transmission lines and waveguides

Small Exam II, February 24th 2023. Answer to all questions.
Jari Kangas

1. (a) A www-page states that
“Transmission lines can be considered as special types of waveguides”.

Evaluate the definition ie. comment on the definition based on what you know about
waveguides and transmission lines.
(3 p.)

(b) Correct or incorrect? To get points, support your answer by an argument or an example.

i. First_a fact: For TM—modes in a cylindrical waveguide one finds solution
E,(r, 6) = CJn(her) cosng

in the cylindrical r, ¢, z coordinate system.
Then the statement: The cutoff frequencies can be calculated if zeros of the Bessel
function are known. (1 p.)

ii. Any wave in a coaxial cavity resonator can be expressed as linear combination of the :
resonance modes. (1 p.)

2. This question is related to Figures 1,2, and 3. Geometry of the structure is the same in all cases.
In one of the pictures the structure is filled with air, in the two others with a dielectric (see the
captions for further info).

(a) Explain the mode shown in Fig. 1. Reason your answer carefully.
(b) Use the Figures to find the relative permittivity of the dielectric inside the cavity.

(c) Use the Figures to find the second longest dimension of the cavity, if the longest dimension
is 0.18 m.

In total 4 p.

3. A collegue of yours asks for advice on rectangular waveguides. The collegue needs to feed an
antenna with the dominant mode. The antenna is supposed to operate at 7.5 GHz (+4300 MHz).

(a) From the list provided find a waveguide suitable for this task (the waveguides are air-filled).
At the bottom of the datasheet you find a picture that shows the geometry parameters.

 

(b) Find the propagation constant and the wavelength of the lowest mode at 4.0 GHz, 7.9 GHz,
and 16.0 GHz. What could you conclude from the results?

(c) Comment on the usable bandwidth of the chosen waveguide.

(d) Outline briefly the analytical process to determine modes in the rectangular waveguide.
HINT: This is rather general question, so aim to focus on the main steps.

In total 5 p.
Constants in free space and some formulas:
¢ dielectric constant ¢ © 8.854 + 1071? F/m
* permeability pio = 4m * 10-7 H/m
e speed of light c + 2.997925 « 108 m/s

e intrinsic impedance = a 1207 2

e* = 1+42 if |} small

Miscellaneous information about rectangular waveguides ete.
ek=w Spe, B= 1? —k2, d= 3,

e —k2 ke + k2 = 0, where k2 = (mz)?, k= a)?

a

For TE-modes fields are related as (where F, is suitable scalar function, for example a trigono-
metric function):

 

 

 

10% 1 OF, _ 10F, H 1 &F,
ee Oy? ° "Oye Ox0z’ ue Ox’ y 4 one OyOz"
e For good conductors skin depth is 1
1
j=-=
a vafuo
e For TEjo1—mode:
_ tfio1poabd(a? + d*) i
Qo = Bias + dB) + ada? +P] where Bs = 55.
e
~1), . os
Hi = 75 BVH. + jwea x VE),
e
-1),.
Ei = 59 (BVE. — jun x ViHz),
where Yi = xe + Ix and
Bx(2,y) = Eosin(™o) sin(=y),
(x,y) = Ho cos( = 2) cosy).
Silver 6.17 * 107 [S/m] Copper 5.80 « 107 [S/m]

* Conduotivities: Brass 1.57 « 10” [S/m] Tron 107 [S/m]
Eigenfrequency=1.1438 GHz Multislice: Electric field, z component (V/m) Arrow Volume: Electric field
A174

150

100

-50
-100

-150

 

Zz

x, v-177

Figure 1: Pattern of a mode in a rectangular cavity resonator filled with a dielectric.

Eigenfrequency=1.3008 GHz Multislice: Electric field, z component (V/m) Arrow Volume: Electric field
Alz71

160

140

120

100

 

Figure 2: Pattern of a mode in a rectangular cavity resonator filled with air.
Eigenfrequency=0.43361 GHz Multislice: Electric field, z component (V/m)
Arrow Volume: Electric field

Alv7l

160

140

 

120

100

 

x, Vv -0.32

Figure 8: Pattern of a mode in a rectangular cavity resonator filled with a dielectric.

 

 
COMM.RF.430 Transmission lines and waveguides

STANDARD RECTANGULAR ouner
FOR QUESTION WAVEGUIDE (SHEET,

 

 

 

 

NUMBER 3

 

 

 

 
  
   
   
 

Waveguide Size

Di A
Europe And USA Internal Dimensions Ax B

 

inches

     
 

165.10x82.55
22 8 109.22x54.61
26 9A 340 86.36x43.18 | 0.11 |3.400x1.700| .005
10 284 72.14x34.04 | 0.08 |2.840x1.340| .004

6.500x3.250

 

    

4.300x2.150

 

 

 

 

1A 229 | 58.17x29.083 | 0.06 | 2.290x1.145] .003
48 12 187 | 47.55x22.149 | 0.05 | 1.872x0.872| .003
58 13 159 | 40.39x20.139 | 0.05 | 1.590x0.795| .002
70 14 137 | 34.85x15.799 | 0.04 |1.372x0.622| .002
84 15 112 | 28.499x12.624| 0.03 | 1.122x0.497| .002
100 16 90 22.86 x10.16 | 0.03 |0.900x0.400) .001
120 17 75 19.05x9.525 | 0.02 |0.750x0.375] .001
140 18 62 15.799x7.899 | 0.02 |0.622x0.311| .0008
180 19 51 12.954x6.477 | 0.02 |0.510x0.255| .0008
220 20 42 10.668x4.318 | 0.02 |0.420x0.170| .0008
280 24 34 8.636x4.318 | 0.02 |0.340x0.170} .0008
320 22 28 7.112x3.556 | 0.02 |0.280x0.140] .0008
400 23 22 5.690x2.845 | 0.02 |0.224x0.112} .0008
500 24 19 4.775x2.388 | 0.02 | 0.188x0.094| .0008
620 25 15 3.759x1.880 | 0.02 |0.148x0.074 | .0008

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

740 26 12 3.099x1.549 | 0.02 |0.122x0.061 | .0008

 

900 27 10 2.540x1.270 | 0.02 |0.100x0.050| .0008

 

 

 

 

 

 

 

 

 

 

Length: 3050 mm (Other upon request)

a a Alloy: 6063 (Other upon request)
Ra
Straightness and twist:
DIN 17615
A
|< —$_____—_—__—_—_——__>

——
COMM.RF.300 Analysis of Electromagnetic Systems

AxB=-BxA
Ax B=(A,B, — A,B,)i— (A,B. — AzBz)j + (ApBy — AyBz)k
A-(Bx C)=B-(Cx A)=C-(A xB)

VF Hise + 556 + ke
Vib+o)=Vo+Vo

Vipd) =oVet¥Ve
V-(A+B)=V-A+V-B

V (WA) =pV-A+A-Vp
Vx (pA)=~VxAtVOxXA
V-(Vx A) =0

V x (Vp) =0

V- (Vo) = VP
V(V-A)-Vx(Vx A) = VA
V- (OVP) = 9&V7b+.V$- Vy

 

r=ia+jy+kz
x! = ig! + jy’ + ke’

f Ver n= bs F tal

. U= -hdaa
vf-dl= (by f(a), dC =b—a
fe
fea) ~ py Het = en

he

VxE=—-8
VxH=J+®2
V: D=p

F = q(E+v xB)
02) = fy aaa ae!

Trer—r]

Bor) = ft Jy ae

e = —1.602 * 1078 C = —1.602 + 10719 Avs

Me = 9.109 * 10-8! ke

€9 = 8.854 « 10- F/m = 8.854 * 107! kg tm. st. A?
Ho = 4a * 10-7 H/m = 4a * 1077 kg. s?- Am

Co = 2.998 * 108 m/s