Priyali Verma
Department of Electronics and Communication Engineering
Jaypee University of Engineering and Technology, Guna, Madhya Pradesh, India
Email: priyaliverma20@yahoo.co.in
Abstract: Microwaves are the electromagnetic waves operating at the frequency ranging from
1 to 30 GHz. It is widely used because it is the safest and secure band for user devices as the devices used in this band are cheaper and compact. This paper aims to analyze transmission lines. It includes parametric study of rectangular waveguides, microstrip lines and coaxial lines. The field equations are used to plot the propagation of waves in each transmission line. Various characteristic curves are plotted and verified for the same. Important conclusions about their operation are drawn after analysis.
I.
INTRODUCTION
Microwave is a descriptive term used to identify electromagnetic waves in the frequency spectrum ranging from 1 Gigahertz
(10 9 ) to 30 Gigahertz. This corresponds to wavelengths from 30 cm to 1 cm. It is widely used because it is the safest and secure band for user devices as the devices used in this band are cheaper and compact. The main characteristic feature of microwaves originates from the small size of wavelengths (1 cm to 30 cm) in relation to the sizes of components or device commonly used. Since the wavelengths are small, the phase varies rapidly with distance. Unlike lower radio frequencies, these waves are not reflected and practically not absorbed by the ionosphere. Therefore, radio astronomers use these frequencies to study electromagnetic radiations originating from stars and other astronomical objects.
Microwaves have large bandwidth which makes it suitable for communication. It is so because the percentage bandwidth remains same even if the operating frequency is transformed from one band to another. They have certain features which make them uniquely applications waveguide. suitable like communication. for radars several and
The term ‘waveguide’ is commonly restricted to indicate what might be more completely called a hollow pipe waveguide or at least a waveguide in which two distinct conductors are not present. The ‘open space’ in waveguide is where the electromagnetic energy finds the path of least resistance and can propagate down the line to get to its destination.
Waveguides are used at micro frequencies ranging from 320 MHz to 333 GHz. They are widely used because of their ease in fabrication than coaxial cables. They suffers from less attenuation and have high power handling capacity as compared to coaxial and microstrip lines. Waveguides require no center conductor and is filled with a dielectric. They use reflection at boundaries of conductor/dielectric to confine waves and travel in a particular direction. Figure 1 shows the cross sectional view of a rectangular satellite
Fig.1. Parameters of a rectangular waveguide useful
A microstrip is an unbalanced transmission line in which the potential at the top and bottom does not remain same. It is one of the most popular types of planar transmission lines, primarily because it can be fabricated by

photolithographic process and is easily integrated with other passive and active microwave devices. There are three main components of a microstrip structure; a top conductor strip, a dielectric substrate and a metal ground plane. The conductor of width w is printed on a thin grounded dielectric substrate of thickness d and relative permittivity 𝝐 r .
The dielectric substance is present to separate the top and bottom portion of the microstrip line. It has mechanical impedance to provide strength to the structure and electrical impedance that decides velocity of propagation through transmission line. Figure 2 shows the cross sectional view of microstrip line.
Fig.2. Cross section of a Microstrip line
A coaxial line has been defined as “A transmission line in which one conductor completely surrounds the other, the two being coaxial and separated by a continuous dielectric or by dielectric spacers.”
Figure 3 shows the cross sectional view of coaxial line.
Fig.3. Coaxial line geometry
Each transmission line is basically a low pass filter. So coaxial lines also have an upper limit. It is so because at very high frequencies, the conduction and dielectric losses increase, also the capacitance between the plates increase, thereby leading to a short and consequently no propagation. The coaxial line geometry is shown in the above figure. The inner conductor is at potential Vo and the outer conductor is at zero potential.
II.
MATHEMATICAL FORMULTION
AND FIELD EQUATIONS

Mathematical formulation for rectangular waveguide
Cutoff wavelength: The free space wavelength corresponding to the cut off frequency is called the cut off wavelength. The cut off wavelength is given as
λ c
=
𝟐𝒂𝒃
√(𝒏𝒃) 𝟐 +(𝒎𝒂) 𝟐
For the dominant TE
10
mode, the cut off wavelength λ c
= (2a).
Cutoff frequency (fc): It is that frequency at which the propagation just begins in the transmission line.
No propagation can ever take place within a waveguide at a
𝒇 𝒄
=
𝟏
√𝝀 𝒄
µ 𝒐 𝜺 𝒐
‘m’ and ‘n’: The general symbol for the mode of operation is TE mn
and TM mn where subscript
‘m’ indicates the number of half wave variations of the electric field along the ‘a’
(wide) dimension of the waveguide and the subscript ‘n’ is the number of half wave variations of the electric field in the ‘b’
(narrow) dimension.
Phase Constant: The phase of the wave varies as it propagates down the waveguide. It is denoted by 𝛃 [5]. 𝜷 = √[𝑲 𝒐
𝟐
− ( 𝒏𝝅 𝒂
)
𝟐
− ( 𝒎𝝅 𝒃
)
𝟐
]
Propagation constant: The propagation constant is a complex quantity. Its real part is the attenuation constant and the imaginary part is the phase constant. At low frequencies, 𝜸 is real [5].

𝜸 = 𝜶 + 𝒋 𝛃; 𝜸 = √𝑨 𝟐 + 𝑩 𝟐 − 𝝎 𝟐 µ𝝐 𝜸 = √( 𝑚𝜋 𝑎
)
𝟐
+ ( 𝑛𝜋 𝑏
)
𝟐
− (𝝎)^𝟐(µ𝝐)
The propagation constant is purely imaginary if the wave propagates successfully in the waveguide without attenuation. When the propagation takes place with attenuation, the propagation constant has both real and imaginary parts as non-zero. Hence, it becomes a complex quantity.
The general wave equations are given by:
Δ 2 E = -j 𝝎 µ 𝝐 E ;
Δ 2 H = -j 𝝎 µ 𝝐H
On solving the above equations, we obtain various components of electric and magnetic field as[3]:
𝑬𝒙 =
−(𝒋𝝎µ)𝝏𝑯𝒛
(𝒉
𝟐
)𝝏𝒚
-
(𝜸)𝝏𝑬𝒛
(𝒉
𝟐
)𝝏𝒙
; 𝑬𝒚 =
+(𝒋𝝎µ)𝝏𝑬𝒛
(𝒉
𝟐
)𝝏𝒙
-
(𝜸)𝝏𝑬𝒛
(𝒉
𝟐
)𝝏𝒚
𝑯𝒙 =
+(𝒋𝝎𝝐)𝝏𝑯𝒛
(𝒉 𝟐 )𝝏𝒚
-
(𝜸)𝝏𝑯𝒛
(𝒉 𝟐 )𝝏𝒙
; 𝑯𝒚 =
−(𝒋𝝎𝝐)𝝏𝑬𝒛
-
(𝒉 𝟐 )𝝏𝒚
(𝜸)𝝏𝑯𝒛
(𝒉 𝟐 )𝝏𝒙
Where, E z
is the component of electric field along z direction, H z
is the component of magnetic field along z direction, µ is the permeability of magnetic substance, ε is the permittivity of dielectric substance and ϓ is the propagation constant which is a complex quantity.

Mathematical formulation for Microstrip line
The expression below shows the dependence of characteristic impedance of a Microstrip
Line on the width and thickness of the
Microstrip[7].
𝒁 𝒐
= 𝜼
𝟐𝝅√𝝐𝒓𝒆
𝒍𝒏 (
𝟖𝒉
𝑾
+
𝟎.𝟐𝟓𝑾
) 𝒉
For w/h <= 1;
𝒁 𝒐
= 𝜼
√𝝐𝒓𝒆
[
𝑾 𝒉
+ 𝟏. 𝟑𝟗𝟑 + 𝟎. 𝟔𝟔𝟕 𝒍𝒏 (
𝑾 𝒉
+ 𝟏. 𝟒𝟒𝟒)] −𝟏
For w/h >= 1
Where η = 120 𝛑 ohm
The magnitude of Quality Factor decides how much energy is concentrated in the cavity or the dielectric. The Quality Factor is defined as
Q = 𝝐𝒓−𝟏
𝟒.𝟔 𝒕
⁄ 𝒉
Where, W is the width of the transmission line, t is the thickness of the copper of the circuit trace, h is the thickness of the dielectric, 𝝐 𝒓 the permittivity of dielectric medium.
is

Mathematical formulation for coaxial cable
For any lossless structure, the phase velocity and characteristic impedance is given by the expressions:
𝟏
Vp =
√𝑳𝑪 𝒄
=
√𝝐𝒓
; Zo = √𝑳𝑪 =
𝟏
𝑽𝒑∗𝑪
Where L is Inductance per unit length of the structure, C is the capacitance per unit length of the structure, c is the velocity of electromagnetic waves in free space, 𝝐 𝒓
is relative permittivity of homogeneously filled dielectric.
When the losses are also being considered, useful relations are given as:
Zo = √(𝑹 + 𝒋𝝎𝑳) (𝑮 + 𝒋𝝎𝑪) 𝜸 = √(𝑹 + 𝒋𝝎𝑳)(𝑮 + 𝒋𝝎𝑪)
Where, ϓ is the propagation constant ( 𝛂 +j 𝛽 ) , 𝝎 is the angular frequency, R is the series resistance per unit length of the structure, G is the shunt conductance per unit length of the structure.
For low loss lines, The characteristic impedance is given as: 𝜼
Zo =
𝟐𝝅 𝒍𝒏 𝒃 𝒂
=
𝟏𝟐𝟎𝝅
𝟐𝝅 𝒍𝒏 𝒃 𝒂
= 60 𝒍𝒏 𝒃 𝒂
ohm.
The series resistance R per unit length is given as:
R =
𝑹𝒔
𝟐𝝅
(
𝟏 𝒃
+
𝟏 𝒂
)
The conductance G per unit length may be written as:
G = ⁄ = 𝟐𝝅𝝎𝝐𝒐𝝐𝒓 𝒕𝒂𝒏𝜹 𝐥𝐧 (𝒃/𝒂)
The capacitance will be:
C = 𝟐𝝅𝝐 𝐥𝐧 (𝒃/𝒂)

And inductance will be given as:
µ
L =
𝟐𝝅 𝐥𝐧 𝒃 𝒂
Where, η is the intrinsic impedance of the dielectric medium,Rs is the surface resistivity of the conductors, 𝝈 is the conductivity, 𝑡𝑎𝑛𝛿 is the Loss tangent.
E r
= V
00
’
𝟏
√𝟐𝝅 𝐥𝐧 𝒂/𝒃
𝟏 𝒓
; H φ = I
00
’
𝟏
√𝟐𝝅 𝐥𝐧 𝒂/𝒃
𝟏 𝒓
E φ = E z
= H r
= H z
= 0
g
Z
0
= √
𝑳
𝑪
Where, Z
0 is the characteristic Impedance of
Microstrip, L is the inductance, C is the capacitance between the two plates, ground and conductor (ε o
A/d).

Dispersion Curve
20
14
12
10
8
18
16
6
4
2
0
0 50 100 150 300 350 400 450 200 250
Phase shift
Fig.4. The Dispersion Curve for TE
10
Mode
From Figure 4 , it can be concluded that as the frequency increases, 𝛽 changes but in a nonlinear fashion. Above fc, 𝛽 becomes real and so the waves can propagate. It is different for different modes and the least dispersion occurs in the dominant mode.
Figure 5, figure 6 and figure 7 shows the propagation of electromagnetic waves in a rectangular waveguide in TE
10
, TE
11
and TE
21 mode respectively.
Fig.5.The front view of propagation of waves in a TE
10
Mode
Fig .6.The front view of propagation of waves in a TE
11
Mode

4000
3000
2000
1000
0
0
7000
Wave Impedance vs Frequency curve of a Rectangular Waveguide in TE mode
6000
TE11
TE10
TE01
5000
0.2
0.4
0.6
0.8
1
Frequency
1.2
1.4
1.6
1.8
2
Fig.9. Wave Impedance versus frequency curve in a TE mode
It is known that for dominant mode, λ = 2a or a = λ/2 where λ = c/f . Hence, there exists an inverse relationship between the cut off frequency and a which is clear from figure 10 .
Cut off frequency vs a curve of a Rectangular Waveguide
150
100
Fig.7.The front view of propagation of waves in a TE
21
Mode
Attenuation is an obvious phenomenon in the propagation of waves inside a waveguide which is expressed in dB/meter. It can take place due to Multipaction effect, variation in the active region formed by using stubs at the terminals of a two wire transmission line, conductor loss and skin effect. From Figure 8 , it can be said that there is no linear relationship between attenuation and frequency. However, higher order modes suffer more attenuation than lower order modes.
0.7
0.6
0.5
0.4
0.3
1
0.9
0.8
0.2
0.1
0
0.2
Attenuation vs Frequency curve of a Rectangular Waveguide
0.4
0.6
0.8
1
Frequency
1.2
1.4
1.6
TM11
TE11
TE10
TE01
1.8
x 10
10
2
Fig.8. Attenuation versus frequency curve
For a rectangular waveguide, the wave impedance comes into picture because voltage current relations cannot be obtained. From figure 9, it is observed that wave impedance decreases as the operating frequency is increased. It is used for the matching purpose.
50
0
0 0.005
0.01
0.02
0.025
0.03
0.015
a
Fig.10.Cut off frequency versus a curve
It is clear from the mathematical expression that the cut off wavelength does not depend on narrower dimension b which is verified from figure 11 .
Cutoff Frequency vs b curve of a Rectangular Waveguide
0.025
0.02
0.015
0.01
0.005
0
4 4.2
4.4
4.6
4.8
5 5.2
Cutoff Frequency
5.4
5.6
5.8
6
Fig.11. Cut off frequency versus b curve
As seen from figure 12 , the magnitude of propagation constant decrease below cutoff frequency as the wave cannot propagate.
Above cut off, propagation constant increase as 𝛽 becomes real.

2.5
2
1.5
1
0.5
0
1
Propagation constant vs frequency curve of a Rectangular Waveguide in TE mode
3.5
3
2 3 4 frequency
5 6 7
Fig.12. Phase Shift versus cut off frequency curve

Microstrip Line
Figure 13 indicates that as the height of the
Microstrip is increased, fringing also increases. The effective field lines are reduced and hence Quality factor decreases.
Q vs h curve of a Microstrip Line
6
5
4
8
7
3
2
1
0 0.1
0.2
0.3
0.4
Height of Microstrip Line
0.5
0.6
0.7
Fig.13. Quality Factor versus Height of the Microstrip
As seen from the figure 14 , it can be said that as the width of the Microstrip is increased, the
Quality Factor is reduced.
Q vs W curve of a Microstrip Line
1.6
x 10
-3
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.1
0.2
0.5
0.6
0.7
0.3
0.4
Quality Factor
Fig14.. Quality Factor versus Width of the Microstrip
If the thickness of the metal plate is increased, there will be less interaction between the electrons of top and bottom of the plate. As a result more charges are shifted down and the
Quality Factor increases. This can be seen in figure 15.
1.5
x 10
-4
1
Q vs t curve of a Microstrip Line
0.5
0
0 0.01
0.02
0.03
0.04
Quality Factor
0.05
0.06
0.07
Fig.15. Quality Factor versus Height of the Microstrip
As we increase the dielectric constant of the material, capacitance also increases. It will furthur reduce the Characteristic impedance. It is verified from figure 16.
40
30
20
80
Dielectric constant vs Characteristic Impedance curve for a Microstrip Line
70
60
50
10
0 5 10 15
Dielectric Constant
20 25 30
Fig.16. Characteristic Impedance versus Dielectric Constant of the Microstrip
Increasing h will increase the separation between the plates. This will reduce the value of capacitance and hence, characteristic impedance will increase as seen in figure 17 .
Height vs Charactristic Impedance curve for a Microstrip Line
100
90
80
70
60
50
40
30
20
10
0
0 1 2 3
Height of Microstrip Line
4 5 6
Fig17. Characteristic Impedance versus Height of the Microstrip
As we increase the width of the metal plate, its area is increased. It will increase the capacitance and hence there will be a reduction in the value of the characteristic impedance. It is verified from figure 18.

Charactristic Impedance vs Width curve of a Microstrip Line
700
600
500
400
300
200
100
0
0 0.2
0.4
0.6
0.8
1
Width of Microstrip Line
1.2
1.4
1.6
x 10
-3
Fig.18. Characteristic Impedance versus Width of the Microstrip

Coaxial Line
Transverse electromagnetic or TEM mode is the dominant, or principal, mode in coaxial guide. Its cut off wavelength is infinite, and hence, λ g
= λ for the dominant mode. Figure
19 shows the propagation of TEM mode in a coaxial transmission line.
Fig.21. A plot to show the propagation of Magnetic field in a
TM mode
Figure 22 shows the propagation of Electric field in a TE mode.
Fig.19. A plot to show the propagation in a TEM mode
Figure 20 shows the propagation of Electric field in a TM mode.
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2 -1 0 1 2 x 10
-3
3
Fig.20. A plot to show the propagation of Electric field in a TM mode
Figure 21 shows the propagation of Magnetic field in a TM mode.
Fig.22. A plot to show the propagation of Electric field in a TE mode
Figure 23 shows the propagation of Magnetic field in a TE mode.
Fig.23. A plot to show the propagation of Magnetic field in a TE mode
If we increase the outer diameter ‘a’, the area of the metal also increases. It will increase the capacitance which will further reduce the

characteristic impedance as seen from the figure 24.
Characteristic Impedance vs a curve for a Coaxial Line
60
50
40
30
90
80
70
20
10
0
1 1.5
2 a
2.5
3 3.5
x 10
-3
Fig.24. Characteristic Impedance versus a
If we increase the outer diameter ‘b’, the separation between the two conductors also increase. It will decrease the capacitance which will further increase the characteristic impedance as seen from the figure 25 .
Characteristic Impedance vs b curve of a Coaxial Line
45
40
35
30
25
1
70
65
60
55
50
1.1
1.2
1.3
1.4
1.5
b
1.6
1.7
1.8
1.9
x 10
-3
2
Fig.25. Characteristic Impedance versus b
From figure 26 , it can be seen that as the magnitude of (b/a) is increased, there is a reduction in the value of characteristic impedance.
Characteristic Impedance vs (b/a) curve of a Coaxial Line
360
350
340
330
1
390
380
370
420
410
400
1.5
2 b/a
2.5
3 3.5
x 10
-3
Fig.26. Characteristic Impedance versus b/a ratio
IV.
CONCLUSION
The work has been carried out in order to do the parametric study of three microwave transmission lines i.e. a Rectangular
Waveguide, a Microstrip and a Coaxial Cable.
The basic study for Transverse Electric and
Transverse Magnetic modes has been done for a Rectangular Waveguide and the results have been plotted. The parameters are varied and correspondingly the changes in the operation of a waveguide are known. It can be concluded that every mode has a certain range of frequencies for propagation. While designing a waveguide, such an operating frequency is chosen in which the wave suffer from least attenuation and dispersion. The Microstrip
Transmission line is very light in weight and is compatible with various Integrated Circuits and Printed Circuit Boards. Its dominant mode is Quasi TEM because of the presence of dielectric between the conducting plate and ground. To design a Microstrip, the results of parametric study should be considered. Care shall be taken while choosing the height, width, thickness and dielectric constant of a microstrip. The coaxial transmission line has comparatively larger bandwidth than a waveguide. It is used mainly to propagate the waves in TEM mode. The parametric study has been done and how the characteristic impedance varies with inner and outer diameter of a coaxial line is known. The effect of inner to outer radius has also been analyzed.
The waveguides have a high power handling capacity and so they are used for high power applications. The only disadvantage is that the entire structure becomes bulky. The Microstrip is lightest of all three transmission lines. It is used when the size of the devices are compact.
Its drawback is its poor power handling capacity. The coaxial transmission lines are a compromise between the waveguides and microstrip. It has neither a very high power handling capacity nor a very broad bandwidth.
REFERENCES
1) Thomas S. Laverghetta, Microwaves and
Wireless Simplified, Artech House Publications, 2005.
2) Robert E. Collin, Foundations for Microwave
Engineering , Mc Graw Hill Publications, 1996.
3) N. Marcuvitz, Waveguide Handbook , Peter
Peregrinus Ltd, 1997.

4) David M. Pozar, Microwave Engineering , John
Wiley & Sons. Inc., 1996.
5) K. C. Gupta, Ramesh Garg, Rakesh Chada,
Computer Aided Design of Microwaves , Artech House
Publications, 1981.
6) T. C. Edwards, Introduction to Microwave
Electronics , Edward Arnold Publications, 1984.
7) K. C. Gupta, Microwaves , Wiley Eastern
Limited, 1979
8) Roger B. Marks, Formulations of the Basic
Network Analyzer Error Model including Switch Terms
9) Ken Wong, Network Analyzer Calibrations –
Yesterday, Today and Tomorrow , Agilent Technologies,
Inc. 1400 Fountaingrove Parkway, Santa Rosa, CA