UNIT-I
MICROWAVE TRANSMISSION
LINES
MICRO WAVES
• Electromagnetic waves whose
frequency ranges from 1 gigahertz
1000 gigahertz
• Microwaves are so called since they
are defined in terms of their
wavelength in the sense that micro
refers to tinyness refer to the
wavelength and the period of cycle of
a cm wave.
MICRO WAVES
• Electromagnetic waves whose
frequency ranges from 1 gigahertz
1000 gigahertz
• Microwaves are so called since they
are defined in terms of their
wavelength in the sense that micro
refers to tinyness refer to the
wavelength and the period of cycle of
a cm wave.
Microwave region and band
designation
Advantages of microwaves
Increased bandwidth availability
Improved directivity properties
Fading effect and reliability
Power requirements
Transparency property of
microwaves
Applications of microwaves
Applications of microwaves
Telecommunication
Radar
Commercials And industrial applications use
heat property of microwaves
Electronic warfare
Identifying objects or personnel by non
contact method
Wave guides
• A hollow metallic tube of uniform cross section
for transmitting electromagnetic waves by
successive reflections from the inner walls of the
tube is called waveguide
• No TEM wave can exist in the waveguide ,but
TE,TM waves can exist
• It is usually coated with other gold or silver to
improve the conductivity and minimize losses
inside the wave get because of roughness. The
waveguides are generally airfield
Types of waveguides
• Rectangular Waveguide is most common.
• Circular waveguide tends to twist waves
as they travel through them. Circular
waveguide are used with rotating
Antennas as in radar.
• Elliptical shape is often preferred in flexible
waveguides
Flexible waveguides
Flexible wave guide will be required whenever the
waveguide section should be capable of movement like
bending stretching or twisting.
They have smaller transverse corrugations and transition
to rectangular waveguides at the ends, which helps
transform a TE11 modes in in the flexible waveguides
into TE10 modes at either ends.
Advantages: Flexible waveguides have comparable
power handling capability ,attenuation and swr as those
of rectangular waveguides.
Ridge waveguides
• Ridging is a convenient method of reducing the
waveguide dimensions and thereby increasing
the critical wavelength
• Disadvantages: increased attenuation, reduced
power handling capacity and introducing
distortions, not used for standard applications
• Single and double Ridge waveguides
• Useful frequency range of waveguide is
increased by ridging ,it also helps in reducing
the phase velocity
Wave guides of various shapes
Propagation of waves in
rectangular waveguide
TE/TM MODE
REPRESENTATION
Field patterns of waveguide
Field patterns
• In figure 4.28 can be seen that voltage varies
from 0 to maxima and Maxima to 0 by across
wide dimension a. this is half one variation.
Hence m=1.
• Across the narrow dimension there is no
variation in voltage v. Hence n=1.
• This mode is TE10 mode.
• The mode having the highest cutoff wavelength
is known as dominant mode of the waveguide
and all other modes are called higher modes.
• For example TE10 mode is a dominant mode for
TE waves
TE/TM MODE
REPRESENTATION
• The electromagnetic wave inside a waveguide
can have infinite number of pattern which are
called modes.
• The fields in the waveguide which make up this
mode patterns must obey certain physical laws
• The electric field must always be perpendicular
to the surface at a conductor.
• The magnetic field is always parallel to the
surface of the conductor.
Maxwell’s Equations
𝛻. 𝐷 = 𝜌
𝜕𝐵
𝛻×𝐸 =−
𝜕𝑡
𝛻. 𝐵 = 0
𝜕𝐷
𝛻×𝐻 =𝐽+
𝜕𝑡
Longitudinal Fields: 𝐸𝑧, 𝐻𝑍
Transverse Fields in terms of longitudinal Fields:
𝛾 𝜕𝐻𝑧 𝑗𝜔𝜀 𝜕𝐸𝑧 𝛾 𝜕𝐸𝑧 𝑗𝜔𝜇 𝜕𝐻𝑧
𝑥 𝐻 = 2− 𝐸 = 2−
𝛾 = 𝛼 + 𝑗𝛽
ℎ2 𝜕𝑥 𝑥
+ℎ 𝜕𝑦 ℎ2 𝜕𝑥 −ℎ 𝜕𝑦
𝑘=𝜔
𝛾 𝜕𝐻𝑧 𝑗𝜔𝜀 𝜕𝐸𝑧 𝛾 𝜕𝐸𝑧 𝑗𝜔𝜇 𝜕𝐻𝑧
𝑦 𝐻
ℎ2 𝜕𝑦
= 2−
−ℎ 𝜕𝑥
𝑦 𝐸
ℎ2 𝜕𝑦
= 2− 𝜇𝜀 ℎ2
+ℎ 𝜕𝑥
= 𝛾2 + 𝑘2
Wave Equations
𝛻 2 𝐸 + 𝜔2𝜇𝜀𝐸 = 0 𝛻 2 𝐻 + 𝜔2𝜇𝜀𝐻 = 0
𝜕 2 𝐸𝑥 𝜕 2 𝐸𝑥 𝜕 2 𝐸𝑥 𝜕2𝐻𝑥 𝜕2𝐻𝑥 𝜕2𝐻𝑥
+ + =−𝜔2𝜇𝜀𝐸 𝑥 2
+ 2
+ 2
= −𝜔2𝜇𝜀𝐻𝑥
𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2 𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕 2 𝐸𝑦 𝜕 2 𝐸𝑦 𝜕 2 𝐸𝑦 𝜕 2𝐻 𝑦 𝜕2𝐻 𝑦 𝜕2𝐻𝑦 2𝜇𝜀𝐻
+ + = −𝜔2𝜇𝜀𝐸 + + = −𝜔 𝑦
𝜕𝑥 2 𝜕𝑦2 𝜕𝑧 2 𝑦 𝜕𝑥 2 𝜕𝑦2 𝜕𝑧 2
𝜕 2 𝐸𝑧 𝜕 2 𝐸𝑧 𝜕 2 𝐸𝑧 𝜕2𝐻𝑧 𝜕2𝐻𝑧 𝜕2𝐻𝑧 2𝜇𝜀𝐻
+ + = −𝜔2𝜇𝜀𝐸 + + = −𝜔 𝑧
𝜕𝑥 2 𝜕𝑦2 𝜕𝑧 2 𝑧 𝜕𝑥 2 𝜕𝑦2 𝜕𝑧 2
Helmholtz Equations
Propagation of TM waves in
rectangular waveguide
Rectangular Waveguide
𝜕 2𝐸 𝜕 2𝐸 𝜕 2𝐸
𝑧 𝑧 𝑧
𝛻 2 𝐸 + 𝑘 2 𝐸 = 0; + + + 𝜔 2𝜇𝜀𝐸 = 0
𝑧
𝑦
𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2 𝑧
𝐸𝑧 𝑥, 𝑦, 𝑧 = 𝑋 𝑥 𝑌 𝑦 𝑍(𝑧)
𝑏
𝑋′′ 𝑌′′ 𝑍′′ ⇒ −𝑘2 − 𝑘2 + 𝛾2 = −𝑘2
+ + = −𝑘2 𝑥 𝑦 𝑥
𝑋 𝑌 𝑍 𝑜𝑟 − 𝑘2 − 𝑘2 − 𝛽2 = −𝑘2 𝑎
𝑥 𝑦
𝑋′′ ⇒ 𝑋′′ + 𝑘2𝑋 =0 ⇒ 𝑋 = 𝐶 cos 𝑘 𝑥
= −𝑘2𝑥 𝑥 1 𝑥 + 𝐶2 sin 𝑘 𝑥 𝑥
𝑋
𝑌′′ ⇒ 𝑌′′ + 𝑘2𝑌 = 0 ⇒ 𝑌 = 𝐶 cos 𝑘 𝑦 + 𝐶4sin(𝑘𝑦𝑦)
= −𝑘2𝑦 𝑦 3 𝑦
𝑌
𝑍′′
= 𝛾2 ⇒ 𝑍′′ − 𝛾2𝑍 = 0 ⇒ 𝑍 = 𝐶5𝑒𝛾𝑧 + 𝐶6𝑒−𝛾𝑧
𝑍
Rectangular Waveguide
𝐸𝑧 𝑥, 𝑦, 𝑧 = 𝐶1 cos 𝑘𝑥𝑥 + 𝐶2 sin 𝑘𝑥𝑥 𝐶3 cos 𝑘𝑦𝑦 + 𝐶4 sin 𝑘𝑦𝑦 𝐶5𝑒𝛾𝑧 + 𝐶6𝑒−𝛾𝑧
Wave Propagation: Along +z direction ⇒ 𝐶5 = 0
𝛾 𝜕𝐻𝑧 𝑗𝜔𝜀 𝜕𝐸𝑧 𝛾 𝜕𝐸𝑧 𝑗𝜔𝜇 𝜕𝐻𝑧
𝑥 𝐻 = 2− 𝑥 𝐸 = 2− 𝑘 = 𝜔 𝜇𝜀
ℎ2 𝜕𝑥 +ℎ 𝜕𝑦 ℎ2 𝜕𝑥 −ℎ 𝜕𝑦
𝛾 𝜕𝐻𝑧 𝑗𝜔𝜀 𝜕𝐸𝑧 𝛾 𝜕𝐸𝑧 𝑗𝜔𝜇 𝜕𝐻𝑧 ℎ2 = 𝛾2 + 𝑘2
𝐻𝑦 = − 2 − 2 𝐸𝑦 = − + 2
ℎ 𝜕𝑦 ℎ 𝜕𝑥 2
ℎ 𝜕𝑦 ℎ 𝜕𝑥
= 𝑘2 + 𝑘2
𝑥 𝑦
TEM Mode
𝐻𝑧 = 0 𝑎𝑛𝑑 𝐸𝑧 = 0
⇒ 𝐸𝑥 = 0; 𝐸𝑦 = 0 𝑎𝑛𝑑 𝐻𝑥 = 0; 𝐻𝑦 = 0
•All field component vanish
•Rectangular Waveguide can not support TEM
mode
TM Mode
𝐻𝑧 = 0; 𝐸𝑧 ≠ 0
General Solution:
𝐸𝑧 𝑥, 𝑦, 𝑧 = (𝐴1 cos 𝑘𝑥𝑥 + 𝐴2 sin 𝑘 𝑥 𝑥 )(𝐴3 cos 𝑘 𝑦 𝑦 + 𝐴4sin(𝑘𝑦𝑦)) 𝑒−𝛾𝑧
Boundary Conditions:
𝑖 𝐴𝑡 𝑥=0, 𝐸𝑧=0 ⇒𝐴1=0
𝑖𝑖 𝐴𝑡 𝑦=0, 𝐸𝑧=0 ⇒𝐴3=0 ቅ ⇒ 𝐸𝑧 𝑥, 𝑦, 𝑧 = 𝐸0 sin 𝑘𝑥𝑥 sin(𝑘𝑦𝑦)) 𝑒−𝛾𝑧
𝑖𝑖𝑖 𝐴𝑡 𝑥 = 𝑎, 𝐸𝑧 = 0 ⇒ sin 𝑘𝑥𝑎 = 0 ⇒ 𝑘𝑥𝑎 = 𝑚𝜋 ⇒ 𝑘𝑥 = 𝑚𝜋/𝑎
𝑖𝑣 𝐴𝑡 𝑦 = 𝑏, 𝐸𝑧 = 0 ⇒ sin 𝑘𝑦𝑏 = 0 ⇒ 𝑘𝑦𝑏 = 𝑛𝜋 ⇒ 𝑘𝑦 = 𝑛𝜋/𝑏
𝑚𝜋 𝑛𝜋
⇒ 𝐸𝑧 𝑥, 𝑦, 𝑧 = 𝐸0 sin 𝑥 sin 𝑦 𝑒−𝛾𝑧
𝑎 𝑏
Propagating and Non-propagating TM Modes
Non-propagating modes:
• 𝑇𝑀00: 𝐻𝑧 = 0; 𝐸𝑧 = 0; 𝐸𝑥 = 0; 𝐸𝑦 = 0; 𝐻𝑥 = 0; 𝐻𝑦 = 0
• 𝑇𝑀𝑚0: 𝐻𝑧 = 0; 𝐸𝑧 = 0; 𝐸𝑥 = 0; 𝐸𝑦 = 0; 𝐻𝑥 = 0; 𝐻𝑦 = 0
• 𝑇𝑀0𝑛: 𝐻𝑧 = 0; 𝐸𝑧 = 0; 𝐸𝑥 = 0; 𝐸𝑦 = 0; 𝐻𝑥 = 0; 𝐻𝑦 = 0
Propagating modes:
•𝑇𝑀𝑚𝑛 ; 𝑚 ≥ 1 𝑎𝑛𝑑 𝑛≥1
Propagating TM Modes
𝑦
𝑇𝑀21 Image Source: Elements
Of Electromagnetics -
Sadiku - 3rd ed
𝑥 𝑧
E- Field
H-Field
𝑇𝑀11 𝑇𝑀12 𝑇𝑀21 𝑇𝑀31
Image Source: Jensen, E. (2016). RF Cavity
Design. 10.5170/CERN-2014-009.405
Propagation of TM waves in
rectangular waveguide
Propagating and Non-propagating TM
Modes
𝑚𝜋 𝑛𝜋
𝐸𝑧 𝑥, 𝑦, 𝑧 = 𝐸0 sin 𝑥 sin 𝑦 𝑒−𝛾𝑧; 𝐻 = 0
𝑧
𝑎 𝑏
𝛾 𝑚𝜋 𝑚𝜋 𝑛𝜋 𝑚𝜋 𝑛𝜋
𝐸𝑥 = − 𝐸 cos 𝑥 sin 𝑦 𝑒−𝛾𝑧; 𝐸 = − 𝛾 𝑛𝜋
𝐸 sin 𝑥 𝑐𝑜𝑠 𝑦 𝑒−𝛾𝑧
2ℎ 0 𝑦 ℎ2 𝑏 0
𝑎 𝑎 𝑏 𝑎 𝑏
𝑗𝜔𝜀 𝑛𝜋 𝑚𝜋 𝑛𝜋 𝑚𝜋 𝑛𝜋 𝑒−𝛾𝑧
𝐻𝑥 = 𝐸 sin
0
𝑥 𝑐𝑜𝑠 𝑦 𝑒−𝛾𝑧; 𝐻 = −
𝑦
𝑗𝜔𝜀 𝑚𝜋
𝐸 0
cos 𝑥 sin 𝑦
ℎ2 𝑏 𝑎 𝑏 ℎ2 𝑎 𝑎 𝑏
Non-propagating modes:
• 𝑇𝑀00: 𝐻𝑧 = 0; 𝐸𝑧 = 0; 𝐸𝑥 = 0; 𝐸𝑦 = 0; 𝐻𝑥 = 0; 𝐻𝑦 = 0
• 𝑇𝑀𝑚0: 𝐻𝑧 = 0; 𝐸𝑧 = 0; 𝐸𝑥 = 0; 𝐸𝑦 = 0; 𝐻𝑥 = 0; 𝐻𝑦 = 0
• 𝑇𝑀0𝑛: 𝐻𝑧 = 0; 𝐸𝑧 = 0; 𝐸𝑥 = 0; 𝐸𝑦 = 0; 𝐻𝑥 = 0; 𝐻𝑦 = 0
Propagating modes:
•𝑇𝑀𝑚𝑛 ;𝑚 ≥ 1 𝑎𝑛𝑑 𝑛≥1
Propagating TM Modes
𝑦
𝑇𝑀2 Image Source: Elements
Of Electromagnetics -
Sadiku - 3rd ed
1
𝑥 𝑧
E- Field
H-Field
𝑇𝑀12
𝑇𝑀11 𝑇𝑀21 𝑇𝑀31
Propagating and Non-propagating TE
Modes
𝑚𝜋 𝑛𝜋
𝐻𝑧 𝑥, 𝑦, 𝑧 = 𝐻0 𝑐𝑜𝑠 𝑥 cos 𝑦 𝑒−𝛾𝑧; 𝐸𝑧 = 0
𝑎 𝑏
𝑗𝜔𝜇 𝑛𝜋 𝑚𝜋 𝑛𝜋 𝑚𝜋 𝑛𝜋
𝐸𝑥 = 𝐻 cos 𝑥 sin 𝑦 𝑒−𝛾𝑧; 𝐸 =− 𝑗𝜔𝜇 𝑚𝜋
𝐻 sin 𝑥 𝑐𝑜𝑠 𝑦 𝑒−𝛾𝑧
2
ℎ 0 𝑦 ℎ2 0
𝑏 𝑎 𝑏 𝑎 𝑎 𝑏
𝑚𝜋 𝑛𝜋 𝑚𝜋 𝑛𝜋
𝐻𝑥 = 𝛾 𝑚𝜋
𝐻0 sin 𝑥 𝑐𝑜𝑠 𝑦 𝑒−𝛾𝑧; 𝐻 = 𝛾 𝑛𝜋
𝐻0 cos 𝑥 sin 𝑦 𝑒−𝛾𝑧
ℎ2 𝑎 𝑦 ℎ2 𝑏
𝑎 𝑏 𝑎 𝑏
Non-propagating modes:
•𝑇𝐸00: 𝐸𝑧 = 0; 𝐻𝑧 ≠ 0; 𝐸𝑥 = 0; 𝐸𝑦 = 0; 𝐻𝑥 = 0; 𝐻𝑦 = 0
Propagating modes:
•𝑇𝐸0𝑛 ; 𝑛 ≥ 1
•𝑇𝐸𝑚0 ; 𝑚 ≥ 1
•𝑇𝐸𝑚𝑛 ; 𝑚 ≥ 1 𝑎𝑛𝑑 𝑛 ≥ 1
Propagating TE Modes
𝑇𝐸10 𝑇𝐸20 𝑇𝐸01
y
E
Field
x
H
Field
Image Source: https://www.cst.com/academia/examples/hollow-rectangular-waveguide
Cut-off Frequency: Rectangular
Waveguide
ℎ2 = 𝛾2 + 𝑘2 = 𝑘2 + 𝑘2 ⇒𝛾= 𝑘2 + 𝑘2 − 𝑘2
𝑥 𝑦 𝑥 𝑦
2 2
Case1 Evanescent: 𝛾 = 𝛼 ⇒ 𝑘2 + 𝑘2 − 𝑘2 > 0⇒ 𝜔2𝜇𝜀 < 𝑚𝜋
𝑥 𝑦
+ 𝑛𝜋
𝑏
2
Case2 Propagation: 𝛾 = 𝑗𝛽 ⇒ 𝑘2 + 𝑘2 − 𝑘2 < 0⇒ 𝜔2𝜇𝜀 > 𝑚𝜋 𝑛𝜋
𝑥 𝑦 2
𝑎 + 𝑏
2 2 2 2
𝑛 1 𝑚𝜋
⇒𝜔> 1 𝑚𝜋 + 𝑜𝑟 𝑓 > + 𝑛𝜋
𝜇𝜀 𝑎 𝑎 𝑏
𝑏 2𝜋 𝜇𝜀
𝑛 2
⇒ 𝑘2 + 𝑘2 − 𝑘2 = 0⇒ 𝜔2𝜇𝜀 = 𝑚𝜋 2
Case3 Cut-off: 𝛾 = 0 𝑥 𝑦 +
𝑏
2 2 2 2
1 𝑚𝜋 𝑛 1 𝑚
⇒ 𝑓𝑐 = + = + 𝑛
2𝜋 𝜇𝜀 𝑎 2 𝜇𝜀 𝑎 𝑏
𝑏
Cut-off Frequency for Different Modes
2 2
𝑣 𝑚 1
𝑓𝑐 = + 𝑛 ; 𝑣=
𝑎
2 𝑏 𝜇𝜀
• Fundamental modes: Modes with lowest cut-off frequency
For TM mode:TM11
• For TE mode: TE01 or TE10
Degenerate modes:
Modes with same cut-off
frequency- TM11 & TE11
Cut-off frequencies of Rectangular waveguide with
TM21 & TE 21; TM31 & TE 31 a=2.5cm and b=1cm
Phase Constant and Intrinsic Impedance
𝛾= 𝑘2 + 𝑘2 − 𝑘2
𝑥 𝑦
2 2 2
𝑚𝜋 𝑓𝑐
Phase Constant: 𝛽 = 𝑘2 − − 𝑛𝜋
=𝜔 𝜇𝜀 1
𝑎 𝑏 𝑓
Intrinsic Impedance: −
𝐸𝑥 𝐸𝑦 𝛽 2 2
=− = 𝑓𝑐 𝑓𝑐
𝜂 𝑇𝑀 = = 1− = 𝜂0 1 −
𝐻𝑦 𝐻𝑥 𝜔𝜀 𝑓 𝑓
𝜀
𝐸𝑥 𝐸𝑦 𝜔𝜇 1 𝜂0
𝜂 𝑇𝐸 = =− = =
𝐻𝑦 = 2
𝐻𝑥 𝛽 𝜀 𝑓𝑐 2
𝑓𝑐
1− 1−
Phase Velocity and Group Velocity
𝜔 𝑣
Phase Velocity: 𝑣 = =
2
𝑓𝑐
1−
1 2
Group Velocity: 𝑣 𝑔 = =𝑣 1− 𝑓𝑐
𝜕𝜔 𝑓
𝜕𝛽
1
𝑣𝑝 𝑣𝑔 = 𝑣 2 𝑣=
𝜇𝜀
Example: Rectangular Waveguide
For an air-filled rectangular waveguide WR430.
(i) Find cut-off frequencies in TE10 and TM21 modes. Dimensions of
WR430:
𝑎 = 4.3′′ = 4.3 × 2.54𝑐𝑚 = 10.922𝑐𝑚; 𝑏 = 𝑎/2 = 5.461𝑐𝑚
2
𝑚 𝑛 2 3×1010 1
Cut-off frequency: 𝑓𝑐 =
𝑐 + = = 1.372 𝐺𝐻𝑧 for TE 10
𝑎
2 𝑏 2 10.922
10 2 2
3×10 2 1
= + = 3.884 𝐺𝐻𝑧 for TM
21
2 10.922 5.461
(ii) If the given waveguide is filled with dielectric with 𝜀𝑟 = 2.2, then find the cut-off
frequencies.
𝑐 𝑚 2 𝑛 2 1.372
Cut-off frequency: 𝑓𝑐 = + = = 0.925 𝐺𝐻𝑧 for TE
2 𝜀𝑟 𝑎 𝑏 2.2 10
3.884
= = 2.619 𝐺𝐻𝑧 for TM 21
2.2
Conclusion
Phase velocity concept in rectangular wave guide.
different propagating mode existences present
inside the wave.
Expression of cut-off frequency and its relation with
wavelength.
Distinguish between phase velocity and group
velocity inside the wave guide.
Propagation of TM waves in
rectangular waveguide
Propagation of TE waves in
rectangular waveguide
Cut-off Frequency: Rectangular
Waveguide
ℎ2 = 𝛾2 + 𝑘2 = 𝑘2 + 𝑘2 ⇒𝛾= 𝑘2 + 𝑘2 − 𝑘2
𝑥 𝑦 𝑥 𝑦
2 2
Case1 Evanescent: 𝛾 = 𝛼 ⇒ 𝑘2 + 𝑘2 − 𝑘2 > 0⇒ 𝜔2𝜇𝜀 < 𝑚𝜋
𝑥 𝑦
+ 𝑛𝜋
𝑏
2
Case2 Propagation: 𝛾 = 𝑗𝛽 ⇒ 𝑘2 + 𝑘2 − 𝑘2 < 0⇒ 𝜔2𝜇𝜀 > 𝑚𝜋 𝑛𝜋
𝑥 𝑦 2
𝑎 + 𝑏
2 2 2 2
𝑛 1 𝑚𝜋
⇒𝜔> 1 𝑚𝜋 + 𝑜𝑟 𝑓 > + 𝑛𝜋
𝜇𝜀 𝑎 𝑎 𝑏
𝑏 2𝜋 𝜇𝜀
𝑛 2
⇒ 𝑘2 + 𝑘2 − 𝑘2 = 0⇒ 𝜔2𝜇𝜀 = 𝑚𝜋 2
Case3 Cut-off: 𝛾 = 0 𝑥 𝑦 +
𝑏
2 2 2 2
1 𝑚𝜋 𝑛 1 𝑚
⇒ 𝑓𝑐 = + = + 𝑛
2𝜋 𝜇𝜀 𝑎 2 𝜇𝜀 𝑎 𝑏
𝑏
Phase Velocity and Group Velocity
Intrinsic Impedance
Power transmission
Power transmission
Conclusion
Power transmission concept in rectangular wave
guide.
Expression of TE wave mode on wave guide
Cut-off frequency Problem based on TE/TM mode .
Power transmission
Power transmission
Power losses
Power losses
Power losses
Microstrip Line
𝜺𝒓 + 𝟏 𝜺𝒓 − 𝟏 𝟏
𝒆 𝜺 =+
𝟐 𝟐 𝒅
𝟏 + 𝟏𝟎 𝑾
60 8𝑑 𝑊
ln + 𝑓𝑜𝑟 𝑊Τ𝑑 ≤ 1
𝜀𝑒 𝑊 4𝑑
𝑍𝑜 =
120𝜋
𝑓𝑜𝑟 𝑊Τ𝑑 ≥ 1
𝜀𝑒 𝑊Τ𝑑 + 1.393 + 0.667ln 𝑊Τ𝑑 + 1.444
Microstripline design
For a given Z0, W/d can be found as:
𝟖𝒆𝑨 𝐖
𝑾 ; 𝒇𝒐𝒓
𝒆𝟐𝑨 − 𝟐
= 𝟐 <𝒅
𝟐
𝒅 𝜺𝒓 − 𝟏 𝟎. 𝟔𝟏
𝑩 − 𝟏 − 𝐥𝐧(𝟐𝑩 − 𝟏) + 𝐥𝐧(𝑩 − 𝟏) + 𝟎. 𝟑𝟗 𝐖
𝝅 𝟐𝜺𝒓 𝜺𝒓
− ;
𝑾𝒉𝒆𝒓𝒆, 𝒇𝒐𝒓
>𝟐
𝒁𝟎 𝜺𝒓 + 𝟏 𝜺𝒓 − 𝟏 𝟎. 𝟏𝟏 𝒅
𝑨= + 𝟎. 𝟐𝟑 +
𝟔𝟎 𝟐 +𝟏
𝜺 𝜺𝒓
𝒓
𝟑𝟕𝟕𝝅
𝑩=
𝟐𝒁 𝒐 𝜺𝒓
Microstrip line design problem
For FR4 substrate (𝝐𝒓 = 𝟒. 𝟒) of height (h) = 1.6 mm, find the value of
microstrip line width (W) for characteristic impedance (Z0) of 100 Ω.
Design:
𝒁𝟎 𝜺𝒓 + 𝟏 𝜺𝒓 − 𝟏 𝟎. 𝟏𝟏 𝟏𝟎𝟎 𝟒. 𝟒 + 𝟏 𝟒. 𝟒 − 𝟏 𝟎. 𝟏𝟏
𝑨= + 𝟎. 𝟐𝟑 + = + 𝟎. 𝟐𝟑 +
𝟔𝟎 𝟐 𝜺𝒓 + 𝟏 𝜺𝒓 𝟐𝟔𝟎 𝟒. 𝟒 + 𝟏 𝟒. 𝟒
= 𝟐. 𝟖𝟗𝟗
−𝟏/𝟐
𝒘 𝟖𝒆𝟐𝑨 𝜺𝒓+𝟏 𝜺𝒓−𝟏 𝟏𝟎𝒉
= = 0.443 ⇒ 𝒘 = 𝟎. 𝟕𝟏𝐦𝐦 ; 𝜺𝒆 = + 𝟏+ = 3.05
𝒅 𝒆𝟐𝑨−𝟐 𝟐 𝟐 𝒘
Verification using analysis equation:
𝟔𝟎 𝟖𝒅
𝒘
<𝟏⇒𝒁𝟎 = 𝒍𝒏 +𝒘 = 𝟗𝟗. 𝟔𝟐Ω
𝒅 √𝜺𝒆 𝒘
𝟒𝒅
Percentage Error in 𝒁𝟎 = 𝟏𝟎𝟎 −𝟗𝟗.𝟔𝟐 × 𝟏𝟎𝟎 = 𝟎. 𝟑𝟖%
𝟏𝟎𝟎
Cavity resonators
Cavity resonators
Cuf-off frequency of resonator
Modes in a cavity resonator
Modes in a cavity resonator
applications cavity resonator
• as a tuned circuit
• in UHF tubes,klystron
amplifiers/oscillators,cavity
magnetron
• in duplexers of radars
• cavity wave meter in measurement of
frequency
